diff --git a/-NE1T4oBgHgl3EQfUgOU/content/2301.03091v1.pdf b/-NE1T4oBgHgl3EQfUgOU/content/2301.03091v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7a43054b70da732cb2f6c20a1cc4110162f2bdff --- /dev/null +++ b/-NE1T4oBgHgl3EQfUgOU/content/2301.03091v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:629bffbca7adc4d57a3982f62c928c49ba8933a4d95de0f560cb5112170c6fd3 +size 288297 diff --git a/-NE1T4oBgHgl3EQfUgOU/vector_store/index.faiss b/-NE1T4oBgHgl3EQfUgOU/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1d2dcb1a5ca25cad6dcf9ea4545651ededbd4e41 --- /dev/null +++ b/-NE1T4oBgHgl3EQfUgOU/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:31ce46e77c89ffd20340e3e2d3f809aff1b11f9721cfb43950b5ae8ec5c6ab09 +size 2687021 diff --git a/-NE1T4oBgHgl3EQfUgOU/vector_store/index.pkl b/-NE1T4oBgHgl3EQfUgOU/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..6813123cc43dcfa619ec6e7fe7d4c5f7da45f5b2 --- /dev/null +++ b/-NE1T4oBgHgl3EQfUgOU/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2bd760e2924de1b7ef5bceb0db70d25faf07b86f0a922de5635d2c55ef796589 +size 90463 diff --git a/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf b/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c33daac882472298848dec2fc6b704f2393c34a1 --- /dev/null +++ b/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:97511271dad5b2fc492cd95305744e5c2f175e58ce2a69b0e9f6bf8cbcce05f4 +size 499082 diff --git a/-dE2T4oBgHgl3EQfmQc9/vector_store/index.faiss b/-dE2T4oBgHgl3EQfmQc9/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3e34f964ce9aeb4c5f9036dcf5df88bc92cd3fb9 --- /dev/null +++ b/-dE2T4oBgHgl3EQfmQc9/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:00d0436e38f49547e816f8861f99e5950bce1bcdbdd1291e78c2a1be348cfb8b +size 1703981 diff --git a/-tFAT4oBgHgl3EQfqR1A/vector_store/index.faiss b/-tFAT4oBgHgl3EQfqR1A/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8c407deff4ca0c0e205f5ebc1c209c0ef43c09d8 --- /dev/null +++ b/-tFAT4oBgHgl3EQfqR1A/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:79b16c352dffd8101d786d1dfdc56d48a54a99a9cc2cb469e0cad8dfa658a27e +size 720941 diff --git a/.gitattributes b/.gitattributes index d35119ff8315ec73249cb5948788972a03fb633e..5b8991c865906a973547c40995d9ffc4debe6261 100644 --- a/.gitattributes +++ b/.gitattributes @@ -2688,3 +2688,66 @@ ItAzT4oBgHgl3EQfVPwp/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex XtE2T4oBgHgl3EQfYgfk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text eNA0T4oBgHgl3EQfHP9d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text G9A0T4oBgHgl3EQfBv-A/content/2301.01980v1.pdf filter=lfs diff=lfs merge=lfs -text +-dE2T4oBgHgl3EQfmQc9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +1dFLT4oBgHgl3EQfpy-5/content/2301.12137v1.pdf filter=lfs diff=lfs merge=lfs -text +_NE1T4oBgHgl3EQfUwPr/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +jdAzT4oBgHgl3EQf4_6S/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +NNAyT4oBgHgl3EQf6_pQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +MdFJT4oBgHgl3EQfzC0_/content/2301.11641v1.pdf filter=lfs diff=lfs merge=lfs -text +A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf filter=lfs diff=lfs merge=lfs -text +MdFJT4oBgHgl3EQfzC0_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf filter=lfs diff=lfs merge=lfs -text +odAyT4oBgHgl3EQfzPlC/content/2301.00697v1.pdf filter=lfs diff=lfs merge=lfs -text +-tFAT4oBgHgl3EQfqR1A/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +EdE1T4oBgHgl3EQfWgTb/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +NdFQT4oBgHgl3EQfWTbR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +hdE_T4oBgHgl3EQf3hyR/content/2301.08347v1.pdf filter=lfs diff=lfs merge=lfs -text +m9AyT4oBgHgl3EQfyvnQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf filter=lfs diff=lfs merge=lfs -text +m9AyT4oBgHgl3EQfyvnQ/content/2301.00691v1.pdf filter=lfs diff=lfs merge=lfs -text +vdAyT4oBgHgl3EQfafdn/content/2301.00243v1.pdf filter=lfs diff=lfs merge=lfs -text +W9FQT4oBgHgl3EQfcjYS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +j9E0T4oBgHgl3EQf7QLf/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +A9FAT4oBgHgl3EQfrx7P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf filter=lfs diff=lfs merge=lfs -text +I9A0T4oBgHgl3EQfCP_M/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +M9E0T4oBgHgl3EQfTABv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +dNAyT4oBgHgl3EQfjPh9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +v9FST4oBgHgl3EQfQjh8/content/2301.13759v1.pdf filter=lfs diff=lfs merge=lfs -text +ctE4T4oBgHgl3EQfQAwO/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +kb_46/content/kb_46.pdf filter=lfs diff=lfs merge=lfs -text +cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf filter=lfs diff=lfs merge=lfs -text +-NE1T4oBgHgl3EQfUgOU/content/2301.03091v1.pdf filter=lfs diff=lfs merge=lfs -text +PtFJT4oBgHgl3EQf2C2D/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf filter=lfs diff=lfs merge=lfs -text +-NE1T4oBgHgl3EQfUgOU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf filter=lfs diff=lfs merge=lfs -text +xdAyT4oBgHgl3EQfa_cC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +4NAyT4oBgHgl3EQf1_nb/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +bdE_T4oBgHgl3EQfzRz6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +dNAyT4oBgHgl3EQfjPh9/content/2301.00411v1.pdf filter=lfs diff=lfs merge=lfs -text +HdFIT4oBgHgl3EQfXith/content/2301.11244v1.pdf filter=lfs diff=lfs merge=lfs -text +1dFLT4oBgHgl3EQfpy-5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +VdE1T4oBgHgl3EQfIwNu/content/2301.02943v1.pdf filter=lfs diff=lfs merge=lfs -text +v9FST4oBgHgl3EQfQjh8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +odAyT4oBgHgl3EQfzPlC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +CNFJT4oBgHgl3EQfASxa/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +4NAyT4oBgHgl3EQf1_nb/content/2301.00745v1.pdf filter=lfs diff=lfs merge=lfs -text +hdE_T4oBgHgl3EQf3hyR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ctE4T4oBgHgl3EQfQAwO/content/2301.04977v1.pdf filter=lfs diff=lfs merge=lfs -text +EdE5T4oBgHgl3EQfUw9y/content/2301.05546v1.pdf filter=lfs diff=lfs merge=lfs -text +CNFJT4oBgHgl3EQfASxa/content/2301.11419v1.pdf filter=lfs diff=lfs merge=lfs -text +wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf filter=lfs diff=lfs merge=lfs -text +EdE5T4oBgHgl3EQfUw9y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf filter=lfs diff=lfs merge=lfs -text +G9A0T4oBgHgl3EQfBv-A/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +CNE1T4oBgHgl3EQfDwNk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf filter=lfs diff=lfs merge=lfs -text +I9A0T4oBgHgl3EQfCP_M/content/2301.01988v1.pdf filter=lfs diff=lfs merge=lfs -text +SNE3T4oBgHgl3EQfzAs2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf filter=lfs diff=lfs merge=lfs -text +NNAyT4oBgHgl3EQf6_pQ/content/2301.00830v1.pdf filter=lfs diff=lfs merge=lfs -text +QtAyT4oBgHgl3EQfU_dt/content/2301.00135v1.pdf filter=lfs diff=lfs merge=lfs -text +BtE5T4oBgHgl3EQfTQ-D/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +wNFLT4oBgHgl3EQfki_9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +NNE0T4oBgHgl3EQf0QK5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/09FAT4oBgHgl3EQfjh3M/content/tmp_files/2301.08606v1.pdf.txt b/09FAT4oBgHgl3EQfjh3M/content/tmp_files/2301.08606v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b27587355c6d587296a2a1ab06ed14282a6a6ea --- /dev/null +++ b/09FAT4oBgHgl3EQfjh3M/content/tmp_files/2301.08606v1.pdf.txt @@ -0,0 +1,1117 @@ +Data Augmentation for Modeling Human +Personality: The Dexter Machine +Yair Neuman, Vladyslav Kozhukhov and Dan Vilenchik +January 23, 2023 +Abstract +Modeling human personality is important for several AI challenges, +from the engineering of artificial psychotherapists to the design of persona +bots. However, the field of computational personality analysis heavily re- +lies on labeled data, which may be expensive, difficult or impossible to +get. This problem is amplified when dealing with rare personality types +or disorders (e.g., the anti-social psychopathic personality disorder). In +this context, we developed a text-based data augmentation approach for +human personality (PEDANT). PEDANT doesn’t rely on the common +type of labeled data but on the generative pre-trained model (GPT) com- +bined with domain expertise. Testing the methodology on three different +datasets, provides results that support the quality of the generated data. +1 +Introduction +Personality concerns the individual’s relatively stable pattern of thoughts, emo- +tions and behaviors [1]. There are various personality theories from the Big Five +[2] to Affective Neuroscience [3] and Mischel’s contextual approach to person- +ality [4]. In this paper, we adhere to the clinical approach represented by the +Psychodynamic Diagnostic Manual (PDM) [5] and SWAP [6], which is highly +relevant for diagnosis, and research [7]. According to the PDM approach, per- +sonality types are stable configurations characterized by key features such as +the individual’s core beliefs about self and others. For instance, a depressive +personality is characterized by self-criticism and accompanied by the belief that +”something is essentially bad about me”. +Current computational personality research is almost exclusively focused on +features’-based data-driven classification involving the prediction of a person- +ality class label. Accomplishing such tasks relies on the availability of a large +amount of high-quality labeled data (e.g., [8, 9, 10]). However, obtaining such +data may be expensive, difficult, or impossible for various reasons. For instance, +the prevalence of the anti-social psychopathic personality disorder in the pop- +ulation is low +[11, 12, 13], and it is currently impossible to gain access to a +massive dataset of labeled texts produced by clinically diagnosed psychopaths. +High-quality diagnostic procedures, such as SWAP, are costly as they require +human expertise and significant time to complete. While self-reported question- +naires for personality assessment are available, they rely on the collaboration +of the diagnosed individual and their ability to provide a valid self-assessment, +1 +arXiv:2301.08606v1 [cs.CL] 20 Jan 2023 + +which in the case of the anti-social personality disorder, for instance, is not +trivial to gain. +In the face of these challenges, a natural solution for data scarcity is data +augmentation, intensively developed in computer vision but ”relatively under- +explored” in NLP, where the generation of effective augmented examples is ”less +obvious” [14, 15]. To illustrate the challenges in textual data augmentation, we +ran the SOTA data-augmentation pipeline LAMBADA [16] to generