Add files using upload-large-folder tool

-dAyT4oBgHgl3EQfdfdm/content/tmp_files/2301.00303v1.pdf.txt

@@ -0,0 +1,1483 @@
+Rethinking with Retrieval: Faithful Large Language Model Inference
+Hangfeng He†∗
+Hongming Zhang‡
+Dan Roth§
+†University of Rochester
+‡Tencent AI Lab, Seattle
+§University of Pennsylvania
+[email protected], [email protected]
+[email protected]
+Abstract
+Despite the success of large language mod-
+els (LLMs) in various natural language pro-
+cessing (NLP) tasks, the stored knowledge
+in these models may inevitably be incom-
+plete, out-of-date, or incorrect.
+This mo-
+tivates the need to utilize external knowl-
+edge to assist LLMs. Unfortunately, current
+methods for incorporating external knowl-
+edge often require additional training or
+fine-tuning, which can be costly and may
+not be feasible for LLMs. To address this
+issue, we propose a novel post-processing
+approach, rethinking with retrieval (RR),
+which retrieves relevant external knowledge
+based on the decomposed reasoning steps
+obtained from the chain-of-thought (CoT)
+prompting. This lightweight approach does
+not require additional training or fine-tuning
+and is not limited by the input length of
+LLMs. We evaluate the effectiveness of RR
+through extensive experiments with GPT-3
+on three complex reasoning tasks: common-
+sense reasoning, temporal reasoning, and
+tabular reasoning. Our results show that RR
+can produce more faithful explanations and
+improve the performance of LLMs.1
+1
+Introduction
+Large language models (LLMs) have shown
+exceptional performance across various tasks
+through in-context learning without task-specific
+training or fine-tuning (Brown et al., 2020;
+Chowdhery et al., 2022; Zhang et al., 2022;
+Ouyang et al., 2022). Recent progress in prompt-
+ing (Wei et al., 2022; Zhou et al., 2022; Kojima
+et al., 2022) and decoding (Wang et al., 2022) has
+made it feasible for LLMs to tackle tasks that de-
+mand complex reasoning.
+∗Part of this work was done while the author was at the
+University of Pennsylvania.
+1Our code is publicly available at https://github.
+com/HornHehhf/RR.
+Query
+Prediction
+LLM
+Query 
+Explanation + Prediction
+LLM
+Query
+Explanation + Prediction
+LLM
+(a)
+(b)
+(c)
+Knowledge
+Chain of thought
+Chain of thought
+Retrieval
+Rethinking
+Figure 1: An overview of three approaches for using
+LLMs: (a) Standard prompting for generating a pre-
+diction in response to a query. (b) Chain-of-thought
+prompting for generating both an explanation and a
+prediction in response to a query. (c) Rethinking with
+retrieval, our proposed approach for using the decom-
+posed reasoning steps obtained from chain-of-thought
+prompting to retrieve relevant external knowledge for
+LLMs, leading to more faithful explanations and im-
+proved predictions in response to a query.
+However, the knowledge stored in LLMs might
+inevitably be incomplete, out-of-date, or incorrect.
+As a result, external sources of knowledge, such
+as Wikipedia, may be essential for the success-
+ful deployment of LLMs for real-world applica-
+tions. Previously, people tried to utilize knowl-
+edge for smaller language models (LMs), such
+as T5 (Raffel et al., 2020), BERT (Devlin et al.,
+2019), and RoBERTa (Liu et al., 2019). However,
+these methods often require additional training or
+fine-tuning, which can be costly and thus imprac-
+tical for LLMs.
+In this paper, we present a post-processing
+approach called rethinking with retrieval (RR)
+for utilizing external knowledge in LLMs. Our
+method begins by using the chain-of-thought
+(CoT) prompting method (Wei et al., 2022) to gen-
+erate a diverse set of reasoning paths, as described
+in Wang et al. (2022).
+We then use each rea-
+soning step in those paths to retrieve relevant ex-
+ternal knowledge, which enables RR to provide
+arXiv:2301.00303v1  [cs.CL]  31 Dec 2022
+
+more faithful explanations and more accurate pre-
+dictions, as illustrated in Figure 1.
+We evaluate the effectiveness of our proposed
+method, RR, on three complex reasoning tasks:
+commonsense reasoning, temporal reasoning, and
+tabular reasoning, using GPT-3 175B (Brown
+et al., 2020) and different external knowledge
+sources:
+Wikipedia, Wikidata (Vrandeˇci´c and
+Krötzsch, 2014), WordNet (Miller, 1995), and
+Conceptnet (Speer et al., 2017).
+The results
+demonstrate that RR consistently outperforms all
+baselines on all three tasks without requiring ad-
+ditional training or fine-tuning, indicating the su-
+periority of our approach in leveraging external
+knowledge to enhance the performance of LLMs.
+2
+Related Work
+Enhancing LMs through retrieval.
+Retrieval-
+enhanced LMs have received significant attention
+as a means of improving performance through the
+incorporation of external knowledge. For exam-
+ple, the k-most similar training contexts can be re-
+trieved to improve the estimation of the next word
+distribution in both the training stage (Borgeaud
+et al., 2021) and the inference stage (Khandelwal
+et al., 2020). Furthermore, search query genera-
+tors have been adopted to generate search queries
+for search engines to retrieve relevant documents
+(Komeili et al., 2022; Shuster et al., 2022; Thop-
+pilan et al., 2022).
+Other approaches have uti-
+lized retrieved documents as the additional con-
+text in generation tasks (Joshi et al., 2020; Guu
+et al., 2020; Lewis et al., 2020). Nakano et al.
+(2021) instead use human feedback in a text-based
+web-browsing environment.
+Among these pre-
+vious works, Khandelwal et al. (2020) is most
+closely related to our approach.
+However, they
+focus on improving local inference by using the
+nearest neighbor datastore constructed from train-
+ing data, whereas we focus on conducting faith-
+ful inference using external knowledge. In con-
+trast to other aforementioned approaches, which
+require training or fine-tuning to incorporate re-
+trieved knowledge, we propose a post-processing
+method for leveraging retrieved knowledge with-
+out additional training or fine-tuning.
+Incorporating external knowledge into LMs.
+Significant effort has been devoted to leveraging
+external knowledge to improve the reasoning abil-
+ity of LMs. Previous work has incorporated exter-
+nal knowledge sources such as WordNet (Miller,
+1995) and ConceptNet (Speer et al., 2017) to en-
+hance LMs for tabular reasoning tasks (Neeraja
+et al., 2021; Varun et al., 2022).
+Explicit rules
+have also been added to inputs to improve rea-
+soning ability over implicit knowledge (Talmor
+et al., 2020). In addition, explicit knowledge from
+Wikidata (Vrandeˇci´c and Krötzsch, 2014) and im-
+plicit knowledge in LLMs have been integrated
+into a transformer (Vaswani et al., 2017) for vi-
+sual question answering (Gui et al., 2021). Nye
+et al. (2021) instead introduces a symbolic reason-
+ing module to improve coherence and consistency
+in LLMs. Among these previous works, Nye et al.
+(2021) is the most relevant to our approach. Still,
+they focus on incorporating logical constraints to
+improve coherence and consistency, whereas we
+aim to improve the faithfulness of explanations
+through the use of external knowledge. In con-
+trast to other aforementioned approaches that in-
+corporate external knowledge before generation
+and require additional training or fine-tuning, our
+proposal leverages external knowledge in a post-
+processing manner to enhance LMs without addi-
+tional training or fine-tuning.
+Uncovering latent Knowledge in LLMs.
+There
+has been a line of work exploring the knowledge
+hidden within LLMs for reasoning. This has in-
+cluded the use of careful prompting to encourage
+LLMs to generate explanations in the reasoning
+process, such as through chain of thought prompt-
+ing in few-shot (Wei et al., 2022) or zero-shot
+(Kojima et al., 2022) learning, or through the use
+of scratchpads for intermediate computation (Nye
+et al., 2022). In addition, various methods based
+on sampling a diverse set of reasoning paths in
+LLMs have been proposed, including training ver-
+ifiers to judge the correctness of model comple-
+tions (Cobbe et al., 2021), calibrating model pre-
+dictions based on the reliability of the explana-
+tions (Ye and Durrett, 2022), and promoting self-
+consistency over diverse reasoning paths (Wang
+et al., 2022). Zelikman et al. (2022) instead it-
+eratively bootstrap the ability of LLMs to gener-
+ate high-quality rationales from a few initial ex-
+amples. Liu et al. (2022) further propose generat-
+ing knowledge from LLMs, which is then used as
+additional input to improve commonsense reason-
+ing. In contrast to this line of work, our proposal
+focuses on leveraging external knowledge to en-
+hance LLMs, while they aim to explore the knowl-
+edge hidden within LLMs.
+
+3
+Rethinking with Retrieval
+LLMs have been shown to generate incorrect sup-
+porting facts from time to time, even when they ac-
+curately capture the perspective needed to answer
+a question. This phenomenon highlights intrinsic
+issues in the way LLMs store and retrieve knowl-
+edge, including (1) the presence of out-of-date,
+incorrect, or missing relevant knowledge in the
+pre-training corpus; (2) incorrect memorization of
+relevant knowledge during pre-training; and (3)
+incorrect retrieval of relevant knowledge during
+the inference stage. To address these issues, we
+propose the use of RR, which leverages external
+knowledge through the retrieval of relevant infor-
+mation based on decomposed reasoning steps.
+Overview.
+Given a query Q, we utilize chain-of-
+thought prompting to generate a diverse set of rea-
+soning paths R1, R2, · · · RN, where each reason-
+ing path Ri consists of an explanation Ei followed
+by a prediction Pi. After that, we retrieve relevant
+knowledge K1, · · · KM from a suitable knowledge
+base KB to support the explanation in each reason-
+ing path, and select the prediction ˆP that is most
+faithful to this knowledge. To better illustrate our
+proposal, we use “Did Aristotle use a laptop?” as
+a running example in this work.
+Chain-of-thought prompting.
+In contrast to
+standard prompting, CoT prompting (Wei et al.,
+2022) includes demonstrations of step-by-step rea-
+soning examples in the prompt to produce a series
+of short sentences that capture the reasoning pro-
+cess. For instance, given the question “Did Aris-
+totle use a laptop?”, CoT prompting aims to gen-
+erate the complete reasoning path “Aristotle died
+in 322 BC. The first laptop was invented in 1980.
+Thus, Aristotle did not use a laptop. So the answer
+is no.” rather than simply outputs “No.” Empirical
+results show that CoT prompting significantly im-
+proves the performance of LLMs on many multi-
+step reasoning tasks. Therefore, we adopt CoT
+prompting to obtain both explanation E and pre-
+diction P for the query Q.
+Sampling diverse reasoning paths.
+Similar to
+Wang et al. (2022), we sample a diverse set of rea-
+soning paths R1, R2, · · · RN rather than only con-
+sidering the greedy path as in Wei et al. (2022).
+For the question “Did Aristotle use a laptop?”, the
+potential reasoning paths can be as follows:
+(R1) Aristotle died in 2000. The first laptop was
+invented in 1980. Thus, Aristotle used a lap-
+top. So the answer is yes.
+(R2) Aristotle died in 322BC. The first laptop was
+invented in 2000. Thus, Aristotle did not use
+a laptop. So the answer is no.
+(R3) Aristotle died in 322BC. The first laptop was
+invented in 1980. Thus, Aristotle did not use
+a laptop. So the answer is no.
+Knowledge
+retrieval.
+Different
+knowledge
+bases can be used to address different tasks. For
+example, to address the question “Did Aristotle
+use a laptop?”, we can use Wikipedia as the ex-
+ternal knowledge base KB. Information retrieval
+techniques can be applied to retrieve the relevant
+knowledge K1, · · · KM from Wikipedia based
+on the decomposed reasoning steps. Ideally, we
+would obtain the following two paragraphs from
+Wikipedia for this question:
+(K1) Aristotle (384–322 BC) was a Greek philoso-
+pher and polymath during the Classical pe-
+riod in Ancient Greece. ...
+(K2) The Epson HX-20, the first laptop computer,
+was invented in 1980. ...
+Faithful inference.
+The faithfulness of each rea-
+soning path Ri can be estimated using a function
+fKB(Ri), which is based on relevant knowledge
+K1, · · · , KM retrieved from the knowledge base
+KB. The final prediction is obtained through the
+application of the following inference procedure2:
+ˆP =
+arg max
+Pi∈{P1,··· ,PN}
+N
+�
+i=1
+1(Pi = P)fKB(Ri), (1)
+where Pi denotes the corresponding prediction in
+the reasoning path Ri. This inference procedure
+is designed to identify the most faithful prediction
+ˆP to the knowledge base among all predictions in
+the N reasoning paths. For instance, in the run-
+ning example, given reasoning paths R1, R2, R3
+and the retrieved knowledge K1, K2, the above in-
+ference procedure would output the prediction “So
+the answer is no.”, as it is supported by both R2
+and R3 and has a higher faithfulness score com-
+pared to the prediction “So the answer is yes.”,
+which is only supported by R1.
+2Note that this is the basic version of faithful inference,
+and further variations can be found in Section 5.3.
+
+4
+Experiments
+In this section, we present the evaluation of our
+proposed method, RR, on three complex reason-
+ing tasks: commonsense reasoning, temporal rea-
+soning, and tabular reasoning.
+4.1
+Baselines
+We compare with the following baselines.
+Zero-shot/few-shot prompting.
+In our experi-
+ments, we consider GPT-3 with standard zero-
+shot/few-shot prompting as baselines, following
+the approach described in Brown et al. (2020), in
+which zero or few in-context exemplars of input-
+output pairs are provided in the prompt.
+Chain-of-thought prompting.
+In addition to
+the standard zero-shot/few-shot prompting, we
+also consider GPT-3 with the CoT prompting pro-
+posed in (Wei et al., 2022) as a baseline in our ex-
+periments. This approach involves feeding LLMs
+step-by-step reasoning examples instead of stan-
+dard input-output examples.
+Self-consistency.
+In addition, we also consider
+self-consistency (Wang et al., 2022) as a baseline
+in our experiments. This approach, proposed as an
+alternative to the naive greedy decoding used in
+CoT prompting (Wei et al., 2022), involves sam-
+pling a diverse set of reasoning paths and select-
+ing the most consistent answer by marginalizing
+the sampled paths.
+4.2
+Commonsense Reasoning
+Dataset description.
+For commonsense reason-
+ing, we consider the StrategyQA dataset (Geva
+et al., 2021), which includes questions that require
+implicit reasoning strategies.
+For example, the
+question “Did Aristotle use a laptop?” requires
+implicit decomposition into reasoning steps, while
+the question “Was Aristotle alive when the laptop
+was invented?” explicitly specifies the reasoning
+process. The StrategyQA dataset includes 2, 290
+training examples, each consisting of a question
+(Q), a yes/no answer (A), a decomposition (D),
+evidence paragraphs (E), and supporting facts (F).
+On average, each question requires about 2.93 rea-
+soning steps and 2.33 evidence paragraphs. In ad-
+dition, a development set is constructed by ran-
+domly sampling 10% of the training examples
+(i.e., 229 examples). The answer distribution is
+roughly balanced, with approximately 47% "yes"
+questions in both the training and development
+sets. Unless otherwise specified, the models are
+evaluated on the development set3 for StrategyQA.
+Implementation details.
+In this part, we uti-
+lize Wikipedia as the external knowledge base
+KB. For each sentence in the explanation of ev-
+ery reasoning path, we first apply BM25 (Robert-
+son et al., 2009) to retrieve the top 10 most rele-
+vant paragraphs from Wikipedia. In particular, we
+use the re-implementation of the sparse retrieval
+BM254 in Karpukhin et al. (2020) from Pyserini
+(Lin et al., 2021). Subsequently, we use the pre-
+trained MPNet model (Song et al., 2020) to se-
+lect the most similar paragraph based on the cosine
+similarity between the sentence embeddings of the
+retrieved paragraph and the sentence.
+We then
+employ a pre-trained natural language inference
+(NLI) model (Nie et al., 2020) to obtain the en-
+tailment and contradiction scores for the sentence,
+treating the most similar paragraph as the premise.
+The faithfulness of each reasoning path is then
+calculated using fKB(·) based on the entailment
+scores, contradiction scores, and MPNet similari-
+ties of all sentences in the explanation of the rea-
+soning path. The final prediction for each ques-
+tion is obtained through faithful inference (Equa-
+tion 1). More details about fKB(·) can be found in
+Appendix A.2.
+4.3
+Temporal Reasoning
+Dataset description.
+In this experiment, we use
+the TempQuestions dataset (Jia et al., 2018) to
+investigate temporal reasoning. This dataset in-
+cludes 1, 271 temporal questions that are divided
+into four classes: explicit temporal, implicit tem-
+poral, temporal answer, and ordinal constraints.
+The questions are paired with their answers from
+Freebase (Bollacker et al., 2008). To examine the
+most challenging aspect of temporal reasoning, we
+focus on the set of implicit temporal questions,
+which contain implicit temporal expressions, in-
+cluding free-text temporal expressions.
+For ex-
+ample, the question “who was governor of oregon
+when shanghai noon was released?” is an implicit
+temporal question. To facilitate our analysis, we
+only consider questions with a single answer, re-
+sulting in a total of 175 examples. Of these ex-
+3As the annotations for the test set are not publicly avail-
+able, we use the development set for evaluation. This allows
+us to perform a more comprehensive analysis.
+4We also experimented with DPR and BM25+DPR, and
+found that BM25 outperformed these methods in our experi-
+ments. More details can be found in Appendix A.3.
+
+Methods
+Commonsense
+Temporal
+Tabular
+GPT-3
+Zero-shot prompting
+58.08
+28.40
+82.00
+Few-shot prompting
+63.32
+29.59
+83.08
+Chain-of-thought prompting
+65.94
+33.14
+83.33
+Self-consistency
+73.36
+37.28
+84.00
+Rethinking with retrieval
+77.73
+39.05
+84.83
+Table 1: Performance of different methods using GPT-3 on three reasoning tasks.
+amples, the first 6 are used for prompting, and the
+remaining 169 are used for evaluation.
+Implementation details.
+In this part, we utilize
+Wikidata (Vrandeˇci´c and Krötzsch, 2014) as the
+external knowledge base KB, as it is the largest
+publicly available knowledge graph, and the data
+from Freebase has been migrated to Wikidata. To
+incorporate this knowledge into our system, we
+apply an entity linking system5 to each sentence
+in the explanation of each reasoning path to iden-
+tify the corresponding Wikidata pages for all enti-
+ties in the sentence. Next, we extract all temporal
+relations from these relevant Wikidata pages and
+use templates to convert these temporal relations
+into sentences. This step generates a set of rele-
+vant knowledge sentences for each sentence in the
+explanation of each reasoning path. The final pre-
+diction is then obtained by applying the procedure
+described in Section 4.2, in which the retrieved
+paragraphs are replaced with the relevant knowl-
+edge sentences from the current part.
+4.4
+Tabular Reasoning
+Dataset
+description.
+We
+consider
+the
+IN-
+FOTABS dataset (Gupta et al., 2020) for tabu-
+lar reasoning, which consists of 23, 738 human-
+written textual hypotheses based on premises in
+the form of tables extracted from 2, 540 unique
+Wikipedia info-boxes. We focus on the develop-
+ment set, which includes 1, 800 hypotheses based
+on 200 tables, and only consider entailed and con-
+tradictory hypotheses as it is tricky to write CoT
+demonstrations for neutral hypotheses. This re-
+sults in a total of 1, 200 hypotheses based on 200
+tables for evaluation, with an equal number of en-
+tailed and contradictory hypotheses.
+Implementation details.
+In this part, we utilize
+WordNet (Miller, 1995) and ConceptNet (Speer
+5We use the spacy entity linker: https://pypi.org/
+project/spacy-entity-linker/.
+et al., 2017) as external knowledge bases. To con-
+vert tables into textual premises, we follow the
+same technique as in Varun et al. (2022). For each
+premise-hypothesis pair, we follow the procedure
+outlined in Varun et al. (2022) to retrieve rele-
+vant word relation triples that connect the premise
+and hypothesis words, such as “married” RelatedTo
+←−−−−→
+“spouse”.
+These triples are then converted into
+sentences using some simple templates. The re-
+sulting sentences, along with the textual premises
+from the tables, serve as relevant knowledge for
+each sentence in the explanation of each reasoning
+path. To obtain the final prediction, the procedure
+described in Section 4.2 is applied, whereby the
+retrieved paragraphs in Section 4.2 are replaced
+with the relevant knowledge from the current part.
+4.5
+Evaluation
+Experimental settings.
+In all experiments, we
+utilize GPT-3 text-davinci-002 unless oth-
+erwise stated. The maximum number of tokens for
+generation during completion is set to 256. For
+zero-shot, few-shot, and chain-of-thought prompt-
+ing, the temperature is fixed at 0.
+For self-
+consistency and rethinking with retrieval, we ran-
+domly sample 10 outputs6 with temperature 0.7.
+Detailed prompts can be found in Appendix A.1.
+We evaluate the performance of different methods
+on commonsense and tabular reasoning using ac-
+curacy, and on temporal reasoning using the exact
+match metric as defined in Rajpurkar et al. (2016).
+Results.
+As shown in Table 1, our proposed
+method, rethinking with retrieval, consistently
+outperforms all baselines on all three reasoning
+tasks without requiring additional training or fine-
+tuning. The results highlight the effectiveness of
+our approach in leveraging external knowledge to
+improve the performance of LLMs.
+6For commonsense reasoning, we sample 9 outputs, as we
+have found that odd numbers of outputs tend to yield better
+voting performance for self-consistency on StrategyQA.
+
+Wrong supporting facts
+Wrong inference
+Question
+Was Lil Jon’s top ranked Billboard song a
+collaboration with a member of The Lox?
+Would the top of Mount Fuji stick out of
+the Sea of Japan?
+Answer
+No
+Yes
+Facts
+Lil Jon’s highest ranked billboard song
+was Yeah; Yeah was a collaboration be-
+tween Lil Jon, Usher, and Ludacris; The
+Lox is a rap trio consisting of: Styles P,
+Sheek Louch, and Jadakiss.
+The average depth of the Sea of Japan is
+5,748 feet (1,752 metres) and its maxi-
+mum depth is 12,276 feet (3,742 metres);
+Mount Fuji is 3,776.24 metres (12,389.2
+ft) tall.
+GPT-3
+Lil Jon’s top ranked Billboard song was
+"Get Low" with the East Side Boyz. The
+Lox is not a member of the East Side
+Boyz.
+Thus, Lil Jon’s top ranked Bill-
+board song was not a collaboration with
+a member of The Lox. So the answer is
+no.
+Mount Fuji is 3,776 meters tall.
+The
+Sea of Japan is about 3,741 meters deep.
+Thus, the top of Mount Fuji would not
+stick out of the Sea of Japan. So the an-
+swer is no.
+Table 2: Examples of incorrect outputs from GPT-3 with CoT prompting.
+5
+Analysis
+In this section, we perform a thorough analysis to
+gain a deeper understanding of RR.
+5.1
+Limitations of LLMs in Reasoning
+In this subsection, we present an analysis of GPT-
+3 with CoT prompting on the StrategyQA dataset.
+Upon closer examination of the outputs of GPT-
+3, we observed that it can provide reasonable ex-
+planations and correct predictions for a number
+of questions. For example, when given the ques-
+tion “Will the Albany in Georgia reach a hundred
+thousand occupants before the one in New York?”,
+GPT-3 produced the following output:
+The Albany in New York has a pop-
+ulation of about 98,000.
+The Albany
+in Georgia has a population of about
+77,000. Thus, the Albany in New York
+is more populous than the Albany in
+Georgia. So the answer is no.
+The above output consists of three components:
+(1) supporting facts (in cyan) that are based on a
+particular perspective, (2) chaining arguments (in
+orange), and (3) a prediction (in green).
+Com-
+ponents (1) and (2) contribute to the explanation.
+Overall, the output exhibits a high level of quality.
+However, we also observed that GPT-3 may occa-
+sionally produce incorrect supporting facts for its
+explanations or make incorrect inferences for its
+Retrieval
+Commonsense
+Tabular
+Query-based
+73.36
+36.69
+Decomposition-based
+77.73
+39.05
+Table
+3:
+Comparison
+of
+query-based
+and
+decomposition-based
+retrieval
+on
+commonsense
+and tabular reasoning.
+predictions, despite generally being able to iden-
+tify suitable perspectives.
+Wrong supporting facts.
+As shown in Table 2,
+GPT-3 provides the incorrect supporting fact for
+Lil Jon’s top-ranked Billboard song, stating that
+it was “Get Low” instead of the correct answer,
+“Yeah”. However, it does have the correct per-
+spective on how to answer the question, “Was Lil
+Jon’s top ranked Billboard song a collaboration
+with a member of The Lox?”.
+Wrong inference.
+As shown in Table 2, GPT-3
+makes an incorrect inference, stating that the top
+of Mount Fuji “would not stick out” of the Sea of
+Japan, rather than the correct answer, “would stick
+out”. However, it does provide correct supporting
+facts based on the appropriate perspective for the
+question, “Would the top of Mount Fuji stick out of
+the Sea of Japan?”.
+5.2
+Ablation Study
+Importance of decomposition-based retrieval.
+In our proposed method, we retrieve relevant ex-
+
+Knowledge
+Tabular
+External
+79.92
+Background
+84.75
+Background + External
+84.83
+Table 4: Performance of RR with different types of
+knowledge on tabular reasoning: external only, back-
+ground only, and a combination of both.
+External
+knowledge refers to WordNet and ConceptNet, while
+background knowledge refers to the tables.
+ternal knowledge based on the decomposed rea-
+soning steps rather than the original query. To fur-
+ther investigate the impact of this choice, we con-
+ducted additional experiments in which we used
+the original query for knowledge retrieval while
+keeping other aspects of our method unchanged.
+As shown in Table 3, the results for these experi-
+ments are poor for both commonsense and tempo-
+ral reasoning, indicating the importance of using
+decomposition-based retrieval in our approach.
+The impact of different types of knowledge.
+For tabular reasoning, we use both external knowl-
+edge (WordNet and ConceptNet) and background
+knowledge (tables) in our experiments.
+In this
+section, we further examine the effect of differ-
+ent types of knowledge on the performance of our
+proposed method. As shown in Table 4, the addi-
+tional improvement gained by incorporating Wiki-
+data and ConceptNet in addition to tables is lim-
+ited, indicating that GPT-3 already captures many
+word-level relations in these external knowledge
+sources. In addition, the observed significant im-
+provement in tabular reasoning from using tables
+alone suggests that our proposed method can also
+effectively leverage background knowledge.
+5.3
+Variations of the Proposed Approach
+Basic approach: Weighting outputs.
+In Sec-
+tion 3, we present a basic version of our proposal
+for taking advantage of external knowledge. Our
+basic approach involves weighting outputs as indi-
+vidual units and using a voting mechanism to se-
+lect the best-supported prediction. We can also di-
+rectly choose the best-supported output, which in-
+cludes both an explanation and a prediction, with-
+out using voting.
+For example, in the running
+example of “Did Aristotle use a laptop?”
+(see
+more in Section 3), the third reasoning path R3 is
+the output most supported by the knowledge para-
+graphs K1 and K2.
+Variant I: Fact selection.
+The first variant of
+our approach involves selecting facts from the out-
+puts of LLMs based on external knowledge. For
+example, consider the running example of “Did
+Aristotle use a laptop?”, where we only have ac-
+cess to the first two reasoning paths, R1 and R2.
+In this case, the first sentence in R2 and the sec-
+ond sentence in R1 are supported by knowledge
+K1 and K2, respectively. Therefore, the first vari-
+ant would output the first sentence in R2 and the
+second sentence in R1 as the supporting facts.
+Variant II: Fact generation.
+The second vari-
+ant of our approach involves generating facts
+based on both the outputs of LLMs and external
+knowledge. For example, consider the running ex-
+ample of “Did Aristotle use a laptop?”, where we
+only have access to the first reasoning path R1.
+The second sentence in R1 is supported by the sec-
+ond knowledge paragraph K2. However, the first
+sentence is not supported by any evidence para-
+graphs. We can generate questions about the first
+sentence, such as “When did Aristotle die?” and
+use the first knowledge paragraph K1 to generate
+a new fact: “Aristotle died in 322BC.”. As a result,
+the second variant would output the generated fact
+“Aristotle died in 322 BC.” and the second sen-
+tence in R1 as the supporting facts.
+Inference with supporting facts.
+For the two
+variants of our approach, we only have the sup-
+porting facts and need to perform a final inference
+step to obtain the corresponding prediction. One
+option for this inference is to use LLMs, but they
+can be costly (Brown et al., 2020) or difficult to
+use (Zhang et al., 2022). An alternative is to use an
+off-the-shelf model for inference with supporting
+facts, such as UnifiedQA (Khashabi et al., 2020,
+2022). As discussed in Appendix A.5, UnifiedQA
+is more robust to noisy supporting facts than GPT-
+3. We thus use the second version of UnifiedQA,
+UnifiedQA-v2 (Khashabi et al., 2022), for the final
+step of inference.
+Experimental settings.
+In this part, we focus
+on commonsense reasoning and use the evidence
+paragraphs provided in StrategyQA as the rele-
+vant knowledge, rather than the retrieved para-
+graphs discussed in Section 4.2. To evaluate the
+quality of the explanations, we adopt the best met-
+ric for factual consistency evaluation in Honovich
+
+1.3B
+2.7B
+6.7B
+13B
+30B
+175B
+Model Size
+0
+20
+40
+60
+80
+Accuracy (%)
+Chain-of-thought prompting
+Rethinking with retrieval
+(a) Accuracy of predictions
+1.3B
+2.7B
+6.7B
+13B
+30B
+175B
+Model Size
+20
+25
+30
+35
+40
+45
+50
+55
+Factuality (%)
+Chain-of-thought prompting
+Rethinking with retrieval
+(b) Faithfulness of explanations
+Figure 2: The effect of LM size on the performance of our proposed method (Variant II) and CoT prompting. We
+use various sizes of OPT models, with the exception of the 175B model, which is GPT-3.
+Methods
+Accuracy (%)
+Faithfulness (%)
+CoT prompting
+65.94
+38.73
+Basic (w/o voting)
+76.86
+50.02
+Variant I
+78.60
+54.11
+Variant II
+78.60
+54.54
+Table 5: Comparison of various variations of RR and
+the CoT prompting baseline on StrategyQA using evi-
+dence paragraphs.
+et al. (2022). For simplicity, we use the pre-trained
+NLI model released by Nie et al. (2020) to com-
+pute the NLI-based metric, rather than fine-tuning
+T5-11B (Raffel et al., 2020) ourselves. The imple-
+mentation details of the two variants can be found
+in Appendix A.4.
+Results.
+Table 5 illustrates that the fact selec-
+tion and fact generation variants of our proposal
+improve the faithfulness of the supporting facts in
+explanations, leading to increased prediction ac-
+curacy compared to the basic approach without
+voting. Across all variations of our proposal, we
+observe significant improvements in both predic-
+tion accuracy and the faithfulness of explanations
+when compared to the CoT prompting baseline.
+The incorporation of a voting mechanism leads
+to an increased prediction accuracy of 79.91% for
+the basic approach. Comparison with the perfor-
+mance (i.e., 77.73%) of the same approach us-
+ing retrieved paragraphs rather than evidence para-
+graphs in Table 1 demonstrates that retrieved para-
+graphs are also effective for our proposal, as both
+significantly outperform the voting baseline, self-
+consistency (i.e., 73.36%), as shown in Table 1.
+It is noteworthy that UnifiedQA performs
+poorly on StrategyQA, achieving an accuracy of
+only 58.95%.
+However, when provided with
+gold supporting facts in StrategyQA, UnifiedQA
+demonstrates excellent performance with an accu-
+racy of 90.83%. This suggests that UnifiedQA is
+suitable for last-step inference, but not effective
+for answering questions in StrategyQA.
+5.4
+Impact of the Size of LMs
+In this subsection, we examine the effect of the
+size of LMs on the performance of our proposed
+method, specifically in the context of the fact gen-
+eration variant. We compare the performance of
+our method using various sizes of OPT models
+(Zhang et al., 2022) in addition to GPT-3 (175B)
+using the same experimental setup as in Sec-
+tion 5.3.
+As shown in Figure 2, our proposed
+method (Variant II) consistently outperforms CoT
+prompting in terms of both prediction accuracy
+and the faithfulness of explanations, even when
+using smaller LMs.
+6
+Conclusion
+In conclusion, the proposed approach is a promis-
+ing solution for utilizing external knowledge to as-
+sist LLMs. Unlike traditional methods, RR does
+not require additional training or fine-tuning, mak-
+ing it a lightweight and feasible option for LLMs.
+Through extensive experiments on three reason-
+ing tasks using GPT-3, we have shown that RR is
+able to produce more faithful explanations and im-
+prove the performance of LLMs. In the future, we
+plan to investigate various variations of RR to en-
+hance its effectiveness and efficiency in augment-
+ing LLMs with external knowledge.
+
+References
+Kurt Bollacker, Colin Evans, Praveen Paritosh,
+Tim Sturge, and Jamie Taylor. 2008. Freebase:
+a collaboratively created graph database for
+structuring human knowledge. In Proceedings
+of the 2008 ACM SIGMOD international con-
+ference on Management of data, pages 1247–
+1250.
+Sebastian Borgeaud, Arthur Mensch, Jordan Hoff-
+mann, Trevor Cai, Eliza Rutherford, Katie Mil-
+lican, George van den Driessche, Jean-Baptiste
+Lespiau, Bogdan Damoc, Aidan Clark, et al.
+2021. Improving language models by retriev-
+ing from trillions of tokens.
+arXiv preprint
+arXiv:2112.04426.
+Tom Brown,
+Benjamin Mann,
+Nick Ryder,
+Melanie Subbiah, Jared D Kaplan, Prafulla
+Dhariwal, Arvind Neelakantan, Pranav Shyam,
+Girish Sastry, Amanda Askell, et al. 2020.
+Language models are few-shot learners.
+Ad-
+vances in neural information processing sys-
+tems, 33:1877–1901.
+Aakanksha Chowdhery, Sharan Narang, Jacob De-
+vlin, Maarten Bosma, Gaurav Mishra, Adam
+Roberts, Paul Barham, Hyung Won Chung,
+Charles Sutton, Sebastian Gehrmann, et al.
+2022. Palm: Scaling language modeling with
+pathways. arXiv preprint arXiv:2204.02311.
+Karl Cobbe, Vineet Kosaraju, Mohammad Bavar-
+ian, Jacob Hilton, Reiichiro Nakano, Christo-
+pher Hesse, and John Schulman. 2021. Training
+verifiers to solve math word problems. arXiv
+preprint arXiv:2110.14168.
+Ido
+Dagan,
+Oren
+Glickman,
+and
+Bernardo
+Magnini. 2005.
+The pascal recognising tex-
+tual entailment challenge. In Machine learning
+challenges workshop, pages 177–190. Springer.
+Daniel Deutsch, Tania Bedrax-Weiss, and Dan
+Roth. 2021.
+Towards question-answering as
+an automatic metric for evaluating the content
+quality of a summary. Transactions of the Asso-
+ciation for Computational Linguistics, 9:774–
+789.
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and
+Kristina Toutanova. 2019. BERT: Pre-training
+of deep bidirectional transformers for language
+understanding.
+In Proceedings of the 2019
+Conference of the North American Chapter
+of the Association for Computational Linguis-
+tics: Human Language Technologies, Volume 1
+(Long and Short Papers), pages 4171–4186.
+Alexander R Fabbri, Chien-Sheng Wu, Wenhao
+Liu, and Caiming Xiong. 2021.
+Qafacte-
+val:
+Improved qa-based factual consistency
+evaluation for summarization.
+arXiv preprint
+arXiv:2112.08542.
+Mor Geva, Daniel Khashabi, Elad Segal, Tushar
+Khot, Dan Roth, and Jonathan Berant. 2021.
+Did aristotle use a laptop? a question answer-
+ing benchmark with implicit reasoning strate-
+gies. Transactions of the Association for Com-
+putational Linguistics, 9:346–361.
+Liangke Gui, Borui Wang, Qiuyuan Huang, Alex
+Hauptmann, Yonatan Bisk, and Jianfeng Gao.
+2021.
+Kat: A knowledge augmented trans-
+former for vision-and-language. arXiv preprint
+arXiv:2112.08614.
+Vivek Gupta, Maitrey Mehta, Pegah Nokhiz, and
+Vivek Srikumar. 2020. Infotabs: Inference on
+tables as semi-structured data.
+In Proceed-
+ings of the 58th Annual Meeting of the As-
+sociation for Computational Linguistics, pages
+2309–2324.
+Kelvin Guu, Kenton Lee, Zora Tung, Panupong
+Pasupat, and Mingwei Chang. 2020. Retrieval
+augmented language model pre-training. In In-
+ternational Conference on Machine Learning,
+pages 3929–3938. PMLR.
+Or Honovich, Roee Aharoni, Jonathan Herzig,
+Hagai Taitelbaum, Doron Kukliansy, Vered Co-
+hen, Thomas Scialom, Idan Szpektor, Avinatan
+Hassidim, and Yossi Matias. 2022. True: Re-
+evaluating factual consistency evaluation.
+In
+Proceedings of the Second DialDoc Workshop
+on Document-grounded Dialogue and Conver-
+sational Question Answering, pages 161–175.
+Or Honovich, Leshem Choshen, Roee Aharoni,
+Ella Neeman, Idan Szpektor, and Omri Abend.
+2021.
+Q2::
+Evaluating factual consistency
+in knowledge-grounded dialogues via question
+generation and question answering.
+In Pro-
+ceedings of the 2021 Conference on Empiri-
+cal Methods in Natural Language Processing,
+pages 7856–7870.
+
+Zhen
+Jia,
+Abdalghani
+Abujabal,
+Rishiraj
+Saha Roy,
+Jannik Strötgen,
+and Gerhard
+Weikum. 2018. Tempquestions: A benchmark
+for temporal question answering. In Compan-
+ion Proceedings of the The Web Conference
+2018, pages 1057–1062.
+Mandar Joshi, Kenton Lee, Yi Luan, and Kristina
+Toutanova. 2020.
+Contextualized representa-
+tions using textual encyclopedic knowledge.
+arXiv preprint arXiv:2004.12006.
+Vladimir Karpukhin, Barlas Oguz, Sewon Min,
+Patrick Lewis, Ledell Wu, Sergey Edunov,
+Danqi Chen, and Wen-tau Yih. 2020.
+Dense
+passage retrieval for open-domain question an-
+swering. In Proceedings of the 2020 Confer-
+ence on Empirical Methods in Natural Lan-
+guage Processing (EMNLP), pages 6769–6781.
+Urvashi Khandelwal, Omer Levy, Dan Juraf-
+sky, Luke Zettlemoyer, and Mike Lewis. 2020.
+Generalization through memorization: Nearest
+neighbor language models.
+In International
+Conference on Learning Representations.
+Daniel Khashabi, Yeganeh Kordi, and Hannaneh
+Hajishirzi. 2022. Unifiedqa-v2: Stronger gen-
+eralization via broader cross-format training.
+arXiv preprint arXiv:2202.12359.
+Daniel Khashabi,
+Sewon Min,
+Tushar Khot,
+Ashish Sabharwal, Oyvind Tafjord, Peter Clark,
+and Hannaneh Hajishirzi. 2020.
+Unifiedqa:
+Crossing format boundaries with a single qa
+system. In Findings of the Association for Com-
+putational Linguistics:
+EMNLP 2020, pages
+1896–1907.
+Takeshi Kojima, Shixiang Shane Gu, Machel
+Reid, Yutaka Matsuo, and Yusuke Iwasawa.
+2022. Large language models are zero-shot rea-
+soners. arXiv preprint arXiv:2205.11916.
+Mojtaba Komeili, Kurt Shuster, and Jason Weston.
+2022. Internet-augmented dialogue generation.
+In Proceedings of the 60th Annual Meeting of
+the Association for Computational Linguistics
+(Volume 1: Long Papers), pages 8460–8478.
+Patrick Lewis, Ethan Perez, Aleksandra Piktus,
+Fabio Petroni, Vladimir Karpukhin, Naman
+Goyal, Heinrich Küttler, Mike Lewis, Wen-tau
+Yih, Tim Rocktäschel, et al. 2020. Retrieval-
+augmented generation for knowledge-intensive
+nlp tasks. Advances in Neural Information Pro-
+cessing Systems, 33:9459–9474.
+Jimmy Lin, Xueguang Ma, Sheng-Chieh Lin,
+Jheng-Hong Yang, Ronak Pradeep, and Rodrigo
+Nogueira. 2021. Pyserini: A Python toolkit for
+reproducible information retrieval research with
+sparse and dense representations. In Proceed-
+ings of the 44th Annual International ACM SI-
+GIR Conference on Research and Development
+in Information Retrieval (SIGIR 2021), pages
+2356–2362.
+Jiacheng Liu,
+Alisa Liu,
+Ximing Lu,
+Sean
+Welleck, Peter West, Ronan Le Bras, Yejin
+Choi, and Hannaneh Hajishirzi. 2022.
+Gen-
+erated knowledge prompting for commonsense
+reasoning. In Proceedings of the 60th Annual
+Meeting of the Association for Computational
+Linguistics (Volume 1:
+Long Papers), pages
+3154–3169.
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei
+Du, Mandar Joshi, Danqi Chen, Omer Levy,
+Mike Lewis, Luke Zettlemoyer, and Veselin
+Stoyanov. 2019.
+Roberta:
+A robustly opti-
+mized bert pretraining approach. arXiv preprint
+arXiv:1907.11692.
+George A Miller. 1995.
+Wordnet:
+a lexical
+database for english.
+Communications of the
+ACM, 38(11):39–41.
+Reiichiro Nakano, Jacob Hilton, Suchir Bal-
+aji, Jeff Wu, Long Ouyang, Christina Kim,
+Christopher Hesse,
+Shantanu Jain,
+Vineet
+Kosaraju, William Saunders, et al. 2021. We-
+bgpt:
+Browser-assisted
+question-answering
+with
+human
+feedback.
+arXiv
+preprint
+arXiv:2112.09332.
+J Neeraja, Vivek Gupta, and Vivek Srikumar.
+2021. Incorporating external knowledge to en-
+hance tabular reasoning. In Proceedings of the
+2021 Conference of the North American Chap-
+ter of the Association for Computational Lin-
+guistics: Human Language Technologies, pages
+2799–2809.
+Yixin Nie, Adina Williams, Emily Dinan, Mohit
+Bansal, Jason Weston, and Douwe Kiela. 2020.
+Adversarial nli: A new benchmark for natu-
+ral language understanding. In Proceedings of
+the 58th Annual Meeting of the Association for
+Computational Linguistics, pages 4885–4901.
+
+Maxwell Nye, Anders Johan Andreassen, Guy
+Gur-Ari, Henryk Michalewski, Jacob Austin,
+David Bieber, David Dohan, Aitor Lewkowycz,
+Maarten Bosma, David Luan, et al. 2022. Show
+your work: Scratchpads for intermediate com-
+putation with language models. In Deep Learn-
+ing for Code Workshop.
+Maxwell Nye, Michael Tessler, Josh Tenenbaum,
+and Brenden M Lake. 2021. Improving coher-
+ence and consistency in neural sequence mod-
+els with dual-system, neuro-symbolic reason-
+ing. Advances in Neural Information Process-
+ing Systems, 34:25192–25204.
+Long Ouyang, Jeff Wu, Xu Jiang, Diogo Almeida,
+Carroll L Wainwright, Pamela Mishkin, Chong
+Zhang, Sandhini Agarwal, Katarina Slama,
+Alex Ray, et al. 2022. Training language mod-
+els to follow instructions with human feedback.
+arXiv preprint arXiv:2203.02155.
+Colin Raffel, Noam Shazeer, Adam Roberts,
+Katherine
+Lee,
+Sharan
+Narang,
+Michael
+Matena, Yanqi Zhou, Wei Li, and Peter J Liu.
+2020. Exploring the limits of transfer learning
+with a unified text-to-text transformer. Journal
+of Machine Learning Research, 21:1–67.
+Pranav Rajpurkar, Jian Zhang, Konstantin Lopy-
+rev, and Percy Liang. 2016. Squad: 100,000+
+questions for machine comprehension of text.
+In Proceedings of the 2016 Conference on Em-
+pirical Methods in Natural Language Process-
+ing, pages 2383–2392.
+Stephen Robertson, Hugo Zaragoza, et al. 2009.
+The probabilistic relevance framework: Bm25
+and beyond. Foundations and Trends® in In-
+formation Retrieval, 3(4):333–389.
+Kurt Shuster, Mojtaba Komeili, Leonard Adolphs,
+Stephen Roller, Arthur Szlam, and Jason We-
+ston. 2022.
+Language models that seek for
+knowledge:
+Modular search & generation
+for dialogue and prompt completion.
+arXiv
+preprint arXiv:2203.13224.
+Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu,
+and Tie-Yan Liu. 2020.
+Mpnet: Masked and
+permuted pre-training for language understand-
+ing. Advances in Neural Information Process-
+ing Systems, 33:16857–16867.
+Robyn Speer, Joshua Chin, and Catherine Havasi.
+2017.
+Conceptnet 5.5: An open multilingual
+graph of general knowledge.
+In Thirty-first
+AAAI conference on artificial intelligence.
+Alon Talmor, Oyvind Tafjord, Peter Clark, Yoav
+Goldberg, and Jonathan Berant. 2020.
+Leap-
+of-thought:
+Teaching pre-trained models to
+systematically reason over implicit knowledge.
+Advances in Neural Information Processing
+Systems, 33:20227–20237.
+Romal Thoppilan, Daniel De Freitas, Jamie Hall,
+Noam Shazeer, Apoorv Kulshreshtha, Heng-
+Tze Cheng, Alicia Jin, Taylor Bos, Leslie
+Baker, Yu Du, et al. 2022. Lamda: Language
+models for dialog applications. arXiv preprint
+arXiv:2201.08239.
+Yerram Varun, Aayush Sharma, and Vivek Gupta.
+2022.
+Trans-kblstm: An external knowledge
+enhanced transformer bilstm model for tabular
+reasoning. In Proceedings of Deep Learning In-
+side Out (DeeLIO 2022): The 3rd Workshop on
+Knowledge Extraction and Integration for Deep
+Learning Architectures, pages 62–78.
+Ashish Vaswani, Noam Shazeer, Niki Parmar,
+Jakob Uszkoreit, Llion Jones, Aidan N Gomez,
+Łukasz Kaiser, and Illia Polosukhin. 2017. At-
+tention is all you need. Advances in neural in-
+formation processing systems, 30.
+Denny Vrandeˇci´c and Markus Krötzsch. 2014.
+Wikidata: a free collaborative knowledgebase.
+Communications of the ACM, 57(10):78–85.
+Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc
+Le, Ed Chi, and Denny Zhou. 2022.
+Self-
+consistency improves chain of thought rea-
+soning in language models.
+arXiv preprint
+arXiv:2203.11171.
+Jason Wei, Xuezhi Wang, Dale Schuurmans,
+Maarten Bosma, Ed Chi, Quoc Le, and Denny
+Zhou. 2022. Chain of thought prompting elic-
+its reasoning in large language models. arXiv
+preprint arXiv:2201.11903.
+Thomas Wolf, Lysandre Debut, Victor Sanh,
+Julien Chaumond, Clement Delangue, Anthony
+Moi, Pierric Cistac, Tim Rault, Rémi Louf,
+Morgan Funtowicz, et al. 2020. Transformers:
+State-of-the-art natural language processing. In
+
+Proceedings of the 2020 conference on empir-
+ical methods in natural language processing:
+system demonstrations, pages 38–45.
+Xi Ye and Greg Durrett. 2022. The unreliability
+of explanations in few-shot in-context learning.
+arXiv preprint arXiv:2205.03401.
+Eric Zelikman, Yuhuai Wu, and Noah D Good-
+man. 2022. Star: Bootstrapping reasoning with
+reasoning. arXiv preprint arXiv:2203.14465.
+Susan Zhang, Stephen Roller, Naman Goyal,
+Mikel Artetxe, Moya Chen, Shuohui Chen,
+Christopher Dewan, Mona Diab, Xian Li,
+Xi Victoria Lin, et al. 2022. Opt: Open pre-
+trained transformer language models.
+arXiv
+preprint arXiv:2205.01068.
+Denny Zhou, Nathanael Schärli, Le Hou, Ja-
+son Wei, Nathan Scales, Xuezhi Wang, Dale
+Schuurmans, Olivier Bousquet, Quoc Le, and
+Ed Chi. 2022. Least-to-most prompting enables
+complex reasoning in large language models.
+arXiv preprint arXiv:2205.10625.
+
+A
+Appendix
+In this section, we provide additional details on
+our experimental setup. Further information can
+be found in our code.
+A.1
+Detailed Prompts
+We adopt the same CoT prompt for commonsense
+reasoning (i.e., StrategyQA) as those presented in
+Wei et al. (2022). The CoT prompt for tempo-
+ral reasoning is provided in Table 6. For tabular
+reasoning, we adopt the method of Brown et al.
+(2020) for converting NLI into QA for RTE (Da-
+gan et al., 2005), and randomly sample 6 examples
+from the training data to construct the prompt, as
+shown in Table 8. The few-shot prompt utilizes
+the same exemplars as the CoT prompt and does
+not involve CoT reasoning processes.
+A.2
+Description of Faithfulness Functions
+For a sentence s, we denote its MPNet similarity,
+entailment score, and contradiction score as M(s),
+E(s), and C(s), respectively. In our experiments,
+the corresponding thresholds for these scores are
+Tm = 0.5, Te = 0.6, and Tc = 0.99. Given the
+entailment scores, contradiction scores, and MP-
+Net similarities of all supporting facts (denoted as
+S) in the explanation of a reasoning path R, differ-
+ent faithfulness functions fKB(·) can be adopted in
+different settings as follows:
+(1) fKB(R) = �
+s∈S[M(s)×(M(s) >= Tm)+
+E(s) × (M(s) < Tm) − C(s)]
+(2) fKB(R) = �
+s∈S[M(s) + E(s)]
+(3) fKB(R) = �
+s∈S[E(s) × (E(s) >= Te) −
+C(s) × (C(s) >= Tc)]
+In Section 4, we employ function (1) for com-
+monsense and tabular reasoning. For temporal rea-
+soning, we use function (2) as the distinct nature of
+sentences converted from temporal relations leads
+to unreliable contradiction scores. In Sections 5.3-
+5.4, we use function (3) for commonsense reason-
+ing with evidence paragraphs, as the high quality
+of the relevant knowledge negates the need for the
+complementary use of the MPNet similarity to im-
+prove the entailment score.
+A.3
+Comparison of Retrieval Systems
+For commonsense reasoning, we utilized different
+retrieval systems in Karpukhin et al. (2020) to re-
+trieve relevant paragraphs from Wikipedia. The
+performance of BM25, DPR, and BM25+DPR
+were 77.73%, 58.52%, and 77.29%, respectively,
+indicating that BM25 is the best choice in our case.
+A.4
+Implementation Details for the Two
+Variants of RR
+Fact selection implementation details.
+In this
+work, we utilize the information present in the top-
+ranked output produced by our basic approach as
+a guide. To this end, we apply a greedy clustering
+algorithm to group the sentences from all outputs
+into distinct topic categories based on the cosine
+similarity of their MPNet sentence embeddings.
+For each fact in the top-ranked output of our ba-
+sic approach, we identify the fact with the highest
+faithfulness within the same topic group and re-
+place it in the output. The faithfulness of a fact is
+calculated using the fKB function by replacing the
+supporting facts with a single fact.
+Fact generation implementation details.
+In
+this part, we generate questions for the named en-
+tities present in each fact of the top-ranked output
+produced by our basic approach, and retrieve the
+corresponding answers from the evidence para-
+graphs using UnifiedQA. We employ the ques-
+tion generation model described in Deutsch et al.
+(2021), which has been shown to be more ex-
+tractive compared to other models as demon-
+strated in Fabbri et al. (2021).
+We adopt the
+question filtering approach proposed in Honovich
+et al. (2021) using an off-the-shelf extractive
+QA model (ktrapeznikov/albert-xlarge-v2-squad-
+v2 from Hugging Face (Wolf et al., 2020)). We
+then use an off-the-shelf model (MarkS/bart-base-
+qa2d from Hugging Face) to convert the generated
+QA pairs into declarative sentences.
+We apply
+simple rules based on the entailment and contra-
+diction scores of the selected facts from the fact se-
+lection variant and the generated declarative sen-
+tences to obtain the final generated facts.
+A.5
+Comparison of Different Inference
+Methods with Supporting Facts
+In our experiments, we utilize UnifiedQA for the
+final step of inference in both variants. However,
+it is worth noting that GPT-3 could also be used
+for this purpose. As shown in Table 7, we observe
+that UnifiedQA performs better at inference with
+generated facts, while GPT-3 with CoT prompt-
+ing performs better with empty or gold facts. This
+suggests that UnifiedQA is more robust to noisy
+
+Q: who was governor of minnesota when maathaad maathaadu mallige was released?
+A: Maathaad Maathaadu Mallige was released on 24 August 2007. Tim Pawlenty served as the 39th gov-
+ernor of Minnesota from 2003 to 2011. Thus, Tim Pawlenty was governor of minnesota when maathaad
+maathaadu mallige was released. So the answer is Tim Pawlenty.
+Q: who was us president during the costa rican civil war?
+A: The Costa Rican civil war was a civil war in Costa Rica from 12 March to 24 April 1948. Harry S.
+Truman was the 33rd president of the United States, serving from 1945 to 1953. Thus, Harry S. Truman
+was us president during the costa rican civil war. So the answer is Harry S. Truman.
+Q: who was governor of oregon when the collector was released?
+A: The Collector premiered at the Cannes Film Festival on May 20, 1965. Mark Hatfield served as the
+29th governor of Oregon from 1959 to 1967. Thus, Mark Hatfield was governor of oregon when the
+collector was released. So the answer is Mark Hatfield.
+Q: who was governor of oregon when shanghai noon was released?
+A: Shanghai Noon was released on May 26, 2000. John Kitzhaber served as the 35th governor of Oregon
+from 1995 to 2003. Thus, John Kitzhaber was governor of oregon when shanghai noon was released. So
+the answer is John Kitzhaber.
+Q: who was us president when john andrew shulze was a teenager?
+A: John Andrew Shulze was born on July 19, 1775. A teenager is someone who is between 13 and 19
+years old. George Washington served as the first president of the United States from 1789 to 1797. Thus,
+George Washington was us president when john andrew shulze was a teenager. So the answer is George
+Washington.
+Q: who was us president during the seventh coalition?
+A: The War of the Seventh Coalition was from 20 March to 8 July 1815. James Madison served as the
+fourth president of the United States from 1809 to 1817. Thus, James Madison was us president during
+the seventh coalition. So the answer is James Madison.
+Table 6: The CoT prompt for temporal reasoning.
+Methods
+Accuracy (%)
+Empty facts
+GPT-3 (zero-shot)
+58.08
+GPT-3 (CoT)
+65.94
+UnifiedQA
+58.95
+Gold facts
+GPT-3 (zero-shot)
+81.66
+GPT-3 (CoT)
+91.70
+UnifiedQA
+90.83
+Generated facts
+GPT-3 (zero-shot)
+69.87
+GPT-3 (CoT)
+76.42
+UnifiedQA
+78.60
+Table 7: Comparison of different inference methods on
+empty, gold, and generated facts.
+inputs compared to GPT-3.
+Additionally, both
+UnifiedQA and GPT-3 with CoT prompting signif-
+icantly outperform GPT-3 with zero-shot prompt-
+ing, indicating that the CoT prompting is also ben-
+eficial for the final step of inference.
+
+Charles Sumner Tainter was Born on April 25, 1854 ( 1854-04-25 ) Watertown, Massachusetts, U.S..
+Charles Sumner Tainter was Died on April 20, 1940 ( 1940-04-21 ) (aged 85) San Diego, California,
+U.S.. The Nationality of Charles Sumner Tainter are American. The Known for of Charles Sumner
+Tainter are Photophone, phonograph Father Of The Speaking Machine.
+Question: Charles Sumner Tainter never left the state of Massachusetts. True or False?
+Answer: Charles Sumner Tainter was died in San Diego, California, U.S.. California is a state. Thus,
+Charles Sumner Tainter has left the state of Massachusetts. So the answer is false.
+The Region of Curitiba are South. The Elevation of Curitiba are 934.6 m (3,066.3 ft). The Density of
+Curitiba are 4,062/km 2 (10,523/sq mi). The Metro density of Curitiba are 210.9/km 2 (546.2/sq mi).
+Question: Curitiba is above sea level. True or False?
+Answer: The elevation of Curitiba are 934.6 m (3,066.3 ft). Elevation is a hypernym of level. Thus,
+Curitiba is above sea level. So the answer is true.
+Charles (Prince of Wales) was Born on 14 November 1948 ( 1948-11-14 ) (age 70) Buckingham Palace,
+London, England. The Spouse of Charles (Prince of Wales) are Lady Diana Spencer ( m. 1981 ; div.
+1996 ) , and Camilla Parker Bowles ( m. 2005 ). The Issue of Charles (Prince of Wales) are Prince
+William, Duke of Cambridge , and Prince Harry, Duke of Sussex.
+Question: Charles was born in 1948 and has been married twice. True or False?
+Answer: Charles (Prince of Wales) was Born on 14 November 1948. The Spouse of Charles (Prince of
+Wales) are Lady Diana Spencer ( m. 1981 ; div. 1996 ) , and Camilla Parker Bowles ( m. 2005 ). Married
+is related to spouse. Thus, Charles was born in 1948 and has been married twice. So the answer is true.
+The Born of Idris Elba are 6 September 1972 (age 46) Hackney, London, England. The Residence of
+Idris Elba are London. The Other names of Idris Elba are DJ Big Driis, Big Driis the Londoner, Big
+Driis, and 7 Dub. The Occupation of Idris Elba are Actor, producer, director, musician, and DJ.
+Question: Idris Elba is an English entertainer. True or False?
+Answer: The residence of Idris Elba is London. English is related to London. The occupation of Idris
+Elba are actor, producer, director, musician, and DJ. Actor is a hyponym of entertainer. Musician is a
+hyponym of entertainer. DJ is an entertainer. Thus, Idris Elba is an English entertainer. So the answer
+is true.
+The Breed of Jean, the Vitagraph Dog are Scotch Collie. The Sex of Jean, the Vitagraph Dog are Female.
+The Born of Jean, the Vitagraph Dog are 1902 Eastport, Maine. The Years active of Jean, the Vitagraph
+Dog are 1909 - 1916.
+Question: Jean, the Vitagraph Dog was a Golden Retriever which perform in circus. True or False?
+Answer: The Breed of Jean, the Vitagraph Dog are Scotch Collie. Collie is a hyponym of dog. Retriever
+is a hyponym of dog. Thus, Jean, the Vitagraph Dog was not a Golden Retriever which perform in circus.
+So the answer is false.
+The Studio of Hydrograd are Sphere Studios, North Hollywood, Los Angeles. The Genre of Hydrograd
+are Hard rock. The Label of Hydrograd are Roadrunner. The Producer of Hydrograd are Jay Ruston.
+Question: Hydrograd is in the rap genre. True or False?
+Answer: The Genre of Hydrograd are Hard rock. Rap is distinct from rock. Thus, Hydrograd is not in
+the rap genre. So the answer is false.
+Table 8: The CoT prompt for tabular reasoning.
+

.gitattributes

@@ -2929,3 +2929,53 @@ CdFQT4oBgHgl3EQf_DcV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex
 a9AyT4oBgHgl3EQfW_fi/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 4NFKT4oBgHgl3EQfRi0P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 y9E4T4oBgHgl3EQfyA3N/content/2301.05263v1.pdf filter=lfs diff=lfs merge=lfs -text
+1NFQT4oBgHgl3EQf1DYE/content/2301.13418v1.pdf filter=lfs diff=lfs merge=lfs -text
+F9FKT4oBgHgl3EQfbS5P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ItFJT4oBgHgl3EQfFyzw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+d9FRT4oBgHgl3EQfUTfv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+xNE3T4oBgHgl3EQf_QuN/content/2301.04833v1.pdf filter=lfs diff=lfs merge=lfs -text
+utAzT4oBgHgl3EQf6_5m/content/2301.01883v1.pdf filter=lfs diff=lfs merge=lfs -text
+XNE1T4oBgHgl3EQfJQOq/content/2301.02950v1.pdf filter=lfs diff=lfs merge=lfs -text
+t9FJT4oBgHgl3EQfdywM/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+19E4T4oBgHgl3EQfzg22/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+HtE2T4oBgHgl3EQf_AnW/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+KdAyT4oBgHgl3EQff_ia/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+I9AyT4oBgHgl3EQffvhT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+nNFPT4oBgHgl3EQf5TUv/content/2301.13196v1.pdf filter=lfs diff=lfs merge=lfs -text
+htE4T4oBgHgl3EQfrg2l/content/2301.05209v1.pdf filter=lfs diff=lfs merge=lfs -text
+8tE2T4oBgHgl3EQflgdv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+h9AzT4oBgHgl3EQf4v4M/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf filter=lfs diff=lfs merge=lfs -text
+ytE0T4oBgHgl3EQftgFE/content/2301.02592v1.pdf filter=lfs diff=lfs merge=lfs -text
+htE4T4oBgHgl3EQfrg2l/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+y9E4T4oBgHgl3EQfyA3N/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ztAyT4oBgHgl3EQfbPfN/content/2301.00260v1.pdf filter=lfs diff=lfs merge=lfs -text
+XNE1T4oBgHgl3EQfJQOq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf filter=lfs diff=lfs merge=lfs -text
+tNE2T4oBgHgl3EQffgdg/content/2301.03927v1.pdf filter=lfs diff=lfs merge=lfs -text
+BNAzT4oBgHgl3EQfhv2_/content/2301.01490v1.pdf filter=lfs diff=lfs merge=lfs -text
+8tE2T4oBgHgl3EQflgdv/content/2301.03989v1.pdf filter=lfs diff=lfs merge=lfs -text
+ptFST4oBgHgl3EQfNzhl/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf filter=lfs diff=lfs merge=lfs -text
+4NFKT4oBgHgl3EQfRi0P/content/2301.11771v1.pdf filter=lfs diff=lfs merge=lfs -text
+tNE2T4oBgHgl3EQffgdg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+WdAyT4oBgHgl3EQfu_n3/content/2301.00625v1.pdf filter=lfs diff=lfs merge=lfs -text
+ytE0T4oBgHgl3EQftgFE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+VNAzT4oBgHgl3EQfJ_sw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+U9AzT4oBgHgl3EQf1P4j/content/2301.01795v1.pdf filter=lfs diff=lfs merge=lfs -text
+99E1T4oBgHgl3EQfCgIZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+pdE1T4oBgHgl3EQfiQRF/content/2301.03249v1.pdf filter=lfs diff=lfs merge=lfs -text
+VdE4T4oBgHgl3EQfnQ2t/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ItE3T4oBgHgl3EQfXAqD/content/2301.04475v1.pdf filter=lfs diff=lfs merge=lfs -text
+VNAzT4oBgHgl3EQfJ_sw/content/2301.01088v1.pdf filter=lfs diff=lfs merge=lfs -text
+ldFAT4oBgHgl3EQfbh3i/content/2301.08559v1.pdf filter=lfs diff=lfs merge=lfs -text
+xNAzT4oBgHgl3EQfQfuL/content/2301.01200v1.pdf filter=lfs diff=lfs merge=lfs -text
+xNAzT4oBgHgl3EQfQfuL/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+WdAyT4oBgHgl3EQfu_n3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+odE3T4oBgHgl3EQfLQlL/content/2301.04361v1.pdf filter=lfs diff=lfs merge=lfs -text
+ptAzT4oBgHgl3EQf5v4s/content/2301.01863v1.pdf filter=lfs diff=lfs merge=lfs -text
+ldFAT4oBgHgl3EQfbh3i/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+O9FJT4oBgHgl3EQf1S22/content/2301.11651v1.pdf filter=lfs diff=lfs merge=lfs -text
+utAzT4oBgHgl3EQf6_5m/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ZdFLT4oBgHgl3EQfWC9a/content/2301.12055v1.pdf filter=lfs diff=lfs merge=lfs -text
+YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf filter=lfs diff=lfs merge=lfs -text

19E4T4oBgHgl3EQfzg22/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c4f96ea874e96776c37c78cea2fc122f2b0e00854eb86e78f3e33e0475f388d5
+size 4063277

1NFQT4oBgHgl3EQf1DYE/content/2301.13418v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5524eab9698a1f69d210e8bfc676704563a0cef63f17b7d22f7b3cf5479a657c
+size 4862595

4NFKT4oBgHgl3EQfRi0P/content/2301.11771v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1151bca4f060499cba2dbd7ebf30f0b3c7385aeb31df04cd2ca8ef686626ea94
+size 3950996

5NE5T4oBgHgl3EQfPA41/content/tmp_files/2301.05501v1.pdf.txt

@@ -0,0 +1,1580 @@
+MNRAS 000, 1–9 (2015)
+Preprint 16 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The rebrightening of a ROSAT-selected tidal disruption event: repeated
+weak partial disruption flares from a quiescent galaxy?
+A. Malyali1★, Z. Liu1, A. Rau1, I. Grotova1, A. Merloni1, A. J. Goodwin2, G. E. Anderson2,
+J. C. A. Miller-Jones2, A. Kawka2, R. Arcodia1, J. Buchner1, K. Nandra1, D. Homan3, M. Krumpe3
+1Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany
+2International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia
+3Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The ROSAT-selected tidal disruption event (TDE) candidate RX J133157.6-324319.7 (J1331), was detected in 1993 as a
+bright (0.2–2 keV flux of (1.0 ± 0.1) × 10−12 erg s−1 cm−2), ultra-soft (𝑘𝑇 = 0.11 ± 0.03 keV) X-ray flare from a quiescent
+galaxy (𝑧 = 0.05189). During its fifth All-Sky survey (eRASS5) in 2022, SRG/eROSITA detected the repeated flaring of
+J1331, where it had rebrightened to an observed 0.2–2 keV flux of (6.0 ± 0.7) × 10−13 erg s−1 cm−2, with spectral properties
+(𝑘𝑇 = 0.115 ± 0.007 keV) consistent with the ROSAT-observed flare ∼30 years earlier. In this work, we report on X-ray, UV,
+optical, and radio observations of this system. During a pointed XMM observation ∼17 days after the eRASS5 detection, J1331
+was not detected in the 0.2–2 keV band, constraining the 0.2–2 keV flux to have decayed by a factor of ≳40 over this period.
+Given the extremely low probability (∼ 5 × 10−6) of observing two independent full TDEs from the same galaxy over a 30 year
+period, we consider the variability seen in J1331 to be likely caused by two partial TDEs involving a star on an elliptical orbit
+around a black hole. J1331-like flares show faster rise and decay timescales (O(days)) compared to standard TDE candidates,
+with neglible ongoing accretion at late times post-disruption between outbursts.
+Key words: accretion, accretion discs – galaxies: nuclei – black hole physics – transients: tidal disruption events
+1 INTRODUCTION
+Benefitting from the latest generation of time-domain surveys, the
+past decade has seen a vast growth in the diversity of observed tran-
+sients originating from galactic nuclei. These events can be crudely
+divided into, and described as, either ‘one-off’ or ‘repeating’ events,
+depending on the observed evolution of their lightcurves.
+‘One-off’ events, characterised by a single epoch of major tran-
+sient behaviour over an observed monitoring campaign, comprise the
+majority of newly reported nuclear transients. These include systems
+where the variability is likely linked to changes in the accretion pro-
+cess onto a supermassive black hole, such as has been reported in
+previously known AGN (e.g. changing-state AGN; Frederick et al.
+2019; Trakhtenbrot et al. 2019b; Ricci et al. 2020, 2021; Frederick
+et al. 2021; short-rise, slowed-decay Bowen accretion flares, Trakht-
+enbrot et al. 2019a), or due to stellar tidal disruption events (TDEs) in
+quiescent galaxies1 (see Saxton et al. 2020; van Velzen et al. 2020,
+2021a; Alexander et al. 2020 for recent reviews of X-ray, optical,
+infrared and radio observations of TDEs, respectively). Other tran-
+sients, which may occur so close to the centres of galaxies that they
+are astrometically indistinguishable from SMBH accretion, have also
+★ E-mail: [email protected]
+1 Strong TDE candidates have also been reported in galaxies showing signs
+of previous AGN activity (e.g. Merloni et al. 2015; Blanchard et al. 2017;
+Liu et al. 2020).
+been reported (e.g. supernovae exploding in the narrow-line region of
+AGN, Drake et al. 2011), or predicted to exist (e.g. stellar collisions
+in nuclear star clusters; Dale et al. 2009).
+Even more recently, the population of known ‘repeating’ events
+has expanded. Several TDE candidates have now shown multiple
+major outbursts, either through their strong, double-peaked optical
+lightcurves (AT 2019avd, Malyali et al. 2021; Chen et al. 2022),
+repeated X-ray outbursts (IC 3599, Grupe et al. 1995, 2001, 2015;
+Campana et al. 2015; eRASSt J045650.3-203750, Liu et al. 2022;
+AT 2018fyk, Wevers et al. 2022), or quasi-periodic optical outbursts
+potentially associated with repeated partial TDEs (ASASSN-14ko,
+Payne et al. 2021). Towards the more extreme end of known re-
+peating transients lie the recently-discovered class of quasi-periodic
+eruptions (QPEs; Miniutti et al. 2019; Giustini et al. 2020; Arcodia
+et al. 2021, 2022), which show large amplitude, ultra-soft X-ray out-
+bursts, with flare duration of the order of hours, and which recur over
+timescales of hours to days.
+In this work, we report on the SRG/eROSITA (Sunyaev et al. 2021;
+Predehl et al. 2021) detection of the repeated flaring of a previously
+reported, ROSAT-selected TDE candidate, RXJ133157.6-324319.7
+(Reiprich & Greiner 2001; Hampel et al. 2022), originating from
+a quiescent galaxy at 𝑧 = 0.05189 (Moretti et al. 2017). In Sec-
+tion 2, we report on the detection of this system with eROSITA and
+follow-up observations performed with NICER (Section 2.2), XMM
+(Section 2.3), and Swift XRT (Section 2.4), as well as archival X-ray
+© 2015 The Authors
+arXiv:2301.05501v1  [astro-ph.HE]  13 Jan 2023
+
+2
+Adam Malyali et al.
+observations (Section 2.5), UV, optical and mid-infrared photometry
+(Section 2.6) and radio observations (Section 2.7). We discuss the
+nature of the system in Section 3, before providing a summary in
+Section 4.
+All magnitudes are reported in the AB system and corrected for
+Galactic extinction using 𝐴V = 0.142 mag, obtained from (Schlafly
+& Finkbeiner 2011), 𝑅V = 3.1 and a Cardelli extinction law (Cardelli
+et al. 1989), unless otherwise stated. The effective wavelength for
+each filter was retrieved from the SVO Filter Profile Service2. All
+dates/times will be reported in universal time (UT).
+2 RE-DISCOVERY AND FOLLOW-UP
+eRASSt J133157.9-324321 (herein J1331) was detected on 2022-01-
+20 as a bright new X-ray point source in a systematic search for TDE
+candidates during the fifth eROSITA All-Sky survey (eRASS5). The
+eROSITA Science Analysis Software (eSASS; Brunner et al. 2022)
+inferred source position was (RAJ2000, DecJ2000)=(13h31m57.9s, -
+32◦43′21.2′′), with a 1𝜎 positional uncertainty of 1.6′′. No X-ray
+point source was detected within 60" of this position in each of the
+previous four eRASS. The eROSITA source position is consistent
+with a quiescent host galaxy at 𝑧 = 0.05189, with total stellar mass,
+log(𝑀★/𝑀⊙) = 10.15 ± 0.09, and an inferred black hole mass,
+log(𝑀BH/𝑀⊙) = 6.5 ± 0.2 (appendix A). The quiescent nature of
+the host is suggested by both the optical spectrum of its host galaxy
+(appendix B; see also Hampel et al. 2022) and its AllWISE (Wright
+et al. 2010; Mainzer et al. 2014) mid-infrared colour, W1-W2=0.05±
+0.05 mag, far below the threshold of ≳0.7 for mid-infrared AGN
+selection (Stern et al. 2012; Assef et al. 2018). After selecting J1331
+as a promising TDE candidate, it was also realised that the host galaxy
+of J1331 was the same as that identified for the ROSAT-selected
+TDE candidate, RXJ133157.6324319.7, first detected in outburst
+in 1993, and recently presented in Hampel et al. (2022), with the
+finder chart for these transients presented in Fig. A1. The eRASS5
+detection of J1331 thus suggested the remarkable rebrightening of
+a previously known TDE candidate, ∼29 years after the outburst
+detected by ROSAT.
+2.1 eROSITA
+Using the eSASS task SRCTOOL (eSASSusers_211214; Brunner et al.
+2022), source (and background) spectra and lightcurves were ex-
+tracted from a 60" radius source region centred on the eRASS5
+inferred position, with background counts extracted from a circular
+annulus with inner and outer radii of 140" and 240", respectively.
+eROSITA scanned the position of J1331 eight times during
+eRASS5, with each scan separated by ∼4 hours, thus spanning a
+∼28 hour window in total. During this time, J1331 was observed
+to be persistently bright (Fig. D2), as opposed to showing a short-
+lived flaring, and was clearly detected above background in each
+observation.
+The eRASS5 X-ray spectra were then fitted using the Bayesian X-
+ray Analysis software (BXA; Buchner et al. 2014), which connects
+the nested sampling algorithm UltraNest (Buchner 2021) with the
+fitting environment XSPEC (Arnaud 1996). The source and back-
+ground spectra were jointly fit with a source plus background model,
+with the latter using the Principal Component Analysis (PCA) back-
+ground modelling first described in Simmonds et al. (2018), and as
+2 http://svo2.cab.inta-csic.es/theory/fps/
+also applied to AT 2019avd in Malyali et al. (2021). The eRASS5
+spectrum is well fitted by a tbabs*zbbody model (Fig. D1), with the
+Galactic equivalent neutral hydrogen column density, 𝑁H, fixed to
+3.84 × 1020 cm−2, the value along the line of sight to J1331 in HI4PI
+Collaboration: et al. (2016), and 𝑘𝑇 = 0.115+0.007
+−0.007 keV. A fit with a
+power-law (tbabs*zpowerlaw) leaves large residuals between the
+observed data and model above 1 keV. When using the best fitting
+tbabs*zbbody model described above, the eRASS5 observed (un-
+absorbed) 0.2–2 keV flux for J1331 is (6.0±0.7)×10−13 erg s−1 cm−2
+((8 ± 1) × 10−13 erg s−1 cm−2), translating to an unabsorbed 0.2–
+2 keV luminosity of (5.5 ± 0.7) × 1042 erg s−1.
+J1331 was not detected in eRASS1–4, with 2𝜎 upper limits on
+the 0.2–2 keV count rate of 0.016, 0.03, 0.07 and 0.03 cts s−1 in
+each successive eRASS (see Table D1 for a full log of the X-ray
+observations of J1331). These count rate upper limits were then
+converted to 0.2–2 keV flux upper limits using the best fitting spectral
+parameters to the eRASS5 spectrum described above.
+2.2 NICER XTI
+Follow-up observations of J1331 were obtained with the X-ray Tim-
+ing Instrument (XTI) on board the Neutron Star Interior Composition
+Explorer observatory (NICER; Gendreau et al. 2016) through pre-
+approved ToOs (PI: Z. Liu). NICER observations commenced ∼4
+days after the last eRASS5 observation, and continued for the next
+15 days on a near daily basis (Table D1). We first generated cleaned
+and screened event files using the nicerl2 task (with default recom-
+mended parameters), before using nibackgen3C50 (Remillard et al.
+2022) to generate total and background spectra for each observation
+ID (GTIs were filtered out using hbgcut=0.05 and s0cut=2, as
+recommended in Remillard et al. 2022). ARF and RMF files were
+subsequently generated using the tasks nicerarf and nicerrmf,
+and the X-ray spectra were binned using the Kaastra & Bleeker
+(2016) method to a minimum of 20 counts per bin. The total and
+background count rates were then estimated in the 0.4–2 keV band3.
+J1331 is not detected at 2 sigma above background in each OBSID
+(Fig. D3), with 2𝜎 upper limits on the source count rates, inferred
+using 𝐶𝑅tot +2𝜎, with 𝐶𝑅tot the total measured count rate, and 𝜎 the
+estimated error on 𝐶𝑅tot. The 0.4–2 keV count rates were converted
+to 0.2–2 keV fluxes (Table D1) assuming the eRASS5 spectral model
+(section 2.1). NICER observations rule out a further brightening be-
+yond eRASS5, or a persistently bright source that rapidly ‘cuts-off’
+in brightness by the time of the XMM observation (section 2.3).
+2.3 XMM
+J1331 was later observed by XMM (P.I. Z. Liu) on 2022-02-07 (de-
+noted XMM1), ∼16 days after the last eRASS5 observation, and
+also on 2022-08-06 (denoted XMM2). Observations were carried
+out with the medium filter on PN, MOS1 and MOS2. The XMM data
+were reduced using HEASOFT v6.29, SAS version 20211130_0941,
+and the latest calibration data files (CALDB v20210915). Follow-
+ing standard XMM data reduction procedures, calibrated event files
+were first generated from the Observation Data Files (ODF) using
+the SAS tasks emproc and epproc for the MOS and PN cameras
+respectively. Then, periods of high background flaring were filtered
+3 The 0.4 keV lower bound here was chosen to reduce contamination from
+any incompletely modelled optical loading.
+MNRAS 000, 1–9 (2015)
+
+Repeated partial tidal disruption flares from a quiescent galaxy
+3
+10
+41
+10
+42
+10
+43
+LX [erg s
+1]
+48000
+50000
+52000
+54000
+56000
+58000
+60000
+MJD
+10
+14
+10
+13
+10
+12
+FX [erg s
+1 cm
+2]
+49005
+49010
+49015
+10
+13
+10
+12
+1992
+1996
+2000
+2004
+2008
+2012
+2016
+2020
+Year
+ROSAT
+XRT
+eROSITA
+XMM
+Figure 1. Long-term 0.2–2 keV lightcurve of J1331, with circular and triangle markers representing observed fluxes and 2𝜎 upper limits, respectively. The
+initial outburst was detected by ROSAT in 1993, before being observed by eROSITA in 2022 to have rebrightened to a similar 0.2–2 keV observed flux. The
+X-ray spectra remained ultra-soft in each observation where the source was detected. For plotting clarity, we include the time-averaged flux measurement for
+eRASS5, and omit the NICER upper limits.
+out4. For XMM1 (XMM2), this resulted in only 4.1ks (25.7 ks),
+12.8 ks (30.7 ks) and 11.8 ks (30.2 ks) of usable exposure time
+for PN, MOS1 and MOS2, respectively. In the subsequent analysis,
+only events with PATTERN<=4 and FLAG==0 were extracted for PN,
+whilst PATTERN<=12 and FLAG==0 filtering was applied for MOS1
+and MOS2.
+For XMM1, no source is detected within 30" of the host galaxy
+position in PN and MOS1 with detection likelihood, DETML, above
+3, when running the standard XMM source detection pipeline in the
+0.2–2 keV band on the PN, MOS1, and MOS2 images. However, a
+source was detected in MOS2 at (RAJ2000, DecJ2000)=(13h31m58s,
+-32◦43′19′′), with a 1𝜎 positional uncertainty of 2′′, consistent with
+the ROSAT and eROSITA positions (Fig. A1). The DETML for this
+source is low (10.3), and the estimated observed 0.2–2 keV flux in
+the emldetect output is (8±3) ×10−15 erg s−1 cm−2, ∼75× fainter
+than the eRASS5 observed flux.
+Given the uncertain detection of the system across all three EPIC
+cameras, we computed a 2𝜎 upper limit on the 0.2–2 keV count
+rate using the SAS task eupper. This was done using the 0.2–2 keV
+band images, exposure and background maps for each camera, and
+a 30" radius circular extraction region for the source counts (centred
+on the Gaia position of the host galaxy). For XMM1, this yielded
+upper limits of 0.006 ct s−1, 0.0014 ct s−1 and 0.002 ct s−1 for
+PN, MOS1 and MOS2, respectively. We conservatively estimate the
+upper limit for the XMM observation to that inferred from the MOS2
+data, which corresponds to a 0.2–2 keV observed (unabsorbed) flux
+of 1 × 10−14 erg s−1 cm−2 (2 × 10−14 erg s−1 cm−2), assuming the
+4 https://www.cosmos.esa.int/web/xmm-newton/
+sas-thread-epic-filterbackground
+spectral model inferred from the eRASS5 observation. The same
+procedure was repeated for XMM2, where we inferred upper limits
+of 0.003 ct s−1, 0.0014 ct s−1 and 0.0010 ct s−1 for PN, MOS1 and
+MOS2, respectively, translating to 2𝜎 upper limits on the observed
+(unobserved) flux of 6×10−15 erg s−1 cm−2 (1×1014 erg s−1 cm−2).
+2.4 Swift XRT
+Additional Swift XRT (Burrows et al. 2005) observations of J1331
+were performed between 2022-02-27 and 2022-08-245. The XRT
+observations were performed in photon counting mode, with the
+data analysed using the UK Swift Science Data Centre’s (UKSSDC)
+online XRT product building tool (Evans et al. 2007, 2009). No
+source was detected in the 0.3–2 keV band at the position of J1331 in
+any follow-up observation.The 0.3–2 keV count rates were converted
+to 0.2–2 keV fluxes using webPIMMs6, assuming the same spectral
+model as from the eROSITA eRASS5 detection, with the fluxes
+presented in Table D1.
+2.5 Archival X-ray observations
+A
+detailed
+analysis
+of
+the
+ultra-soft
+outburst
+from
+RXJ133157.6324319.7,
+detected
+by
+pointed
+ROSAT
+PSPC
+observations in the early 1990s, was previously performed in
+Hampel et al. (2022). In summary, the flare was characterised by an
+5 The delay between the eRASS5 and Swift observations stemmed from the
+January 2022 reaction wheel failure on-board the Swift observatory.
+6 https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/
+w3pimms.pl
+MNRAS 000, 1–9 (2015)
+
+4
+Adam Malyali et al.
+8x increase in the 0.1–2.4 keV flux, relative to a 2𝜎 upper limit, over
+an 8 day period (and a net increase in the same band by a factor of
+at least 40 relative to the deepest upper limit available). The X-ray
+spectrum at peak observed brightness was well fitted by a blackbody
+with 𝑘𝑇 = 0.11 ± 0.03 keV. The system was then not detected in two
+PSPC observations ∼165 days later, where it had faded by a factor
+of at least 30 relative to the peak observed ROSAT flux.
+To construct a long-term 0.2–2 keV lightcurve, the 0.1–2.4 keV
+ROSAT PSPC lightcurve data in Table 1 of Hampel et al. (2022) was
+converted into 0.2–2 keV band fluxes using webPIMMS, assuming
+the best fitting spectral model to the ROSAT spectrum found in
+Hampel et al. (2022). Then, the 2𝜎 upper limits from ROSAT Survey,
+XMM Slew and Swift XRT observations were computed using the
+High-Energy Lightcurve Generator server (HILIGT; Saxton et al.
+2021; König et al. 2021); the archival fluxes are presented in Fig. 1
+and Table D1.
+2.6 UV, optical and mid-infrared photometry
+J1331 was observed both before (Section 2.5) and after (Section 2.4)
+the eRASS5-detected outburst by Swift XRT and UVOT (UVM2
+filter; Roming et al. 2005). To search for transient UV emission,
+aperture photometry was performed on the level 2 UVOT sky im-
+ages (downloaded from the UKSSDC) using the uvotsource task
+(HEASOFT v6.29, CALDB v20201215). Source counts were ex-
+tracted from a circular aperture of 5′′ radius, centred on the Gaia
+position of the host of J1331, and background counts were extracted
+from a source-free region of radius 15′′. The measured UVM2 mag-
+nitudes in the follow-up observations are consistent with the archival
+measured UVM2 magnitudes on the 2018-04-18, 2018-04-22, 2018-
+04-26 (Table E1).
+No significant optical variability is seen in the ∼6 years before
+the eRASS5 outburst (57500≲ MJD ≲59500) in the forced photom-
+etry lightcurve provided by ATLAS (Tonry et al. 2018) (Fig. E1).
+Lastly, we note that no major variability is detected above the host
+galaxy emission within the NEOWISE mid-infrared lightcurve be-
+tween MJD∼56680 and 59400 (Fig. E1), which was generated using
+the procedure described in section 3.2 of Malyali et al. (2021).
+2.7 Radio
+We observed the coordinates of J1331 on 2022 Mar 02 with the
+Australia Telescope Compact Array (ATCA) radio telescope in 6 km
+configuration, using the 4cm dual receiver with central frequencies
+5.5/9 GHz, each with a 2 GHz bandwidth split into 2049×1 MHz
+spectral channels, and for a total of 150 min on source. Data were
+reduced following standard procedures in the Common Astronomy
+Software Applications (McMullin et al. 2007; CASA-TEAM et al.
+2022). We used 1934-638 for flux and bandpass calibration and 1336-
+260 for phase calibration. Images of the target field were created using
+the CASA task tclean. No source was detected at the location of
+J1331 at either frequency band with a 3𝜎 upper limit of 73.5𝜇Jy/bm
+at 5.5 GHz and 54𝜇Jy/bm at 9 GHz. Additionally, no source was
+detected in a stacked 5.5 and 9 GHz image, with a 3𝜎 upper limit of
+57.9𝜇Jy/bm at a central frequency of 7.3 GHz.
+3 DISCUSSION
+Comparing the X-ray lightcurve of J1331 with other ultra-soft nu-
+clear transients (Fig. D4) from galaxies that were recently quiescent,
+or hosted low luminosity AGN, then J1331 decays faster than the
+majority of other X-ray bright TDEs7, but decays over much longer
+timescales than the bursts typically seen in QPEs (burst durations
+≲30 ks, or ≲0.3 days; Miniutti et al. 2019; Giustini et al. 2020;
+Arcodia et al. 2021, 2022).
+Given the quiescent nature of the host galaxy, and the ultra-soft
+X-ray spectrum, an AGN origin for J1331 is disfavoured. We also
+rule out a mechanism similar to that producing the X-ray flares ob-
+served in Sgr A* (e.g. Neilsen et al. 2013; Ponti et al. 2015; Yuan &
+Wang 2016; Ponti et al. 2017; Mossoux et al. 2020), as the latter are
+clearly observationally distinct to J1331, with respect to the flaring
+timescales (Sgr A* flare durations ≲ 104 s; Mossoux et al. 2020),
+spectral properties (flaring X-ray emission in Sgr A* is hard and
+likely synchrotron, e.g. Ponti et al. 2017), and peak observed lumi-
+nosity (bolometric luminosity of Sgr A* is ∼ 1036 erg s−1; Genzel
+et al. 2010). Arguments against a Galactic origin for this system have
+previously been presented in Hampel et al. (2022).
+Ultra-soft X-ray flares from quiescent galaxies have previously
+been considered as a reliable signature of a TDE (e.g. Zabludoff et al.
+2021). However, the current theoretically predicted TDE rates are
+≳ 10−4 yr−1 galaxy−1 (Stone et al. 2020), so it would be exceptionally
+unlikely to have observed two independent tidal disruption flares
+occuring within the same galaxy over a ∼30 year timescale (Poisson
+probability ∼ 5 × 10−6; Fig. C2); a more exotic class of TDE would
+need to be invoked to explain J1331.
+One such possibility, discussed in Hampel et al. (2022), is that
+J1331 was produced by a TDE involving a supermassive black hole
+binary (SMBHB). This scenario was partly proposed in an attempt
+to explain the fast X-ray brightening observed by ROSAT, since
+such TDEs may have highly non-monotonic decays of their X-ray
+lightcurves. This stems from the gravitational interaction between
+the companion BH and the debris streams, which may cause large
+perturbations to the orbits of the less bound debris and cause their
+chaotic evolution, as well as a complex evolution of the accretion
+rate over time. Liu et al. (2014); Ricarte et al. (2016); Coughlin et al.
+(2017) predict these systems to show sharp dips and rises in the X-
+ray lightcurve rate (of ∼1–2 orders of magnitude), on timescales of
+the order of the binary orbital period (Liu et al. 2014; Ricarte et al.
+2016), although Coughlin et al. (2017) find highly variable accretion
+rates between different simulation runs and over timescales shorter
+than the SMBHB orbital periods (i.e. there still seems to be quite
+large uncertainties in the theoretically predicted lightcurves of TDEs
+involving SMBHBs).
+Under the SMBHB scenario, both the eROSITA and ROSAT obser-
+vations would have had to have sampled a ‘dipping’, or ‘brightening
+from a dip’, phase of the X-ray lightcurve, respectively. For binary
+orbital periods of the order of ∼months, assuming ∼mpc binary sep-
+aration as in Liu et al. (2014), then it would be quite fortuitous for us
+to have observed such behaviour. Furthermore, there is importantly
+no evidence for late time X-ray rebrightening episodes in the months
+after each outburst, as seen by XMM and Swift (Fig. 1), which one
+might expect to have observed given that the accretion rate is pre-
+dicted to eventually revert back to the 𝑡−5/3 decay following ‘dips’
+(e.g. Fig. 12 in Coughlin et al. 2017). We would therefore disfavour
+J1331 being caused by a full TDE around a SMBHB, given the fine
+tuning needed in order to match observations.
+A more feasible scenario is that both outbursts were driven by a
+partial tidal disruption event (pTDE), potentially of the same object.
+Unless the pTDE rate is orders of magnitude larger than currently
+7 Ignoring short timescale flaring behaviour seen in some TDE candidates,
+such as AT 2019ehz (van Velzen et al. 2021b).
+MNRAS 000, 1–9 (2015)
+
+Repeated partial tidal disruption flares from a quiescent galaxy
+5
+10
+40
+10
+41
+10
+42
+10
+43
+LX [erg s
+1]
+10
+1
+10
+2
+MJD - 59581
+10
+15
+10
+14
+10
+13
+10
+12
+FX [erg s
+1 cm
+2]
+t
+5/3
+t
+9/4
+t
+4
+Figure 2. Zoom-in on the first eROSITA-detected outburst in 2022, along
+with multiple power-law decay slopes plotted in grey dashed lines. The decay
+slope appears to be much steeper than the canonical 𝑡−5/3 decay predicted
+for TDEs with a uniform distribution of specific energies, and appears more
+consistent with a 𝑡−4 decay, as predicted in Ryu et al. (2020). We assume a
+peak MJD of 59593 for the X-ray outburst, and roughly estimate the MJD of
+disruption to be 59581 (section C). The markers follow the same legend as
+for Fig. 1.
+estimated in the literature (Stone & Metzger 2016; Chen & Shen
+2021; Zhong et al. 2022), then both outbursts would likely be re-
+lated to the same star being disrupted by the same black hole (i.e.
+the star should have survived the initial encounter). Considering that
+the recurrence timescale of J1331 is ≲ 30 years, then it is also diffi-
+cult to reconcile this with theoretical predictions for the recurrence
+timescales of flares in pTDEs where the star was initially scattered
+onto a parabolic orbit around the black hole (≳ 400 years, e.g. Ryu
+et al. 2020). Instead, the flaring may have been driven by the repeated
+stripping of a star on an elliptical orbit by the disrupting SMBH (see
+Hayasaki et al. 2013 for a discussion on potential origins for such
+stars). This scenario would be further supported by both the relatively
+small amount of inferred energy emitted in the eROSITA-detected
+outburst8 of (5+6
+−3) × 1049 erg, corresponding to an accreted mass of
+(5+7
+−2) × 10−4(𝜖/0.05)−1 M⊙, where 𝜖 is the radiative efficiency of
+accretion, and also by the extremely low 𝐿X at late-times (as sug-
+gested by the non-detection and deep upper limits in XMM2), since
+elliptical TDEs are predicted to produce short-lived, finite accretion
+bursts (Hayasaki et al. 2013). Given this, and that the radio obser-
+vations were taken ∼40 days after the eRASS5 flare (section 2.7),
+then we note that we may have missed any associated jet or out-
+flow launched in this event, as seen in other TDE candidates (e.g.
+Goodwin et al. 2022).
+The case for a repeated pTDE is further enhanced by the fast rise
+and decay timescales seen with ROSAT and eROSITA. Compared
+with full disruptions, pTDEs only strip the outermost layers of the
+star, with the specific energy distribution of the debris, d𝑀/d𝐸,
+differing from full TDEs (e.g. Coughlin & Nixon 2019; Miles et al.
+2020; Ryu et al. 2020). Since the mass fallback rate, �𝑀fb(𝑡), scales
+∝ d𝑀/d𝐸, then �𝑀fb(𝑡) is also predicted to differ between full and
+pTDEs. Ryu et al. (2020) find that the narrower spreads in d𝑀/d𝐸
+for pTDEs can yield �𝑀fb(𝑡) ∝ 𝑡−𝑝, where 𝑝 ∼ 2−5, more consistent
+with what is observed in J1331 (Fig. 2), and much steeper than a
+canonical 𝑡−5/3 decline predicted for the mass fallback rate in full
+TDEs (Rees 1988; Phinney 1989).
+Lastly, although the mass fallback in weak pTDEs may evolve
+over shorter timescales relative to full TDEs, the viscous timescale,
+8 Assuming a similar temporal evolution for both the eROSITA-detected and
+ROSAT-detected outbursts- see section C.
+𝑡visc, still needs to be shorter than the minimum orbital period of the
+stellar debris so that the X-ray luminosity traces the mass fallback
+rate (assuming a constant radiative efficiency, negligible obscuration
+of the soft X-rays, and negligible disc cooling). Considering 𝑡visc ∼
+𝛼−1(𝐻/𝑅)−2Ω−1(𝑟), where 𝛼 is the viscosity parameter (Shakura
+& Sunyaev 1973), 𝐻 and 𝑅 the scale height and width of the disc,
+and Ω−1(𝑟) the orbital period at distance 𝑟 from the black hole,
+then 𝑡visc ∼ 0.4(𝛼/0.1)−1(𝐻/𝑅)−2 days at the circularisation radius
+(∼ 2𝑅tidal/𝛽, where 𝑅tidal and 𝛽 are the tidal radius and impact
+parameter for the disruption). A geometrically thick disc (𝐻/𝑅 ∼ 1),
+as may be expected to form for super-Eddington mass fallback rates,
+would be needed to reproduce accretion timescales of the order ∼days
+as seen in J1331. However, it is currently unclear how the stellar
+debris might circularise so efficiently in a weak pTDE (see Bonnerot
+& Stone 2021 for a review on accretion flow formation in TDEs),
+and we also highlight here that similar concerns have recently been
+raised for explaining the short X-ray flare durations observed in QPEs
+via an accretion origin (e.g. Krolik & Linial 2022; Lu & Quataert
+2022). Although future simulations would likely be needed to explore
+the debris circularisation in J1331-like events, alternative origins for
+the X-ray emission may be from compression shocks of the debris
+streams at pericentre (e.g. Steinberg & Stone 2022), or circularisation
+shocks from debris stream collisions (Krolik & Linial 2022; Lu &
+Quataert 2022).
+4 SUMMARY
+J1331 is a repeating X-ray transient associated to a quiescent galaxy
+at 𝑧 = 0.05189, which we consider to be consistent with a scenario
+involving two weak pTDEs. Whilst several previously reported pTDE
+candidates have occurred in galaxies hosting an AGN, we highlight
+that the host of J1331 is quiescent. The main properties of J1331 can
+be summarised as follows:
+(i) J1331 was first detected by ROSAT in 1993 (Hampel et al.
+2022), where it had shown an ultra-soft (𝑘𝑇 = 0.11 ± 0.03 keV)
+flaring by a factor of at least 40 relative to a previous 2𝜎 upper limit.
+The outburst also showed a fast rise, where it had brightened by a
+factor of eight over an 8 day period. The system was subsequently not
+detected in a deep pointed ROSAT observation ∼165 days afterwards,
+as well as in XMM Slew, and Swift XRT observations performed
+between 2006 and 2018 (Table D1).
+(ii) After not being detected by eROSITA in its first four eRASS,
+J1331 was observed to have brightened in eRASS5 to a 0.2–2 keV
+flux of (6.0 ± 0.7) × 10−13 erg s−1 cm−2. The eRASS5 spectrum
+is ultra-soft (𝑘𝑇 = 0.115+0.007
+−0.007 keV), and is consistent with the 𝑘𝑇
+inferred from the ROSAT-observed flare in 1993.
+(iii) J1331 was not detected during pointed XMM observations
+and Swift XRT observations when followed up after the eRASS5
+detection; the first (second) XMM observation constrains the 0.2–
+2 keV flux to decay by a factor of ≳40 (≳100) over a 17 (∼200)
+day period after the eRASS5 observation. The faint 0.2–2 keV X-ray
+luminosities (< 7×1040 erg s−1, unabsorbed) at ∼ 200 days post-peak
+brightness, inferred via the second XMM observation (Table D1),
+may be due to a late-time drop off in the mass fallback rate once the
+disruption episode is over.
+(iv) Combined with the fast rise timescale seen by ROSAT, then
+J1331-like outbursts are short lived (rise and decay timescales of
+6+1
+−1 days and 3.9+0.1
+−0.1 days, respectively; appendix C) and evolve over
+shorter timescales relative to full TDEs.
+(v) J1331 has only been observed to show transient emission in
+MNRAS 000, 1–9 (2015)
+
+6
+Adam Malyali et al.
+the 0.2–2 keV band, with no transient optical, UV, or radio emission
+observed in follow-up observations.
+We conclude by noting that J1331 appears to fill in the continuum
+of observed soft X-ray outbursts from quiescent galaxies, lying in be-
+tween QPEs and TDEs with respect to its rise and decay timescales
+(Fig. D4), although the recurrence timescales are much longer than
+in the current sample of QPEs. Additional follow-up observations
+will be scheduled in order to more tightly constrain the recurrence
+timescales of outbursts from J1331. Future planned X-ray missions
+geared towards exploiting the X-ray transient sky, such as the Einstein
+Probe (Yuan et al. 2018), will likely be sensitive towards detecting
+similar partial disruptions; for these missions, the eROSITA All-Sky
+survey data may play an important role by providing a long-term
+baseline towards which new candidates can be identified. Given the
+faster decay timescales of J1331-like systems, then we would advo-
+cate promptly triggering high-cadence X-ray follow-up in order to
+better constrain the evolution of the accretion rate in future candi-
+dates.
+ACKNOWLEDGEMENTS
+AM thanks Taeho Ryu for very useful discussions whilst preparing
+the manuscript. AM acknowledges support by DLR under the grant
+50 QR 2110 (XMM_NuTra, PI: Z. Liu). This work was supported by
+the Australian government through the Australian Research Council’s
+Discovery Projects funding scheme (DP200102471). We would like
+to thank the referee for a constructive report that improved the quality
+of the paper.
+This work is based on data from eROSITA, the soft X-ray instru-
+ment aboard SRG, a joint Russian-German science mission supported
+by the Russian Space Agency (Roskosmos), in the interests of the
+Russian Academy of Sciences represented by its Space Research In-
+stitute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt
+(DLR). The SRG spacecraft was built by Lavochkin Association
+(NPOL) and its subcontractors, and is operated by NPOL with sup-
+port from the Max Planck Institute for Extraterrestrial Physics (MPE).
+The development and construction of the eROSITA X-ray instru-
+ment was led by MPE, with contributions from the Dr. Karl Re-
+meis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg),
+the University of Hamburg Observatory, the Leibniz Institute for
+Astrophysics Potsdam (AIP), and the Institute for Astronomy and
+Astrophysics of the University of Tübingen, with the support of DLR
+and the Max Planck Society. The Argelander Institute for Astronomy
+of the University of Bonn and the Ludwig Maximilians Universität
+Munich also participated in the science preparation for eROSITA.
+The eROSITA data shown here were processed using the eSASS
+software system developed by the German eROSITA consortium.
+This work made use of data supplied by the UK Swift Science
+Data Centre at the University of Leicester.
+The Australia Telescope Compact Array is part of the Aus-
+tralia Telescope National Facility (https://ror.org/05qajvd42)
+which is funded by the Australian Government for operation as a
+National Facility managed by CSIRO. We acknowledge the Gomeroi
+people as the traditional owners of the Observatory site.
+The Legacy Surveys consist of three individual and complemen-
+tary projects: the Dark Energy Camera Legacy Survey (DECaLS;
+Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the
+Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID #2015A-0801;
+PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Sur-
+vey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS
+and MzLS together include data obtained, respectively, at the Blanco
+telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIR-
+Lab; the Bok telescope, Steward Observatory, University of Arizona;
+and the Mayall telescope, Kitt Peak National Observatory, NOIR-
+Lab. Pipeline processing and analyses of the data were supported by
+NOIRLab and the Lawrence Berkeley National Laboratory (LBNL).
+The Legacy Surveys project is honored to be permitted to conduct
+astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with
+particular significance to the Tohono O’odham Nation.
+NOIRLab is operated by the Association of Universities for Re-
+search in Astronomy (AURA) under a cooperative agreement with
+the National Science Foundation. LBNL is managed by the Regents
+of the University of California under contract to the U.S. Department
+of Energy.
+This project used data obtained with the Dark Energy Camera
+(DECam), which was constructed by the Dark Energy Survey (DES)
+collaboration. Funding for the DES Projects has been provided by the
+U.S. Department of Energy, the U.S. National Science Foundation,
+the Ministry of Science and Education of Spain, the Science and
+Technology Facilities Council of the United Kingdom, the Higher
+Education Funding Council for England, the National Center for
+Supercomputing Applications at the University of Illinois at Urbana-
+Champaign, the Kavli Institute of Cosmological Physics at the Uni-
+versity of Chicago, Center for Cosmology and Astro-Particle Physics
+at the Ohio State University, the Mitchell Institute for Fundamental
+Physics and Astronomy at Texas A&M University, Financiadora de
+Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Fi-
+nanciadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
+de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Na-
+cional de Desenvolvimento Cientifico e Tecnologico and the Minis-
+terio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungs-
+gemeinschaft and the Collaborating Institutions in the Dark Energy
+Survey. The Collaborating Institutions are Argonne National Labo-
+ratory, the University of California at Santa Cruz, the University of
+Cambridge, Centro de Investigaciones Energeticas, Medioambien-
+tales y Tecnologicas-Madrid, the University of Chicago, University
+College London, the DES-Brazil Consortium, the University of Ed-
+inburgh, the Eidgenossische Technische Hochschule (ETH) Zurich,
+Fermi National Accelerator Laboratory, the University of Illinois at
+Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC),
+the Institut de Fisica d’Altes Energies, Lawrence Berkeley National
+Laboratory, the Ludwig Maximilians Universitat Munchen and the
+associated Excellence Cluster Universe, the University of Michigan,
+NSF’s NOIRLab, the University of Nottingham, the Ohio State Uni-
+versity, the University of Pennsylvania, the University of Portsmouth,
+SLAC National Accelerator Laboratory, Stanford University, the Uni-
+versity of Sussex, and Texas A&M University.
+BASS is a key project of the Telescope Access Program (TAP),
+which has been funded by the National Astronomical Observatories
+of China, the Chinese Academy of Sciences (the Strategic Prior-
+ity Research Program “The Emergence of Cosmological Structures”
+Grant # XDB09000000), and the Special Fund for Astronomy from
+the Ministry of Finance. The BASS is also supported by the Exter-
+nal Cooperation Program of Chinese Academy of Sciences (Grant
+# 114A11KYSB20160057), and Chinese National Natural Science
+Foundation (Grant # 12120101003, # 11433005).
+The Legacy Survey team makes use of data products from the
+Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE),
+which is a project of the Jet Propulsion Laboratory/California Insti-
+tute of Technology. NEOWISE is funded by the National Aeronautics
+and Space Administration.
+The Legacy Surveys imaging of the DESI footprint is supported
+by the Director, Office of Science, Office of High Energy Physics
+MNRAS 000, 1–9 (2015)
+
+Repeated partial tidal disruption flares from a quiescent galaxy
+7
+of the U.S. Department of Energy under Contract No. DE-AC02-
+05CH1123, by the National Energy Research Scientific Comput-
+ing Center, a DOE Office of Science User Facility under the same
+contract; and by the U.S. National Science Foundation, Division of
+Astronomical Sciences under Contract No. AST-0950945 to NOAO.
+M.K. acknowledges support from DFG grant KR 3338/4-1. D.H.
+is supported by DLR grant FKZ 50OR2003.
+DATA AVAILABILITY
+The eRASS1-4 data taken within the German half of the eROSITA
+sky is currently planned to be made public by Q2 2024, whilst
+the eRASS5 data is scheduled to become public by Q2 2026. The
+Swift data is available to download through the UK Swift Data Sci-
+ence website9, whilst the NICER data is accessible through NASA’s
+HEASARC interface10. Publicly available ATLAS data can be ac-
+cessed through the ATLAS forced photometry service11, and NEO-
+WISE lightcurves can be accessed through the IRSA web portal12.
+ATCA data are stored in the Australia Telescope Online Archive13,
+and will become publicly accessible 18 months from the date of ob-
+servation. The XMM data will become public after the propietory
+period expires (2023-08-30). Follow-up optical spectra will likely
+remain private at least until the release of the forthcoming eROSITA-
+selected TDE population paper, but could be made available upon
+reasonable request.
+REFERENCES
+Alexander K. D., van Velzen S., Horesh A., Zauderer B. A., 2020, Space
+Science Reviews, 216, 81
+Antonini F., Lombardi J. C., Merritt D., 2011, The Astrophysical Journal,
+731, 128
+Arcodia R., et al., 2021, Nature, 592, 704
+Arcodia R., et al., 2022, Astronomy & Astrophysics, 662, A49
+Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society
+of the Pacific Conference Series Vol. 101, Astronomical Data Analysis
+Software and Systems V. p. 17
+Assef R. J., Stern D., Noirot G., Jun H. D., Cutri R. M., Eisenhardt P. R. M.,
+2018, The Astrophysical Journal Supplement Series, 234, 23
+Blanchard P. K., et al., 2017, The Astrophysical Journal, 843, 106
+Bonnerot C., Stone N. C., 2021, Space Science Reviews, 217, 16
+Bright J. S., et al., 2018, Monthly Notices of the Royal Astronomical Society,
+475, 4011
+Brown T. M., et al., 2013, Publications of the Astronomical Society of the
+Pacific, 125, 1031
+Brunner H., et al., 2022, Astronomy & Astrophysics, 661, A1
+Buchner J., 2021, UltraNest – a robust, general purpose Bayesian inference
+engine, http://arxiv.org/abs/2101.09604
+Buchner J., et al., 2014, Astronomy & Astrophysics, 564, A125
+Burrows D. N., et al., 2005, Space Science Reviews, 120, 165
+CASA-TEAM et al., 2022, CASA, the Common Astronomy Software Ap-
+plications for Radio Astronomy, doi:10.1088/1538-3873/ac9642, http:
+//arxiv.org/abs/2210.02276
+Campana S., Mainetti D., Colpi M., Lodato G., D’Avanzo P., Evans P. A.,
+Moretti A., 2015, Astronomy & Astrophysics, 581, A17
+9 https://www.swift.ac.uk/archive/index.php
+10 https://heasarc.gsfc.nasa.gov/docs/nicer/nicer_archive.
+html
+11 https://fallingstar-data.com/forcedphot/
+12 https://irsa.ipac.caltech.edu/applications/wise/
+13 https://atoa.atnf.csiro.au/
+Cannizzaro G., et al., 2021, Monthly Notices of the Royal Astronomical
+Society, 504, 792
+Cardelli J. A., Clayton G. C., Mathis J. S., 1989, \apj, 345, 245
+Chen J.-H., Shen R.-F., 2021, The Astrophysical Journal, 914, 69
+Chen J.-H., Dou L.-M., Shen R.-F., 2022, The Astrophysical Journal, 928, 63
+Childress M. J., Vogt F. P. A., Nielsen J., Sharp R. G., 2014, Astrophysics
+and Space Science, 349, 617
+Coughlin E. R., Nixon C. J., 2019, The Astrophysical Journal, 883, L17
+Coughlin E. R., Armitage P. J., Nixon C., Begelman M. C., 2017, Monthly
+Notices of the Royal Astronomical Society, 465, 3840
+Dale J. E., Davies M. B., Church R. P., Freitag M., 2009, Monthly Notices of
+the Royal Astronomical Society, 393, 1016
+Dopita M., et al., 2010, Astrophysics and Space Science, 327, 245
+Drake A. J., et al., 2011, The Astrophysical Journal, 735, 106
+Evans P. A., et al., 2007, Astronomy & Astrophysics, 469, 379
+Evans P. A., et al., 2009, Monthly Notices of the Royal Astronomical Society,
+397, 1177
+Frederick S., et al., 2019, The Astrophysical Journal, 883, 31
+Frederick S., et al., 2021, The Astrophysical Journal, 920, 56
+Gaia Collaboration et al., 2021, Astronomy & Astrophysics, 649, A6
+Gendreau
+K.
+C.,
+et
+al.,
+2016.
+Edinburgh,
+United
+Kingdom,
+p.
+99051H,
+doi:10.1117/12.2231304,
+http://proceedings.
+spiedigitallibrary.org/proceeding.aspx?doi=10.1117/
+12.2231304
+Genzel R., Eisenhauer F., Gillessen S., 2010, Reviews of Modern Physics,
+82, 3121
+Giustini M., Miniutti G., Saxton R. D., 2020, Astronomy & Astrophysics,
+636, L2
+Goodwin A. J., et al., 2022, Monthly Notices of the Royal Astronomical
+Society, 511, 5328
+Grupe D., Beuermann K., Mannheim K., Bade N., 1995, Astronomy & As-
+trophysics, 299, L5
+Grupe D., Thomas H.-C., Beuermann K., 2001, Astronomy & Astrophysics,
+367, 470
+Grupe D., Komossa S., Saxton R., 2015, The Astrophysical Journal, 803, L28
+HI4PI Collaboration: et al., 2016, Astronomy & Astrophysics, 594, A116
+Hampel J., Komossa S., Greiner J., Reiprich T. H., Freyberg M., Erben T.,
+2022, Research in Astronomy and Astrophysics, 22, 055004
+Hayasaki K., Stone N., Loeb A., 2013, Monthly Notices of the Royal Astro-
+nomical Society, 434, 909
+Hinkle J. T., et al., 2020, Monthly Notices of the Royal Astronomical Society,
+500, 1673
+Kaastra J. S., Bleeker J. A. M., 2016, Astronomy & Astrophysics, 587, A151
+Kettlety T., et al., 2018, Monthly Notices of the Royal Astronomical Society,
+473, 776
+Krolik J. H., Linial I., 2022, Quasi-Periodic Erupters: A Stellar Mass-Transfer
+Model for the Radiation, http://arxiv.org/abs/2209.02786
+König O., et al., 2021, HILIGT, Upper Limit Servers II – Implementing the
+data servers, http://arxiv.org/abs/2111.13563
+Liu F. K., Li S., Komossa S., 2014, The Astrophysical Journal, 786, 103
+Liu Z., Li D., Liu H.-Y., Lu Y., Yuan W., Dou L., Shen R.-F., 2020, The
+Astrophysical Journal, 894, 93
+Liu Z., et al., 2022, arXiv:2208.12452 [astro-ph]
+Lu W., Quataert E., 2022, Quasi-periodic eruptions from mildly eccentric un-
+stable mass transfer in galactic nuclei, http://arxiv.org/abs/2210.
+08023
+Mainzer A., et al., 2014, The Astrophysical Journal, 792, 30
+Malyali A., et al., 2021, Astronomy & Astrophysics, 647, A9
+McMullin J. P., Waters B., Schiebel D., Young W., Golap K., 2007, in Shaw
+R. A., Hill F., Bell D. J., eds, Astronomical Society of the Pacific Confer-
+ence Series Vol. 376, Astronomical Data Analysis Software and Systems
+XVI. p. 127
+Merloni A., et al., 2015, Monthly Notices of the Royal Astronomical Society,
+452, 69
+Miles P. R., Coughlin E. R., Nixon C. J., 2020, The Astrophysical Journal,
+899, 36
+Miniutti G., et al., 2019, Nature, 573, 381
+Moretti A., et al., 2017, Astronomy & Astrophysics, 599, A81
+MNRAS 000, 1–9 (2015)
+
+8
+Adam Malyali et al.
+Mossoux E., Finociety B., Beckers J.-M., Vincent F. H., 2020, Astronomy &
+Astrophysics, 636, A25
+Neilsen J., et al., 2013, The Astrophysical Journal, 774, 42
+Payne A. V., et al., 2021, The Astrophysical Journal, 910, 125
+Phinney E. S., 1989, in Morris M., ed., The Center of the Galaxy. Springer
+Netherlands, Dordrecht, pp 543–553
+Ponti G., et al., 2015, Monthly Notices of the Royal Astronomical Society,
+454, 1525
+Ponti G., et al., 2017, Monthly Notices of the Royal Astronomical Society,
+468, 2447
+Predehl P., et al., 2021, Astronomy & Astrophysics, 647, A1
+Rees M. J., 1988, Nature, 333, 523
+Reines A. E., Volonteri M., 2015, The Astrophysical Journal, 813, 82
+Reiprich T. H., Greiner J., 2001, in Kaper L., Heuvel E. P. J. V. D.,
+Woudt P. A., eds, Black Holes in Binaries and Galactic Nuclei. p. 168,
+doi:10.1007/10720995_31
+Remillard R. A., et al., 2022, The Astronomical Journal, 163, 130
+Ricarte A., Natarajan P., Dai L., Coppi P., 2016, Monthly Notices of the Royal
+Astronomical Society, 458, 1712
+Ricci C., et al., 2020, The Astrophysical Journal, 898, L1
+Ricci C., et al., 2021, The Astrophysical Journal Supplement Series, 255, 7
+Roming P. W. A., et al., 2005, Space Science Reviews, 120, 95
+Ryu T., Krolik J., Piran T., Noble S. C., 2020, The Astrophysical Journal, 904,
+100
+Saxton R., Komossa S., Auchettl K., Jonker P. G., 2020, Space Science
+Reviews, 216, 85
+Saxton R., et al., 2021, arXiv:2111.14238 [astro-ph]
+Schlafly E. F., Finkbeiner D. P., 2011, The Astrophysical Journal, 737, 103
+Shakura N. I., Sunyaev R. A., 1973, Astronomy & Astrophysics, 24, 337
+Simmonds C., Buchner J., Salvato M., Hsu L.-T., Bauer F. E., 2018, Astron-
+omy & Astrophysics, 618, A66
+Steinberg E., Stone N. C., 2022, arXiv:2206.10641 [astro-ph]
+Stern D., et al., 2012, The Astrophysical Journal, 753, 30
+Stone N. C., Metzger B. D., 2016, Monthly Notices of the Royal Astronomical
+Society, 455, 859
+Stone N. C., Vasiliev E., Kesden M., Rossi E. M., Perets H. B., Amaro-Seoane
+P., 2020, Space Science Reviews, 216, 35
+Sunyaev R., et al., 2021, Astronomy & Astrophysics, 656, A132
+Tonry J. L., et al., 2018, Publications of the Astronomical Society of the
+Pacific, 130, 064505
+Trakhtenbrot B., et al., 2019a, Nature Astronomy, 3, 242
+Trakhtenbrot B., et al., 2019b, The Astrophysical Journal, 883, 94
+Wevers T., et al., 2022, arXiv:2209.07538 [astro-ph]
+Wright E. L., et al., 2010, The Astronomical Journal, 140, 1868
+Yuan Q., Wang Q. D., 2016, Monthly Notices of the Royal Astronomical
+Society, 456, 1438
+Yuan
+W.,
+et
+al.,
+2018,
+in
+den
+Herder
+J.-W.
+A.,
+Nakazawa
+K.,
+Nikzad S., eds, Space Telescopes and Instrumentation 2018: Ul-
+traviolet
+to
+Gamma
+Ray.
+SPIE,
+Austin,
+United
+States,
+p.
+76,
+doi:10.1117/12.2313358,
+https://www.spiedigitallibrary.
+org/conference-proceedings-of-spie/10699/2313358/
+Einstein-Probe--a-lobster-eye-telescope-for-monitoring-the/
+10.1117/12.2313358.full
+Zabludoff A., et al., 2021, Space Science Reviews, 217, 54
+Zhong S., Li S., Berczik P., Spurzem R., 2022, The Astrophysical Journal,
+933, 96
+van Velzen S., Holoien T. W.-S., Onori F., Hung T., Arcavi I., 2020, Space
+Science Reviews, 216, 124
+van Velzen S., Pasham D. R., Komossa S., Yan L., Kara E. A., 2021a, Space
+Science Reviews, 217, 63
+van Velzen S., et al., 2021b, The Astrophysical Journal, 908, 4
+APPENDIX A: HOST GALAXY PROPERTIES
+Using the correlation reported in Kettlety et al. (2018) between
+galaxy total stellar mass, 𝑀★, and luminosity in the WISE 𝑊1-band,
+13
+h32
+m00
+s
+31
+m58
+s
+57
+s
+56
+s
+-32°43'00"
+15"
+30"
+45"
+J2000
+J2000
+Figure A1. Legacy Survey DR10 (early) 𝑔-band cutout image of the sky
+region surrounding eRASSt J133158-324321. The dark orange circle is the
+error circle for RXJ133157.6324319.7 inferred from ROSAT pointed obser-
+vations in Hampel et al. (2022), whilst the red and blue circles denote the 3𝜎
+error circles on the source position inferred from eROSITA and XMM MOS2
+observations (although the detection of J1331 in the first XMM observation is
+uncertain and we quote upper limits on the count rates for this in section 2.3,
+we include it in this finder chart for completeness). The cyan star marks the
+Gaia EDR3 (Gaia Collaboration et al. 2021) position of the host galaxy.
+𝐿W1, then we infer log(𝑀★/𝑀⊙) = 10.15 ± 0.09 for the host galaxy.
+Combining this with 𝑀BH − 𝑀★ relation in Reines & Volonteri
+(2015), suggests a black hole mass of log(𝑀BH/𝑀⊙) = 6.5 ± 0.2.
+The finder chart for J1331 is presented in Fig A1.
+APPENDIX B: OPTICAL SPECTROSCOPY
+LCO spectrum (2022-02-12): J1331 was observed with the low
+dispersion FLOYDS spectrograph on the LCOGT 2m telescope at
+Siding Spring Observatory operated by the Las Cumbres Observa-
+tory (LCO; Brown et al. 2013) on 2022 February 12 (proposal ID
+CON2022A-001, PI: M. Salvato). We obtained an exposure of 1800
+seconds using the “red/blu” grism and the 2” slit oriented along the
+parallactic angle. The spectrum has a wavelength range of 3200-
+10000A with dispersions of 3.51A/pixel and 1.74 A/pixel in the blue
+(3200-5700A) and red (5400-10000A) bands, respectively. The data
+were reduced and calibrated using the automatic FLOYDS pipeline.
+The HgAr and Zn lamps were used for wavelength calibration and
+a Tungsten-Halogen + Xenon lamp for flat fielding. A sensitivity
+function from the FLOYDS archive was used for flux calibration.
+WiFeS spectrum (2022-05-09): We observed J1331 with the Wide
+Field Spectrograph (WiFeS; Dopita et al. 2010) on the ANU 2.3m
+telescope at Siding Spring Observatory on 2022 May 08 (proposal
+ID 2220157, PI Miller-Jones). We obtained 2x2400 s exposures us-
+ing the R3000 and B3000 gratings and a NeAr arc lamp exposure
+immediately following the target exposures. The data were reduced
+using standard procedures including the PyWiFeS reduction pipeline
+(Childress et al. 2014). LTT4364 was used as the flux standard and
+a quartz-iodine lamp was used for flat-fielding. We then chose the
+slitlets with the most significant flux from the calibrated spectra
+MNRAS 000, 1–9 (2015)
+
+Repeated partial tidal disruption flares from a quiescent galaxy
+9
+5500
+6000
+6500
+7000
+7500
+8000
+Rest Wavelength [Å]
+0.0
+0.5
+1.0
+1.5
+2.0
+F  [10
+16 erg cm
+2 s
+1 Å
+1]
+2022-02-12: LCO
+2022-05-09: WiFeS
+Figure B1. Optical spectra of J1331, with the first follow-up spectrum being
+obtained on 2022-02-12, ∼23 days after the last eRASS5 detection.
+obtained from the pipeline and performed background subtraction,
+resulting in a spectrum with spectral range 3500 to 9000 Å.
+Each follow-up optical spectrum appears to be consistent with a
+quiescent host galaxy (Fig. B1), with no TDE-like optical emission
+features detected, nor any transient features relative to the NOT spec-
+trum taken on 1999-01-26 and presented in Hampel et al. (2022).
+APPENDIX C: INFERRING THE OUTBURST
+PROPERTIES
+To obtain a coarse reconstruction of the 2022 outburst, we perform a
+joint fit of the rising lightcurve from 1993, observed by ROSAT, and
+the decay lightcurve from 2022, observed by eROSITA and XMM,
+using:
+𝐹X(𝑡) = 𝐹X,max ×
+�
+exp
+�
+−(𝑡 − 𝑡peak,1)2/2𝜎2�
+if 𝑡 < 𝑡peak,1
+exp
+�
+−(𝑡 − 𝑡peak,2)/𝜏
+�
+if 𝑡 > 𝑡peak,2
+(C1)
+where the free parameters of this model are 𝜎 (the rise timescale),
+𝑡peak,1 and 𝑡peak,2 (the peak time of the ROSAT and eROSITA out-
+bursts, respectively), 𝜏 (the decay timescale), and 𝐹X,max (the peak
+flux of both outbursts), with the priors on these parameters listed in
+Table C1. We assume that the upper bound on the peak luminosity
+must be less than the Eddington luminosity for the SMBH, and that
+both outbursts have the same peak luminosity. We then assume that
+the rise for 2022 outburst was similar to the 1993 outburst (see below),
+and use its modelled rise to approximate that of the unobserved rise of
+the 2022 outburst. From this fittedlightcurve model (Fig. C1), we then
+computed the integrated 0.2–2 keV luminosity, and corrected this to
+a bolometric luminosity using the best fitting X-ray spectral model.
+The inferred energy emitted in each outburst is (5+6
+−3) × 1049 erg,
+corresponding to an accreted mass of (5+7
+−2) × 10−4(𝜖/0.05)−1 M⊙,
+where 𝜖 is the radiative efficiency of accretion, whilst the inferred
+peak MJD for each outburst are 49024+6
+−6 and 59593+3
+−2. The inferred
+𝜎 and 𝜏 are 6+1
+−1 days and 3.9+0.1
+−0.1 days, respectively, and we roughly
+estimate the MJD of disruption to be 59593 − 2 ∗ 𝜎 ∼ 59581.
+It is of course extremely important to consider that these estimates
+are subject to a number of caveats, mainly related to our observations
+not covering the rise of the 2022 outburst, such that the estimated
+values here should be treated with caution. For example, it is assumed
+that the outburst can be well modelled by equation C1, and that
+both the 1993 and 2022 outbursts are similar, whereas the actual
+Table C1. Priors adopted in the fitting of the 1993 and 2022 outbursts. The
+rise and decay timescales are in units of days. 𝑡peak,1 and 𝑡peak,2 are in MJD,
+whilst 𝐹max is the maximum 0.2–2 keV flux of each outburst (with upper
+bound set by the Eddington luminosity of the system).
+Parameter
+Prior
+log[𝜎]
+∼ U(0, log[50])
+𝑡peak,1
+∼ U(49006, 49178)
+𝑡peak,2
+∼ U(58450, 58650)
+log[𝜏]
+∼ U(0, log[50])
+log[𝐹X,max]
+∼ U(log[5 × 10−13], log[4 × 10−11])
+10
+40
+10
+42
+10
+44
+LX [erg s
+1]
+59560
+59580
+59600
+59620
+59640
+MJD
+10
+16
+10
+14
+10
+12
+FX [erg s
+1 cm
+2]
+Figure C1. Inferred full outburst (red) for the flaring observed by eROSITA
+in 2022, assuming the model described in equation C1. The markers follow
+the same legend as for Fig. 1. The darker and lighter shaded red bands enclose
+the inner 68% and 98% of the posterior.
+lightcurve may have had an extended plateau phase prior to the
+eROSITA detection (so our estimated fluence and accreted mass
+would be underestimated).
+However, if the 2022 outburst does evolve relatively closely to the
+functional form in equation C1, then it may be reasonable to consider
+that the rise timescale for the flare in 1993 is similar to that observed
+in 2022 (under a tidal disruption scenario), due to the approximately
+constant eccentricity of the stellar remnant after repeated partial
+disruptions (Antonini et al. 2011), and the weak dependence of the
+period of the most bound debris on the stellar mass (Hayasaki et al.
+2013).
+APPENDIX D: ADDITIONAL X-RAY INFORMATION
+The BXA fitted model to the eRASS5 spectrum is shown in Fig. D1,
+and the eRASS5 lightcurve is shown in Fig. D2. The NICER count
+rate lightcurve is plotted in Fig. D3, whilst the full X-ray lightcurve of
+J1331 is presented in Table D1. A comparison of the X-ray lightcurve
+of J1331 with other nuclear transients is presented in Fig D4.
+APPENDIX E: ADDITIONAL PHOTOMETRIC
+INFORMATION
+Table E1 contains the Swift UVOT aperture photometry of the host
+galaxy of J1331, whilst Fig. E1 shows the long term ATLAS and
+NEOWISE lightcurves of J1331.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–9 (2015)
+
+10
+Adam Malyali et al.
+Table D1. X-ray lightcurve table for J1331. The fluxes from the ROSAT
+pointed observations were derived from Hampel et al. (2022). The first four
+eROSITA observations listed, between MJD 58868 and 59419, are upper
+limits estimated from eRASS1, 2, 3 and 4, respectively; eROSITA fluxes
+outside of this window have been computed from the individual visits within
+eRASS5.
+MJD
+Observation
+𝐹0.2−2keV,obs
+𝐹0.2−2keV,unabs
+[10−13 erg cm−2 s−1]
+[10−13 erg cm−2 s−1]
+48260.000
+ROSAT/ RASS
+< 2.9
+< 4.5
+48844.598
+ROSAT/ Pointed
+< 0.2
+< 0.4
+49006.094
+ROSAT/ Pointed
+< 1.2
+< 1.9
+49012.146
+ROSAT/ Pointed
+6.1 ± 0.7
+9.4 ± 1.0
+49012.180
+ROSAT/ Pointed
+8.9 ± 1.9
+13.8 ± 2.9
+49013.591
+ROSAT/ Pointed
+10.0 ± 1.1
+15.5 ± 1.7
+49178.555
+ROSAT/ Pointed
+< 0.7
+< 1.0
+49178.766
+ROSAT/ Pointed
+< 0.3
+< 0.5
+53745.291
+XMM/ Slew
+< 3.8
+< 5.9
+57056.039
+XMM/ Slew
+< 5.4
+< 8.3
+57241.869
+XMM/ Slew
+< 8.3
+< 12.8
+58226.719
+Swift/ XRT
+< 0.9
+< 1.4
+58230.707
+Swift/ XRT
+< 0.5
+< 0.8
+58234.028
+Swift/ XRT
+< 0.8
+< 1.2
+58868.114
+SRG/ eROSITA
+< 0.3
+< 0.4
+59051.625
+SRG/ eROSITA
+< 0.5
+< 0.7
+59229.875
+SRG/ eROSITA
+< 1.3
+< 1.7
+59418.532
+SRG/ eROSITA
+< 0.5
+< 0.7
+59599.448
+SRG/ eROSITA
+10.8 ± 8.0
+14.4 ± 10.6
+59599.614
+SRG/ eROSITA
+3.4 ± 1.7
+4.6 ± 2.2
+59599.781
+SRG/ eROSITA
+5.7 ± 1.5
+7.7 ± 2.0
+59599.948
+SRG/ eROSITA
+4.9 ± 1.2
+6.5 ± 1.6
+59600.114
+SRG/ eROSITA
+5.1 ± 1.3
+6.9 ± 1.7
+59600.281
+SRG/ eROSITA
+2.6 ± 1.1
+3.5 ± 1.5
+59600.448
+SRG/ eROSITA
+9.0 ± 2.4
+12.0 ± 3.2
+59600.614
+SRG/ eROSITA
+6.4 ± 3.5
+8.5 ± 4.6
+59604.892
+NICER/ XTI
+<8.6
+<13.8
+59605.566
+NICER/ XTI
+<10.3
+<16.6
+59606.082
+NICER/ XTI
+<9.5
+<15.3
+59607.533
+NICER/ XTI
+<7.7
+<12.3
+59608.280
+NICER/ XTI
+<6.6
+<10.6
+59609.473
+NICER/ XTI
+<6.3
+<10.1
+59610.119
+NICER/ XTI
+<7.3
+<11.7
+59611.432
+NICER/ XTI
+<6.5
+<10.4
+59612.210
+NICER/ XTI
+<8.1
+<13.0
+59613.305
+NICER/ XTI
+<7.6
+<12.3
+59614.210
+NICER/ XTI
+<8.1
+<13.1
+59615.500
+NICER/ XTI
+<6.9
+<11.1
+59616.889
+NICER/ XTI
+<6.2
+<10.0
+59617.287
+XMM/ Pointed
+< 0.1
+< 0.2
+59617.598
+NICER/ XTI
+<5.5
+<8.8
+59618.630
+NICER/ XTI
+<5.5
+<8.8
+59619.666
+NICER/ XTI
+<5.6
+<9.0
+59620.463
+NICER/ XTI
+<5.8
+<9.4
+59621.229
+NICER/ XTI
+<9.5
+<15.3
+59622.488
+NICER/ XTI
+<9.8
+<15.8
+59623.102
+NICER/ XTI
+<12.2
+<19.6
+59624.362
+NICER/ XTI
+<6.5
+<10.5
+59638.031
+Swift/ XRT
+< 0.8
+< 1.4
+59766.375
+Swift/ XRT
+< 0.7
+< 1.2
+59773.061
+Swift/ XRT
+< 24.6
+< 43.7
+59774.292
+Swift/ XRT
+< 2.2
+< 3.9
+59778.974
+Swift/ XRT
+< 0.8
+< 1.4
+59780.760
+Swift/ XRT
+< 0.8
+< 1.5
+59787.468
+Swift/ XRT
+< 0.8
+< 1.4
+59794.352
+Swift/ XRT
+< 0.8
+< 1.4
+59797.916
+XMM/ Pointed
+< 0.06
+< 0.10
+59801.282
+Swift/ XRT
+< 0.5
+< 1.0
+59808.180
+Swift/ XRT
+< 0.9
+< 1.6
+59815.534
+Swift/ XRT
+< 0.8
+< 1.4
+10
+3
+10
+2
+10
+1
+10
+0
+10
+1
+TDE rate,  [30 yr
+1 gal
+1]
+10
+6
+10
+4
+10
+2
+p(N
+2)| )
+= 0.01
+= 0.05
+= 0.15
+= 0.003
+Figure C2. Poisson probability of 𝑁 ≥ 2 TDEs occurring within a 30 year
+period for a given galaxy. The red dotted lines mark the estimated probability
+for current theoretical estimates for TDE rates (10−4 yr−1 gal−1; Stone et al.
+2020). The grey dashed lines mark out the TDE rates of 0.15, 0.05 and 0.01
+per
+30 yr−1 gal−1, required to produce probabilities of 0.01, 0.001, and
+0.0001, respectively.
+0.3
+1.0
+2.0
+5.0
+Energy [keV]
+10
+4
+10
+2
+100
+Counts s
+1 keV
+1
+Figure D1. BXA fit of a tbabs*zbbody model to the eRASS5 spectrum.
+The solid red line represents the median model fit, whilst the shaded red
+region encloses the inner 98% of the credible region. The X-ray spectrum is
+ultra-soft with 𝑘𝑇 = 0.115+0.007
+−0.007 keV.
+Table E1. Swift UVM2 photometry of the host galaxy of J1331.
+MJD
+Magnitude
+58226.727
+23.1 ±1.0
+58230.747
+22.9 ±0.6
+58234.068
+22.3 ±0.4
+59638.032
+22.2 ±0.3
+59766.376
+23.0 ±0.7
+59774.294
+22.8 ±1.0
+59778.975
+22.5 ±0.6
+59780.762
+22.9 ±0.8
+59794.353
+22.7 ±0.6
+59801.283
+22.5 ±0.4
+59808.181
+22.8 ±0.6
+59815.535
+22.5 ±0.5
+MNRAS 000, 1–9 (2015)
+
+Repeated partial tidal disruption flares from a quiescent galaxy
+11
+0
+5
+10
+15
+20
+25
+30
+t
+teRASS5, 0 [hr]
+10
+3
+10
+2
+10
+1
+10
+0
+Rate [cts/s]
+Figure D2. 0.2–2 keV band eRASS5 lightcurve of J1331. The blue and grey
+markers denote the inferred source and background count rates in the source
+aperture, respectively. Times are measured relative to the start of the earliest
+observation of J1331 in eRASS5, 𝑡eRASS5,0. J1331 is clearly detected above
+background in each visit.
+59605
+59610
+59615
+59620
+59625
+MJD - 0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Rate  0.4-2.0 keV [cts s
+1]
+3C50 background
+Total
+Figure D3. NICER count rate lightcurve in the 0.4-2 keV band, with blue
+markers denoting the total observed count rate (source and background), and
+grey markers representing the estimated background rate inferred using the
+3C50 background model (Remillard et al. 2022). The system is not detected
+at 2𝜎 above background in each NICER OBSID.
+10
+2
+10
+1
+10
+0
+10
+1
+10
+2
+10
+3
+t
+tpeak [days]
+10
+40
+10
+41
+10
+42
+10
+43
+10
+44
+LX [erg s
+1]
+Figure D4. Comparison of the 0.2–2 keV X-ray lightcurve evolution of J1331
+(red markers) with other soft nuclear transients from quiescent galaxies (or
+those recently hosting low luminosity AGN). J1331 decays in 𝐿X over longer
+timescales than QPEs (orange for eROQPE1; Arcodia et al. 2021), but still
+over much shorter timescales than previously reported TDEs in the literature,
+such as ASAS-SN 14li (grey, Bright et al. 2018), AT 2019azh decay phase
+(blue, Hinkle et al. 2020), AT 2019dsg (pink, Cannizzaro et al. 2021). The
+𝑡peak for J1331 was set to MJD=59592.9, following the assumptions described
+in Section C.
+MNRAS 000, 1–9 (2015)
+
+12
+Adam Malyali et al.
+57500
+58000
+58500
+59000
+59500
+MJD
+50
+0
+50
+100
+150
+200
+F  [Jy]
+ATLAS o
+ATLAS c
+57000
+57500
+58000
+58500
+59000
+59500
+MJD
+13.8
+14.0
+14.2
+14.4
+14.6
+Vega Magnitude
+W1
+W2
+Figure E1. No major variability is seen within the ATLAS forced photometry
+generated on the difference imaging (top), nor within the NEOWISE lightcurve
+(bottom).
+MNRAS 000, 1–9 (2015)
+

6dE1T4oBgHgl3EQf7AUK/content/tmp_files/2301.03528v1.pdf.txt

@@ -0,0 +1,717 @@
+Multi-point Padè for the study of phase transitions: from
+the Ising model to lattice QCD
+Francesco Di Renzo∗ and Simran Singh
+Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma
+and INFN, Gruppo Collegato di Parma, I-43100, Parma, Italy
+E-mail: [email protected], [email protected]
+The Bielefeld Parma collaboration has recently put forward a method to investigate the QCD phase
+diagram based on the computation of Taylor series coefficients at both zero and imaginary values
+of the baryonic chemical potential. The method is based on the computation of multi-point Padé
+approximants. We review the methodological aspects of the computation and, in order to gain
+confidence in the approach, we report on the application of the method to the two-dimensional
+Ising model (probably the most popular arena for testing tools in the study of phase transitions).
+Besides showing the effectiveness of the multi-point Padé approach, we discuss what these results
+can suggest in view of further progress in the study of the QCD phase diagram. We finally report
+on very preliminary results in which we look for Padé approximants at different temperatures and
+fixed values of the (imaginary) baryonic chemical potential.
+The 39th International Symposium on Lattice Field Theory (Lattice2022),
+8-13 August, 2022
+Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.03528v1  [hep-lat]  9 Jan 2023
+
+Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+1.
+How it all began: from Taylor expansions on thimbles to imaginary 𝜇𝐵 LQCD
+The QCD phase diagram is still to a large extent elusive: in particular, due to the so-called sign
+problem, the lattice (the non-perturbative tool which would be supposed to provide valuable insight)
+cannot probe the relevant regions in the 𝑇 − 𝜇𝐵 (Temperature-baryonic chemical potential) plane.
+In the last couple of years, the Bielefeld-Parma collaboration put forward a method to compute
+finite-density QCD thermodynamic observables in the region to which access would be precluded
+by the sign problem; this approach is also able to probe the singualrity structure of the theory in the
+complex 𝜇𝐵 plane [1–4]. The method is based on the computation of Taylor series coefficients at
+both zero and imaginary values of the baryonic chemical potential, which enables the computation
+of multi-point Padé approximants. This work aims to assess the effectiveness of the method by
+making use of it in the context of a very standard playground for the physics of phase transitions (e.g.
+the 2d Ising model). At the same time, we present (very) preliminary results on new applications
+in the context of finite-density QCD.
+Before entering the main subject, it is useful to recall when the idea of applying multi-point
+Padé rational approximants first came to our mind; that was in the context of thimble regularisation.
+The latter [5, 6] was introduced to solve (or at least tame) the sign problem by re-expressing the
+path integral as a sum of integrals computed on manifolds different from the original one. After
+complexifying the degrees of freedom, one considers the so-called Lefschetz thimbles, i.e. the
+manifolds that are the union of the steepest ascent paths stemming from the various stationary
+points of the action. On such manifolds the imaginary part of the action stays constant, so that
+the sign problem reduces to the so-called residual phase which is there due to the Jacobian of
+the change of variables. There is a thimble attached to each stationary point and in principle all
+can give a contribution to the path integral. This is referred to as the thimble decomposition. To
+make a long story short, we recall that (a) not all the thimbles give a non-null contribution, (b)
+this picture changes in different regions of the parameters space of the theory (i.e. a given thimble
+can contribute to the path integral in a region and not in another one) and (c) there are cases in
+which a single thimble (usually the so called dominant one, attached to the stationary point with
+the lowest action) is enough to compute the answer one is interested in. The latter observation
+gave raise to the single thimble dominance hypothesis, which was shown to hold in a few cases,
+but failed in others. The first example of a failure was provided by the 1-D Thirring model [7, 8],
+where it was clearly shown that a single thimble is not enough to account for the known analytic
+result. It is nevertheless important to remark that there are regions in which one single thimble is
+enough, and this was the logical starting point for the success of a computation based on multi-point
+Padé rational approximants. The success of such approach [9] can be recognised in Fig. 1. On
+the left, we display the known analytic result for the chiral condensate ¯𝜒𝜒 of the 1-D Thirring
+model (𝐿 = 8, 𝑚 = 1, 𝛽 = 1) at various values of the chemical potential by mass ratio 𝜇
+𝑚. This
+is plotted together with the numerical results which we got: triangles are results computed on one
+single thimble at points where we are able to show that this is enough; dots are results taken from
+the multi-point Padé method that we will better describe in the next section. Here it is enough to
+say that a few Taylor expansion coefficients were computed at the points marked by triangles and
+from those the multi-point Padé approximant was computed. The right panel of the figure shows
+2
+
+Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+how the singularity pattern of the solution was reconstructed: the rational approximant displayed a
+singularity which falls on top of the analytic one. Convergence radii of the Taylor expansions we
+computed can be spotted, showing that there is an intersection of convergence disks, validating the
+procedure of bridging the two regions where we were able to compute single thimble results: all in
+all, while the thimble decomposition is discontinuous, the physical observable is not. The figure
+refers to a given choice of lattice size, mass and 𝛽-value; we were able to show [10] that the method
+can successfully account for the extraction of the continuum limit.
+μ. We can obtain a dimensionless quantity by taking the
+ratio μ
+m ¼ ˆμ
+ˆm. Since the analytic result is known, the single
+thimble approximation was shown not to account for the
+correct result on the entire μ
+m axis. In our new approach the
+problem is solved and in Fig. 2 we display the essential
+features of our results: as an example, we show results for
+the chiral condensate h¯χχi (parameters are L ¼ 8, β ¼ 1,
+m ¼ 2). We can argue that all the requirements of the
+program that we sketched above can be met. There is a
+preliminary point we have to make. For real β a Stokes
+phenomenon is potentially present up to a given value of μ
+m:
+this involves the dominant thimble pσ0 and another critical
+point. We denote the latter pσ¯0, following the notation of
+[19]. The problem can be easily solved by adding a small
+imaginary part to β: in this way a Stokes phenomenon does
+not take place, a thimble decomposition is in place and
+while pσ¯0 could in principle give a contribution to the
+result, this is de facto negligible due to the huge difference
+SRðpσ¯0Þ ≫ SRðpσ0Þ. This solves the problem and any
+further reference to this point will be omitted in the
+following.
+(1) A first value of
+μ
+m for which only the dominant
+thimble pσ0 accounts for the correct result can be
+found in a very fundamental, yet simple way. The
+range of values SI can take on the real axis depends
+on the values of ˆμ and ˆm and, below a given value of
+μ
+m, this range is limited. By explicit computation of
+the SðσÞ
+I ðμ
+mÞ we can show that no unstable thimble
+associated to a critical point pσ other that the
+dominant one can intersect the original domain of
+integration below a given value μ0
+m.7 Thus for μ
+m < μ0
+m
+we can easily select a first point at which the
+dominant thimble provides the only contribution
+to the result. We picked μ
+m ¼ 0.4 and computed the
+Taylor expansion up to the second derivative.
+We now need to find a second value of μ
+m at which
+the dominant thimble accounts for the complete
+result and compute the Taylor expansion on it. In
+principle we could study the crossing mechanism
+between the different curves SðσÞ
+I ðμ
+mÞ (see subsec-
+tion II B). In practice there is a much simpler way to
+proceed. First of all, we point out that the asymptotic
+value of h¯χχi is known: for large enough values of μ
+the chiral condensate is zero. We notice that for μ
+m ¼
+1.4 the value of h¯χχi computed on the dominant
+thimble is very close to zero. By inspecting the
+values of SRðpσÞ for thimbles other than the funda-
+mental one, we find that, for μ
+m ¼ 1.4, SRðpσÞ ≫
+SRðpσ0Þ for all the critical points but three, that we
+denote σ1, σ¯1, σ¯2.8 Two of them (σ¯1 and σ¯2) have
+values of the real action which are lower than Smin,
+which is the minimum value SR takes on the original
+domain of integration: because of this, the unstable
+thimbles associated to them can’t intersect the
+original domain of integration. As for σ1, in this
+simple model it does not take that much to show that
+the unstable thimble attached to it does not intersect
+the original domain of integration (see the left panel
+of Fig. 2). We conclude that the dominant thimble σ0
+can account for the complete result at this value of μ
+m.
+We have thus selected the second point we were
+looking for; at this point the series has been
+computed up to the fifth derivative. One might
+object that we made use of the explicit query for
+intersections between the original domain of inte-
+gration and a given unstable thimble, which thing is
+FIG. 2.
+(Left panel) The flow lines highlighting the thimbles structure of the 1-dim Thirring model at μ
+m ¼ 1.4: stable thimbles are
+depicted in blue, unstable thimbles in magenta. The dominant thimble is associated to the critical point sitting at ℜðzÞ ¼ 0. The critical
+point σ1 is the closest to the latter to the right (there is a mirror image to the left as well): notice that the unstable thimble associated to it
+does not intersect the original domain of integration (which is on the real axis). (Center panel) The chiral condensate as obtained from
+the analytic solution (continuous black line) and from our Pad´e approximant (we plot points instead of a continuum line so that the size
+of errors are easier to spot.). The points providing input to the evaluation of Pad´e are marked as triangles. (Right panel) Singularity of the
+solution in the complex plane: red point computed from the analytic solution, green point is the only pole of our Pad´e approximant. We
+plot the radii of convergence which are relevant for the expansions at hand: our analytic continuation indeed stands on firm ground.
+7The value of ˆm is held fixed.
+8We once again adhere to the notation of [19].
+F. DI RENZO, S. SINGH, and K. ZAMBELLO
+PHYS. REV. D 103, 034513 (2021)
+034513-6
+Figure 1: Left panel: (continuum line) analytic solution for the condensate ¯𝜒𝜒 of the 1-D Thirring model
+(𝐿 = 8, 𝑚 = 1, 𝛽 = 1) at various values of the chemical potential by mass ratio 𝜇
+𝑚; (triangles) numerical
+results obtained on one single thimble; (dots) numerical results taken from the rational approximant. Right
+panel: we plot in the complex 𝜇
+𝑚 plane the singularity we got from the rational approximant; it is depicted
+on top of the known analytic one.
+2.
+Multi-point Padè method for finite density Lattice QCD
+2.1 Basics of the multi-point Padè method
+Suppose we know a few Taylor expansion coefficients of a given function 𝑓 (𝑧) at different
+points {𝑧𝑘 | 𝑘 = 1 . . . 𝑁}. The basic idea of our multi-point Padé approach is to approximate 𝑓 (𝑧)
+by a rational function 𝑅𝑚
+𝑛 (𝑧), which we call a [𝑚/𝑛] Padé approximant
+𝑅𝑚
+𝑛 (𝑧) = 𝑃𝑚(𝑧)
+˜𝑄𝑛(𝑧)
+=
+𝑃𝑚(𝑧)
+1 + 𝑄𝑛(𝑧) =
+𝑚�
+𝑖=0
+𝑎𝑖 𝑧𝑖
+1 +
+𝑛�
+𝑗=1
+𝑏 𝑗 𝑧 𝑗
+.
+(1)
+𝑅𝑚
+𝑛 (𝑧) (i.e. the 𝑎𝑖, 𝑏 𝑗 coefficients defining it) can be fixed by requiring that it reproduces the values
+of 𝑓 and a few of its derivatives at the given points {𝑧𝑘}. Provided that 𝑛 + 𝑚 + 1 = 𝑁𝑠 ( 𝑓 (𝑠−1)
+being the highest order derivative we computed at each point), this is possible by requiring that
+. . .
+𝑃𝑚(𝑧𝑘) − 𝑓 (𝑧𝑘)𝑄𝑛(𝑧𝑘) = 𝑓 (𝑧𝑘)
+𝑃′
+𝑚(𝑧𝑘) − 𝑓 ′(𝑧𝑘)𝑄𝑛(𝑧𝑘) − 𝑓 (𝑧𝑘)𝑄′
+𝑛(𝑧𝑘) = 𝑓 ′(𝑧𝑘)
+. . .
+(2)
+3
+
+0.6
+1.0
+3.0/
+0.5
+2.0
+0.4
+0.5 
+0.3
+21.0
+m
+0.2
+0.0
+0.0
+0.1
+0.0
+-0.5
+-1.0
+-0.1
+2.0
+-0.2
+-1.0
+3.0
+-2.0
+-1.0
+0.0
+1.0
+2.0
+3.0
+0.0
+0.5
+1.0
+1.5
+2.0
+1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Re z
+μ/m
+Re (μ / m)Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+In Eq. (2) we only wrote 2 out of 𝑠 equations for 1 out of 𝑁 points. It should be clear what the
+overall problem amounts to: we have to solve a linear system, the unknowns being the {𝑎𝑖, 𝑏 𝑗 | 𝑖 =
+1 . . . 𝑚, 𝑗 = 1 . . . 𝑛}. This is not the only possible way to solve for 𝑅𝑚
+𝑛 (𝑧), but for the purpose of
+understanding our approach it suffices (the interested reader can refer to [4] for other alternatives1).
+It should be clear that
+• Not only 𝑅𝑚
+𝑛 (𝑧) can reproduce our input pieces of information; by a natural analytic continu-
+ation it can predict values of 𝑓 in an extended region (to the extent we do not exit the region
+in which the approximation holds, which thing of course deserves care of its own): left panel
+of Fig. 1 is an example.
+• When a zero in the denominator of 𝑅𝑚
+𝑛 (𝑧) is not canceled by a corresponding zero of the
+numerator, we face a singularity of the rational approximation, which is supposed to teach us
+something on the singularity structure of 𝑓 ; quite obviously, singularities live in the complex
+𝑧 plane: right panel of Fig. 1 is an example.
+2.2 First application of the multi-point Padè method to finite density LQCD
+In [4] the Bielefeld Parma collaboration applied the multi-point Padè method to finite density
+LQCD. In the example of section 1 we did not have a way to safely compute the 1D Thirring
+condensate in regions where more than one thimble give a contribution; on the other hand, we
+could safely compute (on a single thimble) at given values of 𝜇
+𝑚. This is the same as in LQCD:
+the sign problem does not allow us to compute observables at real values of the baryonic chemical
+potential 𝜇𝐵, but computations are safe at 𝜇𝐵 = 0 and at imaginary values of 𝜇𝐵 (in particular, we
+can compute a few orders of the Taylor expansion of an observable). For (2+1)-flavor of highly
+improved staggered quarks (HISQ) [11] with imaginary chemical potential, we computed cumulants
+of the net baryon number density, given as
+𝜒𝑛𝐵(𝑇,𝑉, 𝜇𝐵) =
+� 𝜕
+𝜕 ˆ𝜇𝐵
+�𝑛 ln 𝑍(𝑇,𝑉, 𝜇𝑙, 𝜇𝑠)
+𝑉𝑇3
+,
+(3)
+with ˆ𝜇𝐵 = 𝜇𝐵/𝑇 and 𝑙, 𝑠 referring to light and strange flavors. Dependence on masses is not made
+explicit: the light to strange ratio is the physical one. By computing at different imaginary values of
+ˆ𝜇𝐵 (including ˆ𝜇𝐵 = 0) we could implement the program of subsection 2.1. Fig. 2 is the counterpart
+of Fig. 1. We point out that
+• In the left panel we can see how well the rational approximants for the number density 𝜒1𝐵
+describe data at different temperatures. Actually we show two different rational approximants
+(enforcing parity or not): they are both fine. The big spike is expected to be there: it is related
+to the Roberge Weiss transition, and it occurs at the temperature which is supposed to be the
+relevant one (𝑇𝑅𝑊 ). Minor spikes can be also spotted: they are harmless, and they can be
+understood in terms of what we will explain in the next section (partial cancellation of zeros
+between numerator and denominator).
+1Notice that this is the simplest setting also with respect to another point: there is no reason for strictly asking
+knowledge of the same number of derivatives at each point.
+4
+
+Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+0
+1
+2
+3
+4
+5
+Re[µB/T]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Im[µB/T]
+ˆµLY
+RW scaling
+chiral scaling
+CEP scaling
+Figure 2: (Left panel) The number density 𝜒1𝐵 at various values of ˆ𝜇𝐵 and different temperatures 𝑇. Data
+are shown together with two different rational approximants (enforcing parity or not): both describe data very
+well. The big spike is expected: it is the hint for the Roberge Weiss transition. (Right panel) The singularity
+pattern in the complex ˆ𝜇𝐵, highlighting their expected overall compliance with Roberge Weiss, chiral and
+Critical End Point scaling.
+• In the right panel we display the singularities we found at different temperatures, relating them
+to the expected singularity scaling pattern. These are the expected Lee-Yang singularities:
+one expects a given scaling for the singularities connected to the Roberge Weiss transition,
+to the chiral transition and to the QCD Critical End Point. While the last two are still under
+investigation2, one can clearly see a consistent picture for the Roberge Weiss scaling: indeed
+in [4] we were able to show that it is the expected one.
+All in all, results are intriguing. That’s why we now want to show that the machinery is under
+control for the the most popular arena for testing tools in the study of phase transitions, i.e. the
+two-dimensional Ising model.
+3.
+Testing the method on the 2d Ising model
+Lee-Yang theory is one of the possible approach to the study of phase transitions. For an
+example of its application, we refer the interested reader to [12], where the authors study the 2d
+Ising model. We will basically follow their program, but will not rely on the study of many different
+cumulants (as they do). We will instead make use of our multi-point Padè method and study only
+two different cumulants at different values of temperature and magnetic field. The hamiltonian is
+the well-known one, based on interactions between nearest neighbours and with external magnetic
+field ℎ
+𝐻 = −𝐽
+∑︁
+<𝑖, 𝑗>
+𝜎𝑖𝜎𝑗 − ℎ
+∑︁
+𝑖
+𝜎𝑖
+(4)
+2Indeed we now have an estimate for the CEP Temperature.
+5
+
+RataprxST=167MeV
+0.8
+RataprxNST=167MeV
+RataprxST=186MeV
+0.6
+RataprxNST=186MeV
+RataprxSTRW
+0.4
+RataprxNSTRW
+Nt4T=167MeVdata
+Nt4T=186MeVdata
+0.2
+Nt4TRWdata
+0
+-0.2
+-0.4
+-0.6
+-0.8
+1
+0
+1
+2
+3
+4
+5
+6
+Im[μg/T]Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+with the only possible values 𝜎𝑖 = ±1. In the following 𝐽 will be set to 𝐽 = 1. The partition function
+can be written in terms of its zeros {𝛽𝑘}
+𝑍(𝛽, ℎ) = 𝑍(0, ℎ) 𝑒 𝛽𝑐 �
+𝑘
+(1 − 𝛽
+𝛽𝑘
+)
+(5)
+𝑐 being a constant. If we define thermal cumulants by
+⟨⟨𝑈𝑛⟩⟩ =
+𝜕𝑛
+𝜕(−𝛽)𝑛 ln 𝑍(𝛽, ℎ)
+it is easy to show that they can be expressed as
+⟨⟨𝑈𝑛⟩⟩ = (−1)(𝑛−1) ∑︁
+𝑘
+(𝑛 − 1)!
+(𝛽𝑘 − 𝛽)𝑛
+(𝑛 > 1)
+(6)
+Furthermore, scaling relations describe the approach of leading zeros to critical inverse temperature
+|𝛽0 − 𝛽𝑐| ∼ 𝐿−1/𝜈
+Im(𝛽0) ∼ 𝐿−1/𝜈.
+(7)
+In Eq. (7) 𝛽0 is the Fisher zero, that is the closest zero of the partition function to the real axis,
+resulting in the closest singularity of cumulants to the real axis3, 𝛽𝑐 is the critical inverse temperature
+and 𝜈 is the relevant critical exponent.
+Our program now entails four steps: (1) we compute the 𝑛 = 2 thermal cumulant (i.e. the specific
+heat) at various inverse temperatures 𝛽 and lattice sizes 𝐿; (2) for each 𝐿 we compute the rational
+approximant 𝑅𝑚
+𝑛 (𝛽) by our multi-point Padè method; (3) at each 𝐿 we find the Fisher zero 𝛽0, which
+is obtained as the the closest singularity of the cumulant to the real axis; (4) we study the finite size
+scaling of the values of 𝛽0. The result of the procedure can be inspected in Fig. 3.
+Figure 3: (Left panel) The scaling in 1/𝐿 of Im(𝛽0), i.e. the imaginary part of the Fisher zero, detected as
+that the closest singularity of the cumulant to the real axis. The correct critical exponent 𝜈 = 1 is got with
+fairly good accuracy. (Right panel) Once 𝜈 has been recognised to be the right one, one can fit the value of
+the critical inverse temperature 𝛽𝑐, which is reconstructed to per mille accuracy.
+3𝛽0 shows up together with its complex conjugate 𝛽∗
+0.
+6
+
+V = 1.03(3)
+0.08
+0.07
+0.06
+0.05
+Im(βo)
+0.04
+0.03
+0.02
+0.01
+0
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+1/Lβ。= 0.4405(5)
+0.45
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1
+0.05
+0
+-0.05
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+1/ LMulti-point Padè for the study of phase transitions
+Francesco Di Renzo
+• In the left panel we display the scaling in 1/𝐿 of Im(𝛽0). Errors are computed by varying
+results with respect to statistical errors for the cumulant and functional form for the rational
+approximant. As one can see, the value of the relevant critical exponent 𝜈 = 1 is got with
+fairly good accuracy (1.03(3)).
+• Once 𝜈 = 1 has been recognised, we can fit the scaling of the real part Re(𝛽0) (right panel),
+thus finding the value of the critical inverse temperature. We get the very accurate result
+𝛽𝑐 = 0.4405(5).
+Once the critical inverse temperature is known, one can sit on top of it and study the scaling in 𝐿
+of Im(ℎ0), ℎ0 being the Lee Yang zero, that is the closest singularity of a magnetic cumulant to
+the real axis. Explicitly, our program again entails four steps: (1) we compute the 𝑛 = 1 magnetic
+cumulant (i.e. the magnetisation) at 𝛽 = 𝛽𝑐 and various values of external magnetic field ℎ and
+lattice size 𝐿; (2) for each 𝐿 we compute the rational approximant 𝑅𝑚
+𝑛 (ℎ) for the magnetisation by
+our multi-point Padè method; (3) at each 𝐿 we find the Lee Yang zero ℎ0, which is the singularity
+of the rational approximant for the magnetisation which is the closest to the real axis; (4) we study
+the finite size scaling of the values of Im(ℎ0) (as we will see, ℎ0 always sits at Re(ℎ0) = 0).
+Before we inspect this scaling behaviour, it is useful to have a closer look at the singularity pattern
+in the complex ℎ plane at given values of 𝐿. In Fig 4 we depict the zeros of the numerator (blue
+crosses) and of the denominator (red circles) of our 𝑅𝑚
+𝑛 (ℎ) at different values of the lattice size 𝐿,
+i.e. 𝐿 = 15 (left panel) and 𝐿 = 30 (right panel). We can easily make a couple of key observations.
+• A few zeros of the denominator are canceled by corresponding zeros of the numerator. These
+are not genuine pieces of information: actually their location vary when varying e.g. the order
+of the Padé approximant [𝑚, 𝑛]. On the other hand, genuine pieces of information (i.e. actual
+zeros and poles) stay constant to a very good precision. Notice that this is the explanation for
+the small spikes in Fig. 2: they are simply the shadow of cancellations which are indeed very
+good, but not good enough to be invisible when plotting the rational approximant.
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+Figure 4: (Left panel) Zeros of the numerator (blue crosses) and of the denominator (red circles) of the
+rational approximant 𝑅𝑚
+𝑛 (ℎ) for the magnetisation on 𝐿 = 15 (left panel) and 𝐿 = 30 (right panel). We
+highlight the closest singularity to the real axis, which is getting closer to the real axis itself as 𝐿 gets larger,
+with real parts being Re(ℎ0) = 0. Plots are in the complex ℎ plane.
+7
+
+Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+• We can clearly see that, as the lattice size 𝐿 gets larger, the closest singularity (Lee Yang
+zero, highlighted in the plot) gets closer to the real axis, with real parts being Re(ℎ0) = 0.
+Finally, in Fig. 5 we plot the finite size scaling of Im(ℎ0). As one can see, the critical exponent in
+is got with very good accuracy (this time, less than percent: −1.880(16) vs −1.875). The steps we
+could take in the (much simpler) case of the Ising model would be the preferred conceptual path to
+follow also for LQCD. Needless to say, it will take time before we can be in a position to do that.
+4.
+Back to LQCD: a T-Padé application
+We finally go back to LQCD for a (very) preliminary account of a new application. Till now
+we have seen multi-point Padè approximants from data taken at a given temperature 𝑇 and different
+values of ˆ𝜇𝐵: with this we mean that we obtained different 𝑅𝑚
+𝑛 ( ˆ𝜇𝐵) at different 𝑇 values. With
+the very same data, we can think of going the other way around, that is we can obtain 𝑅𝑚
+𝑛 (𝑇) at
+different ˆ𝜇𝐵 values. Fig. 6 is an example of what we can get following this path. Of course, this
+time singularities emerge in the complex 𝑇 plane.
+5.
+Conclusions
+The multi-point Padè method for the study of phase transitions has already proved to be quite
+effective in the case of LQCD. Here we showed how the approach can provide very accurate results
+when collecting a rich statistics is not such a hard numerical task (as it was the case for the 2d Ising
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+L1/8-2
+10-3
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Im(h0)
+1.880(16)
+Figure 5: Finite size scaling of Im(ℎ0). To guide the eye, we plot data versus 𝐿1/8−2, where the correct
+critical exponent is taken. As the figure title we report the absolute value of the one we got, which turns out
+to be a very accurate estimate, to less than percent.
+8
+
+Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+Figure 6: (Top-left panel) An example of 𝑅𝑚
+𝑛 (𝑇) for 𝜒1𝐵 at a given value of ˆ𝜇𝐵 on top of data taken at
+different temperatures 𝑇 at the same given value of ˆ𝜇𝐵. (Top-right) Actual measurements of 𝜒1𝐵( ˆ𝜇𝐵) at a
+given temperature 𝑇 plotted together with interpolating data obtained from 𝑅𝑚
+𝑛 (𝑇). Everything looks pretty
+smooth; we plot in a different colour the only data point possibly not falling smoothly on top of actual data.
+(Bottom-left) Zeros of denominator (red) and zeros of numerator (blue) of 𝑅𝑚
+𝑛 (𝑇) in the complex 𝑇 plane at
+a low value of ˆ𝜇𝐵. (Bottom-right) The same plot at a value of ˆ𝜇𝐵 close to ˆ𝜇𝐵 = 𝑖𝜋 (𝑇 is expressed in GeV)
+model). This is at same time a proof of concept of the reliability of the method and a stimulus to
+do better in the case of finite density LQCD.
+Acknowledgements
+This work has received funding from the European Union’s Horizon 2020 research and inno-
+vation programme under the Marie Skłodowska-Curie grant agreement No. 813942 (EuroPLEx).
+We also acknowledge support from I.N.F.N. under the research project i.s. QCDLAT. This work
+benefits from the HPC facility of the University of Parma, Italy.
+References
+[1] C. Schmidt, J. Goswami, G. Nicotra, F. Ziesché, P. Dimopoulos, F. Di Renzo et al.,
+Net-baryon Number Fluctuations, Acta Physica Polonica B Proceedings Supplement 14
+(2021) 241.
+9
+
+0.4
+0.35
+0.3
+0.25
+B
+Im(X1
+0.2
+0.15
+0.1
+0.05
+0.12
+0.13
+0.14
+0.15
+0.16
+0.17
+0.18
+0.19
+T0.2
+0.15
+0.1
+Im(X1B)
+0.05
+0$
+-0.05
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+μ/T0.06
+0.04
+0.02
+(L)w)
+-0.02
+-0.04
+-0.06
+0.115
+0.12
+0.125
+0.13
+0.135
+0.14
+0.145
+0.15
+0.155
+Re(T)0.06
+0.04
+0.02
+(D)w)
+-0.02
+-0.04
+-0.06
+0.11
+0.12
+0.13
+0.14
+0.15
+0.16
+0.17
+0.18
+0.19
+0.2
+0.21
+Re(T)Multi-point Padè for the study of phase transitions
+Francesco Di Renzo
+[2] S. Singh, P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, G. Nicotra et al., Lee-Yang edge
+singularities in lattice QCD : A systematic study of singularities in the complex 𝜇𝐵 plane
+using rational approximations, Proceedings of The 38th International Symposium on Lattice
+Field Theory — PoS(LATTICE2021) (2022) 544.
+[3] G. Nicotra, P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, C. Schmidt et al., Lee-Yang
+edge singularities in 2 + 1 flavor QCD with imaginary chemical potential, Proceedings of The
+38th International Symposium on Lattice Field Theory — PoS(LATTICE2021) (2022) 260.
+[4] P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, G. Nicotra, C. Schmidt et al., Contribution
+to understanding the phase structure of strong interaction matter: Lee-Yang edge
+singularities from lattice QCD, Phys. Rev. D 105 (2022) 034513 [2110.15933].
+[5] AuroraScience collaboration, New approach to the sign problem in quantum field theories:
+High density QCD on a Lefschetz thimble, Phys. Rev. D 86 (2012) 074506 [1205.3996].
+[6] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano, Hybrid Monte Carlo on
+Lefschetz thimbles - A study of the residual sign problem, JHEP 10 (2013) 147 [1309.4371].
+[7] H. Fujii, S. Kamata and Y. Kikukawa, Monte Carlo study of Lefschetz thimble structure in
+one-dimensional Thirring model at finite density, JHEP 12 (2015) 125 [1509.09141].
+[8] A. Alexandru, G. Basar, P.F. Bedaque, G.W. Ridgway and N.C. Warrington, Sign problem
+and Monte Carlo calculations beyond Lefschetz thimbles, JHEP 05 (2016) 053
+[1512.08764].
+[9] F. Di Renzo, S. Singh and K. Zambello, Taylor expansions on Lefschetz thimbles, Phys. Rev.
+D 103 (2021) 034513 [2008.01622].
+[10] F. Di Renzo and K. Zambello, Solution of the Thirring model in thimble regularization, Phys.
+Rev. D 105 (2022) 054501 [2109.02511].
+[11] HPQCD, UKQCD collaboration, Highly improved staggered quarks on the lattice, with
+applications to charm physics, Phys. Rev. D 75 (2007) 054502 [hep-lat/0610092].
+[12] A. Deger and C. Flindt, Determination of universal critical exponents using Lee-Yang theory,
+Phys. Rev. Research. 1 (2019) 023004 [1905.02379].
+10
+

8tE2T4oBgHgl3EQflgdv/content/2301.03989v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e1eb2bc820c24a2b52a8a12cb9da07ad468f742ae980ebfffca5e38a8ccfc10b
+size 9508107

8tE2T4oBgHgl3EQflgdv/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ee54a6e93162b0dc91a369177be2914fe6328af01d425cf102bfa243d5e9bfaf
+size 4849709

99E1T4oBgHgl3EQfCgIZ/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b62628a00e53b227adb06cba3c53410559509b13635fc0fe1c48e9d86c5ad3d7
+size 5767213

ANAzT4oBgHgl3EQfTPxG/content/tmp_files/2301.01245v1.pdf.txt

@@ -0,0 +1,934 @@
+RegTraffic: A Regression Based Traffic Simulator
+for Spatiotemporal Traffic Modeling, Simulation
+and Visualization
+Sifatul Mostafi, Taghreed Alghamdi, Khalid Elgazzar
+IoT Research Lab, ECSE, Ontario Tech University, Oshawa, ON, Canada
+{sifatul.mostafi, Taghreed Alghamdi, khalid.elgazzar}@ontariotechu.ca
+Abstract—Traffic simulation is a great tool to demonstrate
+complex traffic structures which can be extremely useful for
+the planning, development, and management of road traffic
+networks. Current traffic simulators offer limited features when
+it comes to interactive and adaptive traffic modeling. This paper
+presents RegTraffic, a novel interactive traffic simulator that
+integrates dynamic regression-based spatiotemporal traffic anal-
+ysis to predict congestion of intercorrelated road segments. The
+simulator models traffic congestion of road segments depending
+on neighboring road links and temporal features of the dynamic
+traffic flow. The simulator provides a user-friendly web interface
+to select road segments of interest, receive user-defined traffic
+parameters, and visualize the traffic for the flow of correlated
+road links based on the user inputs and the underlying correlation
+of these road links. Performance evaluation shows that RegTraffic
+can effectively predict traffic congestion with a Mean Squared
+Error of 1.3 Km/h and a Root Mean Squared Error of 1.71
+Km/h. RegTraffic can effectively simulate the results and provide
+visualization on interactive geographical maps.
+Index Terms—Road traffic, simulator, regression, visualization,
+software
+I. INTRODUCTION
+With the advancement of computer technologies and soft-
+ware engineering, computer-based traffic simulation has be-
+come a popular approach for traffic analysis in support of the
+evaluation and design of Intelligent Transport Systems (ITS)
+[2]. Traffic simulation software supported by the ability to
+emulate the variability of spatial and temporal components in
+traffic flows is a practical tool for capturing and explaining
+complex traffic systems [6].
+The purpose of developing Traffic simulation tools is to
+experiment with varieties of strategies in traffic modeling [3].
+Traffic simulation software tools and models built on real-
+life traffic data are widely applied to support real-time traffic
+decisions and management solutions.
+Regression analysis in the traffic domain is a well-
+established approach that facilitates traffic modeling and pre-
+diction [1]. Regression-based traffic modeling helps in ana-
+lyzing complex traffic structures which is a useful method for
+the development and planning of traffic systems and networks.
+Hence, traffic congestion estimation and computerized simula-
+tion are suitable options for policymakers to analyze different
+complex traffic scenarios and take actions accordingly [5].
+A lot of microscopic and macroscopic traffic simulators
+have been developed including SUMO [10], Aimsun [13],
+Traffsim [14], SUMMIT [15], SifTraffic [16] and VISSIM [7].
+These simulators have practical use cases in traffic analysis
+including traffic flow measurement, multi-agent simulation,
+particle-based simulation, and so on. Although these state-of-
+the-art simulators have many practical traffic use cases, they
+face challenges in the application of simulating road traffic
+congestion in heterogeneous road transportation networks with
+a small amount of real-time data [3]. Also, these simulators
+lack the feature to adopt regression-based traffic modeling and
+simulate traffic congestion of a road link depending on traffic
+congestion of neighboring road links. Some of the simulators
+do not provide visualization for the simulation results in
+interactive geographical maps.
+We aim to develop a regression-based traffic simulator for
+spatiotemporal traffic modeling to predict traffic congestion
+of a road link depending on neighboring road segments and
+provide features to simulate and visualize the results using
+interactive geographical maps.
+The remainder of this paper is organized as follows. Section
+II briefly reviews the state-of-the-art traffic simulators and
+their scope in traffic modeling and simulation. The traffic
+modeling approach of RegTraffic is described in Section III.
+Section IV outlines the processing pipeline of the different
+components of RegTraffic. Section V provides a step-by-step
+simulation scenario of a traffic use case. Section VI shows
+the performance analysis of RegTraffic. Lastly, Section VII
+concludes this work and provides future research directions.
+II. BACKGROUND AND RELATED WORK
+Traffic simulator software is commonly divided into two
+categories: microscopic traffic simulators [7], [10] and macro-
+scopic traffic simulators [8].
+FreeSim [9] is a traffic simulator designed to conduct real-
+time freeway traffic simulation. SUMO (Simulation of Urban
+Mobility) [10] is a microscopic traffic simulator that is devel-
+oped to process complex and large road networks. SUMO is
+widely used in many applications including traffic flow model-
+ing [11] and color mapping Google Maps routes [12]. Aimsun
+[13] is a traffic simulator for modeling smart mobility. Traffsim
+[14] simulator is widely used for modeling isolated traffic
+control strategies. SUMMIT [15] provides functionalities to
+simulate urban driving in large traffic scenarios with massive
+and mixed traffic. SifTraffic [16] is a practical software tool to
+arXiv:2301.01245v1  [cs.NI]  23 Nov 2022
+
+TABLE I
+COMPARISON OF TRAFFIC SIMULATORS
+Comparison Category Simulator
+RegTraffic
+FreeSim [9]
+SUMO [10]
+Aimsun [13]
+TraffSim [14]
+SUMMIT [15]
+SimTraffic [16]
+VISSIM [7]
+Spatiotemporal Traffic Modeling
+Yes
+Yes
+Yes
+Yes
+Yes
+Yes
+Yes
+Yes
+Regression Modeling
+Yes
+No
+No
+No
+No
+No
+No
+No
+Interactive Geographical Maps
+Yes
+No
+Yes
+Yes
+No
+Yes
+Yes
+Yes
+Web Interface
+Yes
+No
+No
+No
+No
+No
+No
+No
+conduct simulations of practical traffic applications. VISSIM
+[7] is a microscopic traffic simulator for behavior-based multi-
+purpose traffic flow simulation.
+Wang et al. [17] explored different methods of correct-
+ing the traffic simulation models based on linear regression.
+Golovnin et al. [20] took a web-oriented approach to simulate
+road traffic, especially in urban settings. Mizuta et al. [21]
+evaluated the traffic flow near intersections of a metropolitan
+city to understand how agent-based traffic simulators work to
+approximate vehicle behaviors.
+A comparison among the existing traffic simulators along
+with RegTraffic is listed in Table I in terms of some key
+characteristics and features.
+III. MATHEMATICAL MODELING
+A. Spatial Feature
+Figure 1 shows a traffic road intersection. In this intersec-
+tion, we consider a road link as the spatial road feature that is
+dependent on one or several connected spatial road features.
+For example, for the intersection shown in Figure 1, the road
+link ˆy is a spatial feature and is modeled as a dependent
+variable in our regression modeling. The road links xs
+1, xs
+2 up
+to the road link xs
+n are the independent spatial features. It’s
+worth noting that the dependent road link ˆy is an outbound
+while all the independent road links xs
+1, xs
+2, ..., xs
+n inbound
+to the intersection. Our proposed traffic modeling approach
+described in [18] indicates that the dependent spatial feature
+must be an outbound road link and the independent spatial
+features must be inbound road links. The model incorporates
+a set of temporal features that can be extracted from both in-
+dependent and dependent spatial features through exploratory
+data analysis. The specific number of temporal features and
+independent spatial features are arbitrary and dependent on the
+specific road intersection and their orientation.
+Here, XS is defined as a set of independent spatial features
+�
+xs
+1, xs
+2, .., xs
+ns
+�
+as shown in Eq. (1).
+XS =
+�
+xs
+1, xs
+2, .., xs
+ns
+�
+(1)
+The cardinality of set XS is defined as ns as shown in Eq.
+(2).
+ns = |XS|
+(2)
+Fig. 1. Traffic Road intersection
+B. Temporal Feature Extraction
+In our modeling, we convert temporal features into categori-
+cal features using one hot encoding. To simplify our modeling,
+temporal features are encoded using only two values. Here,
+XT is a set of temporal features
+�
+xt
+1, xt
+2, .., xt
+nt
+�
+as shown in
+Eq. (3).
+XT =
+�
+xt
+1, xt
+2, .., xt
+nt
+�
+(3)
+The cardinality of set XT is defined as nt as shown in Eq.
+(4).
+nt = |XT |
+(4)
+The set of temporal features is extracted from spatial fea-
+tures using exploratory data analysis. Here, XT is the output
+of function f which takes in the set of spatial features XS as
+input. The function f is a many to many function that takes in
+a set of spatial features and conducts exploratory data analysis
+to extract a set of temporal features as shown in Eq. (5)
+XT = fns→nt(XS)
+(5)
+We define the set X as a union of the temporal features XT
+and spatial features XS as shown in Eq. (6).
+
+X = XT ∪ XS
+(6)
+C. Regression Modeling
+1) Regression Formation: RegTraffic forms a regression
+model through a linear combination of both temporal and
+spatial explanatory features to explain the dependent spatial
+feature ˆy as shown in Eq. (7). In this equation, all the
+independent features are associated with their corresponding
+regression coefficient. α indicates the bias and ϵ refers to the
+error term.
+ˆy =
+nt
+�
+i=1
+βt
+ixt
+i +
+ns
+�
+i=1
+βs
+i xs
+i + α + ϵ
+(7)
+In the regression Eq. (7), every explanatory temporal feature
+from setting XT is associated with a regression coefficient
+from set βT as shown in Eq. (8).
+βT =
+�
+βt
+1, βt
+2, .., βt
+nt
+�
+(8)
+Similarly, in the regression Eq. (7), every explanatory spatial
+feature from set XS is associated with a regression coefficient
+from set βS as shown in Eq. (9).
+βS =
+�
+βs
+1, βs
+2, .., βs
+ns
+�
+(9)
+Here, β is defined as the union of set βT and βS
+β = βT ∪ βS
+(10)
+2) Posterior Probability Distribution:
+We use a novel
+Bayesian linear regression approach for spatiotemporal traffic
+modeling of a road link proposed in [18]. Bayesian linear
+regression formulates a posterior probability distribution of
+the model parameters rather than just finding a single point
+estimate. The response variable is drawn from a probability
+distribution instead of a single value estimation. A Bayesian
+linear regression model samples the response variable from a
+normal distribution as shown in Eq. (11).
+y ∼ N(βT X, σ2I)
+(11)
+In Eq. (11), the response variable y is generated from a
+Gaussian normal distribution, which is characterized by a
+mean and variance. Eq. (12) refers to the Bayes Theorem
+which is the fundamental building block of Bayesian linear
+regression. Here, P(β | ˆy, X) is the posterior probability
+distribution of the model parameters, P(ˆy | β, X) is the
+likelihood of the data, P(β | X) is the prior probability of the
+parameters and P(ˆy | X) is the normalization constant. The
+posterior distribution of the model parameters is proportional
+to the multiplication of the likelihood of the data and the prior
+probability of the parameters. A detailed description of the
+model is described in [18].
+P(β | ˆy, X) = P(ˆy | β, X) ∗ P(β | X)
+P(ˆy | X)
+(12)
+Once the regression model is built, the user can provide
+new observations for independent spatial features XS and
+independent temporal features XT into the model. Based on
+the new observation, the model incorporates the regression
+coefficients associated with the explanatory variables and
+predicts the output for the dependent variable ˆy. An event
+can be associated with a specific value as an input for any
+independent spatial feature.
+D. Event Integration
+Here, XE is defined as a set of events
+�
+XE
+1 , XE
+2 , .., XE
+nE
+�
+as shown in Eq. (13).
+XE =
+�
+XE
+1 , XE
+2 , .., XE
+nE
+�
+(13)
+The cardinality of set XE is defined as nE as shown in Eq.
+(14).
+nE = |XE|
+(14)
+After event integration, the independent spatial features
+associated with an event are integrated into Eq. (7). If any
+independent spatial feature is associated with an event, we
+need to replace the value for the independent spatial feature
+xS with the events xE. Spatial features which are not affected
+by any specific event are represented by xS′ along with their
+model parameter βS′ as shown in Eq. (16). The amount of
+spatial features unaffected by any specific event is denoted as
+shown in Eq. (15).
+nS′ = nS − nE
+(15)
+ˆy =
+nt
+�
+i=1
+βt
+ixs
+i +
+nS′
+�
+i=1
+βS′
+i xS′
+i +
+nE
+�
+i=1
+βE
+i xE
+i + α + ϵ
+(16)
+However, we add a time constraint in association with the
+temporal components for adding a specific event into the
+regression equation. For any specific event XE occurred at
+time TE, the value of spatial feature XS will be replaced by
+the value of XE if TE ⊂ XT .
+IV. PROCESSING PIPELINE
+The processing pipeline of RegTraffic is shown in Figure
+- 2. The spatial feature consists of a unique name of the
+feature, the corresponding time series dataset of that spatial
+feature, and a set of latitude and longitude as the waypoints
+of the route of that spatial feature. The traffic data extraction
+is described in [19]. In this process, a user selects the starting
+and ending points of the route of interest and specifies the
+time range. The traffic data extraction tool gathers time-series
+information of the “congestion index” of that road link every
+15 minutes throughout the time range from Google Maps.
+The congestion index is defined by the average speed of that
+road link in terms of kilometers per hour. At the end of the
+process, the tool generates a time series dataset that has a
+
+Fig. 2. Processing Pipeline.
+unique name as provided by the user when adding a spatial
+feature in RegTraffic.
+RegTraffic also constructs a temporal feature component
+with three core input values. These are the unique name of
+the temporal feature, the corresponding time series dataset,
+and the time range of that temporal feature. RegTraffic takes
+a set of input preferences from the user as part of the model
+selection. It also allows users to choose the dependent feature
+for the regression model. Once the regression model is built,
+RegTraffic passes the regression coefficients to a visualization
+interface where a user can input new observations for the
+independent features that can be both spatial and temporal.
+V. SIMULATION
+A. Spatial Feature
+We conduct our experiment on four connected road links in
+Oshawa, Ontario, Canada as shown in Figure 3(a). The ending
+point of road links 2, 3 and 4 are connected with the origin of
+the road link 1. A connected road network is formed by these
+road links. We represent the traffic congestion level of these 4
+road links as Road1, Road2, Road3, and Road4, respectively.
+Road1 is the dependent link where Road2, Road3 and Road4
+are the independent links that collectively affect Road1 during
+a specific time of the day.
+We collect the average traffic speed of each road link every
+15 minutes for an entire week from 12:00 am March 01, 2020,
+to 11:45 pm March 07, 2020. As a result, there are a total
+of 672 observations over 7 days of time-series data for each
+road link. Figure 3(b) shows the time series of the average
+traffic speed of all four road links for the first two days. The
+y axis represents the average traffic speed in km/h, which is
+considered the traffic congestion index in our analysis. We can
+see that the time series has a cycle as the average traffic speed
+shows regular and predictable changes that recur every day
+within a certain time interval. The higher the average speed,
+the low the traffic congestion, and vice versa.
+B. Temporal Feature Extraction
+Figure 3(c) shows the hourly mean of the average speed for
+each road link. The mean values show very little variance
+(a) Intersection of Simcoe and Conlin
+Road in Oshawa
+(b) Time series data of 4 road links
+(c) Hourly average speed throughout a
+day
+(d) Identifying threshold for Peakhour
+Fig. 3. Average speed throughout a day
+compared to each other as they seem to move together
+throughout the day. The average of the different means of all
+road links is plotted in Figure 3(d). The horizontal line at a
+speed of 11.75 km/h divides the plot evenly and intersects with
+the total average speed at two points, one at daytime 8:00 and
+the other one at 23:00. From this exploratory data analysis,
+a new categorical feature called Peakhour is extracted that
+indicates a certain time interval during a day where the average
+traffic speed remains below 11.75 km/h. From 9:00 am to
+12:00 pm, the value of Peakhour would be 1, otherwise 0.
+Another temporal component is considered in the analysis as
+a categorical variable which is AM. The value of AM would
+be 1 when the meridiem is AM and 0 when it is PM.
+C. Regression Modeling
+The outcome of our Bayesian linear regression is the distri-
+bution of the model parameters. The model does not provide
+an exact estimate for a feature, but the mean value of the
+distribution can be considered as an estimate for the feature.
+The benefit of having a posterior probability distribution is
+that the model also provides an entire range of values that
+shows the uncertainty of the true values. The mean of a
+posterior probability distribution is taken as the best estimate
+of that model parameter. These mean estimates of these model
+parameters are put together to derive a new Eq. (17).
+Road1 = 7.4163 ∗ Intercept + 1.7561 ∗ AM
+−2.7517 ∗ Peakhour − 0.0477 ∗ Road2
+−0.0479 ∗ Road3 + 0.7139 ∗ Road4 + 1.7003 ∗ SD
+(17)
+
+Name
+Name
+Model Selection
+Time Series
+Spatial
+Temporal
+Time Series
+Data
+Feature
+Feature
+Data
+Regression
+Waypoints
+Time Range
+Modelling
+User Input
+User Input
+Visualization
+for
+for
+Scheme
+Spatial Features
+Temporal FeaturesN
+W
+Road 1
+S
+2
+Road 2
+Con
+CopperBrang
+Vegan · ss
+Road 4
+Road 3
+Subway
+Sandwich shop
+S
+C
+S
+SmileRoadl
+20
+Road2
+Road3
+18
+Road4
+16
+Speed (Km/h)
+14
+12
+10
+8
+6
+01-Mar
+06:00
+12:00
+18:00
+02-Mar
+06:00
+12:00
+18:00
+00:00
+00:00
+2020
+Datetime20
+18
+16
+Speed (Km/h)
+14
+12
+10
+Roadl
+Road2
+8
+Road3
+Road4
+6
+5
+10
+15
+20
+0
+Hours (0-23)18
+Hourly Mean
+- Threshold Speed
+16
+14
+Speed (Km/h)
+Peakhour
+Peakhour
+starts
+ends
+12
+S
+10
+8 
+0
+5
+10
+15
+20
+Hours (0-23)Fig. 4. Regression Analysis in RegTraffic Simulator
+D. Visualization
+Figure 4 describes a sample simulation procedure of a road
+intersection where Road1 is considered as a dependent road
+link and Road2, Road3 and Road4 are independent road
+links. Based on the spatial features, two new temporal features
+are extracted which are Peakhour and AM. RegTraffic shows
+the location of the road links on an interactive geographical
+map where the user can provide new observations for indepen-
+dent road links and temporal features to predict the outcome of
+the dependent road link. As shown in the figure, the user sets
+the congestion index of Road2, Road3, and Road4 to 18.05,
+4.4, and 10.45 kilometers per hour, respectively. The user also
+needs to provide the specific time as an input for the temporal
+features Peakhour and AM. RegTraffic calculates the value
+for the temporal features from the time input provided by the
+user and incorporates these values along with the input values
+for independent spatial features to predict the congestion index
+of dependent road link Road1. Based on the input values
+provided by the user, RegTraffic predicts the congestion index
+of the road link Road1, which is 13.3 kilometers per hour in
+this case.
+VI. PERFORMANCE EVALUATION
+A. Test Observations
+To evaluate the performance, the model is tested on a
+testing dataset of traffic observations. Figure 5 shows four
+random test observations from the testing dataset along with
+the probability density function of Road1. The true value of
+Road1 is represented by the dotted line and the mean of the
+probability distribution is represented by the straight line. The
+mean of the probability distribution is considered as the best
+estimate for the distributions. The estimated value provided by
+the model is very close to the true value in Figures 5(a), 5(b),
+5(c) and 5(d).
+B. New Observations
+To see how the model performs for new and modified obser-
+vations, we test the model with a set of new observations with
+random values for both the spatial and temporal components
+as shown in Figure 6. For every new observation, the model
+TABLE II
+MODEL COMPARISON BASED ON DIFFERENT FEATURES
+Mean Absolute
+Error
+Root Mean Squared
+Error
+Multiple Linear Regression
+1.31269
+1.71981
+Elastic Net Regression
+1.33501
+1.91345
+Bayesian Linear Regression
+1.3123
+1.71962
+Baseline
+3.75357
+5.09258
+(a)
+(b)
+(c)
+(d)
+Fig. 5. Test observations
+provides a new posterior distribution with the mean estimate.
+The vertical straight line represents the mean estimate of
+the posterior probability distribution for a new observation.
+We can see the highest probability density near the mean
+estimation of all posterior probability distributions as shown
+in Figures 6(a), 6(b), and 6(c) and 6(d).
+C. Comparison With Other Approaches
+The performance of the Bayesian linear regression model
+is compared in terms of Mean Absolute Error (MAE) and
+Root Mean Squared Error (RMSE) with two state-of-the-art
+frequentist models: Multiple Linear Regression and Elastic Net
+Regression as shown in Table II. We also develop a comparison
+baseline which is the mean of all possible observations of
+the traffic congestion. Here, Bayesian linear regression out-
+performs the state-of the-art-approaches in terms of accuracy
+as it has the lowest MAE and RMSE values.
+
+Estimated Dist.
+0.25
+True Value
+Mean Estimate
+0.20 -
+0.1
+Density
+0.10
+0.05
+0.00
+4
+6
+8
+10
+12
+14
+Speed (Km/h)Estimated Dist.
+0.25
+True Value
+Mean Estimate
+0.20 -
+0.15
+Density
+0.10
+0.05
+0.00
+4
+6
+8
+10
+12
+14
+16
+18
+Speed (Km/h)RegTraffic
+@loT Research Lab,Ontario Tech University
+Provide Time Input:
+Admin Panel
+3
+14:54
+Show Regressions
+【2]
+3
+16]
+Submit
+ShowCorrelations
+Winchester Rd E
+33
+Feature
+Value
+Winchester Rd W
+Winchester Golf Club
+Peakhour
+1
+Windfiel
+Farms
+Shopping
+entre
+Kedron Dells
+Golf Course
+AM
+KingMeadow
+0
+adoGolf Club
+Road1:Dependent
+13.3Km/h
+33
+Associate Event
+The Fields
+ofConlin
+Windfields
+Road2:Independent
+18.05Km/h
+33
+Road4:Independent
+10.45Km/h
+[16]
+CampSamac
+Fresh Food
+Road3:Independent
+Conlin Rd
+4.4Km/h
+Gar
+Sobeys
+P
+F
+35
+[26]
+The Waltzing Weasel
+WI
+Mili Express
+MetroEstimated Dist.
+0.25
+True Value
+Mean Estimate
+0.20 -
+0.1
+Density
+0.10
+0.05
+0.00
+12
+14
+16
+18
+20
+22
+24
+Speed (Km/h)Estimated Dist.
+0.25
+True Value
+Mean Estimate
+0.20 -
+0.15
+Density
+0.10 -
+0.05
+0.00
+2
+4
+6
+8
+10
+12
+14
+16
+Speed (Km/h)(a)
+(b)
+(c)
+(d)
+Fig. 6. New observations
+VII. CONCLUSION
+This paper presents RegTraffic, a new dynamic traffic sim-
+ulator for spatiotemporal traffic modeling for intercorrelated
+road links. RegTraffic builds a regression-based spatiotemporal
+traffic model to predict traffic congestion of a road link
+depending on neighboring road links and temporal compo-
+nents extracted through exploratory data analysis. RegTraffic
+provides a dynamic interface for a user to provide new obser-
+vations for independent features of the regression model and
+provides visualization on interactive geographical maps. The
+Mean Absolute Error and Root Mean Squared Error metrics
+are used to evaluate the performance of the regression-based
+predictive model integrated into RegTraffic. Performance eval-
+uation shows that RegTraffic can effectively predict traffic
+congestion of intercorrelated road links. In the current version
+of RegTraffic, we apply a Bayesian linear regression model for
+better interpretation and uncertainty evaluation. In the future,
+we plan to enhance RegTraffic by supporting other regression-
+based spatiotemporal traffic modeling approaches.
+REFERENCES
+[1] X. Yan and X. Su, Linear regression analysis: theory and computing.
+Singapore, Hackensack, NJ: World Scientific, 2009.
+[2] J. Barcelo, Ed., Fundamentals of Traffic Simulation. New York: Springer,
+2010
+[3] A. Pell, A. Meingast, and O. Schauer, “Trends in real time Traffic Sim-
+ulation,” in Transportation Research Procedia, vol. 25, pp. 1477–1484,
+2017.
+[4] G. Kotusevski and K. A. Hawick, ”A review of traffic simula-
+tion software”. 2009, Accessed: Jul. 30, 2021. [Online]. Available:
+https://mro.massey.ac.nz/handle/10179/4506
+[5] M. M. Mubasher and J. Syed Waqar ul Qounain, ”Systematic literature
+review of vehicular traffic flow simulators,” in International Conference
+on Open Source Software Computing (OSSCOM), 2015, pp. 1-6.
+[6] P. M. Ejercito, K. G. E. Nebrija, R. P. Feria and L. L. Lara-Figueroa,
+”Traffic simulation software review”. in 8th International Conference on
+Information, Intelligence, Systems & Applications (IISA), 2017, pp. 1-4.
+[7] M. Fellendorf, and P. Vortisch, “Microscopic traffic flow simulator
+VISSIM,��� in Fundamentals of Traffic Simulation, J. Barcel´o, Ed. New
+York, NY: Springer, 2010, pp. 63–93.
+[8] H. Ramadhan and I. G. B. B. Nugraha, ”Web based macroscopic road
+traffic simulator,” in 11th International Conference on Telecommunica-
+tion Systems Services and Applications (TSSA), 2017, pp. 1-6.
+[9] J. Miller and E. Horowitz, ”FreeSim - a free real time freeway traffic
+simulator,” 2007 IEEE Intelligent Transportation Systems Conference,
+2007, pp. 18-23.
+[10] P. A. Lopez et al., “Microscopic Traffic Simulation using SUMO,”
+in 21st International Conference on Intelligent Transportation Systems
+(ITSC), Maui, HI, Nov. 2018, pp. 2575–2582.
+[11] S. Haddouch, H. Hachimi and N. Hmina, ”Modeling the flow of road
+traffic with the SUMO simulator,” in 4th International Conference on
+Optimization and Applications (ICOA), 2018, pp. 1-5.
+[12] R. Mena, F. Zum´arraga, L. Urquiza and X. Calder´on, ”Google Maps
+Route Color Mapping with SUMO Simulator,” in International Confer-
+ence on Information Systems and Software Technologies (ICI2ST), 2019,
+pp. 92-99.
+[13] J. Casas, J. L. Ferrer, D. Garcia, J. Perarnau, and A. Torday, “Traffic
+Simulation with Aimsun,” in Fundamentals of Traffic Simulation, J.
+Barcel´o, Ed. New York, NY: Springer, 2010, pp. 173–232.
+[14] M. Lindorfer, C. Backfrieder, C. F. Mecklenbr¨auker and G. Ostermayer,
+”Modeling Isolated Traffic Control Strategies in TraffSim,” in UKSim-
+AMSS 19th International Conference on Computer modeling & Simula-
+tion (UKSim), 2017, pp. 143-148.
+[15] P. Cai, Y. Lee, Y. Luo and D. Hsu, ”SUMMIT: A Simulator for Urban
+Driving in Massive Mixed Traffic,” in IEEE International Conference
+on Robotics and Automation (ICRA), 2020, pp. 4023-4029.
+[16] D. K. Sorenson and J. Collins, “Practical Applications Of Traffic Simula-
+tion Using SifTraffic,” presented at the Compendium of Papers. Institute
+of Transportation Engineers 2000, District 6 Annual MeetingInstitute of
+Transportation Engineers (ITE), 2000. Accessed: Jul. 30, 2021. [Online].
+Available: https://trid.trb.org/view/671329
+[17] L. Wang, Z. Liu and Z. Liu, ”Research on correction method of
+traffic simulation model based on linear regression,” in International
+Conference on Anti-Counterfeiting, Security and Identification, 2010,
+pp. 192-194.
+[18] S. Mostafi, T. Alghamdi and K. Elgazzar, ”A Bayesian Linear Regression
+Approach to Predict Traffic Congestion,” in 2020 IEEE 6th World Forum
+on Internet of Things (WF-IoT), 2020, pp. 1-6.
+[19] S. Mostafi and K. Elgazzar, ”An Open Source Tool to Extract Traffic
+Data from Google Maps: Limitations and Challenges,” in International
+Symposium on Networks, Computers and Communications (ISNCC),
+2021.
+[20] O. K. Golovnin, K. V. Pupynin and A. S. Privalov, ”A Web-Oriented
+Approach for Urban Road Traffic Simulation,” in International Multi-
+Conference on Industrial Engineering and Modern Technologies (Far-
+EastCon), 2019, pp. 1-4.
+[21] H. Mizuta, ”Evaluation of metropolitan traffic flow with agent based
+traffic simulator and approximated vehicle behavior model near intersec-
+tions,” in Winter Simulation Conference (WSC), 2015, pp. 3925-3936.
+
+0.25
+Estimated Dist.
+Mean Estimate
+0.20 -
+0.15
+Density
+0.10 :
+0.05
+0.00
+4
+6
+8
+10
+12
+14
+16
+18
+Speed (Km/h)0.25
+Estimated Dist.
+Mean Estimate
+0.20 -
+0.15
+Density
+0.10
+0.05
+0.00
+4
+6
+8
+10
+12
+14
+16
+Speed (Km/h)0.25
+Estimated Dist
+Mean Estimate
+0.20 -
+0.15
+Density
+0.10
+0.05
+0.00
+10
+12
+14
+16
+18
+20
+22
+24
+Speed (Km/h)Estimated Dist.
+0.25 -
+Mean Estimate
+0.20 
+0.15
+Density
+0.10
+0.05
+0.00
+8
+10
+12
+14
+16
+18
+20
+Speed (Km/h)

BNAzT4oBgHgl3EQfhv2_/content/2301.01490v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1ee1f19af1edd495f07f26fc316fa13beb50a6ce9d133cd148d331e662c9b362
+size 7690522

BtE4T4oBgHgl3EQfFQxm/content/tmp_files/2301.04884v1.pdf.txt

@@ -0,0 +1,1067 @@
+Performance of an ultra-pure NaI(Tl) detector
+produced by an indigenously-developed
+purification method and crystal growth for the
+COSINE-200 experiment
+H. Lee 1,2,B.J. Park 1,2,J.J. Choi 2,3, O. Gileva 2, C. Ha 4, A. Iltis 5, E.J. Jeon 2,1,
+D.Y. Kim 2, K.W. Kim 2, S.H. Kim 2, S.K. Kim 3, Y.D. Kim 2,1, Y.J. Ko 2, C.H. Lee 2,
+H.S. Lee 2,1, I.S. Lee 2,∗, M.H. Lee 2,1, S.J. Ra 2, J.K. Son 2, K.A. Shin 2
+1IBS School, University of Science and Technology (UST), Daejeon 34113, Republic
+of Korea
+2 Center for Underground Physics, Institute for Basic Science (IBS), Daejeon 34126,
+Republic of Korea
+3 Department of Physics and Astronomy, Seoul National University, Seoul 08826,
+Republic of Korea
+4 Department of Physics, Chung-Ang University, Seoul 06973, Republic of Korea
+5 Damavan Imaging, Troyes 10430, France
+Correspondence*:
+I.S. Lee
+[email protected]
+ABSTRACT
+The COSINE-100 experiment has been
+operating with 106 kg of low-background
+NaI(Tl) detectors to test the results from the
+DAMA/LIBRA experiment, which claims to
+have observed dark matter. However, since the
+background of the NaI(Tl) crystals used in the
+COSINE-100 experiment is 2–3 times higher
+than that in the DAMA detectors, no conclusion
+regarding the claimed observation from the
+DAMA/LIBRA experiment could be reached.
+Therefore, we plan to upgrade the current
+COSINE-100 experiment to the next phase,
+COSINE-200, by using ultra-low background
+NaI(Tl) detectors. The basic principle was
+already proved with the commercially available
+Astro-grade NaI powder from Sigma-Aldrich
+company. However, we have developed a
+mass production process of ultra-pure NaI
+powder at the Center for Underground Physics
+(CUP) of the Institute for Basic Science (IBS),
+Korea, using the direct purification of the raw
+NaI powder. We plan to produce more than
+1,000 kg of ultra-pure powder for the COSINE-
+200 experiment.
+With our crystal grower
+installed at CUP, we have successfully grown
+a low-background crystal using our purification
+technique for the NaI powder.
+We have
+assembled a low-background NaI(Tl) detector.
+In this article, we report the performance of this
+ultra-pure NaI(Tl) crystal detector produced at
+IBS, Korea.
+Keywords:
+NaI(Tl) crystal;
+Dark matter;
+COSINE-200;
+Low-
+background detector; Purification
+1
+INTRODUCTION
+Numerous astronomical observations support the
+theory that most of the matter in universe is the
+invisible dark matter, although an understanding of
+its nature and interactions remains elusive (1, 2, 3).
+Even though tremendous efforts have been made
+to search for dark matter, no definitive signals
+have been observed (4, 5). The only exception is
+the DAMA experiment, which has observed an
+annual modulation of event rates using an array of
+NaI(Tl) detectors (6, 7). This observation could be
+interpreted as dark matter-nuclei interactions (8, 9).
+However, this result has been the subject of a
+continuing debate because no other experimental
+searches have observed similar signals (5, 10).
+1
+arXiv:2301.04884v1  [physics.ins-det]  12 Jan 2023
+
+H. Lee et al.
+Several experimental efforts using the same
+NaI(Tl) target materials are currently underway (11,
+12, 13, 14, 15, 16, 17).
+The COSINE-100
+experiment is one such effort presently operating
+at the Yangyang underground laboratory in Korea,
+which has provided several exciting physics
+results (9, 18, 19, 20, 21). However, due to
+the approximately 2–3 times higher background
+level, an unambiguous conclusion regarding the
+observation in the DAMA experiment using the
+same annual modulation signal has not been
+observed yet (22, 23).
+As an effort to upgrade the ongoing COSINE-
+100 experiment for the next-phase COSINE-
+200 experiment, we have conducted an R&D
+program aimed at producing a low-background
+NaI(Tl) detector to conclude on the observed
+signals from DAMA/LIBRA unambiguously. It
+includes the chemical purification of the raw NaI
+powder (24, 25), its crystal growth (26), and
+detector assembly (27). We have already proved
+the principle of a low-background NaI(Tl) detector
+using the commercially available low-background
+Astro-grade NaI powder from Sigma-Aldrich (28).
+As a next step, we have grown an NaI(Tl) crystal
+using our own NaO power produced using the mass
+purification process at IBS, Korea (29). This article
+reports the characteristics and performance of this
+indigenously-produced NaI(Tl) crystal.
+2
+NAI PURIFICATION AND CRYSTAL
+GROWTH
+The COSINE-200 detector requires extremely
+low levels of radioactive contamination in the
+materials used in the detector production. The
+major contributors to the background are the
+decays of 40K and 210Pb in the bulk NaI(Tl)
+crystal (30, 31). Because of the similarity in
+its chemical properties to those of Na, which
+is in the same periodic table group, K is the
+primary impurity contaminant, and its selective
+extraction from NaI powder is challenging. We
+found that the fractional recrystallization method
+effectively reduces the K and Pb impurities (24). In
+addition, using this method, the Ba concentration
+was significantly reduced, indicating a reduction
+of Ra impurities (24). Thus,we constructed a
+mass production facility at IBS, Daejeon, Korea,
+for producing ultra-pure NaI powder using the
+fractional recrystallization method on-site (25).
+The facility has been operated with a maximum
+production rate of 35 kg of ultra-pure powder
+in a single processing cycle of two weeks (29).
+Using our purification facility, we have performed
+mass purification of the fractional recrystallization
+process using raw NaI powder from Merck
+(99.99(5)% purity). In this mass purification process
+of the NaI power, we have achieved a concentration
+of K of 6.4 ppb and that of Pb below 0.3 ppb (29),
+which are consistent with contamination levels of
+the Astro-grade powder.
+The ultra-pure crystal was grown using a small-
+volume Kyropouls grower (32), which is the same
+grower used for growing the proof of principle low-
+background NaI(Tl) crystals using the commercial
+Astro-grade powder (28). In growing the crystal,
+1.7 kg of the purified NaI powder was loaded in
+a 12 cm diameter, 10 cm high quartz crucible. An
+NaI(Tl) crystal ingot, as shown in Figure 1(a), of
+∼70 mm diameter and ∼80 mm high and having
+a 1.1 kg mass, was grown in ∼24 h . During the
+crystal growth, N2 gas was continuously flushed
+using a thallium trap with a flow rate of 10 L/m.
+3
+EXPERIMENTAL SETUP
+3.1
+NaI(Tl) crystal
+The growth of the NaI(Tl) crystal (named NaI-
+037) was completed on January 18, 2021, using
+NaI powder purified IBS (24, 32). The top and
+bottom sections of the crystal ingot were cut using
+a diamond bandsaw, as shown in Figure 1(b). After
+cutting the top and bottom, the NaI-037 crystal is
+70 mm in diameter and 51 mm in height. The flat
+top and bottom surfaces and a barrel-shaped side
+surface were polished using aluminum oxide films
+ranging from 400 to 8000 grit. After polishing, the
+barrel was wrapped with a polytetrafluoroethylene
+(PTFE) film in several layers as a diffusive reflector.
+Frontiers
+2
+
+H. Lee et al.
+Figure 1a. NaI(Tl) Crystal ingot
+Figure 1b. Cut and polished NaI(Tl)
+crystal
+Figure 1. Bare NaI(Tl) (NaI-037) crystal
+A 3 mm thick copper casing with quartz windows at
+each end was encapsulated the crystal hermetically.
+Hamamatsu 3 inch photomultiplier tubes (PMTs),
+selected for high quantum efficiency (R12669SEL),
+were coupled via an optical interface to each end
+of the crystal. The entire assembly was performed
+in a glovebox, where the humidity was maintained
+to be less than 10 ppm (H2O) using Ar gas and
+a molecular sieve trap. Before the assembly, all
+parts were cleaned using diluted Citranox liquid
+with sonication and baked in a vacuum oven for
+more than 12 h. After assembly, the detector was
+delivered to the Yangyang underground laboratory
+(Y2L), which has ∼700 m of rock overburden (33).
+From the crystal growth to Y2L delivery, it took less
+than three weeks and minimized the cosmogenic
+activation in the crystal.
+3.2
+Shielding structure
+The background contamination levels of the
+NaI-037 crystal were evaluated using, the same
+experimental apparatus as that used for the NaI(Tl)
+crystal R&D at the Y2L (28, 34). It includes an
+array of 12 CsI(Tl) crystals surrounded by 10 cm
+copper, 5 cm polyethylene, 15 cm lead, and 30 cm
+liquid scintillator-loaded mineral oil (35, 36) as a
+radiation shield. The detector was installed inside
+the CsI(Tl) array, as shown in Figure 2.
+Figure 2. A schematic view of the Y2L setup. The
+NaI-037 crystal (red circle) was installed inside the
+CsI(Tl) crystal array (blue squares).
+3.3
+Electronics
+The PMTs attached to the NaI-037 crystal had
+two readouts each, a high-gain signal from the
+anode and a low-gain signal from the fifth-stage
+dynode. The anode signal was amplified by a
+factor of 30, whereas the dynode signal was
+amplified by a factor of 100 using a custom-
+made preamplifier. The amplified signals were
+digitized by 500 MHz, 12-bit flash analog-to-digital
+converters (FADCs). Triggers from the individual
+Frontiers
+3
+
+YPARINEOY
+NOUYPCCsl(T)Crystals
+Nal(m)Crystal
+Copper
+Folvetnviene
+Lead
+MineralOilH. Lee et al.
+channels were generated by the field-programmable
+gate arrays (FPGAs) embedded in the FADC. The
+final trigger was generated in the trigger and clock
+board (TCB) when an anode signal corresponding
+to one or more photoelectrons (PEs) occurred in
+each PMT within a 200 ns time window. The anode
+and dynode signals were recorded whenever the
+anode signal produced a trigger.
+Signals from the CsI(Tl) crystals were amplified
+by a factor of 10 and digitized in a charge-sensitive
+62.5 MHz FADC (SADC). The SADC provided the
+integrated charge and the time of the signal. An
+integration time of 2048 ns was used to record the
+CsI(Tl) signals considering their decay time. The
+SADC channels did not generate triggers.
+If the trigger condition was satisfied, the TCB sent
+trigger signals to the FADC and SADC to store the
+signals from the NaI(Tl) and the CsI(Tl) crystals.
+The FADC stored an 8 µs long waveform starting
+approximately 2.4 µs before the time of the trigger
+position. The SADC stored the maximum integrated
+charge within an 8 µs search window. This system
+is similar to the one used in the COSINE-100 data
+acquisition (37).
+Energy [keV]
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Entries
+0
+50
+100
+150
+200
+Am
+241
+Figure 3. Anode energy distribution obtained
+using a 241Am source.
+3.4
+Energy calibration and light yields
+The energy calibration of the anode signal was
+done using a 59.54 keV X-ray emitted from 241Am.
+Figure 3 shows the anode energy spectrum. A clear
+peak at 59.54 keV resulting from the 241Am source
+is shown together with the 127I X-ray escape peak
+around 30 keV. The dynode signal was calibrated
+using the photopeaks corresponding to 214Bi(609
+keV) and 40K(1460 keV) contaminants in the
+crystal.
+The charge distribution of the single photoelectron
+(SPE) was obtained by identifying the isolated
+clusters at the decay tail of the 59.54 keV X-ray
+signal from the 241Am source (3-5 µs after the
+signal start). The light yield was determined from
+the ratio of the total deposited charge and the mean
+of the SPE charge for the 59.54 keV X-ray data. In
+this crystal, a light yield of 17.8±0.6 number of
+photoelectron (NPE)/keV was obtained. It is similar
+to the result for the NaI-036 crystal, which has the
+highest light yield among the previously developed
+low-background NaI(Tl) crystals using the Astro-
+grade powder (28). This light yield is also larger
+than those of the detectors used in the COSINE-100
+and DAMA/LIBRA experiments, as summarized in
+table 1.
+4
+UNDERSTANDING THE
+BACKGROUND IN THE SPECTRUM
+4.1
+40K background
+40K is one of the most problematic background
+sources in the search for weakly interacting massive
+particles (WIMP) using NaI(Tl) crystals. The X-
+rays/Auger electrons from 40K decays produce
+3.2 keV energy signals, similar to the energy signals
+expected for a WIMP-nuclei interaction (30, 31, 38).
+The 40K decays also emit 1460 keV γ rays, which
+can escape from the NaI(Tl) crystal and hit the
+surrounding CsI(Tl) crystals, leading to a double
+coincidence with the 3.2 keV X-rays.
+Figure 4 shows the tagged low-energy spectra
+from the NaI(Tl) crystal by requiring the detection
+of the 1460 keV γ ray in the CsI(Tl) crystals. The
+Frontiers
+4
+
+H. Lee et al.
+40K background level in the NaI(Tl) crystal was
+determined by comparing the measured coincidence
+rate from a GEANT4-based simulated data, as
+described in Ref. (39). By accumulating more than
+six months of data, the K level was measured to be
+8.3±4.6 ppb, which was compared with the other
+NaI(Tl) crystals listed in Table 1. It is well below
+our goal of 20 ppb, consistent with the results from
+the DAMA/LIBRA crystals (34, 40) and previously
+developed NaI-035 and NaI-036 crystals with the
+Astro-grade powder.
+1
+2
+3
+4
+5
+6
+Energy (keV)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Number of Events
+Data
+Fit(Gaussian+Constant)
+Gaussian Component
+Figure 4. Energy deposition of the 3.2 keV
+40K coincidence events in the NaI-037 crystal.
+The model of the energy spectrum assumes a
+combination of a Gaussian 40K signal and a constant
+background.
+4.2
+α analysis
+Alpha-induced events inside the NaI(Tl) crystals
+can be identified using the fast decay times of
+their corresponding signals. The charge-weighted
+duration time, called the meantime, is defined as
+⟨t⟩ = ΣiAiti
+ΣiAi
+,
+(1)
+where A and t are the charge and time of the i-th
+digitized bin of a signal waveform, respectively.
+The meantime is estimated within 1500 ns from the
+pulse starting timing. Figure 5 shows a scatter plot
+of the energy versus the meantime for the NaI-037
+0.2
+0.25
+0.3
+0.35
+s)
+µ
+Meantime (
+1
+2
+3
+4
+Energy (MeV)
+Figure 5. Scatter plot of the meantime versus the
+energy distribution events measured over 7.8 days
+for the NaI-037 crystal. The α events (red dots) and
+the γ/β events (black dots) are separated clearly.
+crystal. In the figure, the populations of γ/β and
+α events can be separated clearly due to the faster
+decay times of the α-induced events.
+4.3
+210Pb background
+In the NaI(Tl) crystal experiments, the dominant
+background source in the low-energy signal region
+is from 210Pb (31, 41, 42). The 210Pb activity can be
+estimated from the alpha-decay studies, because the
+α decays of 210Po originate from the β decays of
+210Pb. Due to the decay time of 200 days of 210Po,
+the amount of 210Pb produced can be estimated
+using a time-dependent fit of the alpha rate as
+follows:
+N = NPb210
+�
+1 − e−(t−t0)/τP o210
+�
++ C,
+(2)
+where N is the total alpha rate, NPb210 is the
+amount of 210Pb at the equilibrium state, t0 is the
+time difference between 210Pb contamination and
+the start time of data taking, τPo210 is the mean
+lifetime of 210Po, approximately 200 days, and C
+represents the long-lived components from 238U
+and 232Th chains. Figure 6 shows the measured
+total alpha rates in the NaI-037 crystal over the
+detector running time. The 210Pb activity in the
+crystal was estimated to be 0.38±0.10 mBq/kg,
+Frontiers
+5
+
+H. Lee et al.
+which is lower than the COSINE-100 crystals and
+is consistent with the activity in the NaI-036 crystal
+produced using the Astro-grade powder. However,
+this activity is slightly higher than the DAMA
+crystals and another crystal NaI-036 grown with
+the same Astro-grade powder. The 0.4 mBq/kg
+level contamination of 210Pb is enough to reach
+1 count/kg/keV/day background level, similar to
+the activity in the DAMA/LIBRA detectors, as
+described in Ref. (28).
+100
+150
+200
+Days from crystal growing
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Activity (mBq/kg)
+Data
+Fit
+Asymptotic Line
+Figure 6. The total alpha rate in the NaI-037
+crystal as a function of time, modeled with 210Po
+assuming contamination of 222Rn (and/or 210Pb).
+The asymptotic line corresponds to the rate of total
+alpha events in the equilibrium state.
+4.4
+232Th background
+Contaminants from the 228Th subchain in the
+232Th family can be estimated by deploying the
+time-delayed α–α coincident events of 220Rn and
+216Po. The alpha decay of 216Po has a half-life of
+0.145 s following its production via alpha decay of
+220Rn. Owing to the short half-life of 216Po, it is
+straightforward to select two successive α particles
+with almost no random coincident events.
+The presence of the coincident events is shown
+in figure 7(a) as the distribution of the time gap
+between those two α events. The exponential
+component indicates the contamination from 232Th,
+corresponding to below 0.39 ppt (90% confidence
+level). The 232Th concentration in the NaI-037
+crystal is the lowest among the other NaI(Tl)
+crystals, as summarized in table 1.
+4.5
+238U background
+238U is one of the common radioisotopes because
+of its long half-life . The 238U content in the
+background can be studied using the time-delayed
+β–α coincident events, similar to the calculation of
+the 232Th background. This method exploits the α
+decay of 214Po with a half-life of 164.3 µs, while
+214Bi, the parent particle of 214Po, undergoes β
+decay. Due to the 50 µs dead time of the trigger
+system, the coincident events with delay times
+greater than 50 µs can be tagged. The results are
+shown in figure 7. The 238U activity of NaI-037 was
+1.02±0.58 ppt, similar to that observed for the other
+NaI(Tl) crystals, as given in Table 1.
+4.6
+External Background
+Because of the small size of the NaI-037 crystal
+and no liquid scintillator active veto, a significantly
+higher background contribution is expected from
+the external background compared to those found
+in the COSINE-100 crystals (30, 41). The PMTs
+attached to the NaI(Tl) and the CsI(Tl) crystals are
+the primary sources of external background. In this
+study, the external background contributions were
+simulated using the GEANT4-based simulation
+toolkit used for the COSINE-100 background
+modeling (30, 41).
+4.7
+Cosmogenic radionuclides
+The
+cosmogenic
+production
+of
+radioactive
+isotopes in the NaI(Tl) crystal is mainly due to long-
+lived nuclides such as 3H and 22Na (30, 44). The
+NaI-037 crystal was grown in Daejeon, Korea (70
+m in altitude) and delivered underground within a
+month. Based on the previous study, one-month
+exposure time near sea level can produce 0.004
+mBq/kg of 3H and 0.05 mBq/kg of 22Na (44),
+respectively.
+Frontiers
+6
+
+H. Lee et al.
+0
+0.2
+0.4
+0.6
+0.8
+1
+Time (s)
+0
+1
+2
+3
+4
+Number of Events
+Data
+Fit(Exponential+Constant)
+Exponential Component
+Constant
+Figure 7a. Time differen ofbetween two α decays of
+the 220Rn–216Po decay chain.
+200
+400
+600
+800
+1000
+s)
+µ
+Time (
+0
+2
+4
+6
+8
+10
+Number of Events
+Data
+Fit(Exponential+Constant)
+Exponential Component
+Constant
+Figure 7b. Time difference between the 214Po α
+decay and 214Bi β decay.
+Figure 7. Time difference distributions of data (black dots) and the exponential fits to them (red-solid
+line).
+Table 1. Measured radioactive contaminants in the NaI-037 crystal, C6 of COSINE-100 (30), DAMA
+crystals (40, 43), and the previously grown NaI-035 and NaI-036 crystals using the Astro-grade
+powder (28).The upper limits are given at a 90% confidence level.
+Crystal
+Mass (kg)
+LY (NPE/keV)
+40K (ppb)
+210Pb (mBq/kg)
+232Th (ppt)
+238U (ppt)
+NaI-037
+0.71
+17.8±0.6
+8.3±4.6
+0.44±0.09
+0.2±0.3
+1.0±0.6
+NaI-035
+0.61
+11.8±1.8
+<42
+0.01±0.02
+1.7±0.5
+0.9±0.3
+NaI-036
+0.78
+17.1±0.5
+<53
+0.42±0.27
+<4.9
+36.5±3.9
+COSINE-100
+12.5
+14.6±1.5
+16.8±2.5
+1.87±0.09
+0.7±0.2
+<0.02
+DAMA
+9.7
+5–10
+<20
+0.01–0.03
+0.5–7.5
+0.7–10
+5
+BACKGROUND MODELING
+For a quantitative understanding of the background
+in the NaI-037 crystal, GEANT4-based simulation,
+developed for the background modeling of the
+COSINE-100 NaI(Tl) crystals (30, 41) and also
+used in the previously grown crystals using the
+Astro-grade powder (28), was performed. The input
+values of the contamination levels are obtained
+from Table 1. A simultaneous fit was done to
+the single-hit low energy (3–60 keV), single-
+hit high energy (60 keV–3 MeV), multiple-hit
+low energy, and multiple-hit high energy events
+using the log-likelihood method. A multiple-hit
+event corresponds to one or more coincident
+hits in any of the surrounding CsI(Tl) crystals.
+The backgrounds from the PMTs attached to the
+NaI(Tl) and CsI(Tl) crystals were measured using
+a high-purity germanium detector (30, 31). These
+values were constrained to be within 50% of
+the measured result because the exact locations
+of such radioisotopes are uncertain. The long-
+lived cosmogenic radioisotopes were constrained
+to be within 50% of their calculation production
+values whereas the other short-lived cosmogenic
+components were floated. Figure 9 and Table 2
+show the fitted results for the NaI-037 crystal
+on all simulated background components and the
+Frontiers
+7
+
+H. Lee et al.
+Energy [keV]
+10
+20
+30
+40
+50
+60
+Counts/da/kg/keV
+2
+−
+10
+1
+−
+10
+1
+10
+2
+10
+Data
+Internal
+Cosmogenic
+External
+Figure 9a. Single-hit low-energy (2–60 keV)
+Energy [keV]
+500
+1000
+1500
+2000
+2500
+3000
+Counts/da/kg/keV
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+10
+2
+10
+Data
+Internal
+Cosmogenic
+External
+Figure 9b. Single-hit high-energy (60–3000 keV)
+Energy [keV]
+10
+20
+30
+40
+50
+60
+Counts/da/kg/keV
+2
+−
+10
+1
+−
+10
+1
+10
+2
+10
+Data
+Internal
+Cosmogenic
+External
+Figure 9c. Multiple-hit low-energy (2–60 keV)
+Energy [keV]
+500
+1000
+1500
+2000
+2500
+3000
+Counts/da/kg/keV
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+10
+2
+10
+Data
+Internal
+Cosmogenic
+External
+Figure 9d. Multiple-hit high-energy (60–3000 keV)
+Figure 9. Measured single-hit and multiple-hit background spectra of the NaI-037 (black point) crystal
+fitted with the different simulated background components using a simultaneous fit of four channels using
+the log-likelihood method. The external component (purple-hatched area) is the dominant contributor.
+summary of the fitted radioactive contaminants,
+respectively. The overall energy spectra match
+the data for the single-hit and multiple-hit events
+satisfactorily.
+The level of the fitted internal
+components is similar to the previously grown
+NaI-036 crystal (28).
+The expected background level in the COSINE-
+200 crystal can be studied from the simulated
+background by assuming a 12.5 kg detector in the
+COSINE-100 shielding, as described in Ref. (28).
+If the measured backgrounds, given in Table 2 for
+the simulated study, are considered, a background
+level of approximately 0.5 counts/kg/keV/day in the
+1–6 keV energy region is obtained, which is similar
+to the result for the NaI-036 crystal in the previous
+study (28). This is a slightly higher background
+level than observed from the NaI-035 crystal owing
+to the higher 210Pb contamination. However, it
+is still less than 1 count/kg/keV/day, the target
+background level for the COSINE-200 experiment.
+6
+CONCLUSION
+In this article, we presented the performance of the
+first ultra-low background NaI(Tl) crystal produced
+Frontiers
+8
+
+H. Lee et al.
+Table
+2. Summary of the fitted radioactive
+contaminants in the modeling of the NaI-037
+crystal.
+Background source
+Isotope
+Activity (mBq/kg)
+Internal
+238U
+0.025 ± 0.35
+228Th
+0.0065 ± 0.00025
+40K
+0.17 ± 0.047
+210Pb
+0.36 ± 0.11
+Cosmogenic
+125I
+0.40 ± 0.0015
+121Te
+0.80 ± 0.0029
+121mTe
+0.063 ± 0.0096
+123mTe
+0.045 ± 0.099
+125mTe
+0.14 ± 0.011
+127mTe
+0.16 ± 0.10
+109Cd
+0.0071 ± 0.0010
+113Sn
+0.020 ± 0.00094
+22Na
+0.050 ± 0.010
+3H
+0.0037 ± 0.0097
+NaI PMTs
+238U
+48.83 ± 5.90
+228Th
+23.80 ± 5.70
+40K
+58.07 ± 17.82
+CsI PMTs
+238U
+27.64 ± 6.15
+228Th
+24.18 ± 6.10
+40K
+378.28 ± 17.74
+using the direct purification of the NaI powder in
+our facility as a part of a program for the next-
+generation COSINE-200 experiment. The results
+of this study show a similar quantity of internal
+background contamination in the crystals grown
+using commercial Astro-grade powder. It indicates
+that the direct powder purification and crystal
+growth procedures employed at our facility can
+provide suitable NaI(Tl) crystals for the COSINE-
+200 experiment. Based on the experience of
+developing ultra-pure NaI(Tl) crystals, we are
+moving to full-size crystal growth with our purified
+powder.
+ACKNOWLEDGMENTS
+We thank Korea Hydro and Nuclear Power Co., Ltd.
+(KHNP) for providing the underground laboratory
+space at Yangyang. This work is supported by the
+Institute for Basic Science (IBS) under the project
+code IBS-R016-A1.
+Frontiers
+9
+
+H. Lee et al.
+REFERENCES
+1 .Clowe D, et al.
+A direct empirical proof of
+the existence of dark matter. Astrophys. J. 648
+(2006) L109. doi:10.1086/508162.
+2 .Aghanim N, et al.
+Planck 2018 results. VI.
+Cosmological parameters. Astron. Astrophys.
+641 (2020) A6.
+doi:10.1051/0004-6361/
+201833910. [Erratum: Astron.Astrophys. 652,
+C4 (2021)].
+3 .Bertone G, Hooper D. History of dark matter.
+Rev. Mod. Phys. 90 (2018) 045002.
+doi:10.
+1103/RevModPhys.90.045002.
+4 .Undagoitia TM, Rauch L. Dark matter direct-
+detection experiments. J. Phys. G 43 (2016)
+013001. doi:10.1088/0954-3899/43/1/013001.
+5 .Schumann M.
+Direct Detection of WIMP
+Dark Matter: Concepts and Status.
+J. Phys.
+G 46 (2019) 103003. doi:10.1088/1361-6471/
+ab2ea5.
+6 .Bernabei R, et al.
+Final model independent
+result of DAMA/LIBRA-phase1.
+Eur. Phys.
+J. C 73 (2013) 2648.
+doi:10.1140/epjc/
+s10052-013-2648-7.
+7 .Bernabei R, et al.
+First model independent
+results from DAMA/LIBRA-phase2.
+Nucl.
+Phys. Atom. Energy 19 (2018) 307.
+doi:10.
+15407/jnpae2018.04.307.
+8 .Savage
+C,
+et
+al.
+Compatibility
+of
+DAMA/LIBRA
+dark
+matter
+detection
+with other searches. JCAP 0904 (2009) 010.
+doi:10.1088/1475-7516/2009/04/010.
+9 .Ko
+YJ,
+et
+al.
+Comparison
+between
+DAMA/LIBRA and COSINE-100 in the light
+of Quenching Factors. JCAP 1911 (2019) 008.
+doi:10.1088/1475-7516/2019/11/008.
+10 .Workman RL, et al. Review of Particle Physics.
+PTEP 2022 (2022) 083C01. doi:10.1093/ptep/
+ptac097.
+11 .Kim KW, et al. Tests on NaI(Tl) crystals for
+WIMP search at the Yangyang Underground
+Laboratory. Astropart. Phys. 62 (2015) 249.
+doi:10.1016/j.astropartphys.2014.10.004.
+12 .Xu J, et al. SABRE – A test of DAMA with
+high-purity NaI(Tl) crystals. AIP Conf. Proc.
+1672 (2015) 040001. doi:10.1063/1.4927983.
+13 .Adhikari G, et al.
+Initial Performance of
+the COSINE-100 Experiment.
+Eur. Phys.
+J. C 78 (2018) 107.
+doi:10.1140/epjc/
+s10052-018-5590-x.
+14 .Fushimi KI. Low Background Measurement
+by Means of NaI(Tl) Scintillator: Improvement
+of
+Sensitivity
+for
+Cosmic
+Dark
+Matter.
+RADIOISOTOPES 67 (2018) 101. doi:10.3769/
+radioisotopes.67.101.
+15 .Coarasa I, et al. ANAIS-112 sensitivity in the
+search for dark matter annual modulation. Eur.
+Phys. J. C 79 (2019) 233. doi:10.1140/epjc/
+s10052-019-6733-4.
+16 .Amare J, et al. Performance of ANAIS-112
+experiment after the first year of data taking.
+Eur. Phys. J. C 79 (2019) 228. doi:10.1140/
+epjc/s10052-019-6697-4.
+17 .Suerfu B, et al. Growth of Ultra-high Purity
+NaI(Tl) Crystal for Dark Matter Searches. Phys.
+Rev. Research 2 (2020) 013223. doi:10.1103/
+PhysRevResearch.2.013223.
+18 .Adhikari G, et al. An experiment to search for
+dark-matter interactions using sodium iodide
+detectors. Nature 564 (2018) 83. doi:10.1038/
+s41586-018-0739-1.
+19 .Adhikari G, et al.
+Strong constraints from
+COSINE-100 on the DAMA dark matter results
+using the same sodium iodide target. Sci. Adv. 7
+(2021) abk2699. doi:10.1126/sciadv.abk2699.
+20 .Adhikari G, et al. Searching for low-mass dark
+matter via the Migdal effect in COSINE-100.
+Phys. Rev. D 105 (2022) 042006. doi:10.1103/
+PhysRevD.105.042006.
+21 .Adhikari G, et al.
+An induced annual
+modulation signature in COSINE-100 data by
+DAMA/LIBRA’s analysis method (2022). doi:
+10.48550/arXiv.2208.05158.
+22 .Adhikari G, et al. Search for a Dark Matter-
+Induced Annual Modulation Signal in NaI(Tl)
+with the COSINE-100 Experiment.
+Phys.
+Rev. Lett. 123 (2019) 031302.
+doi:10.1103/
+PhysRevLett.123.031302.
+23 .Adhikari G, et al. Three-year annual modulation
+search with COSINE-100. Phys. Rev. D 106
+Frontiers
+10
+
+H. Lee et al.
+(2022) 052005.
+doi:10.1103/PhysRevD.106.
+052005.
+24 .Shin K, et al. Reduction of the radioactivity in
+sodium iodide (NaI) powder by recrystallization
+method. J. Radioanal. Nucl. Chem. 317 (2018)
+1329. doi:10.1007/s10967-018-6006-y.
+25 .Shin K, et al. A facility for mass production
+of ultra-pure NaI powder for the COSINE-200
+experiment. JINST 15 (2020) C07031. doi:10.
+1088/1748-0221/15/07/C07031.
+26 .Ra S, et al. Scintillation crystal growth at the
+CUP. PoS ICHEP2018 (2019) 668. doi:10.
+22323/1.340.0668.
+27 .Choi J, et al. Improving the light collection
+using a new NaI(Tl)crystal encapsulation. Nucl.
+Instrum. Meth. A 981 (2020) 164556. doi:10.
+1016/j.nima.2020.164556.
+28 .Park B, et al. Development of ultra-pure NaI(Tl)
+detectors for the COSINE-200 experiment. Eur.
+Phys. J. C 80 (2020) 814. doi:10.1140/epjc/
+s10052-020-8386-8.
+29 .Shin K, et al. Mass production of ultra-pure
+NaI powder for COSINE-200 (2023). Paper in
+preparation.
+30 .Adhikari P, et al.
+Background model for
+the NaI(Tl) crystals in COSINE-100.
+Eur.
+Phys. J. C 78 (2018) 490. doi:10.1140/epjc/
+s10052-018-5970-2.
+31 .Adhikari G, et al. Understanding NaI(Tl) crystal
+background for dark matter searches.
+Eur.
+Phys. J. C 77 (2017) 437. doi:10.1140/epjc/
+s10052-017-5011-6.
+32 .Ra SJ, et al. Status of ultra-pure scintillating
+crystal growth for rare process experiments by
+CUP. JPCS 1468 (2020) 012144. doi:10.1088/
+1742-6596/1468/1/012144.
+33 .Prihtiadi H, et al.
+Muon detector for
+the COSINE-100 experiment.
+JINST 13
+(2018) T02007. doi:10.1088/1748-0221/13/02/
+T02007.
+34 .Adhikari P, et al.
+Understanding internal
+backgrounds in NaI(Tl) crystals toward a 200
+kg array for the KIMS-NaI experiment. Eur.
+Phys. J. C 76 (2016) 185. doi:10.1140/epjc/
+s10052-016-4030-z.
+35 .Lee HS, et al. First limit on WIMP cross section
+with low background CsI(Tl) crystal detector.
+Phys. Lett. B 633 (2006) 201. doi:10.1016/j.
+physletb.2005.12.035.
+36 .Lee H, et al. Development of low-background
+CsI(Tl) crystals for WIMP search.
+Nucl.
+Instrum. Meth. A 571 (2007) 644–650. doi:10.
+1016/j.nima.2006.10.325.
+37 .Adhikari G, et al.
+The COSINE-100 Data
+Acquisition System. JINST 13 (2018) P09006.
+doi:10.1088/1748-0221/13/09/P09006.
+38 .Amare J, et al. Analysis of backgrounds for
+the ANAIS-112 dark matter experiment. Eur.
+Phys. J. C 79 (2019) 412. doi:10.1140/epjc/
+s10052-019-6911-4.
+39 .Kim KW, et al. Tests on NaI(Tl) crystals for
+WIMP search at the Yangyang Underground
+Laboratory. Astropart. Phys. 62 (2015) 249.
+doi:10.1016/j.astropartphys.2014.10.004.
+40 .Bernabei R, et al.
+The DAMA/LIBRA
+apparatus. Nucl. Instrum. Meth. A 592 (2008)
+297–315. doi:10.1016/j.nima.2008.04.082.
+41 .Adhikari G, et al. Background modeling for
+dark matter search with 1.7 years of COSINE-
+100 data. Eur. Phys. J. C 81 (2021) 837. doi:10.
+1140/epjc/s10052-021-09564-0.
+42 .Fushimi K, et al.
+Development of highly
+radiopure NaI(Tl) scintillator for PICOLON
+dark matter search project. PTEP 2021 (2021)
+043F01. doi:10.1093/ptep/ptab020.
+43 .Bernabei R, et al. Performances of the new high
+quantum efficiency pmts in dama/libra. JINST
+7 (2012) P03009. doi:10.1088/1748-0221/7/03/
+P03009.
+44 .de Souza EB, et al.
+Study of cosmogenic
+radionuclides in the COSINE-100 NaI(Tl)
+detectors. Astropart. Phys. 115 (2020) 102390.
+doi:10.1016/j.astropartphys.2019.102390.
+Frontiers
+11
+

F9E1T4oBgHgl3EQf-wbm/content/tmp_files/2301.03574v1.pdf.txt

@@ -0,0 +1,1963 @@
+arXiv:2301.03574v1  [math.NA]  9 Jan 2023
+SHARP PREASYMPTOTIC ERROR BOUNDS FOR THE
+HELMHOLTZ h-FEM
+J. GALKOWSKI∗ AND E. A. SPENCE†
+Abstract.
+In the analysis of the h-version of the finite-element method (FEM), with fixed
+polynomial degree p, applied to the Helmholtz equation with wavenumber k ≫ 1, the asymptotic
+regime is when (hk)pCsol is sufficiently small and the sequence of Galerkin solutions are quasioptimal;
+here Csol is the norm of the Helmholtz solution operator, normalised so that Csol ∼ k for nontrapping
+problems. The preasymptotic regime is when (hk)2pCsol is sufficiently small, and (for physical data)
+one expects the relative error of the Galerkin solution to be controllably small.
+In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-
+coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or
+combinations of these) and with the radiation condition approximated either by a radial perfectly-
+matched layer (PML) or an impedance boundary condition. Previously, such bounds for p > 1 were
+only available for Dirichlet obstacles with the radiation condition approximated by an impedance
+boundary condition.
+Our result is obtained via a novel generalisation of the “elliptic-projection”
+argument (the argument used to obtain the result for p = 1) which can be applied to a wide variety
+of abstract Helmholtz-type problems.
+AMS subject classifications. 35J05, 65N15, 65N30, 78A45
+Key words. Helmholtz, FEM, high order, pollution effect, preasymptotic, perfectly-matched
+layer, elliptic projection.
+1. Introduction.
+1.1. Informal statement of the main result. We consider the h-version of
+the finite-element method (h-FEM), where accuracy is increased by decreasing the
+meshwidth h while keeping the polynomial degree p constant, applied to the Helmholtz
+equation.
+Theorem 1.1 (Informal statement of the main result).
+Let u be the solution to
+the variable-coefficient Helmholtz equation, with wavenumber k > 0, in the exterior
+of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and
+with the radiation condition approximated either by a perfectly-matched layer (PML)
+or an impedance boundary condition. Let Csol be the norm of the solution operator,
+normalised so that Csol ∼ k for nontrapping problems.
+Under the natural regularity assumptions on the domain and coefficients, if
+(1.1)
+(hk)2pCsol is sufficiently small
+then the Galerkin solution uh exists, is unique, and satisfies
+∥u − uh∥H1
+k(Ω) ≤ C
+�
+1 + hk + (hk)pCsol
+�
+min
+vh∈Hh ∥u − vh∥H1
+k(Ω) ,
+(1.2)
+∥u − uh∥L2(Ω) ≤ C
+�
+hk + (hk)pCsol
+�
+min
+vh∈Hh ∥u − vh∥H1
+k(Ω) .
+(1.3)
+Furthermore, if the data is k-oscillatory (in a sense made precise below), then
+(1.4)
+∥u − uh∥H1
+k(Ω)
+∥u∥H1
+k(Ω)
+≤ C
+�
+1 + hk + (hk)pCsol
+�
+(hk)p;
+∗Department of Mathematics, University College London, 25 Gordon Street, London, WC1H
+0AY, UK, [email protected]
+†Department
+of
+Mathematical
+Sciences,
+University
+of
+Bath,
+Bath,
+BA2
+7AY,
+UK,
+[email protected]
+1
+
+i.e., the relative H1
+k error can be made controllably small by making (hk)2pCsol suffi-
+ciently small.
+The norm ∥ · ∥H1
+k(Ω) in the bounds above is defined by
+(1.5)
+∥v∥2
+H1
+k(Ω) := k−2 ∥∇v∥2
+L2(Ω) + ∥v∥2
+L2(Ω) .
+The fact that, for oscillatory data, the relative H1
+k error for the Helmholtz h-FEM
+is controllably small if (hk)2pCsol is sufficiently small was famously identified for 1-d
+nontrapping problems by the work of Ihlenburg and Babuˇska [25, 26]. The bounds
+(1.2) and (1.3) have previously been obtained (i) for the Dirichlet obstacle problem
+with impedance boundary conditions approximating the radiation condition [12, 40]
+and (ii) for PML with constant-coefficients, no obstacle, and p = 1 [32].
+The present paper proves the bounds (1.2), (1.3), and (1.4) assuming only that
+the sesquilinear form is continuous, satisfies a G˚arding inequality, and satisfies certain
+standard elliptic-regularity assumptions, therefore covering a variety of scatterers and
+methods for truncating the exterior domain (to approximate the radiation condition).
+Regarding the latter: in this paper we consider truncating with a PML or an imped-
+ance boundary condition, but truncating with the exact Dirichlet-to-Neumann map
+is also, in principle, covered by the abstract framework; see Remark 5.4 below.
+1.2. Statement of the main abstract result. Let H ⊂ H0 ⊂ H∗ be Hilbert
+spaces with H0 identified with its dual and H ⊂ H0 compact. Let a : H × H → C be
+a continuous sesquilinear form, i.e.,
+(1.6) |a(u, v)| ≤ Ccont ∥u∥H ∥v∥H
+and
+a(λu, µv) = λ¯µa(u, v)
+for all u, v ∈ H,
+satisfying the G˚arding inequality
+(1.7)
+ℜa(v, v) ≥ CG1 ∥v∥2
+H − CG2 ∥v∥2
+H0
+for all v ∈ H
+for some CG1, CG2 > 0. We assume further that Ccont, c, C and all the other constants
+in this section are independent of k.
+Assumption 1.2 (“Elliptic regularity” assumptions on a).
+Let Z0 = H0, Z1 =
+H, and Zj ⊂ Zj−1 for j = 2, . . . , ℓ + 1 such that Zj is dense in Zj−1, and assume
+that for all u ∈ H with
+sup
+v∈H, ∥v∥(Zj−2)∗=1
+|a(u, v)| < ∞,
+u ∈ Zj and
+(1.8)
+∥u∥Zj ≤ C
+�
+∥u∥H0 +
+sup
+v∈H, ∥v∥(Zj−2)∗=1
+|a(u, v)|
+�
+,
+j = 2, . . . , ℓ + 1.
+Assume further that for any w ∈ H such that
+sup
+w∈H, ∥v∥(Zj−2)∗=1
+|(ℜa)(u, v)| < ∞,
+w ∈ Zj with
+(1.9)
+∥w∥Zj ≤ C
+�
+∥u∥H0 +
+sup
+v∈H, ∥v∥(Zj−2)∗=1
+|(ℜa)(u, v)|
+�
+,
+j = 2, . . . , ℓ + 1,
+2
+
+where the sesquilinear form ℜa is defined by
+(1.10)
+(ℜa)(u, v) := 1
+2
+�
+a(u, v) + a(v, u)
+�
+.
+Remark 1.3. Note that ℜa in (1.7) and (1.10) could be replaced by ℜ(eiωa), so
+long as one uses the same value of ω in both conditions. Remark 4.4 below describes
+a situation where this is useful.
+Given g ∈ H∗, suppose that u ∈ H satisfies
+(1.11)
+a(u, v) = ⟨g, v⟩
+for all v ∈ H.
+Given a sequence of finite dimensional subspace {Hh}h>0 with Hh ⊂ H, the
+sequence of Galerkin approximations of u, {uh}h>0, are defined by
+(1.12)
+a(uh, vh) = ⟨g, vh⟩ for all vh ∈ Hh.
+Example 1.4. For the Helmholtz equation outside a Dirichlet obstacle with PML
+truncation and Ω the truncated exterior domain, H0 = L2(Ω), H = H1
+0(Ω), and
+Zj = Hj(Ω) ∩ H1
+0(Ω). Assumption 1.2 is then elliptic regularity for the Helmholtz
+PML operator and its real part, which both hold if the coefficients of the Helmholtz
+equation are in Cℓ−1,1, the PML scaling function is Cℓ,1, and ∂Ω is Cℓ,1 (see Lemma
+4.7 below).
+Theorem 1.5 (Abstract generalisation of the elliptic-projection argument).
+Let a : H × H → C satisfy (1.6), (1.7), and Assumption 1.2.
+Suppose that
+R∗ : H∗ → H defined by
+(1.13)
+a(w, R∗v) = ⟨w, v⟩
+for all w ∈ H, v ∈ H∗,
+is well defined and let
+(1.14)
+η(Hh) :=
+sup
+g∈H0,g̸=0
+∥(I − Π)R∗g∥H
+∥g∥H0
+,
+where Π : H → Hh is the orthogonal projection. Then the solution, u, to (1.11) exists
+and is unique and there exist C1, C2, C3 > 0 such that if h satisfies
+(1.15)
+η(Hh)∥I − Π∥Zℓ+1→H ≤ C1,
+then the solution uh to (1.12) exists, is unique, and satisfies
+∥u − uh∥H ≤ C2
+�
+1 + η(Hh)
+�
+min
+wh∈Hh ∥u − vh∥H ,
+(1.16)
+∥u − uh∥H0 ≤ C3 η(Hh) min
+wh∈Hh ∥u − vh∥H .
+(1.17)
+If, in addition,
+(1.18)
+∥g∥Zℓ−1 ≤ C ∥g∥H∗
+for some C > 0, then there exists C4 > 0 such that if h satisfies (1.15) then
+(1.19)
+∥u − uh∥H
+∥u∥H
+≤ C4
+�
+1 + η(Hh)
+�
+∥I − Π∥Zℓ+1→H ;
+i.e.,
+the
+relative
+error
+in
+H
+can
+be
+made
+controllably
+small
+by
+making
+η(Hh) ∥I − Π∥Zℓ+1→H sufficiently small.
+3
+
+Theorem 1.5 includes the result that the sequence of Galerkin solutions are qua-
+sioptimal with constant independent of k if η(Hh) is sufficiently small – with this the
+so-called asymptotic regime (see the discussion in §1.3).
+The bounds (1.16), (1.17), and (1.19) and the meshthreshold (1.15) in Theorem
+1.5 all involve the quantity η(Hh), which measures how well solutions of the adjoint
+problem are approximated in the space Hh. Bounds on η(Hh) are given in [37, 38, 36,
+13, 6, 29, 19, 20, 3]; see the discussion in §1.3. The following bound on η(Hh) is proved
+using the ideas in [6] (although the end result is phrased in a different way there); we
+include it here both for completeness, and because it holds under the assumptions of
+Theorem 1.5 (in fact, it only requires the regularity assumption (1.8) and not (1.9)).
+Theorem 1.6 (Bound on η(Hh)). Under the assumptions of Theorem 1.5, there
+exists C > 0 such that
+(1.20)
+η(Hh) ≤ C
+� ⌊ℓ/2⌋−1
+�
+j=0
+∥(I − Π)∥Z2(j+1)→H + ∥(I − Π)∥Zℓ+1→H
+�
+1 + ∥R∗∥H0→H
+��
+.
+Example 1.7. In §4 and §5 below we show how Helmholtz problems with the ra-
+diation condition approximated by either a PML or an impedance boundary condition,
+respectively, fit into the abstract framework of Theorems 1.5 and 1.6. In both these
+cases, the norm of the adjoint solution operator, i.e., ∥R∗∥H0→H, is the same as the
+norm of the solution operator of the original (non-adjoint) problem, which we denote
+by Csol.
+Furthermore, with {Hh}h>0 corresponding to the standard finite-element
+spaces of piecewise degree-p polynomials on shape-regular simplicial triangulations,
+indexed by the meshwidth h,
+∥(I − Π)∥Zm+1→H ≤ C(hk)m
+for 0 ≤ m ≤ p.
+The meshthreshold (1.15) then becomes that (hk)2ℓCsol is sufficiently small. Recall
+that ℓ is a parameter in the elliptic-regularity assumptions (Assumption 1.2). If the
+polynomial degree p is taken to be ℓ then (1.15) becomes (1.1). The bounds (1.16) and
+(1.17) then become (1.2) and (1.3), respectively.
+1.3. Discussion of the context, novelty, and ideas behind Theorem 1.5.
+The work of Ihlenburg and Babuˇska in 1-d. The celebrated work of [25, 26] studied
+the h-FEM applied to the constant-coefficient Helmholtz equation in 1-d (a nontrap-
+ping problem), and split the behaviour of the finite-element solutions as a function of
+h into the so-called asymptotic and preasymptotic regimes.
+The asymptotic regime is when h is small enough, as a function of k, for the
+sequence of Galerkin solutions to be quasi-optimal uniformly in k, i.e.,
+∥u − uh∥H1
+k(Ω) ≤ C min
+vh∈Hh ∥u − vh∥H1
+k(Ω)
+with C > 0 independent of k. [26, Theorem 3.5] showed that a sufficient condition to
+be in the asymptotic regime is “hk2/p sufficiently small”, with later work (discussed
+below) then showing that a sufficient condition for nontrapping problems (when Csol ∼
+k) is “(hk)pk sufficiently small”, with this condition then indicated to be necessary
+by numerical experiments. Therefore, the pollution effect for the h-FEM, i.e., the
+fact that one needs h ≪ k−1 to maintain accuracy, becomes less pronounced as p
+increases.
+4
+
+The preasymptotic regime is when the relative H1
+k error is controllably small, uni-
+formly as k → ∞, provided that the data is k-oscillatory, in the sense that it satisfies
+the bound (1.18) 1. [26, Corollary 3.2] used the explicit form of the Helmholtz Green’s
+function in 1-d to prove that if (hk)2pk sufficiently small then the finite-element solu-
+tion is in the preasymptotic regime, with the numerical experiments in [26, Table 2]
+(for p = 1, . . . , 6) indicating that this condition is also necessary. [26] also studied the
+phase difference between the exact and finite-element solutions (following [23, 43]),
+with [26, Theorem 3.2] showing that the difference between the true wavenumber and
+the numerical wavenumber is bounded by C(hk)2pk. Thus the condition “(hk)2pk
+sufficiently small” also controls this phase difference; see also [1, Equation 3.5].
+Error bounds in the asymptotic regime using the Schatz argument.. We now out-
+line the argument that gives the condition “(hk)pCsol sufficiently small” for quasiop-
+timality, with this argument also used in the proof of Theorem 1.5. We work in the
+setting of Examples 1.4 and 1.7; i.e., the PML approximation to the Helmholtz exte-
+rior Dirichlet problem, so that H0 = L2(Ω) and H = H1
+0(Ω). The G˚arding inequality
+(1.7) is then
+ℜa(w, w) ≥ CG1 ∥w∥2
+H1
+k(Ω) − CG2 ∥w∥2
+L2(Ω)
+for all w ∈ H1
+0(Ω)
+for CG1, CG2 > 0 (see Corollary 4.6 below). Combining the G˚arding inequality with
+the Galerkin orthogonality
+(1.21)
+a(u − uh, vh) = 0
+for all vh ∈ Hh,
+we find that, for all vh ∈ Hh,
+∥u − uh∥2
+H1
+k(Ω) ≤ C−1
+G1
+��a(u − uh, u − vh)
+�� + C−1
+G1CG2 ∥u − uh∥2
+L2(Ω)
+≤ C−1
+G1Ccont ∥u − uh∥H1
+k(Ω) ∥u − vh∥H1
+k(Ω) + C−1
+G1CG2 ∥u − uh∥2
+L2(Ω) ,
+(1.22)
+where Ccont is the continuity constant of the sesquilinear form a. Therefore, (1.22)
+implies that a sufficient condition for quasioptimality is that the L2 error is sufficiently
+small relative to the H1
+k error.
+By the definition of R∗ (1.13) (recalling that H = H1
+0(Ω) here) and Galerkin
+orthogonality (1.21), for any vh ∈ Hh,
+∥u − uh∥2
+L2(Ω) = a
+�
+u − uh, R∗(u − uh)
+�
+= a
+�
+u − uh, R∗(u − uh) − vh
+�
+≤ Ccont ∥u − uh∥H1
+k(Ω)
+��R∗(u − uh) − vh
+��
+H1
+k(Ω),
+(1.23)
+and thus, by the definition of η(Hh) (1.14) (recalling that H0 = L2(Ω)),
+(1.24)
+∥u − uh∥L2(Ω) ≤ Ccontη(Hh) ∥u − uh∥H1
+k(Ω) .
+Combining this last inequality with (1.22), we see that a sufficient condition for qua-
+sioptimality is that η(Hh) is sufficiently small. Schatz [42] was the first to use the
+Aubin-Nitsche-type bound (1.24) with the G˚arding inequality, and thus the argument
+above is often called the Schatz argument. The “adjoint approximability” concept,
+and associated definition of η(Hh), was introduced by Sauter in [41].
+1The relative error can only be small for a certain subclass of data, since, given a finite-
+dimensional subspace Hh, one can choose data such that the solution v ∈ H is orthogonal to Hh.
+Then ∥u − uh∥2
+H = ∥u∥2
+H + ∥uh∥2
+H ≥ ∥u∥2
+H.
+5
+
+The bound
+(1.25)
+η
+�
+Hh
+�
+≤ C
+�
+hk + (hk)pCsol
+�
+under sufficient regularity of the coefficients and obstacle has now been proved for a
+wide variety of Helmholtz problems, with this bound sharp by the recent results of [17].
+The bound (1.25) therefore gives the sufficient condition “(hk)pCsol sufficiently small”
+for quasioptimality, with this condition observed sharp for nontrapping problems in,
+e.g., [6, Figures 3, 5, and 8] for p = 1, 2, 3, 4.
+For p = 1, the bound (1.25) can be proved using only H2 regularity of the
+Helmholtz solution, with the condition “hk2 sufficient small” for quasiopimality ob-
+tained for 1-d problems in [2, Theorem 3.1], [11, Lemma 2.6], [27, Theorem 3], and
+[33, Theorem 3.2], 2-d problems in [35, Proposition 8.2.7], and variable-coefficient
+problems in 2- and 3-d in [22, 21].
+For p > 1 the bound (1.25) is proved by a judicious splitting of the solution in
+[37, 38, 13, 36] for constant-coefficient problems and [6, 29, 19, 20, 3] for variable-
+coefficient problems. All these papers apart from [6] make the constant C in (1.25)
+explicit in p under suitably analyticity/smoothness assumptions on the obstacle and
+coefficients, and thus give results about the hp-FEM (showing that quasioptimality
+holds if hk/p is sufficiently small and p/ log k is sufficiently large). In addition, all these
+papers apart from [6] split the solution into “high-” and “low-” frequency components.
+In constrast, [6] instead expands the solution in a series whose terms increase with
+regularity, and with only the remainder satisfying a bound involving Csol; see Lemma
+2.2 below.
+Bounds in the preasymptotic regime. Numerical experiments indicate that, at
+least for nontrapping problems, the condition “(hk)2pCsol sufficiently small” for the
+relative H1
+k error to be controllably small is necessary and sufficient for 2- and 3-d
+Helmholtz problems; see, e.g., [12, Figure 3]. Nevertheless, despite the fact that sharp
+asymptotic error bounds have now been obtained for a variety of Helmholtz problems
+in 2- and 3-d and for arbitrary p ∈ Z+, until now the sharp preasymptotic error bounds
+were obtained only in the following cases.
+1. p = 1, the constant-coefficient Helmholtz equation with an impedance bound-
+ary condition [44, Theorem 6.1] or PML (and no obstacle) [32, Theorem
+4.4], the variable-coefficient Helmholtz equation with truncation via the ex-
+act Dirichlet-to-Neumann map [28, Theorem 4.1].
+2. p ∈ Z+, the constant-coefficient Helmholtz equation with no obstacle and
+an impedance boundary condition approximating the radiation condition [12,
+Theorem 5.1],
+3. p ∈ Z+, the variable-coefficient Helmholtz equation in the exterior of a Dirich-
+let obstacle with an impedance boundary condition approximating the radi-
+ation condition [40, Theorem 2.39].
+The bounds in Point 1 for p = 1 come from the so-called elliptic projection argument,
+which proves error bounds under the condition “(hk)p+1Csol is sufficiently small”; i.e.,
+the sharp condition when p = 1, but not when p > 1. The initial ideas behind this
+argument were introduced in the Helmholtz context in [15, 16] for interior-penalty
+discontinuous Galerkin methods, and then further developed for the standard FEM
+and continuous interior-penalty methods in [44, 45].
+The bounds in Point 2 used an error-splitting argument (with this idea called
+“stability-error iterative improvement”, and used earlier in [16, 44]) together with the
+idea of using discrete Sobolev norms in the duality argument. The bounds in Point 3
+6
+
+for variable-coefficients were obtained by repeating the constant-coefficient arguments
+in Point 2, but now keeping track of how the constants depend on the coefficients.
+The elliptic-projection argument. Theorem 1.5 is proved by generalising the
+elliptic-projection argument, allowing it to prove error bounds under the sharp condi-
+tion “(hk)2pCsol sufficiently small” for p > 1. We therefore recap the main ideas of the
+elliptic-projection argument here, and then we explain below how we generalise this
+argument. Here, and in the rest of the paper, C is used for a constant, independent
+of h and k, but dependent on p, whose value may change line by line.
+The bounds (1.2) and (1.3) come from the bounds
+(1.26)
+∥u − uh∥H1
+k(Ω) ≤ C
+�
+1 + η(Hh)
+�
+min
+vh∈Hh ∥u − vh∥H1
+k(Ω)
+and
+(1.27)
+∥u − uh∥L2(Ω) ≤ Cη(Hh) min
+vh∈Hh ∥u − vh∥H1
+k(Ω)
+and the bound (1.25) on η(Hh).
+Observe that, by the consequence (1.22) of the
+G˚arding inequality, the bound (1.26) follows from the bound (1.27).
+To prove (1.27), the elliptic-projection argument writes (1.23) as
+∥u − uh∥2
+L2(Ω) = a
+�
+u − uh, R∗(u − uh) − vh
+�
+= �a
+�
+u − uh, R∗(u − uh) − vh
+�
+−
+�
+(1 + c−2)(u − uh), R∗(u − uh) − vh
+�
+L2(Ω),
+(1.28)
+where
+�a(u, v) :=
+�
+Ω
+k−2A∇u · ∇v + u v.
+Let �Π : H1
+0(Ω) → Hh be the solution of the variational problem
+�a(wh, �Πv) = �a(wh, v)
+for all wh ∈ Hh.
+Since �a is coercive on H1
+0(Ω) and the continuity and coercivity constants of �a in
+∥ · ∥H1
+k(Ω) are independent of k, �Π is well-defined by the Lax–Milgram theorem and
+(1.29)
+��(I − �Π)v
+��
+H1
+k(Ω) ≤ C min
+wh∈Hh ∥v − wh∥H1
+k(Ω)
+with C > 0 independent of k by C´ea’s lemma. The definition of �Π implies the Galerkin
+orthogonality
+(1.30)
+�a
+�
+wh, (I − �Π)v
+�
+= 0
+for all wh ∈ Hh.
+We now choose vh = �ΠR∗(u − uh) in (1.28) so that, by (1.30), for all wh ∈ Hh,
+∥u − uh∥2
+L2(Ω) = �a
+�
+v − wh, (I − �Π)R∗(u − uh)
+�
+−
+�
+(1 + c−2)(u − uh), (I − �Π)R∗(u − uh)
+�
+L2(Ω).
+(1.31)
+For the first term on the right-hand side of (1.31) we use the continuity of �a, (1.29),
+and the definition of η(Hh) (1.14) to bound this term by
+C ∥v − wh∥H1
+k(Ω) η(Hh) ∥u − uh∥L2(Ω) .
+7
+
+The second term on the right-hand side of (1.31) is bounded by
+C ∥u − uh∥L2(Ω)
+��(I − �Π)R∗(u − uh)
+��
+L2(Ω).
+Using the Schatz argument for �a, one can show that
+(1.32)
+��(I − �Π)R∗(u − uh)
+��
+L2(Ω) ≤ Chk
+��(I − �Π)R∗(u − uh)
+��
+H1
+k(Ω)
+and then (1.29) and the definition of η(Hh) (1.14) imply that the second term on the
+right-hand side of (1.31) is bounded by
+(1.33)
+Chk η(Hh) ∥u − uh∥2
+L2(Ω) ,
+which can be absorbed into the left-hand side if hk η(Hh) is sufficiently small, giving
+the result (1.27).
+The ideas behind the proof of Theorem 1.5. We generalise the elliptic-projection
+argument based on the observation that if �a(u, v) = a(u, v) + (Su, v)L2(Ω) with S a
+self-adjoint smoothing operator, then the second term on the right-hand side of (1.31)
+is replaced by
+(1.34)
+�
+u − uh, S∗(I − �Π)R∗(u − uh)
+�
+L2(Ω)
+(see (2.14) below). Using the Schatz argument for �a and the smoothing property of
+S, the modulus of this term is bounded by
+(1.35)
+��S∗(I − �Π)R∗(u − uh)
+��
+L2(Ω) ≤ C(hk)p��(I − �Π)R∗(u − uh)
+��
+H1
+k(Ω)
+(see (2.16) below). Provided that �Π still satisfies (1.29), the term (1.34) is therefore
+bounded by
+(1.36)
+C(hk)pη(Hh) ∥u − uh∥2
+L2(Ω) .
+Comparing (1.32) and (1.35), and also (1.33) and (1.36), we see that this new argument
+replaces the condition “hkη(Hh) sufficiently small” in the standard elliptic-projection
+argument by the condition “(hk)pη(Hh) sufficiently small”, which is the condition
+(hk)2pCsol sufficiently small” after using the bound (1.25) on η(Hh).
+The challenge now is to ensure that the smoothing operator S is such that the
+projection �Π is well-defined and satisfies (1.29). This is achieved in Lemma 2.1 below,
+where a suitable S such that �a(u, v) = a(u, v) + (Su, v)L2(Ω) is coercive is construc-
+ted. S is defined by an expansion in terms of the eigenfunctions of the (self-adjoint)
+operator associated with the real part of the sesquilinear form a (defined by (1.10)).
+2. Proofs of the main results (Theorems 1.5 and 1.6).
+2.1. Construction of a regularizing operator that produces coercivity
+when added to a.
+Lemma 2.1. Suppose that a : H × H → C satisfies (1.6), (1.7), and Assumption
+1.2. Then there exists S : H0 → H0 self adjoint and c, C > 0 such that, with
+(2.1)
+�a(u, v) := a(u, v) + ⟨Su, v⟩H0,
+(2.2)
+ℜ�a(v, v) ≥ c ∥v∥2
+H
+for all v ∈ H,
+8
+
+(2.3)
+∥S∥H0→Zj ≤ C,
+j = 0, . . . , ℓ + 1
+and �R : H∗ → H defined by
+�a( �Rf, u) = ⟨f, u⟩
+for all u ∈ H, f ∈ H∗,
+(2.4)
+is well defined with
+(2.5)
+∥ �R∥Zj−2→Zj ≤ C,
+2 ≤ j ≤ ℓ + 1.
+The proof of Lemma 2.1 uses the spectral theorem for bounded self-adjoint op-
+erators, B : H → H∗, which we recap here. With H0 and H as in §1.2, let b be a
+sesquilinear form on H satisfying b(u, v) = b(v, u), with associated operator B; i.e.,
+b(u, v) = ⟨Bu, v⟩ for all u, v ∈ H. If b satisfies the G˚arding inequality (1.7) (with
+a replaced by b) then there exist an orthonormal basis (in H0) of eigenfunctions of
+B, {φj}∞
+j=1, with associated eigenvalues satisfying λ1 ≤ λ2 ≤ . . . with λj → ∞ as
+j → ∞. Furthermore, for all u ∈ H,
+(2.6)
+Bu =
+∞
+�
+j=1
+λj⟨φj, u⟩φj
+(where the sum converges in H∗); see, e.g., [34, Theorem 2.37]. Given a bounded
+function f, we define f(B) : H0 → H0 by
+(2.7)
+f(B)u :=
+∞
+�
+j=1
+f(λj)⟨φj, u⟩φj,
+so that
+∥f(B)∥H0→H0 ≤
+sup
+λ∈[λ1,∞)
+|f(λ)|.
+Proof of Lemma 2.1. Let P : H → H∗ be the operator associated with the
+sesquilinear form ℜa defined by (1.10), i.e., (ℜa)(u, v) = ⟨Pu, v⟩ for all u, v ∈ H;
+observe that P is self-adjoint. Since (ℜa) also satisfies the G˚arding equality satis-
+fied by a (1.7), the spectral theorem recapped above applies. Let {λj}∞
+j=1 be the
+eigenvalues of P, let ψ ∈ C∞
+comp(R; [0, ∞)) be such that
+(2.8)
+x + ψ(x) ≥ 1
+for x ≥ −λ1,
+and let S := ψ(P), in the sense of (2.7).
+We now use (1.9) to prove that S : H0 → Zj satisfying (2.3). Since ψ has compact
+support, the function t �→ tmψ(t) is bounded for any m ≥ 0. Thus (2.7) implies that,
+for any m ≥ 0,
+(2.9)
+∥Pmψ(P)∥H0→H0 ≤ Cm.
+By (1.9),
+∥ψ(P)∥H0→Zj ≤ Cℓ
+�
+∥ψ(P)∥H0→H0 + ∥Pψ(P)∥H0→Zj−2
+�
+,
+j = 2, . . . , ℓ + 1,
+so that, by induction and (2.9),
+∥S∥H0→Zℓ+1 = ∥ψ(P)∥H0→Zℓ+1 ≤ Cℓ
+⌈(ℓ+1)/2⌉
+�
+j=0
+��Pjψ(P)
+��
+H0→H0 ≤ Cℓ.
+9
+
+We now show that �a satisfies (2.2). By the definitions of P and S, (2.6), (2.7),
+and the inequality (2.8), for all v ∈ H,
+ℜ�a(v, v) = ℜa(v, v) + ⟨ψ(P)v, v⟩ = ⟨(P + ψ(P))v, v⟩ ≥ ∥v∥2
+H0 .
+Since ψ ≥ 0, S is positive, and thus ℜ�a(v, v) ≥ ℜa(v, v) for all v ∈ H, for any ǫ > 0
+and for all v ∈ H,
+ℜ�a(v, v) ≥ ǫℜa(v, v) + (1 − ǫ)ℜ�a(v, v) ≥ ǫCG1 ∥v∥2
+H − CG2ǫ ∥v∥2
+H0 + (1 − ǫ)∥v∥2
+H0,
+so that, choosing ǫ = min(
+1
+2CG2 , 1
+2), we have
+ℜ�a(v, v) ≥ CG1
+2
+min
+� 1
+CG2
+, 1
+�
+∥v∥2
+H + 1
+2 ∥v∥2
+H0 ;
+i.e., �a is coercive. The existence of �R : H∗ → H satisfying (2.4) and ∥ �R∥H∗→H ≤ C
+then follows from the Lax–Milgram theorem. Finally, to see that
+∥ �R∥Zj−2→Zj ≤ C,
+2 ≤ j ≤ ℓ + 1,
+observe that, since S is self-adjoint and satisfies (2.3), for v ∈ (Zj−2)∗,
+|a( �Rg, v)| = |�a( �Rg, v) − ⟨S �Rg, v⟩| ≤ |�a( �Rg, v)| + |⟨S �Rg, v⟩|
+≤ |⟨v, g⟩| + ∥v∥(Zj−2)∗∥S∥H→Zj−2∥( �R)∗∥H∗→H∥g∥H∗
+≤ ∥v∥(Zj−2)∗(∥g∥Zj−2 + C∥g∥H∗),
+and the claim follows from (1.8).
+2.2. Proof Theorem 1.5 using Lemma 2.1. We claim it is sufficient to prove
+the bounds (1.16) and (1.17) under the assumption of existence. Indeed, by uniqueness
+of the variational problem (1.11), either of the bounds (1.16) or (1.17) under the
+assumption of existence implies uniqueness of uh, and uniqueness implies existence
+for the finite-dimensional Galerkin linear system.
+We next show that the bound (1.16) follows from (1.17). Now, by the G˚arding
+inequality (1.7), Galerkin orthogonality (1.21), and (1.17), for any vh ∈ Hh,
+∥u − uh∥2
+H ≤ C
+���a(u − uh, u − vh)
+�� + ∥u − uh∥2
+H0
+�
+≤ C
+�
+∥u − uh∥H ∥u − vh∥H +
+�
+η(Hh) min
+wh∈Hh ∥u − wh∥H
+�2�
+.
+(2.10)
+The bound (1.16) on the error in H then follows by using the inequality 2ab ≤ ǫa2 +
+b2/ǫ for all a, b, ǫ > 0 in the first term on the right-hand side of (2.10), and then using
+the inequality a2 + b2 ≤ (a + b)2 for a, b > 0.
+We now prove (1.17) (using the ideas outlined in §1.3). By the definition of R∗,
+Galerkin orthogonality (1.21), and the definition of �a (2.1)
+∥u − uh∥2
+H0 = a
+�
+u − uh, R∗(u − uh)
+�
+= a
+�
+u − uh, R∗(u − uh) − vh
+�
+= �a
+�
+u − uh, R∗(u − uh) − vh
+�
+−
+�
+S(u − uh), R∗(u − uh) − vh
+�
+H0.
+(2.11)
+Let �Π : H → Hh be the solution of the variational problem
+�a(wh, �Πv) = �a(wh, v)
+for all wh ∈ Hh.
+10
+
+Since �a is continuous and coercive, with constants independent of k (see (2.2), (1.6),
+and (2.3)), by the Lax–Milgram lemma and C´ea’s lemma given k0 > 0 there exists
+C > 0 such that for all k ≥ k0 and v ∈ H, �Π is well-defined with
+(2.12)
+��(I − �Π)v
+��
+H ≤ C min
+wh∈Hh ∥v − wh∥H .
+The definition of �Π implies the Galerkin orthogonality
+(2.13)
+�a
+�
+wh, (I − �Π)u
+�
+= 0
+for all wh ∈ Hh.
+We now choose vh = �ΠR∗(u − uh) in (2.11) so that, by (2.13), for all wh ∈ Hh,
+(2.14)
+∥u − uh∥2
+H0
+= �a
+�
+u − wh, (I − �Π)R∗(u − uh)
+�
+−
+�
+u − uh, S∗(I − �Π)R∗(u − uh)
+�
+H0
+≤ C ∥u − wh∥H
+��(I − �Π)R∗(u − uh)
+��
+H + ∥u − uh∥H0
+��S∗(I − �Π)R∗(u − uh)
+��
+H0.
+By (2.12) and the definition of η(Hh) (1.14),
+(2.15)
+��(I − �Π)R∗(u − uh)
+��
+H ≤ C min
+wh∈Hh ∥R∗(u − uh) − wh∥H ≤ Cη(Hh) ∥u − uh∥H0 .
+We now claim that the bound (1.17) follows if we can prove that, for all v ∈ H,
+(2.16)
+��S∗(I − �Π)v
+��
+H0 ≤ C∥I − Π∥Zℓ+1→H
+��(I − �Π)v
+��
+H.
+Indeed, we use (2.15) in the first term on the right-hand side of (2.14), and then (2.16)
+with v = R∗(u − uh) in the second term on the right-hand side of (2.14) to obtain
+∥u − uh∥2
+H0 ≤ Cη(Hh) ∥u − wh∥H ∥u − uh∥H0
++ C∥I − Π∥Zℓ+1→H
+��(I − �Π)R∗(u − uh)
+��
+H ∥u − uh∥H0 .
+By (2.15), the last term on the right-hand side is ≤ C∥I−Π∥Zℓ+1→H η(Hh)∥u−uh∥2
+H0
+and (1.17) follows.
+We now prove (2.16) by using the duality argument described in §1.3 (as part of
+the Schatz argument). By the definition of �R (2.4) and Galerkin orthogonality (2.13),
+for all wh ∈ Hh,
+��S∗(I − �Π)v
+��2
+H0 =
+�
+SS∗(I − �Π)v, (I − �Π)v
+�
+H0 = �a
+� �RSS∗(I − �Π)v − wh, (I − �Π)v
+�
+.
+Then, by the bounds (2.5) and (2.3),
+��S∗(I − �Π)v
+��2
+H0 ≤ C min
+wh∈Hh
+�� �RSS∗(I − �Π)v − wh
+��
+H
+��(I − �Π)v
+��
+H
+≤ ∥I − Π∥Zℓ+1→H
+�� �RSS∗(I − �Π)v
+��
+Zℓ+1
+��(I − �Π)v
+��
+H,
+≤ C∥I − Π∥Zℓ+1→H
+��SS∗(I − �Π)v
+��
+Zℓ−1
+��(I − �Π)v
+��
+H,
+≤ C∥I − Π∥Zℓ+1→H
+��S∗(I − �Π)v
+��
+H0
+��(I − �Π)v
+��
+H
+which implies the bound (2.16), and hence (1.17).
+11
+
+Finally, we prove (1.19). By (1.11), (1.18), and the abstract elliptic-regularity
+assumption (1.8), u ∈ Zℓ+1 with
+∥u∥Zℓ+1 ≤ C
+�
+∥u∥H0 + ∥g∥Zℓ−1
+��
+≤ C
+�
+∥u∥H0 + ∥g∥H∗
+�
+.
+The variational problem (1.11) implies that
+∥g∥H∗ =
+sup
+v∈H∗,v̸=0
+|a(u, v)|
+∥v∥H∗
+≤ C ∥u∥H ,
+and thus ∥u∥Zℓ+1 ≤ C ∥u∥H. The bound (1.16) then implies that
+∥u − uh∥H ≤ C2
+�
+1 + η(Hh)
+�
+∥I − Π∥Zℓ+1→H ∥u∥Zℓ+1
+and (1.19) follows.
+2.3. Proof of Theorem 1.6. The following lemma is essentially [6, Theorem
+2.6], rewritten in the abstract notation in §1.2.
+Lemma 2.2. Under the assumptions of Theorem 1.5, let u = R∗g with R∗ defined
+by (1.13) and g ∈ H0. Let um ∈ H, m = 0, . . . , ⌊ℓ/2⌋, be defined by
+(2.17)
+�a(v, u0) = ⟨v, g⟩
+for all v ∈ H,
+and
+(2.18)
+�a(v, um) = ⟨Sv, um−1⟩
+for all v ∈ H, m = 1, . . . , ⌊ℓ/2⌋.
+Then
+(2.19)
+um ∈ Z2(m+1) with
+∥um∥Z2(m+1) ≤ C ∥g∥H0
+for m = 0, . . . , ⌊ℓ/2⌋ − 1,
+and
+(2.20)
+u⌊ℓ/2⌋ ∈ Zℓ+1 with
+��u⌊ℓ/2⌋
+��
+Zℓ+1 ≤ C ∥g∥H0 .
+Furthermore, with
+(2.21)
+rm := u −
+m−1
+�
+j=0
+uj,
+(2.22)
+rm ∈ Z2(m+1) with
+∥rm∥Z2(m+1) ≤
+�
+1+∥R∗∥H0→H
+�
+∥g∥H0
+for m = 0, . . . , ⌊ℓ/2⌋−1,
+and
+(2.23)
+r⌊ℓ/2⌋ ∈ Zℓ+1 with
+��r⌊ℓ/2⌋
+��
+Zℓ+1 ≤
+�
+1 + ∥R∗∥H0→H
+�
+∥g∥H0 .
+Proof. We first prove (2.19) by induction. By the definition of u0 (2.17), conti-
+nuity and coercivity of �a, and boundedness of S (2.3), ∥u0∥H ≤ C ∥g∥H0. Then, by
+(1.8) with j = 2,
+∥u0∥Z2 ≤ C
+�
+∥u0∥H0 + ∥g∥H0
+�
+≤ C ∥g∥H0 ,
+12
+
+which is (2.19) with m = 0.
+Assume that (2.19) holds with m = q. By the definition of uq+1 (2.18), continuity
+and coercivity of �a, and boundedness of S (2.3),
+(2.24)
+∥uq+1∥H ≤ C ∥uq∥H∗ .
+By (1.8) with j = 2(q + 1) and the definition of uq+1 (2.18)
+∥uq+1∥Z2(q+1) ≤ C
+�
+∥uq+1∥H0 +
+sup
+v∈H, ∥v∥(Z2q )∗ =1
+|⟨Sv, uq⟩|
+�
+.
+(2.25)
+By duality
+∥S∥(Zj)∗→H0 ≤ C
+j = 0, . . . , ℓ + 1,
+and thus
+(2.26)
+sup
+v∈H, ∥v∥(Z2q )∗ =1
+|⟨Sv, uq⟩| ≤ ∥S∥(Z2q)∗→H0 ∥uq∥H0 ≤ C ∥uq∥H0 .
+Combining (2.25), (2.26), and (2.24), we find that
+∥uq+1∥Z2(q+2) ≤ C
+�
+∥uq+1∥H0 + ∥uq∥H0
+�
+≤ C ∥uq∥H .
+Using (2.19) with m = q, we obtain (2.19) with m = q + 1, and the induction is
+complete.
+If ℓ is odd, i.e., ℓ + 1 is even, then this establishes both (2.19) and (2.20) since
+2(⌊ℓ/2⌋ + 1) = ℓ + 1 (i.e., the highest-order case is even, and can be reached by
+increasing the regularity at each step by two). If ℓ is even, i.e., ℓ + 1 is odd, then
+the argument above establishes (2.19). The bound for u⌊ℓ/2⌋ (i.e., (2.20)) then follows
+from elliptic regularity, using that u⌊ℓ/2⌋−1 = uℓ/2−1 ∈ Zℓ ⊂ Zℓ−1 (i.e., at the last
+step, we only increase the regularity by one).
+For the proof that rm ∈ Z2(m+1) and satisfies (2.22), observe that the definition
+of rm (2.21) and the definition of um (2.18) implies that r0 = u and
+�a(v, rm) = ⟨Sv, rm−1⟩
+for all v ∈ H, m = 1, . . . , ⌊ℓ/2⌋.
+The proof of (2.22) is then very similar to the proof of (2.19), with the first step being
+that, by (1.8), the fact that u = R∗g, and the definition of R∗ (1.13),
+∥r0∥Z2 = ∥u∥Z2 ≤ C
+�
+∥u∥H0 + ∥g∥H0
+�
+≤ C
+�
+1 + ∥R∗∥H0→H
+�
+∥g∥H0 .
+Proof of Theorem 1.6 using Lemma 2.2. As in Lemma 2.2, given g ∈ H0, let u =
+R∗g. By (2.21),
+∥(I − Π)R∗g∥H ≤
+⌊ℓ/2⌋−1
+�
+j=0
+∥(I − Π)∥Z2(j+1)→H ∥uj∥Z2(j+1) + ∥(I − Π)∥Zℓ+1
+��r⌊ℓ/2⌋
+��
+Zℓ+1
+so that, by the bounds (2.19), (2.20), and (2.23),
+∥(I − Π)R∗g∥H ≤ C
+� ⌊ℓ/2⌋−1
+�
+j=0
+∥(I − Π)∥Z2(j+1)→H
++ ∥(I − Π)∥Zℓ+1→H
+�
+1 + ∥R∗∥H0→H
+��
+∥g∥H0 ;
+the result (1.20) then follows from the definition of η(Hh) (1.14).
+13
+
+3. Elliptic-regularity results. This section collects the elliptic-regularity re-
+sults that are used to verify that Assumption 1.2 holds for Helmholtz problems with
+truncation of the exterior domain either by a PML (in §4) or an impedance boundary
+condition (in §5). Let
+Lu = −k−2∇ · (A∇u) − c−2u,
+with associated sesquilinear form
+a(u, v) =
+�
+Ω
+�
+k−2(A∇u) · ∇v − c−2u v
+�
+,
+where Ω be a bounded Lipschitz domain with outward-pointing unit normal vector
+n. The conormal derivative ∂n,Au is defined for u ∈ H2(Ω) by ∂n,Au := n · (A∇u);
+recall that ∂n,Au can be defined for u ∈ H1(Ω) with Lu ∈ L2(Ω) by Green’s identity;
+see, e.g., [34, Lemma 4.3].
+Assumption 3.1. For all x ∈ Ω, Ajℓ(x) = Aℓj(x) and
+ℜ
+d
+�
+j=1
+d
+�
+ℓ=1
+Ajℓ(x)ξkξj ≥ c|ξ|2
+for all ξ ∈ Cd.
+Theorem 3.2 (Local elliptic regularity near a Dirichlet or Neumann boundary).
+Let Ω be a Lipschitz domain and let G1, G2 be open subsets of Rd with G1 ⋐ G2 and
+G1 ∩ ∂Ω ̸= ∅. Let
+(3.1)
+Ωj := Gj ∩ Ω, j = 1, 2,
+and
+Γ2 := G2 ∩ ∂Ω.
+Suppose that A satisfies Assumption 3.1, A, c ∈ Cm,1(Ω2), Γ2 ∈ Cm+1,1, u ∈ H1(Ω2),
+and Lu ∈ Hm(Ω2) for some m ∈ N, and either u = 0 or ∂n,Au = 0 on Γ2. Then
+(3.2)
+∥u∥Hm+2
+k
+(Ω1) ≤ C
+�
+∥u∥H1
+k(Ω2) + ∥Lu∥Hm
+k (Ω2)
+�
+.
+Proof. In unweighted norms, this follows from, e.g., [34, Theorems 4.7 and 4.16];
+the proof in the weighted norms (4.11) is very similar.
+Theorem 3.3 (Local elliptic regularity for the transmission problem).
+Let Ωin
+be a Lipschitz domain, and let Ωout := Rd \ Ωin. Let G1, G2 be open subsets of Rd
+with G1 ⋐ G2 and G1 ∩ ∂Ωin ̸= ∅. Let
+Ωin/out,j := Gj ∩ Ωin/out,
+j = 1, 2,
+and Γ2 := G2 ∩ ∂Ωin.
+Suppose that A satisfies Assumption 3.1, A|Ωin/out,2, c|Ωin/out,2 ∈ Cm,1(Ωin/out,2), Γ2 ∈
+Cm+1,1, uin/out ∈ H1(Ωin/out), and Lu ∈ Hm(Ωin/out,2) for some m ∈ N. Suppose
+further that
+uin = uout
+and
+∂n,Auin = β∂n,Auout
+on Γ2
+for some β > 0. Then
+∥uin∥Hm+2
+k
+(Ωin,1) + ∥uout∥Hm+2
+k
+(Ωout,1)
+≤ C
+�
+∥uin∥H1
+k(Ωin,2) + ∥uout∥H1
+k(Ωout,2) + ∥Luin∥Hm
+k (Ωin,2) + ∥Luout∥Hm
+k (Ωout,2)
+�
+.
+(3.3)
+14
+
+Proof. In unweighted norms, this is, e.g., [10, Theorem 5.2.1(i)] (and [34, The-
+orems 4.7 and 4.16] when β = 1); the proof in the weighted norms (4.11) is very
+similar.
+Theorem 3.4 (Local elliptic regularity for the impedance problem). Let Ω be a
+Lipschitz domain and let G1, G2 be open subsets of Rd with G1 ⋐ G2 and G1∩∂Ω ̸= ∅.
+Let Ωj and Γ2 be defined by (3.1). Suppose that, for some m ∈ N, Γ2 ∈ Cm+1,1,
+u ∈ H1(Ω2), and ∆u ∈ Hm(Ω2), and (k−1∂n − i)u = 0 on Γ2. Then
+(3.4)
+∥u∥Hm+2
+k
+(Ω1) ≤ C
+�
+∥u∥H1
+k(Ω2) +
+��k−2∆u
+��
+Hm
+k (Ω2)
+�
+.
+Proof. When m = 0, the result can be obtained from [7, Lemma 4.1] by multiply-
+ing by k−2 to switch to weighted norms, and using that the trace operator has norm
+bounded by Ck1/2 from H1
+k to L2 (which can be obtained from, e.g., [39, Theorem
+5.6.4] since the weighted norms there are, up to a constant, the weighted norms (1.5)).
+The proof that (3.4) follow for m > 0 is then standard and can be found e.g.
+in [14, §6.3.2, Theorem 5]. We repeat it here in the context of impedance boundary
+conditions for completeness.
+We now prove that if the bound holds for m = q, then it holds for m = q + 1
+(assuming the appropriate regularity of the coefficients and the domain). Without
+loss of generality, we can change coordinates and work with U := B(0, s) ∩ {xd > 0}
+and V := B(0, t) ∩ {xd > 0} for some 0 < t < s. In these coordinates
+Lu := (−k−2aij∂xi∂xj −k−2(bi∂xi−c))u = f,
+(−k−1∂xd−i)u = 0 on {xd = 0}∩U.
+Suppose that for some q ≥ 0, for any 0 < t < s,
+(3.5)
+∥u∥Hq+2
+k
+(V ) ≤ Ct
+�
+∥u∥L2(U) + ∥f∥Hq
+k(U)
+�
+.
+Now suppose that f ∈ Hq+1
+k
+(U) and a, b, c ∈ Cq+1,1(U), and let W := B(0, r)∩{xd >
+0} with t < r < s. By (3.5),
+(3.6)
+∥u∥Hq+2
+k
+(W) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq
+k(U)
+�
+,
+and, by interior elliptic regularity, u ∈ Hq+3
+loc (U).
+The next step is to bound tangential derivatives of u: let |α| = q + 1 with αd = 0
+(so that ∂α
+x is a tangential derivative). Let
+�f := L
+�
+k−|α|∂α
+x u
+�
+so that
+�f = [L, k−|α|∂α
+x ]u + k−|α|∂α
+x f
+(where [A, B] := AB − BA) and, by (3.6) and the fact that the coefficients of L are
+Cq+1,1(U),
+(3.7)
+∥ �f∥L2(W) ≤ C
+�
+∥u∥Hq+2(W) + ∥f∥Hq+1
+k
+(W)
+�
+≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+.
+Furthermore
+(−k−1∂xd − i)k−|α|∂α
+x u|xd=0 = k−|α|∂α
+x
+�
+(−k−1∂xd − iu)|xd=0
+�
+= 0,
+so that, by the analogue of (3.5) with q = 0 and U replaced by W, (3.6), and (3.7),
+��k−|α|∂α
+x u
+��
+H2
+k(V ) ≤ C
+���k−|α|∂α
+x u
+��
+L2(W) +
+�� �f
+��
+L2(W)
+�
+≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+.
+15
+
+Therefore, by the definition of α,
+��k−|β|∂β
+xu
+��
+L2(V ) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+for all |β| = q + 3 with βd ∈ {0, 1, 2}.
+(3.8)
+To prove that the bound (3.5) holds with q replaced by q + 1, i.e.,
+∥u∥Hq+3
+k
+(V ) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+,
+it is sufficient to prove that
+��k−|β|∂β
+xu
+��
+L2(V ) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+for all |β| = q + 3 with βd ∈ {0, . . . , q + 3}.
+We therefore now prove by induction that if
+(3.9)
+��k−|β|∂β
+xu
+��
+L2(V ) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+for any |β| = q + 3 with βd ∈ {0, . . . , j} for some j ∈ {2, . . . , q + 2}, then (3.9) holds
+for |β| = q + 3 with βd = j + 1. Combined with (3.8), this completes the proof.
+We therefore assume that |β| = q + 3 with βd = j + 1. Then, putting β = γ + δ
+with δ = (0, . . . , 0, 2) and |γ| = q + 1, and using that u ∈ Hq+3
+loc (U), we have
+(3.10)
+k−|γ|∂γLu = addk−|β|∂βu + Bu
+in V,
+where
+Bu =
+�
+|α|≤q+3, αd≤j
+aαk−|α|∂α
+x u.
+By the induction hypothesis (3.9),
+∥Bu∥L2(V ) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+.
+Dividing (3.10) by add, taking the L2(V ) norm, and using that 1/add is bounded, we
+have
+∥k−|β|∂βu∥L2(V ) ≤ C
+�
+∥u∥L2(U) + ∥f∥Hq+1
+k
+(U)
+�
+;
+i.e., we have proved that (3.9) holds for |β| = q + 3 with βd = j + 1, and the proof is
+complete.
+4. Theorem 1.5 applied to the PML problem.
+4.1. Definition of the PML problem.
+Obstacles and coefficients for Dirichlet/Neumann/penetrable obstacle problem.
+Let Ωp, Ω− ⊂ BR0 := {x : |x| < R0} ⊂ Rd, d = 2, 3, be bounded open sets with
+Lipschitz boundaries, Γp and Γ−, respectively, such that Γp ∩ Γ− = ∅, and Rd\Ω− is
+connected. Let Ωout := Rd\Ω− ∪ Ωp and Ωin := (Rd\Ω−) ∩ Ωp.
+Let Aout ∈ C0,1(Ωout, Rd×d) and Ain ∈ C0,1(Ωin, Rd×d) be symmetric positive
+definite, let cout ∈ L∞(Ωout; R), cin ∈ L∞(Ωin; R) be strictly positive, and let Aout
+and cout be such that there exists Rscat > R0 > 0 such that
+Ω− ∪ supp(I − Aout) ∪ supp(1 − cout) ⋐ BRscat.
+16
+
+The obstacle Ω− is the impenetrable obstacle, on which we impose either a zero
+Dirichlet or a zero Neumann condition, and the obstacle Ωin is the penetrable obstacle,
+across whose boundary we impose transmission conditions.
+For simplicity, we do not cover the case when Ω− is disconnected, with Dirichlet
+boundary conditions on some connected components and Neumann boundary con-
+ditions on others, but the main results hold for this problem too (at the cost of
+introducing more notation).
+Definition of the radial PML. Let Rtr > RPML,− > Rscat and let Ωtr ⊂ Rd be a
+bounded Lipschitz open set with BRtr ⊂ Ωtr ⊂ BCRtr for some C > 0 (i.e., Ωtr has
+characteristic length scale Rtr). Let Ω := Ωtr ∩ Ω+ and Γtr := ∂Ωtr. For 0 ≤ θ < π/2,
+let the PML scaling function fθ ∈ C3([0, ∞); R) be defined by fθ(r) := f(r) tan θ for
+some f satisfying
+(4.1)
+�
+f(r) = 0
+�
+=
+�
+f ′(r) = 0
+�
+=
+�
+r ≤ RPML,−
+�
+,
+f ′(r) ≥ 0,
+f(r) ≡ r on r ≥ RPML,+;
+i.e., the scaling “turns on” at r = RPML,−, and is linear when r ≥ RPML,+. We
+emphasize that Rtr can be < RPML,+, i.e., we allow truncation before linear scaling
+is reached. Indeed, RPML,+ > RPML,− can be arbitrarily large and therefore, given
+any bounded interval [0, R] and any function �f ∈ C3([0, R]) satisfying
+� �f(r) = 0
+�
+=
+� �f ′(r) = 0
+�
+=
+�
+r ≤ RPML,−
+�
+,
+�f ′(r) ≥ 0,
+our results hold for an f with f|[0,R] = �f. Given fθ(r), let
+(4.2)
+α(r) := 1 + if ′
+θ(r)
+and
+β(r) := 1 + ifθ(r)/r.
+and let
+(4.3)
+A :=
+
+
+
+
+
+Ain
+in Ωin,
+Aout
+in Ωout ∩ BRPML,−,
+HDHT
+in (BRPML,−)c
+and 1
+c2 :=
+
+
+
+
+
+c−2
+in
+in Ωin,
+c−2
+out
+in Ωout ∩ BRPML,−,
+α(r)β(r)d−1
+in (BRPML,−)c,
+where, in polar coordinates,
+(4.4)
+D =
+�
+β(r)α(r)−1
+0
+0
+α(r)β(r)−1
+�
+and
+H =
+�
+cos θ
+− sin θ
+sin θ
+cos θ
+�
+for d = 2,
+and
+(4.5)
+D =
+
+
+β(r)2α(r)−1
+0
+0
+0
+α(r)
+0
+0
+0
+α(r)
+
+ and H =
+
+
+sin θ cos φ
+cos θ cos φ
+− sin φ
+sin θ sin φ
+cos θ sin φ
+cos φ
+cos θ
+− sin θ
+0
+
+
+for d = 3 (observe that then Aout = I and c−2
+out = 1 when r = RPML,− and thus A and
+c−2 are continuous at r = RPML,−).
+We highlight that, in other papers on PMLs, the scaled variable, which in our
+case is r+ifθ(r), is often written as r(1+i�σ(r)) with �σ(r) = σ0 for r sufficiently large;
+see, e.g., [24, §4], [4, §2]. Therefore, to convert from our notation, set �σ(r) = fθ(r)/r
+and σ0 = tan θ.
+Let
+(4.6)
+H := H1
+0(Ω)
+or
+{v ∈ H1(Ω) : v = 0 on Γtr},
+17
+
+with the former corresponding to zero Dirichlet boundary conditions on Ω− and the
+latter corresponding to zero Neumann boundary conditions on Ω−.
+Definition 4.1 (A variational formulation of the PML problem). Given G ∈
+(H)∗ and β > 0,
+(4.7)
+find u ∈ H such that a(u, v) = G(v) for all v ∈ H,
+where
+(4.8)
+a(u, v) :=
+��
+Ω∩Ωout
++ 1
+β
+�
+Ω∩Ωin
+� �
+k−2(A∇u) · ∇v − c−2uv
+�
+.
+When
+(4.9)
+G(v) :=
+��
+BRPML,− ∩Ωout
++ 1
+β
+�
+Ω∩Ωin
+�
+c−2gv
+for g ∈ L2(Ω+) with supp g ⊂ BRPML,−, the variational problem (4.7) is a weak form
+of the problem
+(4.10)
+k−2c2
+out∇ · (Aout∇uout) + uout = −g
+in Ωout,
+k−2c2
+in∇ · (Ain∇uin) + uin = −g
+in Ωin,
+uin = uout
+and
+∂n,Ainuin = β∂n,Aoutuout
+on ∂Ωin,
+either
+uin = 0
+or
+∂n,Ainuin = 0
+on ∂Ω−,
+and with the Sommerfeld radiation condition approximated by a radial PML ((4.7) is
+obtained by multiplying the PDEs above by c−2
+in/outαβd−1 and integrating by parts).
+Using the fact that the solution of the true scattering problem exists and is unique
+with Aout, Ain, cout, cin, Ω−, and Ωin described above, the solution of (4.7) exists and
+is unique (i) for fixed k and sufficiently large Rtr − R1 by [30, Theorem 2.1], [31,
+Theorem A], [24, Theorem 5.8] and (ii) for fixed Rtr > R1 and sufficiently large k by
+[18, Theorem 1.5].
+For the particular data G (4.9), it is well-known that, for fixed k, the error
+∥u−v∥H1
+k(BRPML,− \Ω) decays exponentially in Rtr−RPML,− and tan θ; see [30, Theorem
+2.1], [31, Theorem A], [24, Theorem 5.8]. It was recently proved in [18, Theorems 1.2
+and 1.5] that the error ∥u − v∥H1
+k(BRPML,− \Ω) also decreases exponentially in k.
+4.2. Showing that the PML problem fits in the abstract framework
+used in Theorem 1.5. Recall that H is defined by (4.6) and let H0 = L2(Ω). We
+work with the norm ∥ · ∥H1
+k(Ω) (1.5) on H, and use below the higher-order norms
+(4.11)
+∥v∥2
+Hm
+k (Ω) :=
+�
+0≤|α|≤m
+k−2|α| ∥∂αv∥2
+L2(Ω) .
+The rationale for using these norms is that if a function v oscillates with frequency k,
+then |(k−1∂)αv| ∼ |v| for all α; this is true, e.g., if v(x) = exp(ikx · a). We highlight
+that many papers on the FEM applied to the Helmholtz equation use the weighted H1
+norm |||v|||2 := ∥∇v∥2
+L2(Ω)+k2 ∥v∥2
+L2(Ω); we work with (1.5) instead, because weighting
+the jth derivative with k−j is easier to keep track of than weighting the jth derivative
+with k−j+1.
+We first check that the sesquilinear form a (4.8) is continuous and satisfies a
+G˚arding inequality, with constants uniform for ǫ ≤ θ ≤ π/2 − ǫ.
+18
+
+Lemma 4.2 (Bounds on the coefficients A and c).
+Given A and c as in (4.3), a
+scaling function f(r) satisfying (4.1), and ǫ > 0 there exist A+ and c− such that, for
+all ǫ ≤ θ ≤ π/2 − ǫ, x ∈ Ω, and ξ, ζ ∈ Cd,
+|(A(x)ξ, ζ)2| ≤ A+∥ξ∥2∥ζ∥2
+and
+1
+|c(x)|2 ≥ 1
+c2
+−
+.
+Proof. This follows from the definitions of A and c in (4.3), the definitions of α
+and β in (4.2), and the fact that fθ(r) := f(r) tan θ.
+Continuity of a (1.6) with Ccont := max{A+, c−2
+− } then follows from the Cauchy-
+Schwarz inequality and the definition of ∥ · ∥H1
+k(Ω) (1.5).
+Assumption 4.3. When d = 3, fθ(r)/r is nondecreasing.
+Assumption 4.3 is standard in the literature; e.g., in the alternative notation
+described above it is that �σ is non-decreasing – see [4, §2].
+Remark 4.4. As noted above, the variational problem (4.7) is obtained by multi-
+plying the PDEs in (4.10) by c−2
+in/outαβd−1 and integrating by parts (as in [9, §3]). If
+one integrates by parts the PDEs directly (as in, e.g., [24, Lemma 4.2 and Equation
+4.8]), the resulting sesquilinear form satisfies Assumption 1.2 after multiplication by
+eiω, for some suitable ω (see Remark 1.3), without the need for Assumption 4.3.
+Lemma 4.5. Suppose that fθ satisfies Assumption 4.3. With A defined by (4.3),
+given ǫ > 0 there exists A− > 0 such that, for all ǫ ≤ θ ≤ π/2 − ǫ,
+ℜ
+�
+A(x)ξ, ξ
+�
+2 ≥ A−∥ξ∥2
+2
+for all ξ ∈ Cd and x ∈ Ω+.
+Reference for the proof. See, e.g., [20, Lemma 2.3].
+Corollary 4.6. If fθ satisfies Assumption 4.3 then
+ℜa(w, w) ≥ A−∥w∥2
+H1
+k(Ω) −
+�
+A− + c−2
+min
+�
+∥w∥2
+L2(Ω)
+for all w ∈ H.
+Let R : L2(Ω) → H be defined by a(Rg, v) = (g, v)L2(Ω) for all v ∈ H; i.e., R
+is the solution operator of the PML problem. The definition of a and the facts that
+(with the matrices H and D defined by (4.4), (4.5)) H is real and the matrix D is
+diagonal (and hence symmetric) imply that a(u, v) = a(v, u) for all u, v ∈ H, and thus
+Rg = R∗g. We therefore let
+(4.12)
+Csol := ∥R∥L2(Ω)→H = ∥R∗∥L2(Ω)→H .
+We highlight that (i) Csol is bounded by the norm of the solution operator of the true
+scattering problem (i.e., with the Sommerfeld radiation condition) by [18, Theorem
+1.6], (ii) Csol ∼ k when the problem is nontrapping (with this the slowest-possible
+growth in k), and (iii) an advantage of working with the weighted norms (4.11) is that
+Csol in fact describes the k-dependence of the Helmholtz solution operator between
+Hm
+k and Hm+2
+k
+for any m.
+Lemma 4.7 (The PML problem satisfies Assumption 1.2).
+Suppose that, for
+some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 and fθ ∈ Cℓ,1 on the closures of the domains
+on which they are defined, ∂Ω is Cℓ,1, and fθ satisfies Assumption 4.3. Let
+(4.13)
+Zj =
+�
+v : vout ∈ Hj(Ω ∩ Ωout), vin ∈ Hj(Ωin)
+�
+∩ H
+19
+
+with norm
+(4.14)
+∥v∥2
+Zj := ∥vout∥2
+Hj
+k(Ωout∩Ω) + ∥vin∥2
+Hj
+k(Ωin) .
+where the “out” and “in” subscripts denote restriction to Ωout∩Ω and Ωin, respectively.
+Then a defined by (4.8) satisfies Assumption 1.2 and given ǫ > 0 and k0 > 0 there
+exists C > 0 such the bounds (1.8) and (1.9) hold for all k ≥ k0 and ǫ ≤ θ ≤ π/2 − ǫ.
+Proof. First observe that Assumption 3.1 is satisfied by the definition (4.3) of A.
+Since
+sup
+v∈H, ∥v∥(Zj−2 )∗=1
+|a(u, v)| = ∥Lu∥Zj−2 ,
+the bound (1.9) holds by combining Theorem 3.2 (used near Γ− and Γtr) and Theorem
+3.3 (used near Γp) and using the fact that, by Green’s identity, for u ∈ H1
+0(Ω) with
+Lu ∈ L2(Ω) and ∂n,Ainuin = β∂n,Aoutuout on ∂Ωin,
+∥uin∥H1
+k(Ωin) + ∥uout∥H1
+k(Ωout)
+≤ C
+�
+∥uin∥L2(Ωin) + ∥uout∥L2(Ωout) + ∥Luin∥L2(Ωin) + ∥Luout∥L2(Ωout)
+�
+(so that the H1
+k norms on the right-hand sides of (3.2) and (3.3) can be replaced by
+L2 norms). Since the operator associated with the sesquilinear form ℜa is
+�L + L∗
+2
+�
+u = −k−2∇ ·
+�A + A
+2
+∇u
+�
+−
+�c−2 + c−2
+2
+�
+u
+and the matrix A is symmetric, this operator also satisfies Assumption 3.1.
+The
+bound (1.8) then holds by a very similar argument.
+4.3. Theorem 1.5 applied to the PML problem.
+Assumption 4.8. Given p ∈ Z+, (Hh)h>0 are such that the following holds.
+There exists C > 0 such that, for all h > 0, 0 ≤ j ≤ m+1 ≤ p+1, and v ∈ H∩Hℓ+1(Ω)
+there exists Ih,pv ∈ Hh such that
+��vout − (Ih,pv)out
+��
+Hj(Ωout∩Ω) +
+��vin − (Ih,pv)in
+��
+Hj(Ωin)
+≤ Chm+1−j�
+∥vout∥Hm+1(Ωout∩Ω) + ∥vin∥Hm+1(Ωin)
+�
+.
+(4.15)
+where the “out” and “in” subscripts denote restriction to Ωout∩Ω and Ωin, respectively.
+Assumption 4.8 holds when (Hh)h>0 consists of piecewise degree-p polynomials
+on shape-regular simplicial triangulations, indexed by the meshwidth; see, e.g., [8,
+Theorem 17.1], [5, Proposition 3.3.17].
+Theorem 4.9 (Existence, uniqueness, and error bound in the preasymptotic
+regime for the PML problem).
+Suppose that, for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈
+Cℓ−1,1 and fθ ∈ Cℓ,1 on the closures of the domains where they are defined, ∂Ω is
+Cℓ,1, fθ satisfies Assumption 4.3, and β > 0. Let Csol be defined by (4.12), and as-
+sume that {Hh}h>0 satisfy Assumption 4.8. Given ǫ > 0 and p ∈ Z+ with p ≥ ℓ,
+there exists k0 > 0 and Cj, j = 1, 2, 3, 4, such that the following is true for all k ≥ k0,
+ǫ ≤ θ ≤ π/2 − ǫ, and Rtr > R1 + ǫ.
+The solution u of the PML problem (4.7) exists and is unique, and if
+(4.16)
+(hk)2ℓCsol ≤ C1
+20
+
+then the Galerkin solution uh, exists, is unique, and satisfies
+∥u − uh∥H1
+k(Ω) ≤ C2
+�
+1 + hk + (hk)ℓCsol
+�
+min
+wh∈Hh ∥u − vh∥H1
+k(Ω) ,
+(4.17)
+∥u − uh∥L2(Ω) ≤ C3
+�
+hk + (hk)ℓCsol
+�
+min
+wh∈Hh ∥u − vh∥H1
+k(Ω) .
+(4.18)
+If, in addition, g ∈ Hp−1(Ω) ∩ H (with H defined by (4.6)) with
+(4.19)
+∥g∥Hp−1
+k
+(Ω) ≤ C ∥g∥H∗
+for some C > 0, then there exists C4 > 0 such that if h satisfies (4.16) then
+(4.20)
+∥u − uh∥H1
+k(Ω)
+∥u∥H1
+k(Ω)
+≤ C4
+�
+hk + (hk)ℓCsol
+�
+(hk)ℓ.
+Theorem 4.9 is most interesting when p = ℓ, i.e., the polynomial degree is the
+smallest possible covered by the theorem. In this case, (4.16) becomes the condi-
+tion (1.1), and the bounds (4.17), (4.18), and (4.20) become (1.2), (1.3), and (1.4),
+respectively.
+Proof of Theorem 4.9. By the results in §4.2, a defined by (4.8) satisfies the as-
+sumptions of Theorem 1.5. By (4.15), the definition of ∥·∥Zj (4.14), and the definition
+(4.11) of the weighted norms, ∥I − Π∥Zm+1→H ≤ C(hk)m. This bound along with
+Theorem 1.6 and (4.12) imply that
+η(Hh) ≤ C
+� ⌊ℓ/2⌋−1
+�
+j=0
+(hk)2j+1 + (hk)ℓCsol
+�
+.
+If hk ≤ C, then η(Hh) ≤ C(hk + (hk)ℓCsol); the result then follows from Theorem
+1.5 and the fact that if the condition (4.16) holds, then hk ≤ C (since Csol ≥ Ck).
+5. Theorem 1.5 applied to the impedance problem.
+5.1. Definition of the impedance problem. Let Aout, Ain, cout, cin, Ω−, Ωin,
+and Ωtr be as in §4.1. Let
+A :=
+�
+Ain
+in Ωin,
+Aout
+in Ωout ∩ Ω,
+and
+1
+c2 :=
+�
+c−2
+in
+in Ωin,
+c−2
+out
+in Ωout ∩ Ω.
+Let
+(5.1)
+H := {v ∈ H1(Ω) : v = 0 on ∂Ω−}
+or
+H1(Ω),
+with the former corresponding to zero Dirichlet boundary conditions on Ω− and the
+latter corresponding to zero Neumann boundary conditions on Ω−.
+Definition 5.1 (Variational formulation of the impedance problem). Given G ∈
+(H)∗ and β > 0,
+(5.2)
+find u ∈ H such that a(u, v) = G(v) for all v ∈ H,
+where
+(5.3)
+a(u, v) :=
+��
+Ω∩Ωout
++ 1
+β
+�
+Ω∩Ωin
+� �
+k−2(A∇u) · ∇v − c−2uv
+�
+− ik−1
+�
+Γtr
+uv.
+The solution of this variational problem exists and is unique by, e.g., [22, Theorem
+2.4].
+21
+
+5.2. Showing that the impedance problem fits in the abstract frame-
+work used in Theorem 1.5. The proofs that the sesquilinear form a is continuous
+and satisfies a G˚arding inequality are very similar to those for the PML problem in
+§4.2 (in fact, they are simpler because there is no PML scaling parameter in which
+the bounds need to be uniform).
+Lemma 5.2 (The impedance problem satisfies Assumption 1.2).
+Suppose that,
+for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 on the closures of the domains on which
+they are defined, and ∂�� is Cℓ,1. With Zj and its norm defined by (4.13) and (4.14),
+a defined by (5.3) satisfies Assumption 1.2 and given k0 > 0 there exists C > 0 such
+the bounds (1.8) and (1.9) hold for all k ≥ k0.
+Proof. This is very similar to the proof of Lemma 4.7. The regularity assumption
+(1.8) follows by combining Theorem 3.2 used near ∂Ω−, Theorem 3.3 used near ∂Ωin,
+and Theorem 3.4 used near Γtr. The regularity assumption (1.9) follows by combining
+Theorem 3.2 used near ∂Ω−, Theorem 3.3 used near ∂Ωin, and now Theorem 3.2
+(with Neumann boundary condition) used near Γtr. Indeed, near Γtr, the operator
+associated with (ℜa) is −k−2∆−1 with Neumann boundary conditions (coming from
+Aout = I and cout = 1 near Γtr and the fact that no boundary condition is imposed
+on Γtr in H (5.1)).
+5.3. Theorem 1.5 applied to the impedance problem.
+Theorem 5.3 (Existence,
+uniqueness,
+and error bound in the preasymp-
+totic regime for the impedance problem).
+Suppose that, for some ℓ ∈ Z+,
+Aout, Ain, cout, cin ∈ Cℓ−1,1 on the closures of the domains where they are defined, ∂Ω
+is Cℓ,1, and β > 0. Let Csol be defined by (4.12), and assume that {Hh}h>0 satisfy
+Assumption 4.8. Given p ∈ Z+ with p ≥ ℓ, there exists k0 > 0 and Cj, j = 1, 2, 3, 4,
+such that the following is true for all k ≥ k0.
+The solution u of the impedance problem (5.2) exists and is unique, and if (4.16)
+holds then the Galerkin solution uh, exists, is unique, and satisfies the bounds (4.17)
+and (4.18). If, in addition, g ∈ Hp−1(Ω) ∩ H (with H defined by (5.1)) with (4.19)
+for some C > 0, then there exists C4 > 0 such that if h satisfies (4.16) then the bound
+(4.20) holds.
+Given Lemma 5.2, the proof of Theorem 5.3 is very similar to the proof of Theorem
+4.9.
+Remark 5.4 (Imposing the exact Dirichlet-to-Neumann map on Γtr).
+With the
+exact Dirichlet-to-Neumann map imposed on Γtr, the Helmholtz sesquilinear form is
+continuous and satisfies a G˚arding inequality (see, e.g., [37, Lemma 3.3 and Corollary
+3.4]). To apply Theorem 1.5 to this problem, one therefore only needs to check the
+elliptic-regularity assumptions of Assumption 1.2. Using Theorems 3.2 and 3.3, this
+boils down to knowing the analogue of Theorem 3.4 with the impedance boundary
+condition replaced by k−1∂nu = DtNu (for (1.8)) and also k−1∂nu = (DtN+DtN∗)u/2
+(for (1.9)). When m = 0 (i.e., the lowest-order regularity shift covered in Theorem
+3.4), the first of these regularity results is given by [28, Theorem 6.1]. To prove this
+result for m > 1 one would need to make an argument similar to that in the proof
+of Theorem 3.4 except that, because DtN and DtN∗ do not commute with tangential
+derivatives, one would need to obtain two additional estimates: 1) estimates on u
+with nontrivial boundary data, e.g., when k−1∂nu − (DtN)u = g ∈ Hs
+k and 2) trace
+estimates for u that are needed to bound, e.g., [T, DtN]u where T is a vector field
+tangent to the boundary. The same strategy could also be used to handle higher-order
+22
+
+impedance boundary conditions.
+Acknowledgements. EAS was supported by EPSRC grant EP/R005591/1 and
+JG was supported by EPSRC grants EP/V001760/1 and EP/V051636/1.
+REFERENCES
+[1] M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at
+high wave number, SIAM Journal on Numerical Analysis, 42 (2004), pp. 553–575.
+[2] A. K. Aziz, R. B. Kellogg, and A. B. Stephens, A two point boundary value problem with
+a rapidly oscillating solution, Numer. Math., 53 (1988), pp. 107–121.
+[3] M. Bernkopf, T. Chaumont-Frelet, and J. M. Melenk, Stability and convergence of
+Galerkin discretizations of the Helmholtz equation in piecewise smooth media, arXiv pre-
+print arXiv:2209.03601, (2022).
+[4] J. H. Bramble and J. Pasciak, Analysis of a finite PML approximation for the three dimen-
+sional time-harmonic Maxwell and acoustic scattering problems, Mathematics of Compu-
+tation, 76 (2007), pp. 597–614.
+[5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15
+of Texts in Applied Mathematics, Springer, 3rd ed., 2008.
+[6] T. Chaumont-Frelet and S. Nicaise, Wavenumber explicit convergence analysis for finite
+element discretizations of general wave propagation problem, IMA J. Numer. Anal., 40
+(2020), pp. 1503–1543.
+[7] T. Chaumont-Frelet, S. Nicaise, and J. Tomezyk, Uniform a priori estimates for ellip-
+tic problems with impedance boundary conditions, Communications on Pure & Applied
+Analysis, 19 (2020), p. 2445.
+[8] P. G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of numerical analysis,
+Vol. II, North-Holland, Amsterdam, 1991, pp. 17–351.
+[9] F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM Jour-
+nal on Scientific Computing, 19 (1998), pp. 2061–2090.
+[10] M. Costabel, M. Dauge, and S. Nicaise, Corner Singularities and Analytic Regularity for
+Linear Elliptic Systems. Part I: Smooth domains., (2010). https://hal.archives-ouvertes.
+fr/file/index/docid/453934/filename/CoDaNi Analytic Part I.pdf.
+[11] J. Douglas Jr., J. E. Santos, D. Sheen, and L. S. Bennethum, Frequency domain treatment
+of one-dimensional scalar waves, Mathematical Models and Methods in Applied Sciences,
+3 (1993), pp. 171–194.
+[12] Y. Du and H. Wu, Preasymptotic error analysis of higher order FEM and CIP-FEM for
+Helmholtz equation with high wave number, SIAM J. Numer. Anal., 53 (2015), pp. 782–
+804.
+[13] S. Esterhazy and J. M. Melenk, On stability of discretizations of the Helmholtz equation,
+in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and
+R. Scheichl, eds., Springer, 2012, pp. 285–324.
+[14] L. C. Evans, Partial differential equations, American Mathematical Society Providence, RI,
+1998.
+[15] X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large
+wave number, SIAM J. Numer. Anal., 47 (2009), pp. 2872–2896.
+[16] X. Feng and H. Wu, hp-Discontinuous Galerkin methods for the Helmholtz equation with
+large wave number, Math. Comp., 80 (2011), pp. 1997–2024.
+[17] J. Galkowski, Lower bounds for piecewise polynomial approximations of oscillatory functions,
+arXiv preprint arXiv:2211.04757, (2022).
+[18] J. Galkowski, D. Lafontaine, and E. A. Spence, Perfectly-matched-layer truncation is
+exponentially accurate at high frequency, arXiv preprint arXiv:2105.07737, (2021).
+[19] J. Galkowski, D. Lafontaine, E. A. Spence, and J. Wunsch, Decompositions of high-
+frequency Helmholtz solutions via functional calculus, and application to the finite element
+method, arXiv preprint arXiv:2102.13081, (2021).
+[20] J. Galkowski, D. Lafontaine, E. A. Spence, and J. Wunsch, The hp-FEM applied to the
+Helmholtz equation with PML truncation does not suffer from the pollution effect, arXiv
+preprint arXiv:2207.05542, (2022).
+[21] J. Galkowski, E. A. Spence, and J. Wunsch, Optimal constants in nontrapping resolvent
+estimates, Pure and Applied Analysis, 2 (2020), pp. 157–202.
+[22] I. G. Graham and S. A. Sauter, Stability and finite element error analysis for the Helmholtz
+equation with variable coefficients, Math. Comp., 89 (2020), pp. 105–138.
+23
+
+[23] I. Harari and T. J. R. Hughes, Finite element methods for the Helmholtz equation in an ex-
+terior domain: model problems, Computer methods in applied mechanics and engineering,
+87 (1991), pp. 59–96.
+[24] T. Hohage, F. Schmidt, and L. Zschiedrich, Solving time-harmonic scattering problems
+based on the pole condition II: convergence of the PML method, SIAM Journal on Mathe-
+matical Analysis, 35 (2003), pp. 547–560.
+[25] F. Ihlenburg and I. Babuˇska, Finite element solution of the Helmholtz equation with high
+wave number Part I: The h-version of the FEM, Comput. Math. Appl., 30 (1995), pp. 9–37.
+[26] F. Ihlenburg and I. Babuska, Finite element solution of the Helmholtz equation with high
+wave number part II: the hp version of the FEM, SIAM J. Numer. Anal., 34 (1997),
+pp. 315–358.
+[27] F. Ihlenburg and I. Babuˇska, Dispersion analysis and error estimation of Galerkin finite
+element methods for the Helmholtz equation, Int. J. Numer. Meth. Eng., 38, Issue 22 (1995),
+pp. 3745–3774.
+[28] D. Lafontaine, E. A. Spence, and J. Wunsch, A sharp relative-error bound for the Helmholtz
+h-FEM at high frequency, Numerische Mathematik, 150 (2022), pp. 137–178.
+[29] D. Lafontaine, E. A. Spence, and J. Wunsch, Wavenumber-explicit convergence of the hp-
+FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients, Comp.
+Math. Appl., 113 (2022), pp. 59–69.
+[30] M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML
+equations, Computing, 60 (1998), pp. 229–241.
+[31] M. Lassas and E. Somersalo, Analysis of the PML equations in general convex geome-
+try, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 131 (2001),
+pp. 1183–1207.
+[32] Y. Li and H. Wu, FEM and CIP-FEM for Helmholtz Equation with High Wave Number and
+Perfectly Matched Layer Truncation, SIAM J. Numer. Anal., 57 (2019), pp. 96–126.
+[33] C. H. Makridakis, F. Ihlenburg, and I. Babuˇska, Analysis and finite element methods for
+a fluid-solid interaction problem in one dimension, Mathematical Models and Methods in
+Applied Sciences, 6 (1996), pp. 1119–1141.
+[34] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University
+Press, 2000.
+[35] J. M. Melenk, On generalized finite element methods, PhD thesis, The University of Maryland,
+1995.
+[36] J. M. Melenk, A. Parsania, and S. Sauter, General DG-methods for highly indefinite
+Helmholtz problems, Journal of Scientific Computing, 57 (2013), pp. 536–581.
+[37] J. M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of
+the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp, 79
+(2010), pp. 1871–1914.
+[38] J. M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin dis-
+cretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49 (2011), pp. 1210–1243.
+[39] J. C. N´ed´elec, Acoustic and electromagnetic equations: integral representations for harmonic
+problems, Springer Verlag, 2001.
+[40] O. R. Pembery, The Helmholtz Equation in Heterogeneous and Random Media: Analysis and
+Numerics, PhD thesis, University of Bath, 2020.
+https://researchportal.bath.ac.uk/en/
+studentTheses/the-helmholtz-equation-in-heterogeneous-and-random-media-analysis.
+[41] S. A. Sauter, A refined finite element convergence theory for highly indefinite Helmholtz
+problems, Computing, 78 (2006), pp. 101–115.
+[42] A. H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms,
+Math. Comp., 28 (1974), pp. 959–962.
+[43] L. L. Thompson and P. M. Pinsky, Complex wavenumber Fourier analysis of the p-version
+finite element method, Computational Mechanics, 13 (1994), pp. 255–275.
+[44] H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with
+high wave number. Part I: linear version, IMA J. Numer. Anal., 34 (2014), pp. 1266–1288.
+[45] L. Zhu and H. Wu, Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equa-
+tion with high wave number. Part II: hp version, SIAM J. Numer. Anal., 51 (2013),
+pp. 1828–1852.
+24
+

F9FKT4oBgHgl3EQfbS5P/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a1b3f2319096cc76124bd2d951ba6834477a08b83073129b5a8f9b2ba07593db
+size 5505069

G9E4T4oBgHgl3EQfgQ3v/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:cf513ad255f471ecbccaa43fe2be9378aa8734dde7b2a40e81cd96e4bac61b62
+size 544639

HtE2T4oBgHgl3EQf_AnW/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ad846cb51da3a116a29e02c84dab42968ed1328d0a6acb4a91e3199b075dd2bf
+size 6881325

I9AyT4oBgHgl3EQffvhT/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a588906877a180ebdfc01e80892b7d1ee7831f328500aea77b41d6a5bb1d2312
+size 5505069

ItE3T4oBgHgl3EQfXAqD/content/2301.04475v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6ef7fdda9019dc50d3879502be9e6e69eaec24f0b2bfe55ff514ffe980e277ac
+size 399686

ItE3T4oBgHgl3EQfXAqD/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:84517955f15c9c1512f7650976097cb4ef4a14f559530b8e5d04326b8d766110
+size 215744

ItFJT4oBgHgl3EQfFyzw/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b43806d39a08ab98b3059d4a6030f1b3921cbbb2c83f966647c5a61a759d27d7
+size 5636141

KdAyT4oBgHgl3EQff_ia/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e7d514a30f29a4ac3c1c540280d97460c0cd1984d8e1f5debe2fd07c326d9c6b
+size 8388653

KdAyT4oBgHgl3EQff_ia/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:144c2265d985f2aedb05a2e0af86bab3241c5428b075cefb06a132279c372d1a
+size 240843

LNE1T4oBgHgl3EQfGwM1/content/tmp_files/2301.02917v1.pdf.txt

@@ -0,0 +1,539 @@
+Dido’s Problem: When a myth of
+ancient literature became a problem of
+variational calculus
+Dora Musielak
+Abstract
+When introducing the calculus of variations, we may invoke Dido’s
+problem to illustrate the most fundamental variational problem: to find
+the curve of given perimeter which bounds the greatest area. This type of
+problem led mathematicians to invent solution methods of maxima and
+minima, and the genesis of variational calculus as a distinct branch of anal-
+ysis. Dido’s problem was inspired by the mythical tale of the foundation
+of Carthage (ancient city in North Africa) by a Phoenician princess as told
+independently by Roman poet Virgil, and by Latin historian Justinus in
+the first two centuries B.C. Historians have debated the facts surrounding
+Carthage’s birth; however, contemporary mathematicians have accepted
+the vague events described by Virgil in his Aeneid, adding details to Dido’s
+story to extrapolate a few verses and use as a basis for the isoperimet-
+ric theorem. Was Leonhard Euler or Lord Kelvin who first interpreted
+Virgil’s poem as Dido’s problem of variational calculus? In this article I
+attempt to resolve a question of historical attribution to identify who first
+defined Dido’s problem.
+Keywords: Isoperimetrics, variational calculus, Euler, Lord Kelvin
+1
+Introduction
+In 1937, Karl Menger1 wrote: “The first human being to solve a problem of
+calculus of variations seems to have been Queen Dido of Carthage.” Contempo-
+rary mathematics books2 go much further than that by adding details to Dido’s
+story taken from Virgil’s Aeneid, alleging that Dido established Carthage, an
+ancient city in modern Tunisia, by application of the isoperimetric property of
+the circle to secure the largest area of land she bought upon arrival to North
+Africa. Here are two examples, where I use Italics to highlight details of Dido’s
+story that are not in Virgil’s poem:
+1Menger (1937)
+2See for example, Brunt (2004); Freguglia and Giaquinta (2016); Coppersmith (2017);
+Nahin (2004); Rojo and Bloch (2018).
+1
+arXiv:2301.02917v1  [math.HO]  7 Jan 2023
+
+“Dido was a Carthaginian queen (ca. 850 B.C.?) who came from a dysfunc-
+tional family. Her brother, Pygmalion, murdered her husband (who was also
+her uncle) and Dido, with the help of various gods, fled to the shores of North
+Africa with Pygmalion in pursuit. Upon landing in North Africa, legend has it
+that she struck a deal with a local chief to procure as much land as an oxhide
+could contain. She then selected an ox and cut its hide into very narrow strips,
+which she joined together to form a thread of oxhide more than two and a half
+miles long. Dido then used the oxhide thread and the North African sea coast
+to define the perimeter of her property ... it is clear that Dido sought to enclose
+the maximum area within her ox and the sea. The city of Carthage was then
+built within the perimeter defined by the thread and the sea coast. Dido called
+the place Byrsa meaning hide of bull.”3
+“Dido ... using the seashore (given as straight) as part of the boundary, she
+laid out the hide-strip to enclose the maximum possible area, which she “knew”
+would be in the shape of a semicircle.”4
+If these accounts were based on fact, then Dido would be the first woman
+in humanity’s history to understand a mathematical principle, much before
+the first mathematicians in recorded history. Since Carthage was founded in
+814 B.C., Dido was born centuries before Thales of Miletus (c. 624-548 BC),
+Pythagoras of Samos (c. 570-490 BC), and Euclid of Alexandria (325-265 BC),
+and much earlier than the Greek mathematicians who dealt with isoperimetric
+problems, e.g. Zenodorus (c. 200-140 BC) who wrote On Isoperimetric Fig-
+ures; this work is lost but details are found in the commentaries by Theon of
+Alexandria (335-405 AD), and by Pappus of Alexandria (290-350 AD). In his
+Mathematical Collection, Pappus presented results from ancient isoperimetry
+studies5 but he did not mention Dido.
+Dido’s problem is now taught as the most fundamental isoperimetric prob-
+lem: for a fixed perimeter, determine the shape of the closed, planar curve
+that encloses the maximum area. The answer is the circle, as any grammar
+school child knows, but in variational calculus the solution is determined by an
+analytical method introduced by Leonhard Euler and refined by Joseph-Louis
+Lagrange.
+2
+Dido in Ancient Literature
+The story of Dido and the foundation of Carthage was immortalized by Virgil
+in his Aeneid, and by third century Roman historian Justinus in his Epitoma
+historiarum Philippicarum Pompei Trogi.
+However, in these stories there is
+absolutely no mention of Dido enclosing a circular shape for the purchased land
+3Brunt (2004), pp. 14-15.
+4Nahin (2004), p. 45.
+5According to Pappus, the first proof of the isoperimetric property of the circle (using
+geometric arguments) is due to Zenodorus.
+2
+
+with the string of hide, or of her using the knowledge that the circle encloses
+the largest area.
+Virgil wrote in the Aeneid,6 Book 1, lines 365-368, referring to Dido and her
+people arriving to Africa: “They came to this place, and bought land, where
+you now see the vast walls, and resurgent stronghold, of new Carthage, as much
+as they could enclose with the strips of hide from a single bull, and from that
+they called it Byrsa.”7
+Justinus, who refers to Dido by her Phoenician name Elissa, wrote in Book
+XVIII: “By this means some respite was given to the fugitives; and Elissa,
+arriving in a gulf of Africa, attached the inhabitants of the coast, who rejoiced
+at the arrival of foreigners, and the opportunity of bartering commodities with
+them, to her interest. Having then bargained for a piece of ground, as much
+as could be covered with an ox-hide, where she might refresh her companions,
+wearied with their long voyage, until she could conveniently resume her progress,
+she directed the hide to be cut into the thinnest possible strips, and thus acquired
+a greater portion of ground than she had apparently demanded; whence the
+place had afterwards the name of Byrsa.”8
+Thus, if neither Virgil nor Justin provided the details given by contemporary
+mathematicians about Dido’s problem as we know it, who did? how Dido’s
+mythical tale became Dido’s problem? Surely, only a scientist would have made
+the connecting leap between “bought land, . . . , as much as they could enclose
+with the strips of hide from a single bull . . . ” (Virgil, 1st century BC), and
+interpreted these words as “she laid out the hide-strip to enclose the maximum
+possible area, which she “knew” would be in the shape of a semicircle” (Nahin,
+2004).
+3
+Isoperimetry and Calculus of Variations
+Isoperimetrics provided the roots for the development of the variational method-
+ology, starting with the observation made by ancient scholars that most motion
+appears to be in either straight lines or circles. The definition of a straight line as
+the shortest path between two points was an early expression of a minimization
+principle known to ancient geometers. The isoperimetric problems considered
+in antiquity (e.g. the circle in the plane and the sphere in three-dimensional
+space were known as the least perimeter figures to enclose a given area and a
+given volume, respectively) were solved by geometric means.
+Pappus gave credit to Zenodorus (200-140 BC) for solving for the optimal
+6Written between 29 and 19 BC, this epic poem in 12 books tells the story of the foundation
+of Rome from the ashes of Troy. Virgil describes the foundation of Carthage by Dido in Book
+I: 297-371.
+7Line 365: Devenere locos, ubi nunc ingentia cernis moenia surgentemque novae Karthagi-
+nis arcem, mercatique solum, facti de nomine Byrsam, taurino quantum possent circumdare
+tergo.
+8Marcus Junianus Justinus, Epitoma historiarum Philippicarum Pompei Trogi (Epitome of
+the Philippic History of Pompeius Trogus). Translated by Rev. John Selby Watson. London:
+Henry G. Bohn, Convent Garden (1853).
+3
+
+form of a maximum area surface for a given perimeter. He also expounded the
+work of Hero (or Heron) of Alexandria (c. 10-75 AD) who studied the optics of
+reflection, finding that reflected light travels in a way that minimizes its travel
+time. The law of reflection of light—that the angle of incidence equals the angle
+of reflection—was well established since ancient times. In his Catoptrics, Euclid
+noted that light travels in straight lines and described the law of reflection (300
+BC). Hero showed by a geometrical method that the actual path taken by a ray
+of light reflected from a plane mirror is shorter than any other reflected path
+that might be drawn between the source and point of observation.
+Ancient Greeks first conceived the idea that Nature selects the shortest,
+easiest and most direct path in moving objects between points. In the seven-
+teenth and eighteenth centuries, ideas about the economy of Nature continued
+preoccupying philosophers and scientists. Finding analytic solutions to more
+complicated problems of maxima and minima attracted the greatest mathe-
+maticians such as Fermat, Newton, Leibniz, the Bernoulli brothers (Jacob and
+Johann I), Euler, Lagrange, and Maupertuis.
+Perhaps inspired by Hero’s reflected light minimization problem, Pierre de
+Fermat (1601-1665) showed that the time required for a light ray to traverse a
+neighboring virtual path differs from the time actually taken by a quantity of
+the second order. This is known as Fermat’s principle of least time.
+Newton (1643-1727) examined the motion of bodies in a resisting medium,
+finding the shape of the body that renders its resistance minimal.
+In June of 1696, Johann Bernoulli (1667-1748) posed the following problem
+as a challenge to mathematicians: Given two points A and B in a vertical plane,
+find the path AMB down which a movable point particle M must, by virtue of its
+weight, will traverse in the shortest possible time (assumes that M’s acceleration
+is due only to gravity). This is the famous Brachistochrone (from the Greek
+brachistos, shortest, and chronos, time) problem, later also called the problem
+of least time descent.
+The brachistochrone problem does not have a trivial
+solution; the Bernoulli brothers (Jacob and Johann I), Newton, Leibniz and
+l’Hˆopital solved the problem correctly, each using a different approach.9
+The initial investigations in the maxima and minima principles carried out
+by Leonhard Euler began from a study of the work of these mathematicians,
+especially motivated by the work of Jacob Bernoulli and prompted by his teacher
+Johann Bernoulli. The latter drew his attention to a problem of geodesic lines
+in a letter he sent Euler in St. Petersburg in 1728, which led Euler to conceive
+in early 1729 an analytical method by which, on any surface, whether convex or
+concave, the shortest line can be drawn between two points.10 Euler solved other
+isoperimetric problems, obtaining results to help him establish the analytical
+foundations of the calculus of variations.
+Euler invented variational calculus as a distinct branch of analysis precisely
+to systemize the solution methods of maxima and minima, as brilliantly intro-
+duced in his 1744 book Methodus inveniendi lineas curvas maximi minimive
+9Fregulia and Giaquinta (2016), pp. 3-4.
+10Euler (1732)
+4
+
+proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu
+accepti, the first treatise on calculus of variations.11 With Euler’s approach, the
+calculus of variations yielded a method for finding an extremum of a quantity
+that is expressible as an (variational) integral.
+Euler’s Methodus inveniendi represented a substantial break with the then
+established tradition set for by his predecessors, including his earlier work in
+the subject.12
+In this treatise, Euler formulated the variational principle of
+mechanics, which is the principle of least action now attributed to Maupertuis:
+For a given projected body, denote its mass by M, half the square of its velocity
+by v, the arclength element by ds. Then, among all curves passing through
+the same pair of endpoints, the desired curve is the one that minimizes the
+integral
+�
+Mdsv1/2. Details on how Euler formulated the principle are provided
+by Goldstine (1980), and by Freguglia and Giaquinta (2016).13
+Euler remarked: “Since the structure of the universe was made most perfect
+as designed by the wisest Creator, nothing in the world will occur in which no
+maximum or minimum rule is shining forth; wherefore there is absolutely no
+doubt that all the effects of the world can be equally successfully determined
+from final causes by means of the maximum and least methods, and from the
+efficient causes themselves.”14
+Considered as the first variational treatment of mechanics, Euler’s principle
+of least action contributed significantly to analytic mechanics and ultimately to
+the fundamental underpinnings of twentieth-century physics, including general
+relativity and quantum mechanics.
+Euler was also known as being able to recite Virgil’s Aeneid by heart. Did
+he interpret Dido’s tale as Dido’s isoperimetric problem?
+4
+Defining Dido’s Problem in the Calculus of
+Variations
+A casual survey of the history of mathematics books written in the eighteenth
+and nineteenth century yields no clues as to when or how Dido’s mythical story
+became part of variational calculus. It required a person with mathematical
+brilliance and fertile imagination to connect ancient myth with mathematics.
+Two names emerge as potential candidates: Leonhard Euler (1707-1783), the
+originator of the calculus of variations, and British mathematician, physicist
+and engineer William Thomson, known in physics as Lord Kelvin (1824-1907).
+4.1
+Leonhard Euler
+In Methodus inveniendi, Euler gives the following example to demonstrate his
+analytical method: to find among all admissible curves, enclosing a given area,
+11Euler (1744)
+12Fraser (1993)
+13Goldstine (1980), p. 101; Freguglia and Giaquinta (2016), pp. 181-189.
+14Euler (1744), Additamentum I.
+5
+
+Figure 1: Euler’s sketch from Methodus inveniendi (1744)
+the one of least length. Figure 1 is Euler’s sketch to demonstrate that the curved
+arc of a circle, BM, is minimum. In his own words:15
+“On the axis AP construct the line BM, so that, when the area ABMP of
+a given size is cut off, the curved arc BM corresponding to that area is the
+minimum of all.” After solving his variational integral, Euler shows the solution
+curve to be an arc of a circle with center somewhere on the line AP, for example,
+at C in Fig. 1.
+But neither Euler’s Methodus inveniendi nor his other published memoirs in
+the field ever mention Dido.
+In October 1783, the Marquis de Condorcet16 gave the ´Eloge d’Euler to the
+members of the Acad´emie des Sciences in Paris. In this solemn eulogy, Con-
+dorcet expounded on Euler’s genius and suggested that a verse from the Aeneid
+had given Euler the first idea for a memoir on a question of Mechanics.
+In
+Condorcet’s own words:
+L’´etude de la Litt´erature ancienne et des Langues savantes avait fait partie
+de son ´education ; il en conserva le goˆut toute sa vie, et n’oublia rien de ce qu’il
+avait appris ; mais il n’eut jamais ni le tems ni le d´esir d’ajouter `a ses premi`eres
+´etudes : il n’avait pas lu les Po`etes modernes, et savait par cœur l’Eneide.
+Cependant M. Euler ne perdait pas de vue les Math´ematiques, mˆeme lorsqu’il
+r´ecitait les vers de Virgile ; tout ´etait propre `a lui rappeler cet objet presque
+unique de ses pens´ees, et on trouve dans ses ouvrages un savant M´emoire sur
+une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait
+donn´e la premi`ere id´ee.17 [The study of ancient literature and scholarly lan-
+guages had been part of his education; he retained a taste for it all his life, and
+15Euler (1744), Chapter IV, p.
+135, Exemplum II: 9.
+Super axe AP construere lineam
+BM, ita comparatant, ut, abscissa area ABMP datæ magnitudinis, arcus curvæ BM illi areæ
+respondens sit omnium minimus.
+16Condorcet, Jean-Antoine-Nicolas de Caritat marquis de (1743-1794).
+17 ´Eloge d’Euler Prononc´e `a l’Acad´emie, par de Condorcet, Histoire de l’Acad´emie royale
+des sciences ... 1783, p. 64.
+6
+
+Bforgot nothing he had learned; but he never had either the time or the desire to
+add to his first studies: he had not read the Modern Poets, and knew the Aeneid
+by heart. However, M. Euler did not lose sight of Mathematics, even when he
+recited the verses of Virgil; everything was likely to remind him of this almost
+unique object of his thoughts, and we find in his works a scholarly Memoir on a
+question of Mechanics, of which he said that a verse from the Aeneid had given
+him the first idea.]
+Euler did take verses from the Aeneid poem to use as mottos for his com-
+peting memoirs submitted to the French Academy.18 These are summarized in
+Table 1. Was this to what Condorcet referred to?
+Table 1. Euler’s Memoirs and Mottos taken from Virgil’s Aeneid.
+Year
+Memoir Title
+Motto
+1753
+“On the movement of ships
+Tali remigio navis se
+(E. 413)
+without the wind’s force.”
+tarda movebat.
+7th winning memoir
+Virg. Aeneid Liv. 5
+1759
+“Concernin pitching
+Insequitur clamorque virum
+(E. 415)
+and rolling.”
+stridorque rudentum.
+9th winning memoir
+Virg. Aeneid, Liv. 1
+1770
+“Moon Theory”
+Errantem que canit Lunam
+(E. 485)
+Prize for 1770
+Virg. Aeneid Liv. 1
+10th winning memoir
+1772
+“Improved Moon theory”
+Hic labor extremus, longarum
+(E. 486)
+Prize for 1772
+haec meta viarum hinc jam
+11th winning memoir
+digressi, vestris appellimus oris
+Virg. Aeneid, Liv. 3
+However, the mottos were carefully selected by Euler to match the research
+topic of the competition.19 In addition to using Virgil’s verses, he also quoted
+from other ancient writers such as Marcus Tullius Cicero, Properci, and he
+composed other adages, asking Christian Goldbach for suggestions. Ultimately,
+Condorcet’s statement “et on trouve dans ses ouvrages un savant M´emoire sur
+une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait
+donn´e la premi`ere id´ee” does not mean that Euler was inspired by Virgil to
+define Dido’s problem.
+As a historian, I cannot rely on obituaries to extract factual data, even if
+written by an eminent scholar. The much younger Condorcet never met Euler,
+and the ´Eloge he wrote, as most eulogies are, was based on hearsay, relaying on
+what the French academicians might have recalled about Euler’s life and work.
+18Submissions were anonymously and the memoir identified by a motto; the author’s name
+enclosed in a sealed envelope was opened only for the winning memoir after the judging of
+the contest.
+19For the significance of the mottos that Euler selected, see Musielak (2022).
+7
+
+A contemporary biography (published in 2016) further implies that Euler
+solved Dido’s problem. The author refers to a copy of an eight-page manuscript
+(preserved in Moscow) that is said to contain Euler’s answer.
+Is this the
+manuscript that categorically would give Euler credit for connecting Dido’s story
+to variational calculus? Unfortunately, the manuscript in question is said to be
+“not in Euler’s own handwriting.” Thus, it diminishes its credibility. It is rather
+improbable that Euler, a prolific writer, would be the author of a manuscript
+inscribed by someone else. Besides, he would have included this solution in a
+paper published in 1764, where Euler summarized the results of the Calculus of
+Variations in terms of the variational operator.
+Joseph-Louis Lagrange (1736-1813) expanded the variational calculus. In his
+second letter to Euler dated August 1755, Lagrange outlined his delta-algorithm
+(for solving constrained optimization problems), an approach Euler embraced,
+prompting him to conceive the term calculus of variations. In the abstract of
+a memoir published in 1764, Euler credits Lagrange for enriching the science20
+Their combined work led eventually to the Euler–Lagrange equations, which are
+the equilibrium equations for minima of variational integrals.21
+Five years after Euler died, Lagrange published M´ecanique analytique, his
+compendium on analytical mechanics, using variational ideas to present me-
+chanics from a unified analytic viewpoint. When teaching at the ´Ecole Poly-
+technique in 1799, Lagrange published Le¸cons sur le calcul des fonctions and
+explained the method of variation. Lagrange provided a brief overview of the
+development of problems of maxima and minima, referring only to Greek math-
+ematician Apollonius (262-190 BC), which dealt exclusively with the largest
+and smallest straight lines which can be drawn from given points to the arcs of
+conic sections.22 Dido’s problem is not mentioned here nor in Lagrange’s other
+published works.
+4.2
+William Thomson, Lord Kelvin
+The first instance in which Dido’s name appear in the context of interest is
+found in a public lecture delivered by William Thomson in 1893. A great physi-
+cist known today as Lord Kelvin, his contributions include a major role in the
+development of the second law of thermodynamics; the absolute temperature
+scale (measured in kelvins); the dynamical theory of heat; the mathematical
+analysis of electricity and magnetism, including the basic ideas for the electro-
+magnetic theory of light; and much more. He brought together disparate areas
+of physics—heat, thermodynamics, mechanics, hydrodynamics, magnetism, and
+20Euler (1764). . .
+ex quo Auctori occasio est oblata hanc scientiam novo Calculi genere
+locupletandi, quem Calculum variationum appellat et cuis elementa hic tradere ac dilucide
+explicare constituit.
+21See Freguglia and Giaquinta (2016) for an excellent presentation of the Euler-Lagrange
+equations, including a historical perspective.
+22Lagrange (1806).
+Les questions de maximis et minimis n’ont pas ´et´e incounues aux
+anciens g´eom`etres ; car on a un livre entier d’Apollonius, qui traite presqu’uniquement des
+plus grandes et des plus petites lignes droites qui peuvent ˆetre men´ees de points donn´es aux
+arcs des sections coniques. p. 424.
+8
+
+Figure 2: Dido’s problem as described by Lord Kelvin in 1893.
+electricity. Lord Kelvin played a key role in the final synthesis of 19th-century
+science, which viewed all physical change as energy-related phenomena.23
+Lord Kelvin related Dido’s clever approach to bargaining for land as follows,
+using the sketch in Fig. 2 to illustrate Dido’s problem:
+“. . .
+She cut the ox-hide into an exceedingly long strip, and succeeded in
+enclosing between it and the sea a very valuable territory on which she built
+Carthage. In Dido’s problem the greatest value of land was to be enclosed by a
+line of given length. If the land is all of equal value the general solution of the
+problem shows that her line of ox-hide should be laid down in a circle. It shows
+also that if the sea is to be part of the boundary, starting, let us say, southward
+from any given point, A, of the coast, the inland bounding line must at its far
+end cut the coast line perpendicularly. Here, then, to complete our solution, we
+have a very curious and interesting, but not at all easy, geometrical question to
+answer: What must be the radius of a circular arc, ADC, of given length, and
+in what direction must it leave the point A, in order that it may cut a given
+curve, ABC, perpendicular at some unknown point, C?”24
+Lord Kelvin added that having enough mathematics knowledge, Dido would
+determine that the boundary had to be a circle. Of course, as illustrated in Fig.
+2, she would have given the boundary a different curvature in different parts to
+gain as much as possible of the more valuable parts of the land offered to her,
+“even though difference of curvature in different parts would cause the total
+area enclosed to be less than it would be with a circular boundary of the same
+length.”25
+23Gray (1910)
+24Thomson (1894), p. 572-574.
+25Ibid., p. 574.
+9
+
+CARTHAGToday, taught as introduction to calculus of variation, the solution of Dido’s
+problem requires an extremization solution under constraint, that is, we max-
+imize the area, A =
+�
+ydx, subject to the condition that the arc, L =
+�
+ds is
+of a given length L. In other words, we wish to maximize the integral A sub-
+ject to the condition that another integral L has a given constant value. Note
+this is an optimization problem with constraints where we use Lagrange’s strat-
+egy for finding the local maxima and minima of a function subject to equality
+constraints.
+It is clear that, without an original reliable source, I cannot conclude that
+Euler defined Dido’s Problem for the first time, inspired by Virgil’s Aeneid,
+as Condorcet implied. The evidence points to Lord Kelvin who described the
+problem in 1893. And as he stated, whether severe critics will call Dido’s story
+mythical or allow it to be historic, it is nevertheless full of scientific interest.
+As for me, Dido’s Problem is an excellent example to introduce students to
+the calculus of variations, as it expresses a perfectly definite case of isoperimet-
+rics, illustrating the fundamental principles introduced by Euler and Lagrange
+in the eighteenth century.
+5
+Dido and Ancient Mathematics
+Nothing is known about Dido’s knowledge. Being a Phoenician princess, it is
+highly probable that she was well educated. What we glimpse from Virgil’s
+and Justinus’s tales is that Dido was a formidable woman, smart, ambitious, a
+foreign leader that left the city of Tyre (on the coastline of modern Lebanon)
+with her faithful followers, navigated the waters of the Mediterranean Sea and
+landed in the coast of North Africa. There, she established and ruled Carthage
+(modern day Tunis), an important port city that rose to the height of its power
+in the second century BC, before Rome became supreme and took over that
+region.
+For the ancient cultures that flourished around the Mediterranean, geom-
+etry was fundamental to their development. The Babylonians thriving in the
+Mesopotamian River Valley engaged in commerce through the Mediterranean,
+and this required considerable mathematical skills. Clay tables preserve records
+of what they knew.
+For instance, clay tables from Babylon, located in the
+southern part of Mesopotamia, about fifty miles south of present-day Baghdad
+(Iraq) suggest that the Babylonian had an advanced knowledge of geometry and
+arithmetic.
+In fact, some scholars believe that the Babylonians knew the Pythagorean
+theorem a thousand years before Pythagoras of Samos. At Susa, an ancient
+city over two hundred miles from Babylon, a set of tablets were discovered in
+1936, which contain the ratios of areas and perimeters of regular polygons to
+their respective side lengths. The best known surviving tablet (estimated to be
+from between 1900 and 1600 BC) contains a list of Pythagorean triples. This
+suggests that the Babylonians had knowledge of the Pythagorean theorem, as
+well as certain algebraic identities.
+10
+
+Moreover, that women knew mathematics in ancient times has been exten-
+sively documented.
+For example, the Pythagorean society included women,
+some of which became famous such as mathematician Theano, who was mar-
+ried to Pythagoras. In the dedication of his Introduction to Harmonics, ancient
+mathematician and music theorist Nicomachus of Gerasa (c. 60-120 AD) ad-
+dresses the lessons to a lady, one of his students.26 In this book, also known as
+Manual of Harmonics, Nicomachus dealt with the theory of music, a version of
+Pythagorean harmonics, in which he assigned number and numerical ratios to
+notes and intervals. And of course, we know about Hypatia of Alexandria (c.
+370-415 AD) considered the first woman scholar to attain eminence as mathe-
+matician and astronomer.27
+I believe that Dido was educated in mathematics, and so she used the the-
+orem of isoperimetry to outsmart the king who sold her the piece of land in
+the northern tip of Africa (today’s Tunisia). Therefore, Dido’s Problem should
+be viewed not only to illustrate a fundamental problem of variational calculus
+but also as a lesson in the history of mathematics and the role ancient women
+played in its development.
+References
+Brunt, B. van (2004). The Calculus of Variations. Published by Springer
+New York ISBN: 978-0-387-40247-5 DOI: 10.1007/b97436.
+Coppersmith, J. (2017) The Lazy Universe: An Introduction to the Principle
+of Least Action. Oxford University Press.
+Euler, L. (1732). De linea brevissima in superficie quacunque duo quaelibet
+puncta iungente (On the shortest line joining two points on a surface). Com-
+mentarii academiae scientiarum Petropolitanae, Volume 3, 1732, pp. 110-124.
+(E. 9)
+Euler, L. (1738). Problematis isoperimetrici in latissimo sensu accepti so-
+lutio generalis (On isoperimetric problems in the widest sense). Commenturii
+Academiae Scientiarum Petropolitanae 6 (1732/3), 123-155. Opera Omnia, 125,
+13-40. (E. 27)
+Euler, L. (1741). Curvarum maximi minimive proprietate gaudentium inven-
+tio nova et facilis (New and easy method of finding curves enjoying a maximal
+or minimal property). In Commentarii Academiae Scientiarum Petropolitanae
+8 (1736). 159-190. Reprinted in Euler, L. Opera Omnia, I 25, 54-80. (E. 56)
+Euler, L. (1744). Methodus inveniendi curvas h’neas maximi minimive pro-
+prietate gaudentes sive solution problematis isoperimetrici latissimo sensu ac-
+cepti. Lausanne, Genf: M.-M. Bousquet. Reprinted in Euler, L. Opera Omnia,
+I 24. (E. 65). According to Enestr¨om, Euler completed the manuscript of this
+work by April 1743.
+Euler, L. (1764).
+Elementa calculi variationum (Elements of Calculus of
+Variations). Novi commentarii academiae scientiarum Petropolitanea 10 (1764),
+26Biographical Note of Nicomachus, in Great Books of the Western World, Robert Maynard
+Hutchins, Ed., Vol. 11, p. 807.
+27Musielak (2020), pp. 206-207.
+11
+
+1766, pp. 51-93. This research (E. 296) was presented at the Berlin Academy
+in 1756.
+Freguglia, P. and Giaquinta, M. (2016). The Early Period of the Calculus
+of Variations. Published by Birkh¨auser.
+Gelfand, I. M. and Fomin, S. V. (1963). Calculus of Variations. Revised
+English Edition Translated and Edited by R. A. Silverman Prentice-Hall, Inc.
+Englewood Cliffs, NJ.
+Goldstine, H. (1980). A History of the Calculus of Variations from the 17th
+through the 19th Century. Springer-Verlag.
+Gray, A. (1910).
+The Life of William Thomson, Baron Kelvin of Large.
+Nature 83, 61–65 (1910).
+Lagrange (1806). Le¸cons sur le calcul des fonctions. Nouvelle ´edition, revue,
+corrig´ee et augment´ee par l’auteur [J.-L. Lagrange]. Initially published as lecture
+notes in 1799 when Lagrange was teaching at the Ecole Polytechnique and
+reprinted in 1804. In 1806, Lagrange published a new edition containing two
+new lessons.
+Menger, K. (1937). What is Calculus of Variations and What Are Its Ap-
+plications? The Scientific Monthly 45 (3) (1937), 250-253.
+Musielak, D. (2022). Leonhard Euler and the Foundations of Celestial Me-
+chanics. Springer History of Physics Series. Springer Nature Switzerland. ISBN
+978-3-031-12321-4.
+Musielak, D. (2020). Sophie Germain: Revolutionary Mathematician. Springer
+Biographies. Springer Nature Switzerland. ISBN 978-3030383770.
+Nahin, J.P. (2004). When Least is Best. Princeton University Press, 2004.
+Rojo, A. and Bloch, A. (2018). The Principle of Least Action: History and
+Physics. Cambridge University Press. doi:10.1017/9781139021029
+Thomson, W. (1894). Popular Lectures and Addresses by Sir William Thom-
+son (Baron Kelvin) in Three Volumes. Nature Series, MacMillan and Co. Lon-
+don 1894.
+Dora Musielak; University of Texas at Arlington, 6 January 2023
+12
+

LNE1T4oBgHgl3EQfGwM1/content/tmp_files/load_file.txt

@@ -0,0 +1,372 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf,len=371
+page_content='Dido’s Problem: When a myth of  ancient literature became a problem of  variational calculus  Dora Musielak  Abstract  When introducing the calculus of variations, we may invoke Dido’s  problem to illustrate the most fundamental variational problem: to find  the curve of given perimeter which bounds the greatest area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' This type of  problem led mathematicians to invent solution methods of maxima and  minima, and the genesis of variational calculus as a distinct branch of anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Dido’s problem was inspired by the mythical tale of the foundation  of Carthage (ancient city in North Africa) by a Phoenician princess as told  independently by Roman poet Virgil, and by Latin historian Justinus in  the first two centuries B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Historians have debated the facts surrounding  Carthage’s birth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' however, contemporary mathematicians have accepted  the vague events described by Virgil in his Aeneid, adding details to Dido’s  story to extrapolate a few verses and use as a basis for the isoperimet-  ric theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Was Leonhard Euler or Lord Kelvin who first interpreted  Virgil’s poem as Dido’s problem of variational calculus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In this article I  attempt to resolve a question of historical attribution to identify who first  defined Dido’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Keywords: Isoperimetrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' variational calculus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Euler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Lord Kelvin  1  Introduction  In 1937,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Karl Menger1 wrote: “The first human being to solve a problem of  calculus of variations seems to have been Queen Dido of Carthage.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Contempo-  rary mathematics books2 go much further than that by adding details to Dido’s  story taken from Virgil’s Aeneid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' alleging that Dido established Carthage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' an  ancient city in modern Tunisia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' by application of the isoperimetric property of  the circle to secure the largest area of land she bought upon arrival to North  Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Here are two examples, where I use Italics to highlight details of Dido’s  story that are not in Virgil’s poem:  1Menger (1937)  2See for example, Brunt (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Freguglia and Giaquinta (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Coppersmith (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Nahin (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Rojo and Bloch (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='02917v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='HO]  7 Jan 2023  “Dido was a Carthaginian queen (ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 850 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=') who came from a dysfunc-  tional family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Her brother, Pygmalion, murdered her husband (who was also  her uncle) and Dido, with the help of various gods, fled to the shores of North  Africa with Pygmalion in pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Upon landing in North Africa, legend has it  that she struck a deal with a local chief to procure as much land as an oxhide  could contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' She then selected an ox and cut its hide into very narrow strips,  which she joined together to form a thread of oxhide more than two and a half  miles long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Dido then used the oxhide thread and the North African sea coast  to define the perimeter of her property .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' it is clear that Dido sought to enclose  the maximum area within her ox and the sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The city of Carthage was then  built within the perimeter defined by the thread and the sea coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Dido called  the place Byrsa meaning hide of bull.”3  “Dido .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' using the seashore (given as straight) as part of the boundary, she  laid out the hide-strip to enclose the maximum possible area, which she “knew”  would be in the shape of a semicircle.”4  If these accounts were based on fact, then Dido would be the first woman  in humanity’s history to understand a mathematical principle, much before  the first mathematicians in recorded history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Since Carthage was founded in  814 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=', Dido was born centuries before Thales of Miletus (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 624-548 BC),  Pythagoras of Samos (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 570-490 BC), and Euclid of Alexandria (325-265 BC),  and much earlier than the Greek mathematicians who dealt with isoperimetric  problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Zenodorus (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 200-140 BC) who wrote On Isoperimetric Fig-  ures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' this work is lost but details are found in the commentaries by Theon of  Alexandria (335-405 AD), and by Pappus of Alexandria (290-350 AD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In his  Mathematical Collection, Pappus presented results from ancient isoperimetry  studies5 but he did not mention Dido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Dido’s problem is now taught as the most fundamental isoperimetric prob-  lem: for a fixed perimeter, determine the shape of the closed, planar curve  that encloses the maximum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The answer is the circle, as any grammar  school child knows, but in variational calculus the solution is determined by an  analytical method introduced by Leonhard Euler and refined by Joseph-Louis  Lagrange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  2  Dido in Ancient Literature  The story of Dido and the foundation of Carthage was immortalized by Virgil  in his Aeneid, and by third century Roman historian Justinus in his Epitoma  historiarum Philippicarum Pompei Trogi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  However, in these stories there is  absolutely no mention of Dido enclosing a circular shape for the purchased land  3Brunt (2004), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 14-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  4Nahin (2004), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  5According to Pappus, the first proof of the isoperimetric property of the circle (using  geometric arguments) is due to Zenodorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  2  with the string of hide, or of her using the knowledge that the circle encloses  the largest area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Virgil wrote in the Aeneid,6 Book 1, lines 365-368, referring to Dido and her  people arriving to Africa: “They came to this place, and bought land, where  you now see the vast walls, and resurgent stronghold, of new Carthage, as much  as they could enclose with the strips of hide from a single bull, and from that  they called it Byrsa.”7  Justinus, who refers to Dido by her Phoenician name Elissa, wrote in Book  XVIII: “By this means some respite was given to the fugitives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' and Elissa,  arriving in a gulf of Africa, attached the inhabitants of the coast, who rejoiced  at the arrival of foreigners, and the opportunity of bartering commodities with  them, to her interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Having then bargained for a piece of ground, as much  as could be covered with an ox-hide, where she might refresh her companions,  wearied with their long voyage, until she could conveniently resume her progress,  she directed the hide to be cut into the thinnest possible strips, and thus acquired  a greater portion of ground than she had apparently demanded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' whence the  place had afterwards the name of Byrsa.”8  Thus, if neither Virgil nor Justin provided the details given by contemporary  mathematicians about Dido’s problem as we know it, who did?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' how Dido’s  mythical tale became Dido’s problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Surely, only a scientist would have made  the connecting leap between “bought land, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' , as much as they could enclose  with the strips of hide from a single bull .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' ” (Virgil, 1st century BC), and  interpreted these words as “she laid out the hide-strip to enclose the maximum  possible area, which she “knew” would be in the shape of a semicircle” (Nahin,  2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  3  Isoperimetry and Calculus of Variations  Isoperimetrics provided the roots for the development of the variational method-  ology, starting with the observation made by ancient scholars that most motion  appears to be in either straight lines or circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The definition of a straight line as  the shortest path between two points was an early expression of a minimization  principle known to ancient geometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The isoperimetric problems considered  in antiquity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' the circle in the plane and the sphere in three-dimensional  space were known as the least perimeter figures to enclose a given area and a  given volume, respectively) were solved by geometric means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Pappus gave credit to Zenodorus (200-140 BC) for solving for the optimal  6Written between 29 and 19 BC, this epic poem in 12 books tells the story of the foundation  of Rome from the ashes of Troy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Virgil describes the foundation of Carthage by Dido in Book  I: 297-371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  7Line 365: Devenere locos, ubi nunc ingentia cernis moenia surgentemque novae Karthagi-  nis arcem, mercatique solum, facti de nomine Byrsam, taurino quantum possent circumdare  tergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  8Marcus Junianus Justinus, Epitoma historiarum Philippicarum Pompei Trogi (Epitome of  the Philippic History of Pompeius Trogus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Translated by Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' John Selby Watson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' London:  Henry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Bohn, Convent Garden (1853).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  3  form of a maximum area surface for a given perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' He also expounded the  work of Hero (or Heron) of Alexandria (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 10-75 AD) who studied the optics of  reflection, finding that reflected light travels in a way that minimizes its travel  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The law of reflection of light—that the angle of incidence equals the angle  of reflection—was well established since ancient times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In his Catoptrics, Euclid  noted that light travels in straight lines and described the law of reflection (300  BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Hero showed by a geometrical method that the actual path taken by a ray  of light reflected from a plane mirror is shorter than any other reflected path  that might be drawn between the source and point of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Ancient Greeks first conceived the idea that Nature selects the shortest,  easiest and most direct path in moving objects between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In the seven-  teenth and eighteenth centuries, ideas about the economy of Nature continued  preoccupying philosophers and scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Finding analytic solutions to more  complicated problems of maxima and minima attracted the greatest mathe-  maticians such as Fermat, Newton, Leibniz, the Bernoulli brothers (Jacob and  Johann I), Euler, Lagrange, and Maupertuis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Perhaps inspired by Hero’s reflected light minimization problem, Pierre de  Fermat (1601-1665) showed that the time required for a light ray to traverse a  neighboring virtual path differs from the time actually taken by a quantity of  the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' This is known as Fermat’s principle of least time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Newton (1643-1727) examined the motion of bodies in a resisting medium,  finding the shape of the body that renders its resistance minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  In June of 1696, Johann Bernoulli (1667-1748) posed the following problem  as a challenge to mathematicians: Given two points A and B in a vertical plane,  find the path AMB down which a movable point particle M must, by virtue of its  weight, will traverse in the shortest possible time (assumes that M’s acceleration  is due only to gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' This is the famous Brachistochrone (from the Greek  brachistos, shortest, and chronos, time) problem, later also called the problem  of least time descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  The brachistochrone problem does not have a trivial  solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' the Bernoulli brothers (Jacob and Johann I), Newton, Leibniz and  l’Hˆopital solved the problem correctly, each using a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='9  The initial investigations in the maxima and minima principles carried out  by Leonhard Euler began from a study of the work of these mathematicians,  especially motivated by the work of Jacob Bernoulli and prompted by his teacher  Johann Bernoulli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The latter drew his attention to a problem of geodesic lines  in a letter he sent Euler in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Petersburg in 1728, which led Euler to conceive  in early 1729 an analytical method by which, on any surface, whether convex or  concave, the shortest line can be drawn between two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='10 Euler solved other  isoperimetric problems, obtaining results to help him establish the analytical  foundations of the calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Euler invented variational calculus as a distinct branch of analysis precisely  to systemize the solution methods of maxima and minima, as brilliantly intro-  duced in his 1744 book Methodus inveniendi lineas curvas maximi minimive  9Fregulia and Giaquinta (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  10Euler (1732)  4  proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu  accepti, the first treatise on calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='11 With Euler’s approach, the  calculus of variations yielded a method for finding an extremum of a quantity  that is expressible as an (variational) integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Euler’s Methodus inveniendi represented a substantial break with the then  established tradition set for by his predecessors, including his earlier work in  the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='12  In this treatise, Euler formulated the variational principle of  mechanics, which is the principle of least action now attributed to Maupertuis:  For a given projected body, denote its mass by M, half the square of its velocity  by v, the arclength element by ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Then, among all curves passing through  the same pair of endpoints, the desired curve is the one that minimizes the  integral  �  Mdsv1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Details on how Euler formulated the principle are provided  by Goldstine (1980), and by Freguglia and Giaquinta (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='13  Euler remarked: “Since the structure of the universe was made most perfect  as designed by the wisest Creator, nothing in the world will occur in which no  maximum or minimum rule is shining forth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' wherefore there is absolutely no  doubt that all the effects of the world can be equally successfully determined  from final causes by means of the maximum and least methods, and from the  efficient causes themselves.”14  Considered as the first variational treatment of mechanics, Euler’s principle  of least action contributed significantly to analytic mechanics and ultimately to  the fundamental underpinnings of twentieth-century physics, including general  relativity and quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Euler was also known as being able to recite Virgil’s Aeneid by heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Did  he interpret Dido’s tale as Dido’s isoperimetric problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  4  Defining Dido’s Problem in the Calculus of  Variations  A casual survey of the history of mathematics books written in the eighteenth  and nineteenth century yields no clues as to when or how Dido’s mythical story  became part of variational calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' It required a person with mathematical  brilliance and fertile imagination to connect ancient myth with mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Two names emerge as potential candidates: Leonhard Euler (1707-1783), the  originator of the calculus of variations, and British mathematician, physicist  and engineer William Thomson, known in physics as Lord Kelvin (1824-1907).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='1  Leonhard Euler  In Methodus inveniendi, Euler gives the following example to demonstrate his  analytical method: to find among all admissible curves, enclosing a given area,  11Euler (1744)  12Fraser (1993)  13Goldstine (1980), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 101;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Freguglia and Giaquinta (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 181-189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  14Euler (1744), Additamentum I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  5  Figure 1: Euler’s sketch from Methodus inveniendi (1744)  the one of least length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Figure 1 is Euler’s sketch to demonstrate that the curved  arc of a circle, BM, is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In his own words:15  “On the axis AP construct the line BM, so that, when the area ABMP of  a given size is cut off, the curved arc BM corresponding to that area is the  minimum of all.” After solving his variational integral, Euler shows the solution  curve to be an arc of a circle with center somewhere on the line AP, for example,  at C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  But neither Euler’s Methodus inveniendi nor his other published memoirs in  the field ever mention Dido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  In October 1783, the Marquis de Condorcet16 gave the ´Eloge d’Euler to the  members of the Acad´emie des Sciences in Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In this solemn eulogy, Con-  dorcet expounded on Euler’s genius and suggested that a verse from the Aeneid  had given Euler the first idea for a memoir on a question of Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  In  Condorcet’s own words:  L’´etude de la Litt´erature ancienne et des Langues savantes avait fait partie  de son ´education ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' il en conserva le goˆut toute sa vie, et n’oublia rien de ce qu’il  avait appris ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' mais il n’eut jamais ni le tems ni le d´esir d’ajouter `a ses premi`eres  ´etudes : il n’avait pas lu les Po`etes modernes, et savait par cœur l’Eneide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Cependant M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Euler ne perdait pas de vue les Math´ematiques, mˆeme lorsqu’il  r´ecitait les vers de Virgile ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' tout ´etait propre `a lui rappeler cet objet presque  unique de ses pens´ees, et on trouve dans ses ouvrages un savant M´emoire sur  une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait  donn´e la premi`ere id´ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='17 [The study of ancient literature and scholarly lan-  guages had been part of his education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' he retained a taste for it all his life, and  15Euler (1744), Chapter IV, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  135, Exemplum II: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Super axe AP construere lineam  BM, ita comparatant, ut, abscissa area ABMP datæ magnitudinis, arcus curvæ BM illi areæ  respondens sit omnium minimus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  16Condorcet, Jean-Antoine-Nicolas de Caritat marquis de (1743-1794).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  17 ´Eloge d’Euler Prononc´e `a l’Acad´emie, par de Condorcet, Histoire de l’Acad´emie royale  des sciences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 1783, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  6  Bforgot nothing he had learned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' but he never had either the time or the desire to  add to his first studies: he had not read the Modern Poets, and knew the Aeneid  by heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' However, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Euler did not lose sight of Mathematics, even when he  recited the verses of Virgil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' everything was likely to remind him of this almost  unique object of his thoughts, and we find in his works a scholarly Memoir on a  question of Mechanics, of which he said that a verse from the Aeneid had given  him the first idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=']  Euler did take verses from the Aeneid poem to use as mottos for his com-  peting memoirs submitted to the French Academy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='18 These are summarized in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Was this to what Condorcet referred to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Euler’s Memoirs and Mottos taken from Virgil’s Aeneid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Year  Memoir Title  Motto  1753  “On the movement of ships  Tali remigio navis se  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 413)  without the wind’s force.”  tarda movebat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  7th winning memoir  Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Aeneid Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 5  1759  “Concernin pitching  Insequitur clamorque virum  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 415)  and rolling.”  stridorque rudentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  9th winning memoir  Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Aeneid, Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 1  1770  “Moon Theory”  Errantem que canit Lunam  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 485)  Prize for 1770  Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Aeneid Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 1  10th winning memoir  1772  “Improved Moon theory”  Hic labor extremus, longarum  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 486)  Prize for 1772  haec meta viarum hinc jam  11th winning memoir  digressi, vestris appellimus oris  Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Aeneid, Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 3  However, the mottos were carefully selected by Euler to match the research  topic of the competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='19 In addition to using Virgil’s verses, he also quoted  from other ancient writers such as Marcus Tullius Cicero, Properci, and he  composed other adages, asking Christian Goldbach for suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Ultimately,  Condorcet’s statement “et on trouve dans ses ouvrages un savant M´emoire sur  une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait  donn´e la premi`ere id´ee” does not mean that Euler was inspired by Virgil to  define Dido’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  As a historian, I cannot rely on obituaries to extract factual data, even if  written by an eminent scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The much younger Condorcet never met Euler,  and the ´Eloge he wrote, as most eulogies are, was based on hearsay, relaying on  what the French academicians might have recalled about Euler’s life and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  18Submissions were anonymously and the memoir identified by a motto;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' the author’s name  enclosed in a sealed envelope was opened only for the winning memoir after the judging of  the contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  19For the significance of the mottos that Euler selected, see Musielak (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  7  A contemporary biography (published in 2016) further implies that Euler  solved Dido’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The author refers to a copy of an eight-page manuscript  (preserved in Moscow) that is said to contain Euler’s answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Is this the  manuscript that categorically would give Euler credit for connecting Dido’s story  to variational calculus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Unfortunately, the manuscript in question is said to be  “not in Euler’s own handwriting.” Thus, it diminishes its credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' It is rather  improbable that Euler, a prolific writer, would be the author of a manuscript  inscribed by someone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Besides, he would have included this solution in a  paper published in 1764, where Euler summarized the results of the Calculus of  Variations in terms of the variational operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Joseph-Louis Lagrange (1736-1813) expanded the variational calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In his  second letter to Euler dated August 1755, Lagrange outlined his delta-algorithm  (for solving constrained optimization problems), an approach Euler embraced,  prompting him to conceive the term calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In the abstract of  a memoir published in 1764, Euler credits Lagrange for enriching the science20  Their combined work led eventually to the Euler–Lagrange equations, which are  the equilibrium equations for minima of variational integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='21  Five years after Euler died, Lagrange published M´ecanique analytique, his  compendium on analytical mechanics, using variational ideas to present me-  chanics from a unified analytic viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' When teaching at the ´Ecole Poly-  technique in 1799, Lagrange published Le¸cons sur le calcul des fonctions and  explained the method of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Lagrange provided a brief overview of the  development of problems of maxima and minima, referring only to Greek math-  ematician Apollonius (262-190 BC), which dealt exclusively with the largest  and smallest straight lines which can be drawn from given points to the arcs of  conic sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='22 Dido’s problem is not mentioned here nor in Lagrange’s other  published works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='2  William Thomson, Lord Kelvin  The first instance in which Dido’s name appear in the context of interest is  found in a public lecture delivered by William Thomson in 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' A great physi-  cist known today as Lord Kelvin, his contributions include a major role in the  development of the second law of thermodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' the absolute temperature  scale (measured in kelvins);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' the dynamical theory of heat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' the mathematical  analysis of electricity and magnetism, including the basic ideas for the electro-  magnetic theory of light;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' and much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' He brought together disparate areas  of physics—heat, thermodynamics, mechanics, hydrodynamics, magnetism, and  20Euler (1764).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  ex quo Auctori occasio est oblata hanc scientiam novo Calculi genere  locupletandi, quem Calculum variationum appellat et cuis elementa hic tradere ac dilucide  explicare constituit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  21See Freguglia and Giaquinta (2016) for an excellent presentation of the Euler-Lagrange  equations, including a historical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  22Lagrange (1806).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Les questions de maximis et minimis n’ont pas ´et´e incounues aux  anciens g´eom`etres ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' car on a un livre entier d’Apollonius, qui traite presqu’uniquement des  plus grandes et des plus petites lignes droites qui peuvent ˆetre men´ees de points donn´es aux  arcs des sections coniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  8  Figure 2: Dido’s problem as described by Lord Kelvin in 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Lord Kelvin played a key role in the final synthesis of 19th-century  science, which viewed all physical change as energy-related phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='23  Lord Kelvin related Dido’s clever approach to bargaining for land as follows,  using the sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 2 to illustrate Dido’s problem:  “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  She cut the ox-hide into an exceedingly long strip, and succeeded in  enclosing between it and the sea a very valuable territory on which she built  Carthage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In Dido’s problem the greatest value of land was to be enclosed by a  line of given length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' If the land is all of equal value the general solution of the  problem shows that her line of ox-hide should be laid down in a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' It shows  also that if the sea is to be part of the boundary, starting, let us say, southward  from any given point, A, of the coast, the inland bounding line must at its far  end cut the coast line perpendicularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Here, then, to complete our solution, we  have a very curious and interesting, but not at all easy, geometrical question to  answer: What must be the radius of a circular arc, ADC, of given length, and  in what direction must it leave the point A, in order that it may cut a given  curve, ABC, perpendicular at some unknown point, C?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='24  Lord Kelvin added that having enough mathematics knowledge, Dido would  determine that the boundary had to be a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Of course, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  2, she would have given the boundary a different curvature in different parts to  gain as much as possible of the more valuable parts of the land offered to her,  “even though difference of curvature in different parts would cause the total  area enclosed to be less than it would be with a circular boundary of the same  length.”25  23Gray (1910)  24Thomson (1894), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 572-574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  25Ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  9  CARTHAGToday, taught as introduction to calculus of variation, the solution of Dido’s  problem requires an extremization solution under constraint, that is, we max-  imize the area, A =  �  ydx, subject to the condition that the arc, L =  �  ds is  of a given length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In other words, we wish to maximize the integral A sub-  ject to the condition that another integral L has a given constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Note  this is an optimization problem with constraints where we use Lagrange’s strat-  egy for finding the local maxima and minima of a function subject to equality  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  It is clear that, without an original reliable source, I cannot conclude that  Euler defined Dido’s Problem for the first time, inspired by Virgil’s Aeneid,  as Condorcet implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The evidence points to Lord Kelvin who described the  problem in 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' And as he stated, whether severe critics will call Dido’s story  mythical or allow it to be historic, it is nevertheless full of scientific interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  As for me, Dido’s Problem is an excellent example to introduce students to  the calculus of variations, as it expresses a perfectly definite case of isoperimet-  rics, illustrating the fundamental principles introduced by Euler and Lagrange  in the eighteenth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  5  Dido and Ancient Mathematics  Nothing is known about Dido’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Being a Phoenician princess, it is  highly probable that she was well educated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' What we glimpse from Virgil’s  and Justinus’s tales is that Dido was a formidable woman, smart, ambitious, a  foreign leader that left the city of Tyre (on the coastline of modern Lebanon)  with her faithful followers, navigated the waters of the Mediterranean Sea and  landed in the coast of North Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' There, she established and ruled Carthage  (modern day Tunis), an important port city that rose to the height of its power  in the second century BC, before Rome became supreme and took over that  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  For the ancient cultures that flourished around the Mediterranean, geom-  etry was fundamental to their development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The Babylonians thriving in the  Mesopotamian River Valley engaged in commerce through the Mediterranean,  and this required considerable mathematical skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Clay tables preserve records  of what they knew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  For instance, clay tables from Babylon, located in the  southern part of Mesopotamia, about fifty miles south of present-day Baghdad  (Iraq) suggest that the Babylonian had an advanced knowledge of geometry and  arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  In fact, some scholars believe that the Babylonians knew the Pythagorean  theorem a thousand years before Pythagoras of Samos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' At Susa, an ancient  city over two hundred miles from Babylon, a set of tablets were discovered in  1936, which contain the ratios of areas and perimeters of regular polygons to  their respective side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The best known surviving tablet (estimated to be  from between 1900 and 1600 BC) contains a list of Pythagorean triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' This  suggests that the Babylonians had knowledge of the Pythagorean theorem, as  well as certain algebraic identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  10  Moreover, that women knew mathematics in ancient times has been exten-  sively documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  For example, the Pythagorean society included women,  some of which became famous such as mathematician Theano, who was mar-  ried to Pythagoras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In the dedication of his Introduction to Harmonics, ancient  mathematician and music theorist Nicomachus of Gerasa (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 60-120 AD) ad-  dresses the lessons to a lady, one of his students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='26 In this book, also known as  Manual of Harmonics, Nicomachus dealt with the theory of music, a version of  Pythagorean harmonics, in which he assigned number and numerical ratios to  notes and intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' And of course, we know about Hypatia of Alexandria (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  370-415 AD) considered the first woman scholar to attain eminence as mathe-  matician and astronomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='27  I believe that Dido was educated in mathematics, and so she used the the-  orem of isoperimetry to outsmart the king who sold her the piece of land in  the northern tip of Africa (today’s Tunisia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Therefore, Dido’s Problem should  be viewed not only to illustrate a fundamental problem of variational calculus  but also as a lesson in the history of mathematics and the role ancient women  played in its development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  References  Brunt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' van (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Published by Springer  New York ISBN: 978-0-387-40247-5 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='1007/b97436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Coppersmith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (2017) The Lazy Universe: An Introduction to the Principle  of Least Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Oxford University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1732).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' De linea brevissima in superficie quacunque duo quaelibet  puncta iungente (On the shortest line joining two points on a surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Com-  mentarii academiae scientiarum Petropolitanae, Volume 3, 1732, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 110-124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 9)  Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1738).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Problematis isoperimetrici in latissimo sensu accepti so-  lutio generalis (On isoperimetric problems in the widest sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Commenturii  Academiae Scientiarum Petropolitanae 6 (1732/3), 123-155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Opera Omnia, 125,  13-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 27)  Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1741).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Curvarum maximi minimive proprietate gaudentium inven-  tio nova et facilis (New and easy method of finding curves enjoying a maximal  or minimal property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In Commentarii Academiae Scientiarum Petropolitanae  8 (1736).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 159-190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Reprinted in Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Opera Omnia, I 25, 54-80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 56)  Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1744).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Methodus inveniendi curvas h’neas maximi minimive pro-  prietate gaudentes sive solution problematis isoperimetrici latissimo sensu ac-  cepti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Lausanne, Genf: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Bousquet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Reprinted in Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Opera Omnia,  I 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' According to Enestr¨om, Euler completed the manuscript of this  work by April 1743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1764).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Elementa calculi variationum (Elements of Calculus of  Variations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Novi commentarii academiae scientiarum Petropolitanea 10 (1764),  26Biographical Note of Nicomachus, in Great Books of the Western World, Robert Maynard  Hutchins, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  27Musielak (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 206-207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  11  1766, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 51-93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' This research (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' 296) was presented at the Berlin Academy  in 1756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Freguglia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' and Giaquinta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The Early Period of the Calculus  of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Published by Birkh¨auser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Gelfand, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' and Fomin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Revised  English Edition Translated and Edited by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Silverman Prentice-Hall, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Englewood Cliffs, NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Goldstine, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' A History of the Calculus of Variations from the 17th  through the 19th Century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1910).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  The Life of William Thomson, Baron Kelvin of Large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Nature 83, 61–65 (1910).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Lagrange (1806).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Le¸cons sur le calcul des fonctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Nouvelle ´edition, revue,  corrig´ee et augment´ee par l’auteur [J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Lagrange].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Initially published as lecture  notes in 1799 when Lagrange was teaching at the Ecole Polytechnique and  reprinted in 1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' In 1806, Lagrange published a new edition containing two  new lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Menger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' What is Calculus of Variations and What Are Its Ap-  plications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The Scientific Monthly 45 (3) (1937), 250-253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Musielak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Leonhard Euler and the Foundations of Celestial Me-  chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Springer History of Physics Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Springer Nature Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' ISBN  978-3-031-12321-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Musielak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Sophie Germain: Revolutionary Mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Springer  Biographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Springer Nature Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' ISBN 978-3030383770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Nahin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' When Least is Best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Princeton University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Rojo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' and Bloch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' The Principle of Least Action: History and  Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='1017/9781139021029  Thomson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' (1894).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Popular Lectures and Addresses by Sir William Thom-  son (Baron Kelvin) in Three Volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Nature Series, MacMillan and Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' Lon-  don 1894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content='  Dora Musielak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}
+page_content=' University of Texas at Arlington, 6 January 2023  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'}

LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8a77b5f95172694e5085a5f80a1fca30c5e0be9e260fe63f319e13db64b00838
+size 612596

LNFAT4oBgHgl3EQfwR78/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f049bf62886ce9dbf02f8aee3c977064d5729f134285580e139d5e7f394e81c0
+size 441504

LNFQT4oBgHgl3EQfUDZM/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e6970f2b8ee7f00a5a57cea6641bc6f87adc9d10c8ae4dd97b66f5765c9f27aa
+size 455696

MtFOT4oBgHgl3EQf1jQj/content/tmp_files/2301.12939v1.pdf.txt

@@ -0,0 +1,666 @@
+Data-driven soiling detection in PV modules
+Alexandros Kalimeris1, Ioannis Psarros1, Giorgos Giannopoulos1, Manolis Terrovitis1,
+George Papastefanatos1, and Gregory Kotsis2
+1Athena RC
+2INACCESS Networks
+January 31, 2023
+Abstract
+Soiling is the accumulation of dirt in solar panels which leads to a decreasing trend in solar energy
+yield and may be the cause of vast revenue losses. The effect of soiling can be reduced by washing the
+panels, which is, however, a procedure of non-negligible cost. Moreover, soiling monitoring systems are
+often unreliable or very costly. We study the problem of estimating the soiling ratio in photo-voltaic (PV)
+modules, i.e., the ratio of the real power output to the power output that would be produced if solar panels
+were clean. A key advantage of our algorithms is that they estimate soiling, without needing to train on
+labelled data, i.e., periods of explicitly monitoring the soiling in each park, and without relying on generic
+analytical formulas which do not take into account the peculiarities of each installation. We consider
+as input a time series comprising a minimum set of measurements, that are available to most PV park
+operators. Our experimental evaluation shows that we significantly outperform current state-of-the-art
+methods for estimating soiling ratio.
+Keywords: Solar energy, solar panels, soiling, performance loss, time series analysis
+1
+Introduction
+Soiling is the accumulation of dirt on the surfaces of photo-voltaic (PV) modules, which leads to a loss in
+the power output. Soiling is typically caused by airborne particles, including for example dust, pollen and
+soot. Depending on the location, soiling may also be caused by heavier material such as ice, bird droppings,
+or falling leaves.
+One standard way to quantify soiling is by the soiling ratio SR [iec21], which is defined as the ratio of the
+real power output to the power output that would be produced if solar panels were clean. Soiling loss is then
+defined as 1 − SR, and soiling rate is defined as the (daily) rate of change of the soiling loss. Other metrics
+have been also proposed, e.g., the insolation-weighted soiling ratio [DMM18], aiming to better capture the
+loss induced by soiling.
+To reduce the effect of soiling, PV modules must be cleaned on strategically chosen dates to reduce
+the cost induced by energy loss while taking into account cleaning costs. Detection of time periods during
+which soiling severely affects power output is therefore significant for the efficient scheduling of cleanings.
+What makes the problem challenging is the shortage of labelled data which is caused by the fact that
+soiling monitoring systems are often considered unreliable or costly. For example, soiling stations which
+are the most common commercially available soiling monitoring solution [BMAF21], still require regular
+cleanings and maintenance, which can be expensive, especially in remote locations, and imperfect cleanings
+can result in significant measurement uncertainty [MMMM17]. Therefore, soiling periods must be deduced
+from measurements of a number of reliable variables, e.g., power output, irradiance, temperature.
+Existing methods that detect soiling follow two alternative strategies: a) they train a model on labelled
+data, i.e., data where the soiling of the panels has been logged using specialized sensors and cleaning events
+1
+arXiv:2301.12939v1  [eess.SP]  30 Jan 2023
+
+have been explicitly recorded (e.g., [MMD11, MMDK13]) and b) by using an analytical formula for optimal
+energy output based on environmental readings (e.g., [KMNW06, DMM18, MTL+21]). The former strategy
+is more accurate but requires significant resources to produce the labelled data, which must be produced for
+each different installation. The latter strategy does not take into account the peculiarities of each installation
+and leads to less accurate results (as we demonstrate in Section 4).
+The main advantage of our method,
+is that it is purely data-driven, in the sense that it does not require a generic analytical formula for the
+relation between power output and the commonly used environmental readings, but it learns this relation
+in a self-supervised manner (without the need for labelled data). This way we achieve better results than
+methods that rely on analytical formulas without the cost of methods that need explicitly labelled data.
+We consider as input the monitoring data from the park operation, i.e., a time series with measurements of
+power output, irradiance, and module temperature for a certain array or string of PV modules, precipitation,
+and dates on which the solar panels were manually cleaned for maintenance (if such information exists). The
+soiling ratio over a sequence of timestamps t1, . . . , tn is defined as SR = Pt1
+P ∗
+t1 , . . . , Ptn
+P ∗
+tn , where each Pti is
+the actual power output corresponding to timestamp ti, and P ∗
+ti is the expected power output assuming
+that the solar panels are clean, corresponding to the same timestamp. Our framework trains a regression
+model M which accurately predicts P ∗
+t1, . . . , P ∗
+tn (which are not given as input). This yields an estimate
+for the soiling ratio as SRM = Pt1
+˜
+Pt1 , . . . , Ptn
+˜
+Ptn , where each ˜Pti is the value predicted by M for timestamp
+ti. We aim for M such that SRM ≈ SR. Raining periods (extracted from precipitation measurements),
+and manual cleanings, are used in the “learning” phase of our proposed model. One of our methods can
+run exclusively on rain information, in case manual cleanings are not performed or logged. Our approach is
+robust to misinformation about manual cleanings because it checks each potential cleaning to determine its
+effect on power output. Manual cleanings that are not logged, have a negligible effect; they can only affect
+the quality of the training set positively.
+The main advantages of our method are that they do not require measurements of soiling from specialized
+equipment which can be costly or inaccurate, they do not rely on the accuracy of an analytical formula for
+the optimal energy output of the park, and they agnostic to the type of PV modules employed.
+As a
+purely data-driven approach, it solely depends on the availability of data, and in particular a minimal set
+of generally available variables. Our approach is robust to misinformation about manual cleanings because
+it checks each potential cleaning to determine its effect on power output. Moreover, manual cleanings that
+are not logged, have a negligible effect on our approach; their existence can only affect the quality of the
+training set positively.
+In Section 2, we discuss related work, in Section 3.1 we provide necessary background, in Section 3.2 we
+present a detailed description of our methods, and in Section 4 we present our experimental findings.
+2
+Related work
+PVUSA introduced a method for rating PV systems based on a simple regression model [DG95] which
+employs the simplified assumption that array current depends only on irradiance and that array voltage
+depends only on module temperature. Massi Pavan et al. [MMD11] compare the standard test conditions
+(STC) (irradiance: 1000W/m2, module temperature: 25◦C) performance of a PV park before and after its
+cleaning. In order to determine the performance at STC conditions they use a regression model, suggested
+in [MWPP08], that accepts as input the two main climate features, i.e. the in-plane global irradiance and
+the photo voltaic module temperature. However, their work requires as input labelled data, i.e. time series
+extracted from both clean and soiled PV modules. Massi Pavan et al. [MMDK13] developed four Bayesian
+Neural Network (BNN) models with the aim to calculate the STC performance of two plants before and
+after a complete clean-up of their modules. The idea is that differences between the STC power before and
+after the clean-up represent the losses due to the soiling effect. However, their work also requires as input
+labelled data, i.e. time series extracted from both clean and soiled PV modules.
+Closer to our work are methods which estimate soiling losses based on PV system data. The Fixed Rate
+Precipitation (FRP) method [KMNW06] calculates the daily soiling loss. The method requires as input:
+2
+
+the slope of the performance metric/index during the longest dry period, a cleaning threshold for rains,
+i.e., the minimum amount of daily precipitation required to have a cleaning effect on PV modules, and a
+number of days after a raining period for which no soiling occurs. The method implicitly assumes that the
+soiling rate remains the same throughout time. This requirement can be very restrictive, because of the
+different types of soiling that may occur, depending also on the location or the season. For the same reason,
+it is unrealistic to assume that there is a certain minimum value classifying rains as effective. More recently,
+Deceglie, Micheli, and Muller [DMM18] developed a new method for quantifying soiling loss, which compares
+favourably to FRP. The new method is termed the stochastic rate and recovery (SRR) method. It uses an
+analytical formula, calculated over values for irradiance and module temperature, to compute the expected
+power output, which is then used to compute a performance metric. The method first detects soiling intervals
+in a dataset, and then, based on the observed characteristics of each interval, estimates the total loss. Notice
+that SRR provides an aggregate estimate of soiling loss, calculated for the whole input period, while our
+focus lies on determining soiling loss even on shorter periods of time. Skomedal and Deceglie [SD20] proposed
+the combined degradation and soiling method for further analyzing a performance metric signal. Finally,
+Micheli et al. [MTL+21] consider non-linear degradation in soiling intervals, and they apply various methods
+for changepoints detection to obtain a refined soiling profile. All methods studied there are based on finding
+changepoints on the performance metric curve, as calculated by SRR. On the contrary, our approach detects
+changepoints as an intermediate step towards computing a performance metric. It is apparent from recent
+work that improvements on estimating the expected power output directly translate to improvements on
+various tasks in PV data analysis.
+3
+Methodology
+3.1
+Preliminaries
+3.1.1
+Basic assumptions and definitions
+Our input consists of a multi-variate time series containing measurements for: i) power output, ii) irradiance,
+iii) module temperature, iv) precipitation. Our methods can be further enhanced if we are also given as
+input the dates on which the PV modules were manually cleaned.
+Let R be the set of all rains, defined as follows: [t, t′] ∈ R if and only if there is a rain starting at t
+and ending at t′. Rains are extracted from input as maximal time intervals containing positive precipitation
+values. Similarly, if manual cleanings are provided let C be the set of all such intervals, defined as follows:
+[t, t′] ∈ C if and only if we know that the PV modules were being cleaned between timestamps t and t′. We
+denote by Wp the set of all potential cleaning events, defined as Wp = C ∪ R. We assume that precipitation
+measurements are sufficiently frequent, so that we can accurately detect rains.
+3.1.2
+Regression models
+A basic component of our methods is regression. We fit regression models to represent power output during
+“dirty” or “clean” periods and we use prediction errors to detect performance changes. We consider as feature
+variables the irradiance and the module temperature, and the target outcome corresponds to the power
+output. We apply Ridge Regression with polynomial features, which is parameterized by the degree of the
+regression polynomial, and a regularization strength parameter for the linear least squares function (the loss
+function) where regularization is given by the ℓ2-norm. The parameters were selected during the initial stages
+of the algorithm development process, where we experimented with cross-validation and hyper-parameter
+tuning techniques. The exact values used in our experiments are discussed in Section 4. Our model selection
+was a consequence of preliminary experiments with various (simple) regression models such as Ordinary Least
+Squares, Support Vector Regression, etc., that we executed in a CPU with maximum processor frequency
+at 3.7GHz, and available RAM at 256Gb. In the experiment that we conducted, we randomly choose 100
+time intervals of maximum duration of one month from the time series provided in [MAD+14], which are
+also discussed in Section 4, and we randomly split them into training and testing subsets containing 80%
+3
+
+and 20% of the points respectively. Our choice satisfies a bifold objective: i) good accuracy and ii) fast
+fitting time. The latter is vital in our method which fits one model for each potential cleaning. Table 3
+contains MAPE values and fitting times for four different models. Polynomial features and the polynomial
+kernel used in Support Vector Regression (SVR) are of degree 3. The highest accuracy is achieved by SVR
+with linear kernel and polynomial features, being roughly 0, 4% better than Ridge Regression which is the
+second best. However, the fitting time of SVR is at least one order of magnitude higher than that of Ridge
+Regression.
+Ridge Regression is a simple model that adds only one extra tunable parameter to our learning
+pipeline, and the regularization it provides acts as a measure to prevent overfitting. We also emphasize the
+fact that one can easily plug-in any regression model in our approach.
+Table 1: Evaluation of regression models.
+Model
+MAPE
+Fitting time (s)
+Linear Regression
+with polynomial features
+0.0812
+0.0015
+Ridge Regression
+with polynomial features
+0.0807
+0.0012
+Support Vector Regression
+with polynomial kernel
+1.0648
+0.0177
+Support Vector Regression
+with linear kernel and polynomial features
+0.0770
+0.0666
+Several steps in our approach rely on computing measures for the prediction accuracy of our model. Let
+Y = Yt1, . . . , Ytn, ˜Y = ˜Yt′
+1, . . . , ˜Yt′n be two univariate time series, and let T = {t1, . . . , tn}, T ′ = {t′
+1, . . . , t′
+n}.
+We use a variant of the mean absolute percentage error (MAPE) which is defined over time intervals as
+follows: for any [t, t′] ⊆ T ∩ T ′,
+mape0(Y, ˜Y, [t, t′]) = mean({|Yj − ˜Yj| | j ∈ [t, t′]}
+mean({|Yj| | j ∈ [t, t′]})
+.
+Note that mape0 is robust to zero true values (as long as not all of them are zeroes) since it uses as
+denominator the mean of the values, as opposed to standard MAPE where all actual values appear as
+denominators leading to singularities even if there is only one zero true value. When Y and ˜Y are clear from
+the context, we omit them from our notation and we simply write mape0([t, t′]). We also use the median
+multiplicative error defined as mede(Y, ˜Y) = median
+��
+Yi
+˜Yj | i ∈ T, j ∈ T ′��
+.
+3.2
+Soiling detection
+In this section, we formally describe our methods, which are composed of two main steps. The first step is
+that of detecting cleaning events. Then, using these cleaning events we define training periods for regression
+models aiming to capture the optimal performance of the PV modules.
+In all our methods, we fit regression
+models which capture the dependence of power output on the values of irradiance and module temperature,
+i.e., power output is the dependent variable, while irradiance and module temperature are the feature vari-
+ables. Measurements are scaled to [0, 1] by subtracting the minimum value and dividing by the range of
+values. Figure 1 summarizes the main steps of our methods.
+3.2.1
+Baseline soiling estimator
+We first present our baseline approach for estimating the soiling ratio. Our baseline algorithm is based on
+the following assumption: manual cleanings alone define points in time where the PV modules are clean.
+While these points are not sufficiently many to define a training set, we can extend them to short intervals of
+a user-defined length wtrain. This is the amount of time during which we can safely assume that the panels
+remain clean.
+4
+
+PV data
+precipitation data
+(manual cleaning dates)
+potential cleanings
+cleaning event detection
+fit regression
+model before
+potential
+cleaning
+compare prediction
+errors before/after
+potential cleaning
+cleaning events
+fit regression model
+on periods following
+the cleaning events
+predict optimal per-
+formance and esti-
+mate soiling ratio
+Forward Checking Soiling Estimator (FCSE)
+PV data
+precipitation data
+manual cleaning dates
+potential cleanings
+cleaning event detection
+fit one regression
+model on periods
+following manual
+cleanings
+compare prediction
+errors before/after
+potential cleaning
+cleaning events fit regression model
+on periods following
+the cleaning events
+predict optimal per-
+formance and esti-
+mate soiling ratio
+Backward Checking Soiling Estimator (BCSE)
+Baseline estimator
+Figure 1: Basic steps of our methods. Manual cleanings are optional for FCSE. To detect cleaning events,
+FCSE fits one regression model before each potential cleaning event, while BCSE fits one regression model
+using manual cleaning dates and uses it in classifying all cleaning events.
+We fit a regression model that aims to capture the power output when PV modules are clean. To this
+purpose, we fit a regression model M on the set of input points with timestamps from �
+[t,t′]∈C[t′, t′ +wtrain].
+We define SRM = Pt1
+˜
+Pt1 , . . . , Ptn
+˜
+Ptn as the modelled soiling ratio where each Pti is an input power output value,
+and ˜Pti is the value predicted M.
+3.2.2
+Forward checking soiling estimator (FCSE)
+Our first method examines each potential cleaning event independently and assigns scores which represent
+the significance of the detected change of behavior. Five input parameters are required: the length of the
+training period w1, the length of the validation period w2, the length of the test period w3, a parameter
+q defining the quantile of the scores which classifies events as cleanings, and the length wtrain defining the
+training set for the final regression model used to estimate soiling. For each interval [t, t′] ∈ Wp, we fit a
+regression model in the time interval [t − w1 − w2, t − w2), we validate it in the time interval [t − w2, t) and
+we test it in the time interval (t′, t′ + w3]. We compute the function mape0 on the validation interval and
+if the returned value is greater than 5% then we consider this event invalid and we discard it from further
+consideration. This threshold aims to discard events that we are unable to classify with certainty.
+The
+reasons behind choosing 5% as our threshold are the following. First, due to the nature of our task, the
+regression model is required to make very accurate predictions and detect power deviations at a very small
+scale. This requires high accuracy of our regression models; therefore a tight threshold. On the other hand,
+this threshold must be pragmatic: having an extremely small value as a threshold will lead to unrealistic
+outputs where no cleaning events are detected and, consequently, no soiling estimation can be derived.
+We
+experimentally validate our choice of 5% in Section 4.2.2.
+The intuition is that if the PV modules under-perform due to soiling, for a time period preceding t, then
+the regression model captures this under-performing behaviour and if [t, t′] is a cleaning event then the model
+should underestimate the power output in (t′, t′ + w3]. To compute the score of the potential cleaning event
+[t, t′], we first compute PIval as the sequence of actual power output values divided by the predicted power
+output values for the time interval [t − w2, t), and PItest as the sequence of actual power output values
+divided by the predicted power output values for the time interval (t′, t′ + w3]. Then, the score assigned
+to [t, t′] is mede(PIval, PItest). We define as cleaning events all intervals [t, t′] ∈ Wp with score above the
+qth-quantile of all scores. Let W1 be the set of detected cleaning events. We fit a regression model M on
+the input points with timestamps from �
+[t,t′]∈W1[t′, t′ + wtrain]. The intuition is that cleaning events define
+points in time where the PV modules are clean. Obviously, these points are not sufficiently many to define
+a proper training set. By extending these points to (short) intervals, of length wtrain, we increase the size of
+5
+
+the training set without (significantly) affecting its quality. We define SRM =
+Pt1
+˜
+Pt1 , . . . , Ptn
+˜
+Ptn as the estimated
+soiling ratio where each Pti is an input power output value, and ˜Pti is the value predicted by the regression
+model M.
+Notice that FCSE does not require having the cleaning dates C as input, and we could simply have
+Wp = R.
+3.2.3
+Backward checking soiling estimator (BCSE)
+Our second method builds upon the baseline approach. This method requires five input parameters w1,
+w2, w3, q, wtrain. Parameters w1 and w2 denote the length of the testing period preceding the potential
+cleaning event and the length of the validation period following the potential cleaning event respectively.
+Parameter w3 denotes the length of the time period following each [t, t′] ∈ C such that the modules remain
+clean. Parameter q defines the quantile of the scores which classifies events as cleanings. Parameter wtrain
+is used to define the training set of the final regression model for estimating the soiling ratio. We train
+one regression model on the set of points defined by timestamps in �
+[t,t′]∈C[t′, t′ + w3]. This model aims
+to capture modules’ “clean” performance. For each [t, t′] ∈ Wp, we use our model to make predictions on
+[t − w1, t) and (t′, t′ + w2]. If mape0((t′, t′ + w2]) is greater than 5% then we consider this interval invalid
+and we discard if from further consideration. As in FCSE, this filtering step is to avoid considering events
+that our models fail to classify with a good amount of certainty.
+The intuition is that if [t, t′] is a cleaning event, then the PV modules’ performance during [t′, t′ + w2]
+must resemble the “clean” performance as predicted by our regression model.
+Similarly, if the modules
+under-perform during [t − w1, t), then the induced ratio of the actual power output over the predicted power
+output must be significantly smaller than 1. To compute the score of the potential cleaning event [t, t′], we
+first compute PIbefore as the sequence of actual power output values divided by the predicted power output
+values for the time interval [t − w1, t), and PIafter as the sequence of actual power output values divided
+by the predicted power output values for the time interval (t′, t′ + w2]. Then, the score assigned to [t, t′]
+is mede(PIbefore, PIafter). We define as our threshold parameter thrsh the qth-quantile of all scores. We
+define as cleaning events all intervals [t, t′] ∈ Wp with score above the qth-quantile of all scores. Let W2,
+be the set of detected cleaning events. We fit a regression model M on the input points with timestamps
+from �
+[t,t′]∈W2[t′, t′ + wtrain]. As in FCSE, the intuition is that cleaning events define points in time where
+the PV modules are clean. Obviously, these points are not sufficiently many to define a training set. By
+extending these points to (short) intervals, of length wtrain, we increase the size of the training set without
+(significantly) affecting its quality. We define SRM = Pt1
+˜
+Pt1 , . . . , Ptn
+˜
+Ptn as the estimated soiling ratio where each
+Pti is an input power output value, and ˜Pti is the value predicted by M.
+4
+Experiments
+4.1
+Datasets
+State-of-the-art dataset
+To evaluate our methods, we use a dataset provided in [MAD+14], which
+contains a set of current-voltage (I-V) curves and associated meteorological data for PV modules representing
+all flat-plate PV technologies and for three different locations and climates for approximately one-year
+periods.
+For each location, we are given values for a normalized metric, called soiling derate which is
+computed using measurements for short-circuit current and irradiance from two identical PV modules; one
+that is cleaned during daily maintenance, and one that is not.
+Soiling derate is the result of dividing
+daily values of ampere-hours per kilowatt-hours per square meter Plane of Array (POA) irradiance for the
+not-cleaned PV module, by the corresponding values of the cleaned PV module [MAD+14]. The soiling
+derate aims to provide a performance index analogous to soiling ratio, estimated on real measurements. We
+emphasize that soiling derate is only used for the evaluation of our methods and are not utilized as input (nor
+in SRR). The time granularity is 5 minutes, and measurements are provided for all hours of daylight. The
+6
+
+three locations are Cocoa, Florida, USA; Eugene, Oregon, USA; and Golden, Colorado, USA. PV modules
+in Cocoa and Eugene were cleaned when this was necessary in order to ensure that levels of soiling loss were
+maintained at a reasonable level. PV modules in Golden were not cleaned because frequent rains helped
+maintaining a reasonable level of soiling loss. Cocoa has a minimum soiling derate of 0.985, Eugene has a
+minimum soiling derate of 0.964, and Golden has a minimum soiling derate of 0.977.
+In our methods, we use measurements for the maximum power of the PV module in watts, the amount of
+solar irradiance in watts per square meter received on the PV module surface, the PV module back-surface
+temperature and the accumulated daily total precipitation. The dataset also provides dates on which all PV
+modules were cleaned. We apply our methods on PV modules that were used in estimating the soiling derate,
+and in particular on those that were not cleaned every day. As discussed in Section 3.1.2 our methods utilize
+Ridge Regression models. For those models, we use polynomial features of the 3rd degree and a regularization
+strength parameter alpha = 10−4 during the fitting stages.
+Real-world dataset
+We also consider a real-world scenario, where no ground truth is available.
+We
+test our methods on a dataset from a very different location and of different climate conditions, comprising
+measurements from a solar park located in Greece. We are given values for power output, irradiance, module
+temperature and precipitation on a time granularity of 15 min for a period of approximately 7 years, and 15
+dates of manual cleanings.
+4.2
+Method evaluation and discussion
+4.2.1
+Soiling estimation
+We evaluate our methods, by comparing them to the analogous model used in SRR. To show robustness of our
+methods in different parameter settings, we try various lengths for the periods used in changepoint detection.
+Table 2 lists the respecting values (in days) for parameters w1, w2, w3 in FCSE and w1, w2 in BCSE. The
+rest of the parameters are set as follows: we apply FCSE with parameters q = 0.9, and wtrain = 30 days and
+BCSE with parameters q = 0.9, w3 = 30 days, and wtrain = 30 days. The baseline soiling estimator is applied
+with wtrain = 30 days. Since our methods are unsupervised, classic automated methods fail to optimize the
+above parameters. Essentially, domain expertise is the main lead for selecting parameters appropriately, also
+depending on the properties of each location that affect the rate at which soiling progresses. However, as
+Table 2 indicates, the methods are robust within a range of reasonable values for the parameters. The fixed
+parameters wtrain (and w3 in BCSE) define time periods during which a clean solar panel is likely to remain
+clean. While smaller values for wtrain (resp. w3) seem to provide safer conclusions, larger values provide a
+bigger size and diversity of the induced training set. The parameter q defines a threshold on how important
+a changepoint should be to be considered as a cleaning event. Setting q = 0.9 implies that the top-scored
+10% of potential cleanings will be considered as cleaning events. Factors that must be taken into account
+when setting this parameter include the total number of potential changepoints, parameters w3, wtrain, and
+the size of the dataset. While larger values of q tend to lead to safer conclusions about cleaning events, this
+may lead to a decreased size of the training set, negatively affecting the final regression model.
+We juxtapose our estimated soiling ratio with the ground-truth soiling derate and the performance metric
+used in SRR. We have three different ways of estimating the soiling ratio: our baseline approach, FCSE and
+BCSE, which are
+described in Section 3.2. In our estimates, we map negative values and values greater
+than one to zero and one, respectively. Then, we apply a rolling median with windows of one day.
+For computing the performance metric as in SRR, we rely again on the publicly available RdTools
+package [DNS+22]. We use as input aggregate daily values calculated on measurements taken between 12:00
+and 14:00, with irradiance greater than 500W/m2. We first compute the performance metric as the ratio of
+realized to modelled PV energy yield, where modelled PV energy yield is derived from a standard formula
+which is implemented in pvlib package [HHM22]. Then, we perform a few processing steps as suggested in
+RdTools’ tutorials1. We first normalize the time series with the expected power, we then apply default filters
+to remove clipping effects and outliers, and finally, we resample to one-day values.
+1https://rdtools.readthedocs.io/en/stable/examples/degradation_and_soiling_example_pvdaq_4.html
+7
+
+Figure 2: Soiling ratio predicted by our models, and the performance metric used in SRR, for the Eugene
+dataset. FCSE with parameters w1 = 10, w2 = 5, w3 = 10 and BCSE with parameters w1 = 5, w2 = 10.
+Let SD be the soiling derate time series. We denote by PM the performance metric used in SRR. In
+Figure 2, we plot our estimated soiling ratio, for all three models discussed in Section 3.2, the soiling derate
+and the performance metric used in SRR, for the site of Eugene. Compared to the other datasets, Eugene
+has periods of declining performance which are more apparent. PV modules at the Eugene site were cleaned
+on March 11, July 10, August 14, August 21, and August 26. No significant precipitation is observed during
+July and August, which leads to a rapid drop in the performance.
+We also calculate the root-mean-square error (RMSE) comparing the soiling derate with each modelled
+ratio, for all three sites. Since no manual cleanings were performed in Golden, the baseline algorithm and
+BCSE cannot be executed. We list these results in Table 2. It becomes evident, both from the RMSE values
+and from the visual inspection of the figure, that a better estimation of the soiling ratio can be derived by
+our models, compared to the model based on an analytical formula which is employed by SRR, in a setting
+where a soiling tendency needs to be detected, nearly real-time, on newly incoming data. Further, BCSE
+compares favourably to FCSE, and improves upon the baseline algorithm in the Eugene dataset. On the other
+hand, both the baseline algorithm and BCSE cannot be executed in the Golden dataset, due to the lack of
+manual cleanings. FCSE and BCSE present slightly diverse behaviors, rendering each potentially preferable
+in diverse real-world settings, depending on the exact objective of a solar park operator. Specifically, BCSE
+provides the most accurate method in approximating soiling ratio, thus preferable when small to medium
+soiling events are tolerable by the operator, as long as “false alarms” are minimised. On the other hand,
+FCSE, while slightly missing in accuracy, it is more sensitive in the detection of smaller (potential) soiling
+events, making it ideal in cases when even small soiling events need to be handled. Finally, we can see that
+the formula used in SRR essentially predicts the majority of the considered period as soiling; a behavior
+with small practical use in a real-world deployment scenario.
+8
+
+Baseline
+1DO
+0.98
+0.96
+soiling derate
+estimated soiling ratio
+FCSE
+0.98
+0.96
+soiling derate
+estimated soiling ratio
+EC5E
+1DO
+0.98
+soiling derate
+0.96
+estimated soiling ratio
+LDO
+FeormanceMetc[SRR
+0.98
+0.96
+soiling derate
+Perf. metric (SRR)
+imestampTable 2: Evaluation.
+Model
+RMSE against SD
+Eugene
+Cocoa
+Golden
+Baseline
+0.006
+0.006
+-
+FCSE (w1 = 2, w2 = 1, w3 = 2)
+0.010
+0.006
+0.008
+FCSE (w1 = 10, w2 = 5, w3 = 10)
+0.007
+0.008
+0.008
+FCSE (w1 = 30, w2 = 10, w3 = 30)
+0.009
+0.007
+0.008
+BCSE (w1 = 1, w2 = 2)
+0.008
+0.006
+-
+BCSE (w1 = 5, w2 = 10)
+0.005
+0.007
+-
+BCSE (w1 = 10, w2 = 30)
+0.007
+0.007
+-
+PM used in SRR
+0.019
+0.020
+0.028
+Figure 3: Segmentation and estimated soiling ratio obtained by FCSE.
+4.2.2
+Required accuracy of regression models
+We experimentally justify our choice of 5% as a threshold for validation MAPE of our models in methods
+FCSE (w1 = 10, w2 = 5, w3 = 10), BCSE (w1 = 5, w2 = 10), as discussed in Section 3.2. To be able to
+execute both methods for various thresholds, we employ them on the two datasets that are accompanied by
+manual cleaning information, i.e., Eugene and Cocoa. For both methods, we calculate the mean (over the
+two sites) RMSE against SD. Experiments in Table 3 indicate that the best result is obtained for 5% (or
+above), for BCSE.
+Table 3: Choice of validation MAPE threshold.
+MAPE threshold
+mean RMSE against SD
+FCSE
+BCSE
+3%
+0.007
+0.009
+4%
+0.008
+0.007
+5%, 10%, 15%, 20%
+0.008
+0.006
+9
+
+1DO
+0.95
+0.90
+0.B5
+estimated soiling ratio (smoothed)
+0.:0]
+cleaning events
+imestampFigure 4: Segmentation and estimated soiling ratio obtained by BCSE.
+4.2.3
+Industrial use-case (absence of ground-truth)
+In this section, we test our methods on the dataset described in Section 4.1. First, we apply FCSE for the
+detection of cleaning events. We filter out rains with maximum precipitation of at most 0.1 to remove noise.
+Figure 3 (resp. Figure 4) illustrates the cleaning events detected by FCSE(resp. BCSE) and our modelled
+soiling ratio.
+Within each interval defined by two consecutive changepoints, we compute a line using the
+Theil–Sen method [The92, Sen68] on the estimated soiling ratio (on a 15min granularity). The Theil-Sen
+method is a way of fitting a line to a set of points, which is robust to outliers. The line is chosen by selecting
+the median slope over all lines defined by pairs of points. We plot the lines with negative slope as red dotted
+line segments lying in the corresponding intervals, over the course of 5 years. We also plot a smoothed
+version of our estimated soiling ratio, where we have applied a rolling median of 5 days.
+In both figures, in almost all time periods defined by two consecutive changepoints, we observe that there
+is a decreasing trend in the time series for the detected period, as dictated by the slope of the line fitted by
+the Theil-Sen regression (red-dotted line segments). This decreasing trend ends with rain or manual cleaning,
+illustrated by a blue vertical line, which is detected by our method as a cleaning event. This example is an
+indication of the effectiveness and generalizability of the proposed method. Despite the lack of labels to be
+able to explicitly verify the result, the trend identified is consistent with soiling and it is verifiable through
+the effect of washing.
+5
+Conclusion
+We have described a method for estimating the soiling ratio, which uses a set of easily accessed measurements
+from sensors that are commonly deployed in PV parks. Our method is data-driven, in the sense that it models
+the optimal performance by efficiently learning it from the data, without relying on generic formulas that
+fail to capture the peculiarities of the site.
+Estimating the soiling ratio is useful for PV park administrators since it allows them to schedule cleaning
+procedures more effectively by taking into account the rate of soil accumulation and the effectiveness of past
+cleaning efforts without the need for frequent visual inspections or installing specialized equipment which
+induces extra cost and maintenance efforts.
+Our method effectively estimates the soiling ratio in historical data. Future possible directions include
+extending our method to forecasting soiling losses in the future, which would assist in deciding cleaning
+10
+
+LDO0
+5L60
+S60
+0.925
+0.900
+0.B75
+0.B50
+0.B25
+estimated soiling ratio (smoothed)
+0.B0.0
+cleaning events
+101:2036actions at a short notice.
+6
+Acknowledgements
+The authors were partially supported by the EU’s Horizon 2020 Research and Innovation programme, under
+the grant agreement No. 957345: “MORE”.
+References
+[BMAF21]
+João Gabriel Bessa, Leonardo Micheli, Florencia Almonacid, and Eduardo F. Fernández. Mon-
+itoring photovoltaic soiling: assessment, challenges, and perspectives of current and potential
+strategies. iScience, 24(3):102165, 2021.
+[DG95]
+R N Dows and E J Gough. PVUSA procurement, acceptance, and rating practices for photo-
+voltaic power plants. 9 1995.
+[DMM18]
+Michael G. Deceglie, Leonardo Micheli, and Matthew Muller. Quantifying soiling loss directly
+from PV yield. IEEE J. of Photov., 8(2):547–551, 2018.
+[DNS+22]
+Michael G. Deceglie, Ambarish Nag, Adam Shinn, Gregory Kimball, Daniel Ruth, Dirk Jordan,
+Jiyang Yan, Kevin Anderson, Kirsten Perry, Mark Mikofski, Matthew Muller, Will Vining, and
+Chris Deline. Nrel/rdtools: Version 2.1.3. January 2022.
+[HHM22]
+William F. Holmgren, Clifford W. Hansen, and Mark A. Mikofski. pvlib python: a python
+package for modeling solar energy systems. J. Open Source Soft., 3(29)), 2022.
+[iec21]
+IEC TS 61724-1 Photovoltaic System Performance Part 1: Monitoring. IEC, 2021.
+[KMNW06] A. Kimber, L. Mitchell, S. Nogradi, and H. Wenger.
+The effect of soiling on large grid-
+connected photovoltaic systems in California and the southwest region of the United States. In
+IEEE 4th World Conf. on Photov. Energy, volume 2, pages 2391–2395, 2006.
+[MAD+14]
+Bill Marion, Allan Anderberg, Chris Deline, Joe del Cueto, Matt Muller, Greg Perrin, Jose
+Rodriguez, Steve Rummel, Timothy J. Silverman, Frank Vignola, Rich Kessler, Josh Peterson,
+Stephen Barkaszi, Mark Jacobs, Nick Riedel, Larry Pratt, and Bruce King.
+New data set
+for validating PV module performance models. In IEEE 40th Phot. Specialist Conf., pages
+1362–1366, 2014.
+[MMD11]
+A. Massi Pavan, A. Mellit, and D. De Pieri. The effect of soiling on energy production for
+large-scale photovoltaic plants. Solar Energy, 85(5), 2011.
+[MMDK13] A. Massi Pavan, A. Mellit, D. De Pieri, and S.A. Kalogirou.
+A comparison between BNN
+and regression polynomial methods for the evaluation of the effect of soiling in large scale
+photovoltaic plants. Appl. Energy, 108:392–401, 2013.
+[MMMM17] Matthew Muller, Leonardo Micheli, and Alfredo A. Martinez-Morales. A method to extract
+soiling loss data from soiling stations with imperfect cleaning schedules. In 2017 IEEE 44th
+Photov. Specialist Conf. (PVSC), pages 2881–2886, 2017.
+[MTL+21]
+Leonardo Micheli, Marios Theristis, Andreas Livera, Joshua S. Stein, George E. Georghiou,
+Matthew Muller, Florencia Almonacid, and Eduardo F. Fernández. Improved PV soiling extrac-
+tion through the detection of cleanings and change points. IEEE J. of Photov., 11(2):519–526,
+2021.
+11
+
+[MWPP08]
+Didier Mayer, Lucien Wald, Yves Poissant, and Sophie Pelland. Performance Prediction of
+Grid-Connected Photovoltaic Systems Using Remote Sensing. International Energy Agency,
+March 2008. Photovoltaic Power Systems Programme (IEA - PVPS Task 2), report IEA-PVPS
+T2-07:2008.
+[SD20]
+Åsmund Skomedal and Michael G. Deceglie. Combined estimation of degradation and soiling
+losses in photovoltaic systems. IEEE J. of Photov., 10(6), 2020.
+[Sen68]
+Pranab Kumar Sen. Estimates of the regression coefficient based on Kendall’s tau. J. of the
+American Statistical Association, 63(324):1379–1389, 1968.
+[The92]
+Henri Theil. A Rank-Invariant Method of Linear and Polynomial Regression Analysis, pages
+345–381. Springer Netherlands, Dordrecht, 1992.
+12
+

N9AyT4oBgHgl3EQftPlh/content/tmp_files/2301.00591v1.pdf.txt

@@ -0,0 +1,704 @@
+ANALYSING DISCRETE SELF SUPERVISED SPEECH
+REPRESENTATION FOR SPOKEN LANGUAGE MODELING
+Amitay Sicherman and Yossi Adi
+School of Engineering and Computer Science
+The Hebrew University of Jerusalem, Israel
+ABSTRACT
+This work profoundly analyzes discrete self-supervised speech
+representations through the eyes of Generative Spoken Lan-
+guage Modeling (GSLM). Following the findings of such an
+analysis, we propose practical improvements to the discrete
+unit for the GSLM. First, we start comprehending these units
+by analyzing them in three axes: interpretation, visualization,
+and resynthesis. Our analysis finds a high correlation between
+the speech units to phonemes and phoneme families, while
+their correlation with speaker or gender is weaker. Addition-
+ally, we found redundancies in the extracted units and claim
+that one reason may be the units’ context. Following this
+analysis, we propose a new, unsupervised metric to measure
+unit redundancies.
+Finally, we use this metric to develop
+new methods that improve the robustness of units clustering
+and show significant improvement considering zero-resource
+speech metrics such as ABX. Code and analysis tools are
+available under the following link.
+Index Terms— self supervised learning, generative spo-
+ken language modeling, textless NLP, speech LM
+1. INTRODUCTION
+Recently Self-Supervised Learning (SSL) methods for speech
+have shown great success on plenty of downs stream tasks [1].
+From Automatic Speech Recognition [2, 3, 4] and speaker
+diarization [5], to phone segmentation [6], these models have
+shown remarkable results.
+Specifically, these SSL models allow recent success in
+Generative Spoken Language Modeling (GSLM) [7, 8, 9].
+In GSLM, we aim to learn a discrete representation of the
+speech signal. This is often being done by applying the k-
+means algorithm over the continuous representation obtained
+from a SSL models. Then, we train a unit Language Model
+(uLM) over such representation, and lastly we decode it back
+to a time domain signal using neural vocoder [10]. During
+inference time, we can sample from the uLM conditionally or
+unconditionally.
+Although these models are capable of generating mean-
+ingful and coherent speech utterances, little is known about
+the properties captured but these discrete representations. The
+authors in [11] examined the purity between phonetics ele-
+ments and the discrete units. The authors’ proposed method
+is to analysis the discrete SSL representation considering
+fine-grained linguistic properties, e.g., different articulatory
+classes or closure and release portions. The authors in [12]
+proposed a probing method to analyze the presence of phone
+classes, gender and language information while comparing
+monolingual and bilingual models.
+In this work, we analyze quantitatively and visually dis-
+crete representations obtained by HuBERT and CPC models
+with respect to phoneme classes, gender and speaker identity.
+Next, equipped with such an analysis we provide a metric to
+identify redundancies in the k-means clustering, and propose
+a method to improve upon it.
+We find a high correlation between the units and the
+phonemes, but with many redundancies in the units.
+We
+show that one reason may be the units’ context. In Addition,
+we propose an unsupervised metric to measure these redun-
+dancies and we use it to significant improvement the unit
+clustering.
+2. BACKGROUND
+The general GSLM pipeline is comprised of three main mod-
+ules: (i) Speech-to-unit, (ii) Unit language model, and (iii)
+Unit-to-speech, where each of these modules is trained sepa-
+rately. Speech resynthesis can be achieved while ignoring the
+language model and directly feeding the quantized units into
+the unit-to-speech module [10]
+Speech To Unit (STU) module encodes the raw speech signal
+into a discrete representation. The model first encodes the
+speech into a continuous representation and then quantize the
+representation to a sequence of discrete units [7, 13, 14].
+Formally, denote the domain of audio samples by x ⊂ R.
+The representation for a raw signal is therefore a sequence of
+samples x = (x1, . . . , xT ), where xt ∈ x for all 1 ≤ t ≤ T.
+Consider an encoder network, f, that gets as input the speech
+utterance and outputs a sequence of spectral representations
+sampled at a low frequency as follows f(x) = (v1, . . . , vT ′).
+Note that we do not assume anything about the structure of
+the encoder network f. Since the representations learned by
+such models are usually continuous, a k-means algorithm is
+arXiv:2301.00591v1  [cs.CL]  2 Jan 2023
+
+Fig. 1. Units visualization process.
+applied over the models’ outputs to generate discrete units,
+denoted as z = (z1, . . . , zT ′). Each element zi in z is a pos-
+itive integer, zi ∈ {1, .., K} for 1 ≤ i ≤ T ′, where K is the
+number of discrete units.
+As the quantized representation, z, usually contain units
+repetitions which degrade the performance of the language
+modeling, a common approach is collapse repetitions and
+generate a de-duplicated sequence while additionally storing
+the units’ duration separately.
+For instance, the sequence
+12,12,25,31,31,31 will be converted into 12,25,31
+and the corresponding durations 2,1,3.
+Unit Language Model (ULM) is trained on the extracted
+and deduplicated discrete units, z. The language model can
+be used, for example, to generate speech conditionally or un-
+conditionally.
+Unit To Speech module converts the discrete speech repre-
+sentation, z, to a raw waveform. The authors in [7] used a
+Tacotron2.0 [15] based model followed by WaveGlow [16]
+vocoder. Later, [10] proposed a unit-based vocoder based on
+the HiFi-GAN architecture to directly convert units to speech.
+In this work, we focus on the latter setting.
+3. METHOD
+We analyze representations obtained by either HuEBRT [2]
+or CPC [4] models considering various number of clusters.
+All analysis code and the developed visualization tools will
+be publicly available.
+3.1. Analysis
+Units Interpretation. We start by measuring the mutual in-
+formation between the discrete representation and different
+speech properties (i.e., phonemes, speaker id, and gender),
+using the V-Measure score [17].
+For this purpose, we align each utterance with its corre-
+sponding attribute. To get units-to-phonemes alignment we
+use the TIMIT corpus [18]. The TIMIT dataset contains pairs
+of audio - phonemes, which are time aligned. For speaker and
+gender analysis we use the LibriSpeech corpus as it contains
+large and diverse set of speaker.
+Fig. 2. Circular Resynthesis evaluation metric.
+Units Visualization. An additional point of view of the units
+meaning is the spatial structure of units. For this purpose,
+we create a 2D spatial view that contains information regard-
+ing the relation between the continuous representation, the
+discrete units, and their corresponding phonemes. Specifi-
+cally, we apply the following two steps: (i) We project the
+high-dimensional speech representation into 2d using the T-
+SNE [19] algorithm. T-SNE is a nonlinear dimensionality re-
+duction that intuitively preserves the non-linear distance re-
+lations between neighbors in the high and low dimensions.
+Then, we use the Voronoi diagram [20] that converts the scat-
+ter plot into an area plot. Finally, we have left with a bounded
+area in the 2D space for each unit; (ii) In the second part,
+we create a single label to represents each cluster. We use
+the units-phonemes alignment from the TIMIT (similarly to
+the process in previous paragraph). Then, we assign for each
+cluster the most represented phoneme in it. Finally, we re-
+place the unit id with their corresponding phonemes and color
+the area base on the phoneme and phoneme family. A visual
+description of the proposed method can be seen in Figure 1.
+Units Resynthesis. Next, we analyze the units’ information
+from the opposite direction - that is, through the speech resyn-
+thesis. We decode the units back to speech using a look-up-
+table of the corresponding 20ms speech segments, then we
+transcribe the generated audio and measure the transcription
+error (e.g., the Character Error Rate). Intuitively, in case of
+strong correlation between the units and the phonemes - we
+can take a single “sound” to represent each unit - and apply
+the UTS step using the concatenation of these sound pieces.
+Notice, this approach is different than the one in [10] as there
+is no neural vocoder.
+Formally, let u, l be a sequences of deduplicated units and
+their length obtained by applying STU on the input audio x.
+and let xi be the part in x that is matched to deduped unit, ui.
+Notice, xi can be of arbitrary length.
+Lookup Vocoder defines as :
+LV (u, l) = concat(F(u1, l1), . . . , F(un, ln)),
+F(ui, li) =
+�
+T[Key(ui, li)],
+if Key(ui, li) in T
+xi,
+else
+,
+(1)
+
+K-Means Centers
+T-SNE
+Voronoi Diagram
+N × Multicdimensional Vectors
+Acoustic-PhoneticCorpus
+M
+Units :
+11313...17
+M
+3-M
+4
+Phonemes : A
+wI
+I.. SH
+7 - SH
+SHcR(i)
+32
+UED
+->
+->
+s<- hs
+32 -> 28
+1, 7, 54....
+28Table 1. Units Interpretation results. For phonemes, higher is
+better. While for the speaker and gender, lower score indicates
+that the model manages to hide information about the speaker
+and gender.
+Model
+Size
+Speaker
+Gender
+Phoneme
+CPC
+50
+1.35
+0.66
+47.30
+100
+2.35
+0.54
+48.45
+200
+3.70
+1.62
+47.74
+2000
+10.39
+4.14
+44.06
+HuBERT
+50
+0.73
+0.03
+42.49
+100
+1.41
+0.17
+45.48
+200
+1.95
+0.21
+46.64
+2000
+5.15
+0.65
+43.32
+MFCC
+50
+9.11
+2.90
+8.57
+100
+11.54
+3.97
+8.73
+200
+13.81
+4.59
+8.96
+where T is a Look-up-table that stores for each key the corre-
+sponding xi of the first appearance of this key, and Key maps
+unit and length into key.
+We explore four different types of Key : (i) Local-Single-
+Key(ui) = (ui); (ii) Local-Full- Key(ui) = (ui, li); (iii)
+Context-Single- Key(ui) = (ui−1, ui, ui+1); (iv) Context-
+Full- Key(ui) = (ui−1, ui, ui+1, li).
+3.2. Circular Resynthesis
+We introduce the Circular Resynthesis (CR) method, an ut-
+terly unsupervised evaluation metric that aims to measure the
+redundancies in the discrete units. As described in Figure 2,
+we first perform a full resynthesis procedure, in which we
+encode and decode the speech signal. Then, we apply an ad-
+ditional resynthesis stage and measure the Unit-Edit-Distance
+(UED) between the first and the second units representing the
+speech. This metric was recently proposed by [14] to evalu-
+ate robustness of discrete speech representation against signal
+variations. Intuitively, a high UED indicates redundancies in
+the discrete units. To reach the final CR metric, for each pair
+of units, we calculate the percentage of swaps between them
+over all the dataset’s transcriptions.
+3.3. Robust Clustering
+Equipped with the CR metric, we explore three simple meth-
+ods to improve the k-means clustering quality. In all three
+methods, we start from the standard k-means with k = 2000
+and iterativly merge the clusters to reach the target number
+of clusters. The first method, named Double K-means. In
+which, we apply an additional k-means over the cluster cen-
+torids from the first k-means step. The second method, de-
+noted as K-means with Hierarchical Clustering, we apply
+an an agglomerative clustering over the cluster centorids from
+the first k-means step. The last method, named K-means with
+Weighed Hierarchical Clustering, we use an agglomerative
+Table 2.
+Units Resynthesis results.
+CER for UTS using
+lookup and concatenate methods. The table contains results
+for different lookup key types: Local-Single (L-S),Local-Full
+(L-F) Context-Single (C-S) and Context-Full (C-F).
+Model
+Size
+Hifi-GEN
+Key Type
+C-F
+C-S
+L-F
+L-S
+CPC
+50
+5.95
+9.12
+25.36
+39.57
+60.98
+100
+5.67
+6.52
+15.21
+22.51
+53.59
+200
+5.37
+5.12
+10.16
+15.18
+40.65
+HuBERT
+50
+7.31
+10.31
+14.96
+47.24
+58.42
+100
+4.39
+5.24
+6.26
+26.55
+57.49
+200
+4.10
+4.25
+4.69
+15.56
+19.88
+MFCC
+50
+50.47
+33.85
+57.60
+71.43
+69.22
+100
+44.68
+15.79
+46.55
+67.54
+66.13
+200
+41.67
+6.22
+30.47
+61.46
+61.31
+clustering using a modified version of the euclidean distance,
+weighted by the CR metric. Formally, the distance metric is
+defined as follows:
+D(i, j) = L2(ci, cj) · SWAP(ui, uj),
+SWAP(ui, uj) = 1
+2 [CR(ui, uj) + CR(uj, ui)] ,
+(2)
+while ci, cj are the ith and jth cluster continuous centroids,
+and ui, uj are the ith and jth discrete unit.
+4. RESULTS
+4.1. Datasets
+We use the the Librispeech[21] corpus to learn the k-means
+clustering (train-clean-100), and the test-clean to
+evaluate both the clustering methods and the look-up vocoder.
+Additionally, we use the Librispeech corpus for calculating
+the V-Measure for speaker and gender. For computing the
+V-Measure over phonemes we use the TIMIT benchmark.
+4.2. Units Interpretation
+Table 1 presents the V-Measure results regarding three dif-
+ferent attributes - speaker, gender, and phoneme.
+The V-
+Measure for the speaker and gender scores is lower than the
+score of the phonemes- which indicates of high correlation
+to the phonemes and a low correlation to the speaker or gen-
+der. In addition, when we check the effect of the number
+of the units- while for the speaker/gender, more units lead
+to a higher score, in the phoneme score there is a max point
+both for the HuBERT and CPC configurations. Therefore, we
+claim that redundancies cause this trend in the units. Finally,
+we can see that CPC has a higher score for the phonemes- but
+also a higher score for speaker and gender.
+4.3. Units Visualization
+Figure 3 shows the spatial structure of the units. One can
+see that there is a very consistent structure- first, units that
+
+Fig. 3. 2D view of the units’ centers. Each bounded area represents a single unit and is colored by the unit’s phoneme. We use
+T-SNE and Voronoi diagram to get the units areas. The matching between the units and phonemes was made using the TIMIT
+corpus, while each unit was labeled as a phoneme that represents her most commonly.
+Table 3. Comparing the different clustering methods using ABX and speaker information.For all these metrics, lower is
+better.The methods are : Regular k-means (K), Double K-means (K-K),K-means with Hierarchical Clustering (K-H) and K-
+means with Weighed Hierarchical Clustering (K-WH)
+Model
+Size
+ABX within
+ABX across
+Speaker probing
+K
+K-K
+K-H
+K-WH
+K
+K-K
+K-H
+K-WH
+K
+K-K
+K-H
+K-WH
+CPC
+50
+5.66
+5.38
+9.62
+8.80
+7.83
+6.77
+11.46
+10.56
+42.22
+32.96
+19.26
+18.15
+100
+5.42
+5.44
+6.66
+6.04
+7.07
+7.13
+8.26
+7.49
+52.96
+45.19
+20.37
+15.56
+200
+5.53
+5.27
+5.61
+5.68
+7.35
+7.10
+7.28
+7.13
+63.70
+49.63
+26.30
+22.59
+HuBERT
+50
+7.23
+5.67
+5.94
+6.12
+8.93
+6.83
+7.43
+7.67
+30.37
+36.30
+36.67
+31.85
+100
+5.82
+5.01
+5.30
+5.29
+7.47
+6.50
+6.54
+6.32
+48.15
+48.89
+48.15
+46.67
+200
+5.79
+5.24
+5.18
+5.05
+7.49
+6.42
+6.46
+6.07
+65.19
+61.11
+54.81
+62.96
+represent the same phoneme are usually close to each other.
+Moreover, phonemes from the same family (affricates, frica-
+tives, Etc.’ ) tend also to be close to each other. In addition,
+we can see that while for HuBERT and CPC, the space divide
+between the different phonemes families is generally equal, in
+the MFCC model, almost all the space uses for vowels. No-
+tice, redundancies in the clusters can be also observed from
+such figures.
+4.4. Units Resynthesis
+In Table 2, we shows the results for the units resynthesis.
+We can see that for some configurations, there is slightly dif-
+ference between the HiFi-GAN and the look-up scores- this
+strengthens our understanding that units express fixed sounds
+and are mainly correlative to phonemes. We can see that the
+context of the units critically affects the results, while the
+unit’s length has a lower effect. Finally, this understanding
+may help in understand units’ redundancies, i.e., the same
+phoneme in a different context will represent different units.
+4.5. Robust Clustering
+We evaluate the proposed approach along two different
+axes: (i) phonetic measure in the form of ABX within and
+across [22]; (ii) speaker information in the form of probing
+similarly to [13]. Table 3 summarizes the results. We can see
+that the proposed methods, although they are straightforward,
+improve both the ABX and the speaker results for most of the
+configurations. Furthermore, the best results for ABX-across
+were obtained using CR- this strengthens our claim regarding
+the unit’s redundancies.
+5. CONCLUSION
+In this work, we analyzed the GSLM discrete unit from three
+different and complementary points of view: interpretation,
+visualization, and resynthesis. The analysis showed a strong
+correlation between the units and the phonemes. In addition,
+we found redundancies in the units, which the units’ context
+can explain. Finally, we proposed methods that improve the
+unit’s clustering based on these understandings.
+
+fricatives
+stops
+affricates
+nasals
+semivowels
+vowels
+others
+HuBERT
+CPC
+MFCC
+laaan
+awa
+ao
+aa
+eh
+aw
+ae
+s
+回
+h#
+sh
+Inr
+Ley
+ux
+ow
+layl
+h#
+h#
+可国
+w
+In
+ae
+lae
+sh]
+dcl
+PP
+s
+Th#
+HiyNiy
+Imm
+pcll
+iyng
+Itcl
+ep
+h#
+y
+a
+h#
+aol
+h#
+Aht
+h#6. REFERENCES
+[1] Shu-wen Yang et al.,
+“Superb:
+Speech processing
+universal performance benchmark,”
+arXiv preprint
+arXiv:2105.01051, 2021.
+[2] Wei-Ning Hsu, Benjamin Bolte, Yao-Hung Hubert Tsai,
+Kushal Lakhotia, Ruslan Salakhutdinov, and Abdelrah-
+man Mohamed, “Hubert: Self-supervised speech rep-
+resentation learning by masked prediction of hidden
+units,” IEEE/ACM Transactions on Audio, Speech, and
+Language Processing, vol. 29, pp. 3451–3460, 2021.
+[3] Alexei Baevski et al., “wav2vec 2.0: A framework for
+self-supervised learning of speech representations,” Ad-
+vances in Neural Information Processing Systems, vol.
+33, pp. 12449–12460, 2020.
+[4] Morgane Riviere et al., “Unsupervised pretraining trans-
+fers well across languages,” in ICASSP 2020-2020 IEEE
+International Conference on Acoustics, Speech and Sig-
+nal Processing (ICASSP). IEEE, 2020, pp. 7414–7418.
+[5] Yehoshua Dissen, Felix Kreuk, and Joseph Keshet,
+“Self-supervised speaker diarization,”
+arXiv preprint
+arXiv:2204.04166, 2022.
+[6] Felix
+Kreuk,
+Joseph
+Keshet,
+and
+Yossi
+Adi,
+“Self-supervised
+contrastive
+learning
+for
+unsu-
+pervised phoneme segmentation,”
+arXiv preprint
+arXiv:2007.13465, 2020.
+[7] Kushal Lakhotia, Eugene Kharitonov, Wei-Ning Hsu,
+Yossi Adi, Adam Polyak, Benjamin Bolte, Tu-Anh
+Nguyen, Jade Copet, Alexei Baevski, Abdelrahman Mo-
+hamed, et al., “On generative spoken language model-
+ing from raw audio,” Transactions of the Association
+for Computational Linguistics, vol. 9, pp. 1336–1354,
+2021.
+[8] Tu Anh Nguyen et al.,
+“Generative spoken dialogue
+language modeling,” arXiv preprint arXiv:2203.16502,
+2022.
+[9] Zal´an Borsos, Rapha¨el Marinier, Damien Vincent, Eu-
+gene Kharitonov, Olivier Pietquin, Matt Sharifi, Olivier
+Teboul, David Grangier, Marco Tagliasacchi, and Neil
+Zeghidour, “Audiolm: a language modeling approach
+to audio generation,” arXiv preprint arXiv:2209.03143,
+2022.
+[10] Adam Polyak et al., “Speech resynthesis from discrete
+disentangled self-supervised representations,”
+arXiv
+preprint arXiv:2104.00355, 2021.
+[11] Dan Wells, Hao Tang, and Korin Richmond,
+“Pho-
+netic analysis of self-supervised representations of en-
+glish speech,” Proc. Interspeech 2022, 2022.
+[12] Maureen de Seyssel, Marvin Lavechin, Yossi Adi, Em-
+manuel Dupoux, and Guillaume Wisniewski,
+“Prob-
+ing phoneme, language and speaker information in un-
+supervised speech representations,”
+arXiv preprint
+arXiv:2203.16193, 2022.
+[13] Eugene Kharitonov et al.,
+“textless-lib: a library for
+textless spoken language processing,”
+arXiv preprint
+arXiv:2202.07359, 2022.
+[14] Itai Gat, Felix Kreuk, Ann Lee, Jade Copet, Gabriel
+Synnaeve, Emmanuel Dupoux, and Yossi Adi, “On the
+robustness of self-supervised representations for spoken
+language modeling,” arXiv preprint arXiv:2209.15483,
+2022.
+[15] Jonathan Shen et al., “Natural tts synthesis by condition-
+ing wavenet on mel spectrogram predictions,” in 2018
+IEEE international conference on acoustics, speech and
+signal processing (ICASSP). IEEE, 2018, pp. 4779–
+4783.
+[16] Ryan Prenger et al.,
+“Waveglow: A flow-based gen-
+erative network for speech synthesis,”
+in ICASSP
+2019-2019 IEEE International Conference on Acous-
+tics, Speech and Signal Processing (ICASSP). IEEE,
+2019, pp. 3617–3621.
+[17] Andrew Rosenberg and Julia Hirschberg, “V-measure:
+A conditional entropy-based external cluster evaluation
+measure,” in Proceedings of the 2007 joint conference
+on empirical methods in natural language processing
+and computational natural language learning (EMNLP-
+CoNLL), 2007, pp. 410–420.
+[18] John S Garofolo,
+“Timit acoustic phonetic continu-
+ous speech corpus,” Linguistic Data Consortium, 1993,
+1993.
+[19] Laurens van der Maaten and Geoffrey Hinton, “Visu-
+alizing data using t-sne,” Journal of Machine Learning
+Research, vol. 9, no. 86, pp. 2579–2605, 2008.
+[20] Franz Aurenhammer, “Voronoi diagrams—a survey of
+a fundamental geometric data structure,” ACM Comput-
+ing Surveys (CSUR), vol. 23, no. 3, pp. 345–405, 1991.
+[21] Vassil Panayotov et al.,
+“Librispeech: an asr corpus
+based on public domain audio books,” in 2015 IEEE
+international conference on acoustics, speech and sig-
+nal processing (ICASSP). IEEE, 2015, pp. 5206–5210.
+[22] Jacob Kahn et al., “Libri-light: A benchmark for asr
+with limited or no supervision,” in ICASSP 2020-2020
+IEEE International Conference on Acoustics, Speech
+and Signal Processing (ICASSP). IEEE, 2020, pp.
+7669–7673.
+

N9AyT4oBgHgl3EQftPlh/content/tmp_files/load_file.txt

@@ -0,0 +1,405 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf,len=404
+page_content='ANALYSING DISCRETE SELF SUPERVISED SPEECH  REPRESENTATION FOR SPOKEN LANGUAGE MODELING  Amitay Sicherman and Yossi Adi  School of Engineering and Computer Science  The Hebrew University of Jerusalem, Israel  ABSTRACT  This work profoundly analyzes discrete self-supervised speech  representations through the eyes of Generative Spoken Lan-  guage Modeling (GSLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Following the findings of such an  analysis, we propose practical improvements to the discrete  unit for the GSLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' First, we start comprehending these units  by analyzing them in three axes: interpretation, visualization,  and resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Our analysis finds a high correlation between  the speech units to phonemes and phoneme families, while  their correlation with speaker or gender is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Addition-  ally, we found redundancies in the extracted units and claim  that one reason may be the units’ context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Following this  analysis, we propose a new, unsupervised metric to measure  unit redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Finally, we use this metric to develop  new methods that improve the robustness of units clustering  and show significant improvement considering zero-resource  speech metrics such as ABX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Code and analysis tools are  available under the following link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Index Terms— self supervised learning, generative spo-  ken language modeling, textless NLP, speech LM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' INTRODUCTION  Recently Self-Supervised Learning (SSL) methods for speech  have shown great success on plenty of downs stream tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  From Automatic Speech Recognition [2, 3, 4] and speaker  diarization [5], to phone segmentation [6], these models have  shown remarkable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Specifically, these SSL models allow recent success in  Generative Spoken Language Modeling (GSLM) [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  In GSLM, we aim to learn a discrete representation of the  speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' This is often being done by applying the k-  means algorithm over the continuous representation obtained  from a SSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Then, we train a unit Language Model  (uLM) over such representation, and lastly we decode it back  to a time domain signal using neural vocoder [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' During  inference time, we can sample from the uLM conditionally or  unconditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Although these models are capable of generating mean-  ingful and coherent speech utterances, little is known about  the properties captured but these discrete representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The  authors in [11] examined the purity between phonetics ele-  ments and the discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The authors’ proposed method  is to analysis the discrete SSL representation considering  fine-grained linguistic properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', different articulatory  classes or closure and release portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The authors in [12]  proposed a probing method to analyze the presence of phone  classes, gender and language information while comparing  monolingual and bilingual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  In this work, we analyze quantitatively and visually dis-  crete representations obtained by HuBERT and CPC models  with respect to phoneme classes, gender and speaker identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Next, equipped with such an analysis we provide a metric to  identify redundancies in the k-means clustering, and propose  a method to improve upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  We find a high correlation between the units and the  phonemes, but with many redundancies in the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  We  show that one reason may be the units’ context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' In Addition,  we propose an unsupervised metric to measure these redun-  dancies and we use it to significant improvement the unit  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' BACKGROUND  The general GSLM pipeline is comprised of three main mod-  ules: (i) Speech-to-unit, (ii) Unit language model, and (iii)  Unit-to-speech, where each of these modules is trained sepa-  rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Speech resynthesis can be achieved while ignoring the  language model and directly feeding the quantized units into  the unit-to-speech module [10]  Speech To Unit (STU) module encodes the raw speech signal  into a discrete representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The model first encodes the  speech into a continuous representation and then quantize the  representation to a sequence of discrete units [7, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Formally, denote the domain of audio samples by x ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  The representation for a raw signal is therefore a sequence of  samples x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' , xT ), where xt ∈ x for all 1 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Consider an encoder network, f, that gets as input the speech  utterance and outputs a sequence of spectral representations  sampled at a low frequency as follows f(x) = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' , vT ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Note that we do not assume anything about the structure of  the encoder network f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Since the representations learned by  such models are usually continuous, a k-means algorithm is  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='00591v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='CL]  2 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Units visualization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  applied over the models’ outputs to generate discrete units,  denoted as z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' , zT ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Each element zi in z is a pos-  itive integer, zi ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='., K} for 1 ≤ i ≤ T ′, where K is the  number of discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  As the quantized representation, z, usually contain units  repetitions which degrade the performance of the language  modeling, a common approach is collapse repetitions and  generate a de-duplicated sequence while additionally storing  the units’ duration separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  For instance, the sequence  12,12,25,31,31,31 will be converted into 12,25,31  and the corresponding durations 2,1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Unit Language Model (ULM) is trained on the extracted  and deduplicated discrete units, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The language model can  be used, for example, to generate speech conditionally or un-  conditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Unit To Speech module converts the discrete speech repre-  sentation, z, to a raw waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The authors in [7] used a  Tacotron2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='0 [15] based model followed by WaveGlow [16]  vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Later, [10] proposed a unit-based vocoder based on  the HiFi-GAN architecture to directly convert units to speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  In this work, we focus on the latter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' METHOD  We analyze representations obtained by either HuEBRT [2]  or CPC [4] models considering various number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  All analysis code and the developed visualization tools will  be publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Analysis  Units Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' We start by measuring the mutual in-  formation between the discrete representation and different  speech properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', phonemes, speaker id, and gender),  using the V-Measure score [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  For this purpose, we align each utterance with its corre-  sponding attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' To get units-to-phonemes alignment we  use the TIMIT corpus [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The TIMIT dataset contains pairs  of audio - phonemes, which are time aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' For speaker and  gender analysis we use the LibriSpeech corpus as it contains  large and diverse set of speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Circular Resynthesis evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Units Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' An additional point of view of the units  meaning is the spatial structure of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' For this purpose,  we create a 2D spatial view that contains information regard-  ing the relation between the continuous representation, the  discrete units, and their corresponding phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Specifi-  cally, we apply the following two steps: (i) We project the  high-dimensional speech representation into 2d using the T-  SNE [19] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' T-SNE is a nonlinear dimensionality re-  duction that intuitively preserves the non-linear distance re-  lations between neighbors in the high and low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Then, we use the Voronoi diagram [20] that converts the scat-  ter plot into an area plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Finally, we have left with a bounded  area in the 2D space for each unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' (ii) In the second part,  we create a single label to represents each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' We use  the units-phonemes alignment from the TIMIT (similarly to  the process in previous paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Then, we assign for each  cluster the most represented phoneme in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Finally, we re-  place the unit id with their corresponding phonemes and color  the area base on the phoneme and phoneme family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' A visual  description of the proposed method can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Units Resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Next, we analyze the units’ information  from the opposite direction - that is, through the speech resyn-  thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' We decode the units back to speech using a look-up-  table of the corresponding 20ms speech segments, then we  transcribe the generated audio and measure the transcription  error (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', the Character Error Rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Intuitively, in case of  strong correlation between the units and the phonemes - we  can take a single “sound” to represent each unit - and apply  the UTS step using the concatenation of these sound pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Notice, this approach is different than the one in [10] as there  is no neural vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Formally, let u, l be a sequences of deduplicated units and  their length obtained by applying STU on the input audio x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  and let xi be the part in x that is matched to deduped unit, ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Notice, xi can be of arbitrary length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Lookup Vocoder defines as :  LV (u, l) = concat(F(u1, l1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' , F(un, ln)),  F(ui, li) =  �  T[Key(ui, li)],  if Key(ui, li) in T  xi,  else  ,  (1)  K-Means Centers  T-SNE  Voronoi Diagram  N × Multicdimensional Vectors  Acoustic-PhoneticCorpus  M  Units :  11313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='17  M  3-M  4  Phonemes : A  wI  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='. SH  7 - SH  SHcR(i)  32  UED  >  >  s<- hs  32 -> 28  1, 7, 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='.  28Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Units Interpretation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' For phonemes, higher is  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' While for the speaker and gender, lower score indicates  that the model manages to hide information about the speaker  and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Model  Size  Speaker  Gender  Phoneme  CPC  50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='66  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='30  100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='54  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='45  200  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='62  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='74  2000  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='39  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='14  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='06  HuBERT  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='03  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='49  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='17  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='48  200  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='21  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='64  2000  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='65  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='32  MFCC  50  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='90  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='57  100  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='54  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='97  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='73  200  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='81  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='59  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='96  where T is a Look-up-table that stores for each key the corre-  sponding xi of the first appearance of this key, and Key maps  unit and length into key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  We explore four different types of Key : (i) Local-Single-  Key(ui) = (ui);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' (ii) Local-Full- Key(ui) = (ui, li);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' (iii)  Context-Single- Key(ui) = (ui−1, ui, ui+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' (iv) Context-  Full- Key(ui) = (ui−1, ui, ui+1, li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Circular Resynthesis  We introduce the Circular Resynthesis (CR) method, an ut-  terly unsupervised evaluation metric that aims to measure the  redundancies in the discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' As described in Figure 2,  we first perform a full resynthesis procedure, in which we  encode and decode the speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Then, we apply an ad-  ditional resynthesis stage and measure the Unit-Edit-Distance  (UED) between the first and the second units representing the  speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' This metric was recently proposed by [14] to evalu-  ate robustness of discrete speech representation against signal  variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Intuitively, a high UED indicates redundancies in  the discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' To reach the final CR metric, for each pair  of units, we calculate the percentage of swaps between them  over all the dataset’s transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Robust Clustering  Equipped with the CR metric, we explore three simple meth-  ods to improve the k-means clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' In all three  methods, we start from the standard k-means with k = 2000  and iterativly merge the clusters to reach the target number  of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The first method, named Double K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' In  which, we apply an additional k-means over the cluster cen-  torids from the first k-means step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The second method, de-  noted as K-means with Hierarchical Clustering, we apply  an an agglomerative clustering over the cluster centorids from  the first k-means step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The last method, named K-means with  Weighed Hierarchical Clustering, we use an agglomerative  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Units Resynthesis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  CER for UTS using  lookup and concatenate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The table contains results  for different lookup key types: Local-Single (L-S),Local-Full  (L-F) Context-Single (C-S) and Context-Full (C-F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Model  Size  Hifi-GEN  Key Type  C-F  C-S  L-F  L-S  CPC  50  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='95  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='12  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='36  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='57  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='98  100  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='67  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='52  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='21  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='51  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='59  200  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='37  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='12  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='16  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='18  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='65  HuBERT  50  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='31  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='31  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='96  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='24  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='42  100  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='39  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='24  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='26  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='55  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='49  200  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='69  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='56  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='88  MFCC  50  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='47  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='85  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='60  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='43  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='22  100  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='68  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='79  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='55  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='54  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='13  200  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='67  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='22  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='47  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='46  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='31  clustering using a modi��ed version of the euclidean distance,  weighted by the CR metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Formally, the distance metric is  defined as follows:  D(i, j) = L2(ci, cj) · SWAP(ui, uj),  SWAP(ui, uj) = 1  2 [CR(ui, uj) + CR(uj, ui)] ,  (2)  while ci, cj are the ith and jth cluster continuous centroids,  and ui, uj are the ith and jth discrete unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' RESULTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Datasets  We use the the Librispeech[21] corpus to learn the k-means  clustering (train-clean-100), and the test-clean to  evaluate both the clustering methods and the look-up vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Additionally, we use the Librispeech corpus for calculating  the V-Measure for speaker and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' For computing the  V-Measure over phonemes we use the TIMIT benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Units Interpretation  Table 1 presents the V-Measure results regarding three dif-  ferent attributes - speaker, gender, and phoneme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  The V-  Measure for the speaker and gender scores is lower than the  score of the phonemes- which indicates of high correlation  to the phonemes and a low correlation to the speaker or gen-  der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' In addition, when we check the effect of the number  of the units- while for the speaker/gender, more units lead  to a higher score, in the phoneme score there is a max point  both for the HuBERT and CPC configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Therefore, we  claim that redundancies cause this trend in the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Finally,  we can see that CPC has a higher score for the phonemes- but  also a higher score for speaker and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Units Visualization  Figure 3 shows the spatial structure of the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' One can  see that there is a very consistent structure- first, units that  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 2D view of the units’ centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Each bounded area represents a single unit and is colored by the unit’s phoneme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' We use  T-SNE and Voronoi diagram to get the units areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The matching between the units and phonemes was made using the TIMIT  corpus, while each unit was labeled as a phoneme that represents her most commonly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Comparing the different clustering methods using ABX and speaker information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='For all these metrics, lower is  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='The methods are : Regular k-means (K), Double K-means (K-K),K-means with Hierarchical Clustering (K-H) and K-  means with Weighed Hierarchical Clustering (K-WH)  Model  Size  ABX within  ABX across  Speaker probing  K  K-K  K-H  K-WH  K  K-K  K-H  K-WH  K  K-K  K-H  K-WH  CPC  50  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='66  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='38  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='62  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='80  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='83  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='77  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='46  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='56  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='22  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='96  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='26  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='15  100  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='42  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='44  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='66  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='04  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='07  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='13  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='26  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='49  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='96  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='19  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='37  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='56  200  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='53  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='61  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='68  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='35  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='10  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='28  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='13  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='70  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='63  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='30  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='59  HuBERT  50  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='67  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='94  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='93  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='83  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='43  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='67  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='37  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='30  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='67  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='85  100  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='82  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='01  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='29  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='47  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='50  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='54  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='32  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='15  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='89  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='15  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='67  200  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='79  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='05  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='49  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='42  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='46  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='07  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='19  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='11  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='81  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='96  represent the same phoneme are usually close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  Moreover, phonemes from the same family (affricates, frica-  tives, Etc.’ ) tend also to be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' In addition,  we can see that while for HuBERT and CPC, the space divide  between the different phonemes families is generally equal, in  the MFCC model, almost all the space uses for vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' No-  tice, redundancies in the clusters can be also observed from  such figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Units Resynthesis  In Table 2, we shows the results for the units resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  We can see that for some configurations, there is slightly dif-  ference between the HiFi-GAN and the look-up scores- this  strengthens our understanding that units express fixed sounds  and are mainly correlative to phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' We can see that the  context of the units critically affects the results, while the  unit’s length has a lower effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Finally, this understanding  may help in understand units’ redundancies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', the same  phoneme in a different context will represent different units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Robust Clustering  We evaluate the proposed approach along two different  axes: (i) phonetic measure in the form of ABX within and  across [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' (ii) speaker information in the form of probing  similarly to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Table 3 summarizes the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' We can see  that the proposed methods, although they are straightforward,  improve both the ABX and the speaker results for most of the  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Furthermore, the best results for ABX-across  were obtained using CR- this strengthens our claim regarding  the unit’s redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' CONCLUSION  In this work, we analyzed the GSLM discrete unit from three  different and complementary points of view: interpretation,  visualization, and resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' The analysis showed a strong  correlation between the units and the phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' In addition,  we found redundancies in the units, which the units’ context  can explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Finally, we proposed methods that improve the  unit’s clustering based on these understandings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  fricatives  stops  affricates  nasals  semivowels  vowels  others  HuBERT  CPC  MFCC  laaan  awa  ao  aa  eh  aw  ae  s  回  h#  sh  Inr  Ley  ux  ow  layl  h#  h#  可国  w  In  ae  lae  sh]  dcl  PP  s  Th#  HiyNiy  Imm  pcll  iyng  Itcl  ep  h#  y  a  h#  aol  h#  Aht  h#6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' REFERENCES  [1] Shu-wen Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=',  “Superb:  Speech processing  universal performance benchmark,”  arXiv preprint  arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='01051, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [2] Wei-Ning Hsu, Benjamin Bolte, Yao-Hung Hubert Tsai,  Kushal Lakhotia, Ruslan Salakhutdinov, and Abdelrah-  man Mohamed, “Hubert: Self-supervised speech rep-  resentation learning by masked prediction of hidden  units,” IEEE/ACM Transactions on Audio, Speech, and  Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 3451–3460, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [3] Alexei Baevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', “wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='0: A framework for  self-supervised learning of speech representations,” Ad-  vances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 12449–12460, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [4] Morgane Riviere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', “Unsupervised pretraining trans-  fers well across languages,” in ICASSP 2020-2020 IEEE  International Conference on Acoustics, Speech and Sig-  nal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 7414–7418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [5] Yehoshua Dissen, Felix Kreuk, and Joseph Keshet,  “Self-supervised speaker diarization,”  arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='04166, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [6] Felix  Kreuk,  Joseph  Keshet,  and  Yossi  Adi,  “Self-supervised  contrastive  learning  for  unsu-  pervised phoneme segmentation,”  arXiv preprint  arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='13465, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [7] Kushal Lakhotia, Eugene Kharitonov, Wei-Ning Hsu,  Yossi Adi, Adam Polyak, Benjamin Bolte, Tu-Anh  Nguyen, Jade Copet, Alexei Baevski, Abdelrahman Mo-  hamed, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', “On generative spoken language model-  ing from raw audio,” Transactions of the Association  for Computational Linguistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 1336–1354,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [8] Tu Anh Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=',  “Generative spoken dialogue  language modeling,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='16502,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [9] Zal´an Borsos, Rapha¨el Marinier, Damien Vincent, Eu-  gene Kharitonov, Olivier Pietquin, Matt Sharifi, Olivier  Teboul, David Grangier, Marco Tagliasacchi, and Neil  Zeghidour, “Audiolm: a language modeling approach  to audio generation,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='03143,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [10] Adam Polyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', “Speech resynthesis from discrete  disentangled self-supervised representations,”  arXiv  preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='00355, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [11] Dan Wells, Hao Tang, and Korin Richmond,  “Pho-  netic analysis of self-supervised representations of en-  glish speech,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' Interspeech 2022, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [12] Maureen de Seyssel, Marvin Lavechin, Yossi Adi, Em-  manuel Dupoux, and Guillaume Wisniewski,  “Prob-  ing phoneme, language and speaker information in un-  supervised speech representations,”  arXiv preprint  arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='16193, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [13] Eugene Kharitonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=',  “textless-lib: a library for  textless spoken language processing,”  arXiv preprint  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='07359, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [14] Itai Gat, Felix Kreuk, Ann Lee, Jade Copet, Gabriel  Synnaeve, Emmanuel Dupoux, and Yossi Adi, “On the  robustness of self-supervised representations for spoken  language modeling,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='15483,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [15] Jonathan Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', “Natural tts synthesis by condition-  ing wavenet on mel spectrogram predictions,” in 2018  IEEE international conference on acoustics, speech and  signal processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 4779–  4783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [16] Ryan Prenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=',  “Waveglow: A flow-based gen-  erative network for speech synthesis,”  in ICASSP  2019-2019 IEEE International Conference on Acous-  tics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' IEEE,  2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 3617–3621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [17] Andrew Rosenberg and Julia Hirschberg, “V-measure:  A conditional entropy-based external cluster evaluation  measure,” in Proceedings of the 2007 joint conference  on empirical methods in natural language processing  and computational natural language learning (EMNLP-  CoNLL), 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 410–420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [18] John S Garofolo,  “Timit acoustic phonetic continu-  ous speech corpus,” Linguistic Data Consortium, 1993,  1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [19] Laurens van der Maaten and Geoffrey Hinton, “Visu-  alizing data using t-sne,” Journal of Machine Learning  Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 86, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 2579–2605, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [20] Franz Aurenhammer, “Voronoi diagrams—a survey of  a fundamental geometric data structure,” ACM Comput-  ing Surveys (CSUR), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 345–405, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [21] Vassil Panayotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=',  “Librispeech: an asr corpus  based on public domain audio books,” in 2015 IEEE  international conference on acoustics, speech and sig-  nal processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' IEEE, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' 5206–5210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  [22] Jacob Kahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=', “Libri-light: A benchmark for asr  with limited or no supervision,” in ICASSP 2020-2020  IEEE International Conference on Acoustics, Speech  and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}
+page_content='  7669–7673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'}

O9FJT4oBgHgl3EQf1S22/content/2301.11651v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bf4088e417ceb1a520ec91b83ffb22e2e3da831e4ec3c5223456011126ae3c3d
+size 328299

O9FJT4oBgHgl3EQf1S22/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:65b303a5ec7b14257c3f96fd9bea72b255ac49a9094d08afec274a0b7e736ebd
+size 86245

Q9E4T4oBgHgl3EQf_A6p/content/tmp_files/2301.05368v1.pdf.txt

@@ -0,0 +1,883 @@
+Router for Wireless Power Packet Transmission:
+Design and Application to Intersystem Power Management∗
+Takahiro Mamiya, Shiu Mochiyama, and Takashi Hikihara
+Department of Electrical Engineering, Kyoto University
+Abstract
+Power supply for small-scale battery-powered systems such as electric vehicles (EVs and mobile robots) is
+being actively researched.
+We are particularly interested in energy management, which considers the inter-
+connection of such systems close to each other. This allows for overall redundancy to be maintained without
+assuming excessive redundancy with individual power sources. Its implementation necessitates a high level of
+integration between power management and information and communication technology. As one of these meth-
+ods, this study investigates energy management based on power packetization. When the individual systems
+to be connected have moving parts or are mobile, wireless power transmission is a promising method for power
+sharing. However, power packetization has so far only been considered for wired transmission. In this paper, we
+address the integration of power and information in wireless channels using power packetization. We propose a
+power packet router circuit that can wirelessly transmit power over multiple channels selectively. Furthermore,
+we demonstrate that the developed system can handle both wired intrasystem power management and wireless
+intersystem power sharing in a unified manner.
+1
+Introduction
+Recent days have witnessed widespread use of electric power systems that are equipped with batteries and can thus
+be driven without relying on an external and large power grid. Common examples include electric vehicles (EVs)
+and mobile robots. While much effort has been dedicated to independent power management in such a system,
+another research trend is the management of a network of such systems. We refer to a minimum element of a system
+that can independently operate a local system throughout the paper. Constituting a networked system addresses
+shared redundancy of power source capacity as a whole system, rather than as each individual system. That is,
+when the power demand of one system temporarily increases, power can be supplied not only from the inside power
+sources but also from the power sources of the other connected systems [1–3].
+Because local systems are spatially dispersed and can have a time-dependent supply/load profile, managing such
+a network necessitates advanced sensing, computation, and communication technologies [4–6]. Several proposals for
+power system management with ICTs support have been made [2,7,8]. Among them, a power packet dispatching
+system is an encouraging proposal for the purpose. The system packetizes supplied power; that is, power is divided
+into time segments, each of which is associated with an information tag via a voltage waveform [9, 10]. Power
+packetization ensures that information exchange and power transmission occur concurrently in the physical layer,
+allowing for power management in a network without causing an imbalance in information and physical quantity
+processing. In the previous study, the authors’ group developed a circuit called a power packet router [9]. We
+validated the concept of power packetization and routing with hardware configuration including the routers.
+One advantage of power packetization is the ability to easily attach/detach local systems from a larger network.
+The use of time-division multiplexing and physical tag attachment ensures that each packetized power transfer is
+independent. In other words, power transfers between different pairs do not get mixed up even on the same power
+line but can be differentiated physically. This leads to realizing what could be called a plug-and-play from the
+perspective of power supply.
+One difficulty here is that the power packet dispatching system has so far been developed using a wired connection
+for power transfer. Wireless power transfer (WPT) is a revolutionary technology for supplying power to mobile
+systems [11, 12]. It is beneficial for improved maneuverability of each local system to introduce the WPT to the
+∗This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this
+version may no longer be accessible.
+1
+arXiv:2301.05368v1  [eess.SY]  13 Jan 2023
+
+Packet
+network
+Packet
+network
+WPT
+router
+Load
+Storage
+Router
+Packet
+network
+Wired connection
+Wireless connection
+Router
+Router
+Router
+Router
+Router
+WPT
+router
+WPT
+router
+Power
+source
+Storage
+Storage
+Load
+Load
+Load
+Load
+Local system
+Local
+system
+Local system
+Connected system
+Power packet
+Figure 1: Surplus power supply via wireless power transfer between power packet networks.
+power packet dispatching system for connections at the boundaries of the local systems. However, as discussed in
+Section 3, simply connecting a WPT circuit to a power packet system does not work in conjunction with packet-
+based power management in local systems. This research seeks to achieve on-demand power supply concentration
+and dispersion in the connected network of local systems while ensuring easy attachment/detachment between
+systems via a wireless connection.
+Here, we investigate the following two points as fundamental studies to realize the wireless connection of mul-
+tiple local systems powered by power packets. First, we suggest a dedicated router design in both software and
+hardware configurations to ensure physical packetization with a wireless channel and collaboration with wired power
+management. Then, in a connected system comprised of three local systems with the developed router installed,
+we assert the selective transmission of power packets to only the local system designated by the tag. Second, we
+demonstrate a power-sharing strategy for increasing power capacity redundancy via a wireless connection. With
+wired and wireless connections, the parallel operation of intra- and intersystem power management is demonstrated.
+Among a network of two local systems, each of which supplies a certain demand of its own load via wired connection,
+surplus power at one system is transferred to another.
+Many proposals for the duplex of multiple channels in WPT have been made, including multiplexing in time,
+frequency, and spatial domain [13–15].
+Furthermore, several reports have addressed the simultaneous wireless
+transmission of power and information [4, 16–18]. These proposals essentially assume that a power transmission
+channel has already been established and that information is being transmitted concurrently, or vice versa. Our
+proposal, on the other hand, attempts to go beyond the simple parallel transmission by integrating information that
+manipulates the spatiotemporal distribution of power with power transmission itself at the physical layer. This, in
+theory, eliminates the disparity between physical quantity and information, allowing us to achieve both wired and
+wireless power transmission by cooperating for smart power management.
+2
+Outline of power packet dispatching system
+The basic configuration and operation of power packet dispatching systems are described in this section.
+2.1
+Constitution of power packet
+As depicted in Fig. 1, a power packet is a unit of power management in the system. A power packet comprises
+pulse-shaped electric power called a payload and an information tag, a header, and a footer, which are attached
+just before and after it. The information tag is a logic bitstream realized by a voltage waveform without current.
+The tag can include any information, like the origin, destination, and length of the power packet.
+The physical tag attachment enables power packet transmission to be time-division multiplexed. Power from
+different sources and destinations is transmitted on the same channel while remaining distinct from one another.
+2
+
+owel Packc
+voltage
+01000011
+.1100101
+time
+header
+payload
+: footerrouter input
+router output
+isolator
+controller
+gate driver
+gate driver
+storage1
+storage2
+demand signal
+clock
+input1
+input2
+output1
+output2
+power
+signal
+Figure 2: Circuit example of 2 input 2 output router [9].
+This feature sets the power packet dispatching system apart from conventional systems that treat power as a
+continuous flow.
+2.2
+Network configuration of power packet dispatching system
+Each local system of Fig. 1 denotes a local system configuration example. Routers connect power sources, storage,
+and loads to the network in this system.
+A power packet router is installed as a node that connects multiple
+transmission lines. The router forwards power packets by selecting a transmission line according to the packet’s tag
+information [10].
+A power packet is routed from a source to a specific destination via several routers. The path to the load is not
+required to be unique and can be changed dynamically depending on the situation. This feature facilitates flexible
+power management in conjunction with a dynamic supply relationship. In the following section, we develop a router
+that can perform this function even with a wireless connection.
+2.3
+Routing method for power packets
+Here we characterize the circuit configuration of a router and the principle of its routing operation [9, 19, 20].
+Figure 2 depicts the circuit configuration of a previously proposed, wire-connected router [21]. The circuit consists
+of two sections: an input section that receives power packets from the transmission network and an output section
+that forwards power packets to the transmission network. The operation of the input part is initialized when a
+power packet reaches the router. The input section includes a signal reading circuit for reading the logic signals
+of information tags. When the router recognizes that the incoming power packet is addressed to it, it turns on
+the corresponding semiconductor switches to receive the payload power. For circuit protection, the signal reading
+circuit electrically separates its signal output from the power supply lines using a device such as a photocoupler.
+The incoming power packet is temporarily stored before being forwarded to the next hop. The output part generates
+power packets from the temporal storage in response to the demand. In some cases, the circuit can be reduced to
+just the input or output section. When installed just next to the source, for example, the output section with the
+storage replaced by a power source is sufficient to produce a generated packet. Similarly, a circuit just before a load
+can only be the input part, with the storage replaced by a load.
+To read the logic signals of power packets, clocks corresponding to the one-bit width of a power packet must
+be synchronized among adjacent routers. This can be accomplished by installing an additional wire for a common
+clock input, or by adding another signal to the header for autonomous clock synchronization [22]. In this paper, we
+employ a simple autonomous clock synchronization scheme, in which the clock period is fixed in advance, the first
+three bits of the header are set to 010, and the phase is shifted if the 010 is not detected within a certain period.
+The information tag consists of bits 1–3, which implies 010 for clock synchronization, and bits 4–7, which imply
+the address of the output destination. Bits 8 – 100 correspond to the payload. For simplicity, the packet length is
+fixed at 100 bits and this setting is shared by all routers. In this way, we exclude the footer.
+3
+
+3
+Router design for wireless transmission of power packets
+In this section, we propose a router configuration for wireless transmission of power packets. We employ magnetic
+resonant coupling for the wireless transmission.
+This method is capable of transmitting large power over long
+distances with high-efficiency [11]. This circuit is powered by AC, whereas the power packet dispatching system is
+powered by DC. We must convert the current to incorporate WPT into power packet routing. Figure 3 depicts a
+conceptual diagram of the voltage and magnetic flux density in the wireless power packet transmission. Using a
+magnetic resonant coupling circuit that includes an inverter and a rectifier, DC is converted to AC and then back
+to DC after wireless transmission. The following section describes the router design.
+It should be noted that the inclusion of wireless transmission in the power packet dispatching system was first
+proposed in the authors’ previous report [23].
+In the report, the wireless transmission was not packetized but
+introduced as a one-to-one transmission channel without any tag attachment. In this paper, we propose a novel
+router configuration that bring the functions of physical tag attachment and its reading to the wireless power
+transfer. These functions not only realize the physical packetization of wireless power transmission but also extends
+its use to packet-based power management as introduced in Section 5.
+3.1
+Wireless transmitter of the power packet
+Figure 4 depicts a router circuit for wireless power packet output. The configuration includes an inverter circuit
+connected to the router’s output section, as described in Section 2.3. For DC/AC conversion, a class-E inverter [24]
+is used.
+The output circuit wirelessly transmits both the header signal and the payload power.
+In this paper, the
+inverter’s input is presented as a form of packetized power. The current flowing through the coil and the magnetic
+flux density induced in the coil is thus modulated in an amplitude-shift-keying (ASK) manner according to the shape
+of the power packet, as depicted in the middle of Fig. 3. It should be noted here that the header signal transmission
+must minimize power consumption while the payload transmission must maximize the amount of power transferred.
+The two requirements cannot be met solely through the transmitter’s operation, but rather through the design of
+the receiver side. This point will be covered in greater detail in the following section.
+3.2
+Wireless receiver of the power packet
+To receive a wirelessly transmitted power packet, demodulation of the ASK-modulated header signal and highly
+efficient AC/DC conversion of the payload are necessitated.
+The proposed circuit shown in Fig. 5 meets both
+requirements by dividing the demodulator into two circuits. The signal demodulation circuit reads the header,
+and a class-E rectifier receives the payload. The detailed procedure is provided below. Initially, the switch Sd
+connected to the signal demodulation circuit is turned on, while the SR connected to the rectifier circuit is turned
+off. For signal demodulation, the envelope of the voltage across the secondary circuit’s resonant capacitor is passed
+through an RC low-pass filter. The router’s controller then samples it at a predetermined clock cycle to convert
+it into a logical sequence. The controller activates the switches that connect the coil to the rectifier circuit when
+it determines from the tag that the power packet is addressed to itself. This causes a class-E rectifier to convert
+the wirelessly transmitted payload into DC output. When the power packet is directed at another router, the
+router’s controller disconnects both circuits and opens the coil. The detachment is used to avoid the influence of the
+unintended connection and the resulting impedance change, which may degrade power transmission at the addressed
+connection. At the end of the previous power packet, the controller turns on the switch to the demodulation circuit
+to prepare for the next power packet. The end of a power packet is detected by simply counting the length of the
+payload in 100-bit intervals.
+Of course, simply connecting the signal demodulation circuit and the Class-E rectifier in parallel allows you to
+read the header and receive the payload. However, when receiving the header, the current passes through the Class-
+Encode
+Power packet (DC)
+ASK modulated AC
+Packet's header
+Packet's payload (DC pulse)
+Decode
+on Wire
+on Wire
+Wireless
+Figure 3: A waveform concept during wireless transmission of power packets.
+4
+
+C2
+r1
+L1
+Lm
+Lf1
+C1
+S1
+Controller
+Gate driver
+Source
+Figure 4: Wireless transmitter of the power packet.
+C3
+r2
+L2
+Lm
+D1
+Lf2
+C4
+Cf
+Rectifier
+Rd
+Dd
+Cd2
+Demodulator
+Isolator
+Controller
+Gate driver
+Cd1
+Output
+Sd
+SR
+Figure 5: Wireless receiver of the power packet.
+E rectifier, and when receiving the payload, it passes through the demodulation circuit. Such a current contributes
+nothing to the receiving operation but results in power loss. Because this type of loss is much greater than the loss
+caused by the switching of the two demodulation circuits, the proposed scheme can greatly reduce the loss.
+The frequency of the carrier wave used for magnetic resonant coupling is 1 MHz. The wireless router’s constants
+are determined as shown in Table 1. The design is conducted in the following manner, regarding [24]. The coil has
+a diameter of 100 mm, a wire diameter of 1 mm, several turns of 10, and a thickness of 12 mm. The transmission
+circuit’s rise time was measured to be 25 µs. The rise time is defined as the time required for the output voltage to
+attain 90 % of its steady-state value. The steady-state value was obtained under the test condition where the load
+was 47 Ω resistor and the vertical distance between the coils was 50 mm. Based on this, we determined that the bit
+width of the power packet should be 100 µs, which is sufficiently larger than the rise time. That is, the modulation
+frequency is 10 kHz. The demodulation circuit is designed to demodulate signals with a cutoff frequency of about
+100 kHz.
+Table 1: Design values of circuit constants.
+Primary side
+Secondary side
+Rectifier
+Demodulator
+f
+1 MHz
+L2
+19.2 µH
+Cd1
+1.0 µF
+Lf1
+100 µF
+r2
+0.88 Ω
+Cd2
+820 pF
+C1
+3.3 nF
+C3
+1.56 nF
+Rd
+12 kΩ
+C2
+1.44 nF
+C4
+1.68 nF
+L1
+19.3 µH
+Lf2
+100 µH
+r1
+0.88 Ω
+Cf
+0.47 µF
+Lm
+1.75 µH
+4
+Verification of selective reception of wirelessly transmitted power
+packets
+In this study, we consider one-to-many or many-to-many wireless power sharing among several local systems placed
+close to each other. The packetization and time-division multiplexing methods enable simultaneous supplies between
+different pairs of a transmitter and a receiver while completely distinguishing them. Here, we experiment with three
+5
+
+Local system 0
+Wireless
+packet
+encoder
+V
+Local system 1
+Local system 2
+R1
+Wireless
+packet
+decoder
+Coil 0
+Coil 1
+Coil 2
+50mm
+70mm
+30mm
+R2
+Wireless
+packet
+decoder
+Figure 6: Network configuration with 3 local systems for verification of selectivity of wirelessly transmitted power
+packet.
+Figure 7: Verification of router operation mode for wirelessly transmitted power packets.
+local systems, one transmitting and two receiving nodes. It is demonstrated that the two receivers can selectively
+accept or reject power packets based on the attached information tag. The number of local systems and their
+connection relationship can of course be easily expanded and modified due to packetization.
+4.1
+Experimental setup for selective reception
+The entire network configuration is depicted in Fig. 6. Local system 0 alternately sends power packets to local
+systems 1 and 2, and systems 1 and 2 receive only those that match their addresses. Power packet header addresses
+are set to 0001 and 0010 for systems 1 and 2, respectively. Local system 0 consists of a circuit from Fig. 4 and a
+DC power supply of 12 V. Local systems 1 and 2 comprise a circuit of Fig. 4 with a load resistor of 47 Ω connected
+to the output port.
+Although the transmitting and receiving roles of the local systems are fixed for simplicity, it is possible to
+transmit power packets bidirectionally by modifying the circuit configuration [25]. Therefore, this assumption will
+not lose generality in power sharing.
+To ensure that the router’s operation is not affected by the distance between the coils, the coil positions are set
+as shown in Fig. 6. The coils of local systems 1 and 2 are placed at the same vertical distance 50 mm as the coil of
+local system 0, but the horizontal distance is 30 mm and 70 mm, respectively.
+4.2
+Receiving mode confirmation
+First, we examine the switching behavior between the header signal demodulator and the payload rectifier circuit,
+as designed in Section 3. Figure 7 depicts an internal signal of the router of local system 1 that represents the
+receiver’s operation mode. The router was in the header mode every 10 ms, which corresponded to the transmission
+cycle of the powder packets. Immediately after the header mode, the router switched to the payload mode every two
+power packet deliveries. During the payload mode, power was supplied to the designated load. This suggests that
+the controller received the header while connected to the demodulation circuit and then switched to the rectifier
+6
+
+payload mode
+headermode
+5.0
+2.5
+0.0
+-10
+0
+10
+20
+30
+time/ msFigure 8: Voltages at two loads in local systems 1 and 2.
+Router m2
+Router l2
+Router rx
+Router l1
+Router tx
+Router m1
+part 
+� : Local system 1
+part � : Local system 2
+part � : Wireless power sharing
+V1
+Rl1
+V2
+Rl2
+VCtx
+Vm1
+VCl1
+VCrx
+Vrx
+VCl2
+Vm2
+VRl1
+VRl2
+Wireless
+packet
+encoder
+Wireless
+packet
+reader
+Wireless
+packet
+decoder
+CNTL rx1
+Srx1
+CNTL m1
+Sm1
+CNTL m2
+Sm2
+CNTL tx2
+Stx2
+CNTL rx2
+Srx2
+CNTL l1
+Sl1
+CNTL l2
+Sl2
+CNTL tx1
+Stx1
+Ctx
+Crx
+Cl1
+Cl2
+Figure 9: Configuration of the network with 2 local systems connected wirelessly.
+circuit in payload mode after recognizing the address. This result confirms that the proposed router can correctly
+route power packets on the wireless channel.
+4.3
+Confirmation of selective reception function
+Second, we confirm that, according to the tag information, the local systems received time-division multiplexed
+power packets. The load voltages of the routers of local systems 1 and 2 are depicted in Fig. 8. It can be seen that
+local systems 1 and 2 received power alternately, indicating that they selectively accepted or denied receiving power
+packets based on the attached destination address signal. Here, local system 1’s supply voltage was higher than
+that of local system 2. This is because the output is proportional to the distance between the coils. This means
+that, regardless of whether the output value is larger or smaller, the router’s selective reception is unaffected by the
+difference in distance between the coils.
+5
+Confirmation of power-sharing in the wirelessly connected systems
+Next, in wirelessly connected local systems, we validate power management based on power packetization. We
+consider two local systems where the local power supply is primarily managed via a wired connection. Every local
+system consists of an internal power source, a capacitor, a wireless transmission circuit, and a resistive load. We
+set a wirelessly connected networking system comprising two such local systems, as shown in Fig. 9. While each
+local system supplies its source to its load, wireless power packet transmission compensates for excess or deficient
+power. Each system’s goal is to keep the voltage supplied to the load above a certain level.
+The proposed scheme deals with a connected system whose elements are subject to dynamic changes, such as
+variable distance between local systems and time-dependent connection/disconnection of local systems. Dealing
+with such dynamic changes altogether in a centralized controller is not ideal. Distributed control of power packet
+transmission, however, is an effective method of accommodating such unpredictability. In this paper, we use a
+distributed control scheme of packet-based power management [26], in which power packet transmission is managed
+7
+
+Load voltage/V
+10
+local system 1
+local system 2
+0
+-20
+-10
+0
+10
+20
+time/msonly between adjacent routers. The following section describes the operation flow of the connected systems.
+5.1
+Operation flow in connected systems
+Capacitors are installed in the connected systems to generate and output power packets to the load. Power packets
+are sent so that the voltages of these capacitors exceed a certain threshold.
+The demand signal to the router for on-demand packet transmission can be given by information tags in power
+packets or by using another channel such as radio signals [26]. In this paper, we use an external wire to transmit
+demand signals for simplicity We designed an input high to the controller of the next router when the storage
+voltage falls below the threshold.
+We divide the configuration of Fig. 9 into the following three parts that are managed independently.
+α Transmission from router m1 to router tx and router l1
+β Transmission from router tx to router rx
+γ Transmission from router rx and router m2 to router l2
+The three parts’ basic operation principles are described below.
+In part α, when the voltages across Ctx and Cl1 fall below the threshold, demand signals are transmitted to the
+router m1 respectively. Router m1 generates and sends power packets to the destination from which the demand
+signal is received. In the event of overlapping demand signals, priority is given to router l1 to keep the load voltage
+stable.
+In part β, router rx sends a demand signal to router tx when the voltage across Crx falls below the threshold.
+Router tx generates and sends power packets to router rx based on the demand signal.
+In part γ, when the voltage across Cl2 drops below the threshold, a demand signal is initially sent to router rx.
+If a power packet is not delivered from router rx to router l2 within a certain amount of time, the demand signal
+is sent to router m2, which generates and sends a power packet to router l2.
+Besides the three principles, two constraints are imposed on the operation of routers tx and rx. First, they
+do not output power packets if the voltages across its capacitor, Ctx or Crx, are lower than a certain value. To
+transmit power packets, there must be an adequate potential difference between the source and the destination.
+This constraint guarantees the possible difference between the source and destination capacitors and guarantees
+the reliable transmission of power packets. Second, the routers are not enabled to input and output power packets
+simultaneously. When both switches are switched on simultaneously, the circuits before and after the router are
+linked parallel. In this case, the output impedance measured from the power supply (capacitor) located before the
+router is lower than when only the input switch is turned on. This can result in an overcurrent at the source and
+a rapid drop in capacitor voltage. The second constraint is levied to avoid this situation. This configuration may
+prevent the router rx from emitting power on occasion. Even if this occurs, router m2 can supply power packets to
+keep router l2’s voltage stable.
+5.2
+Verification of connected systems operation
+To test the operation of the connected systems, we set the supply voltages V1 =15 V and V2 =7 V. To create
+a voltage gradient, the threshold voltages of capacitors Cl1, Ctx, Crx and Cl2 are set as 10 V, 9 V, 7 V and 5 V,
+respectively. The parameters linked to wireless power transmission are set as depicted in Table 1
+It is worth noting that the routers’ wired channel switch units have been replaced with unidirectional ones. As
+previously discussed, the symmetry of the circuit allows us to restrict the flow of power packets to one direction
+without sacrificing generality. The circuit generates high by activating switch Sout−s, and low by activating switch
+Sout−p. The diode prevents reverse current from flowing through the body diode of Sout−s.
+5.2.1
+Confirmation of autonomous maintenance of capacitor voltage
+We demonstrate the transmission of power packets and the modifications in voltages of each capacitor installed in
+part α–γ.
+Figure 10 depicts the voltages Vl1 and Vtx of the capacitors Cl1 and Ctx in part α and the gate signal of
+the switches Sl1 and Stx1 that controlled the route of the power packets. It is observed that Vl1 and Vtx were
+sustained above the threshold voltages. The voltages Vl1 and Vtx elevated when switches Sl1 and Stx1 were driven.
+This demonstrates that capacitors Cl1 and Ctx effectively received power packets and were charged. Furthermore,
+8
+
+Figure 10: State of switches and voltage of capacitors in part α.
+Figure 11: State of switches and voltage of capacitors in part β.
+switching operation of Sl1 and Stx1 did not overlap at any time.
+This result correlates to the setup that the
+transmission of power packets to Cl1 is prioritized (see Section 5.1 for the details).
+Figure 11 depicts the voltages Vtx and Vrx of the capacitors Ctx and Crx in part β and the gate signal of the
+switch Srx that controlled the power packet reception of the router rx. Comparing the top and bottom graphs shows
+that Vtx declined and Vrx elevated while Srx was on. This implies that power packets were wirelessly transmitted
+successfully from router tx to router rx. It can also be validated that Srx turned off when Vtx attained the threshold
+voltage. This implies that the system satisfied the constraints defined in Section 5.1, which hampers the output of
+power packets under the threshold voltage.
+Figure 12 depicts the voltages Vrx and Vl2 of the capacitors Crx and Cl2 in part γ and voltage waveforms of power
+packets outputted from routers rx and m2. When Vl2 dropped below the threshold voltage, router rx transferred
+power packets to router m2 so that Vl2 was kept above the threshold. Now let us concentrate on the operation
+around t =25 ms when Vrx attains the threshold voltage. Router rx stopped outputting the power packets, and
+simultaneously, router m2 started sending power packets. These findings suggest that the selective routing protocol
+specified in Section 5.1 worked; the load sent the demand signal to the router rx at first, and if no packet was
+transmitted, then sent to the router m2.
+In Fig. 12, there exists a possible difference between the voltage of the power packet and Vl2. This was induced
+by the forward voltage drop across the diode installed to prevent backflow current. This loss can be repressed by
+using a switch instead of a diode. Thus, there is no impact on the verification of the principle.
+Figure 13 depicts the gate signal of Stx1, output current from Ctx, input current to Crx, and the output voltage
+waveforms of router rx.
+When Ctx was outputting current, Crx was receiving current.
+This implies that the
+transmitted power packet was received without failure. Since router tx did not output power packets when Stx1 was
+9
+
+15
+10
+5
+Vcrx
+V2
+0
+100
+50
+0
+50
+100
+6
+Gate signal / V
+Sr1
+Stx1
+2
+0
+-100
+50
+0
+50
+100
+Time / msOutput voltage/V
+15
+10
+5
+Vctx
+Vcrx
+100
+50
+0
+50
+100
+6
+signal /V
+Srx1
+Gate
+2
+0
+100
+50
+0
+50
+100
+Time/msFigure 12: Power packets and voltage of capacitors in part γ.
+Figure 13: Input / output current and voltage of Ctx and Crx.
+on, Stx1 and Stx2 were driven solely. Similarly, Srx1 and Srx2 were driven solely. From the above findings, it can
+be deduced that the connected system achieves the load voltage maintenance with wireless power supply between
+local systems 1 and 2 by following the control procedure defined in Section 5.1.
+5.2.2
+Association between the percentage of power supply and the utilized power source
+The percentage of power transferred on the wireless channel depends on the distance between the transceiver/receiver
+coils. Hence, in the previous experiment’s setup, increasing the coil gap reduced the power supply capability from
+the local system 1 to 2. The proposed control scheme of the routers can accommodate such a gap change by choosing
+an appropriate supply channel. To test this operation, we compare the amount of wireless power transmission and
+the power source selection in the local system 2 at various distances between the coils. We set three cases with
+different distances: (i) 50 mm (the same as in the previous experiment), (ii) 100 mm, and (iii) > 250 mm. The setup
+in case iii is supposed to be large enough to prevent wireless transmission.
+Figures 14 and 15 depict the voltage Vrx and Vl2 and the power packets output by routers rx and m2 in cases ii
+and iii. Please refer to Fig. 12 for the result in case i. The larger the distance between the coils, the less frequently
+the router rx outputted power packets and the lower its average voltage got. On the other hand, Vl2 maintained
+above the threshold in all cases.
+Table 2 demonstrates the average of the input/output power of router rx and the output power of router m2
+10
+
+Output voltage/V
+15
+Vcrx
+Vi2
+10
+5
+0
+-100
+50
+0
+50
+100
+Output voltage/V
+15
+Vrx
+Vm2
+10
+5
+0
+-100
+50
+0
+50
+100
+Time / ms5
+1.25
+Gate signal of Stx1
+Output current from Ctx
+1.00
+signal
+Current / A
+3
+0.75
+2
+0.50
+Gate
+1
+0.25
+0
+0.00
+100
+-50
+0
+50
+100
+12.5
+1.25
+Vcrx
+10.0
+1.00
+A
+7.5
+0.75
+Current /
+5.0
+0.50
+2.5
+0.25
+0.0
+0.00
+-100
+-50
+0
+50
+100
+Time / msFigure 14: Power packets and voltage of capacitors in part γ of case (ii) : gap 100 mm.
+Figure 15: Power packets and voltage of capacitors in part γ of case (ii) : gap 250 mm.
+during the measured time 250 ms for different distances. The input/output power of router rx fell and the output
+power of router m2 rose as the distance became larger. Meanwhile, the total output power of router rx and router
+m2 had a slight change. This finding implies that the output power of router m2 compensates for the fall in the
+output power of router rx.
+From the above, it is asserted that the load voltage can be sustained autonomously by the proposed distributed
+control scheme. Even when the amount of wireless transmission falls, the local system compensated for it with a
+wired supply.
+6
+Conclusion
+In this paper, we developed a platform for wireless power packet transmission for power management among
+numerous local systems.
+First, we proposed a novel power packet router configuration capable of wireless transmission. The ASK modu-
+lating circuit is installed on the router’s output side for both information and power transmission, with the power
+packet serving as a power source. The input side includes a demodulation circuit for both information and power
+receipt. The circuit shifts between a signal demodulation circuit and a power rectifier circuit to read the header
+and receive the payload power, respectively. Not only does the switching configuration separate the incoming signal
+and power, but it also reduces unnecessary power consumption during the receiving operation.
+Using this router, we then verified the wireless power packet routing following the information tag. Physical tag
+attachment and wireless power packet time-division multiplexing allowed receiving routers to distinguish the power
+packet based on its destination address. The result shows that the proposed configuration allows for the selective
+11
+
+outputvoltage/V
+15
+Vrx
+Vi2
+10
+5
+0
+100
+50
+0
+50
+100
+output voltage/V
+15
+Power packet from Crx
+Power packet from V2
+10
+0
+-100
+50
+0
+50
+100
+time / msoutputvoltage/V
+.5
+Vrx
+Vi2
+10
+100
+50
+0
+50
+100
+outputvoltage/V
+15
+Power packet from Crx
+Powerpacketfrom V2
+5
+-100
+50
+0
+50
+100
+time / msTable 2: Input/output power of the routers in local system 2 at each gap.
+Case
+Gap
+Router rx
+Router rx
+Router m2
+Total
+input
+output
+output
+output
+i
+50 mm
+0.50 W
+0.46 W
+0.73 W
+1.19 W
+ii
+100 mm
+0.20 W
+0.17 W
+0.94 W
+1.11 W
+iii
+> 250 mm
+0.00 W
+0.00 W
+1.13 W
+1.13 W
+transmission of wireless power packets between multiple nearby local systems. This prevents interference with the
+irrelevant power supply.
+Next, we considered flexible coordination of inter- and intrasystem power management. The former was ac-
+complished through the wireless transmission of power packets, while the latter was accomplished through a wired
+supply. For this purpose, we created a distributed control scheme for the routers. A local system transmitted power
+packets wirelessly to another when it had enough power while keeping the voltage of its load as a top priority. The
+experiments revealed that the two types of operation were coordinated successfully. Furthermore, the proposed
+distributed control scheme chose an appropriate supply channel based on the power interaction availability between
+the local systems. We validated this operation by altering the gap between the coils of the two local systems,
+demonstrating that the inter- or intrasystem power management was successfully chosen to satisfy the local loads’
+demand.
+From the above verifications, we deduce that wireless power packet transmission can improve power management
+capability in a connected power packet dispatching system by selectively cooperating wired and wireless power packet
+transmission.
+Acknowledgments
+This work was partially supported by JSPS KAKENHI 20H02151, JST-OPERA Program no. JPMJOP1841, and
+SIP Cross Ministerial Strategic Innovation Promotion Program no.18088028.
+References
+[1] E. Dialynas and N. D. Hatziargyriou, “Impact of microgrids on service quality,” 2007 IEEE Power Engineering
+Society General Meeting, PES, pp. 1–5, 2007.
+[2] M. M. He, E. M. Reutzel, X. Jiang, R. H. Katz, S. R. Sanders, D. E. Culler, and K. Lutz, “An architecture
+for local energy generation, distribution, and sharing,” in Proc. IEEE Energy 2030 Conf., Atlanta, GA, USA,
+Nov. 2008, pp. 1–6.
+[3] M. Farhadi and O. Mohammed, “Adaptive Energy Management in Redundant Hybrid DC Microgrid for Pulse
+Load Mitigation,” IEEE Trans. Smart Grid, vol. 6, no. 1, pp. 54–62, Jan. 2015.
+[4] H. Wu, H. Tian, G. Nie, and P. Zhao, “Wireless Powered Mobile Edge Computing for Industrial Internet of
+Things Systems,” IEEE Access, vol. 8, pp. 101539–101549, 2020.
+[5] T. L. Vandoorn, B. Zwaenepoel, J. D. M. De Kooning, B. Meersman, and L. Vandevelde, “Smart microgrids
+and virtual power plants in a hierarchical control structure,” in 2011 2nd IEEE PES Int. Conf. Exhibition on
+Innovative Smart Grid Technologies.
+IEEE, Dec. 2011, pp. 1–7.
+[6] S. F. Tie and C. W. Tan, “A review of energy sources and energy management system in electric vehicles,”
+Renewable and Sustainable Energy Rev., vol. 20, pp. 82–102, 2013.
+[7] H. Sugiyama, M. Chatani, R. Simizu, and K. Yasui, “Pulsed power network with inherent operating procedure
+and multiple relaying of power routers,” in Proc. 2017 IEEE 6th Glob. Conf. Consumer Electronics, Japan,
+Oct. 2017, pp. 1–2.
+[8] E. Gelenbe and E. T. Ceran, “Energy Packet Networks With Energy Harvesting,” IEEE Access, vol. 4, pp.
+1321–1331, 2016.
+12
+
+[9] R. Takahashi, K. Tashiro, and T. Hikihara, “Router for power packet distribution network: Design and
+experimental verification,” IEEE Trans. Smart Grid, vol. 6, no. 2, pp. 618–626, Mar. 2015.
+[10] R. Takahashi, S. Azuma, M. Hasegawa, H. Ando, and T. Hikihara, “Power Processing for Advanced Power
+Distribution and Control,” IEICE Trans. Commun., vol. E100.B, no. 6, pp. 941–947, Jun. 2017.
+[11] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljaˇci´c, “Wireless Power Transfer via
+Strongly Coupled Magnetic Resonances,” Science, vol. 317, no. 5834, pp. 83–86, Jul. 2007.
+[12] A. Karalis, J. D. Joannopoulos, and M. Soljaˇci´c, “Efficient Wireless Non-radiative Mid-range Energy Transfer,”
+Annu. Physics, vol. 323, no. 1, pp. 34–48, 2008.
+[13] X. Hou, Z. Wang, Y. Su, Z. Liu, and Z. Deng, “A Dual-Frequency Dual-Load Multirelay Magnetic Coupling
+Wireless Power Transfer System Using Shared Power Channel,” IEEE Trans. Power Electronics, vol. 37, no. 12,
+pp. 15717–15727, 2022.
+[14] H. Ota, J. Liu, Y. Miura, Y. Yanagisawa, A. Wada, S. Sakabe, H. Bevrani, and T. Ise, “Multiphase Direct
+AC Wireless Power Transfer System: Comparative Proposals Using Frequency and Amplitude Modulations,”
+IEEE Journal of Emerging and Selected Topics in Industrial Electronics, vol. 2, no. 2, pp. 101–112, 2021.
+[15] N. Shinohara, “History and Innovation of Wireless Power Transfer via Microwaves,” IEEE Journal of Mi-
+crowaves, vol. 1, no. 1, pp. 218–228, 2021.
+[16] A. Kawamura, K. Ishioka, and J. Hirai, “Wireless Transmission of Power and Information through One High-
+frequency Resonant AC Link Inverter for Robot Manipulator Applications,” IEEE Trans. Industry Applica-
+tions, vol. 32, no. 3, pp. 503–508, 1996.
+[17] M. Mase, N. Shinohara, T. Mitani, and S. Ishino, “Evaluation of efficiency and isolation in wireless power
+transmission using orbital angular momentum modes,” 2021 IEEE Wireless Power Transfer Conf. (WPTC),
+pp. 1–4, 2021.
+[18] M. A. Hossain, R. M. Noor, K. L. A. Yau, I. Ahmedy, and S. S. Anjum, “A Survey on Simultaneous Wireless
+Information and Power Transfer with Cooperative Relay and Future Challenges,” IEEE Access, vol. 7, pp.
+19166–19198, 2019.
+[19] S. Inagaki, S. Mochiyama, and T. Hikihara, “Electric power processing using logic operation and error
+correction,” Royal Society Open Science, vol. 8, no. 7, p. 202344, Jul. 2021.
+[20] S. Mochiyama and T. Hikihara, “Packet-based Feedback Control of Electrical Drive and Its Application to
+Trajectory Tracking of Manipulator,” International Journal of Circuit Theory and Applications, vol. 47, no. 4,
+pp. 612–632, 2019.
+[21] K. Tashiro, R. Takahashi, and T. Hikihara, “Feasibility of Power Packet Dispatching at In-home DC
+DistributionNnetwork,” in Proc. 2012 IEEE 3rd Int. Conf. Smart Grid Commun. (SmartGridComm).
+IEEE,
+nov 2012, pp. 401–405.
+[22] Y. Zhou, R. Takahashi, N. Fujii, and T. Hikihara, “Power Packet Dispatching with Second-order Clock
+Synchronization,” Int. Journal of Circuit Theory and Applications, vol. 44, no. 3, pp. 729–743, Mar. 2016.
+[23] T. Mamiya, S. Mochiyama and T. Hikihara, “An Experimental Study on Time Division Multiplexing of Wired
+and Wireless Power Transfer by Power Packets,” in Proc. 2021 IEEE 10th Glob. Conf. Consumer Electronics,
+Japan, Oct. 2021, pp. 882–884.
+[24] M. K. Kazimierczuk and C. Dariusz, Resonant Power Converters, 2nd ed.
+New Jersey: John Wiley & Sons,
+2011.
+[25] T. Hosotani and I. Awai, “A novel analysis of ZVS wireless power transfer system using coupled resonators,”
+2012 IEEE MTT-S Int. Microwave Workshop Series on Innovative Wireless Power Transmission: Technolo-
+gies, Systems, and Applications,2012, pp. 235–238.
+[26] S. Katayama and T. Hikihara, “Energy-on-Demand Control for Power Packet Dispatching via Single Power
+Line,” in Proc. 2018 IEEE Int. Telecommun. Energy Conf. (INTELEC).
+IEEE, Oct. 2018, pp. 1–5.
+13
+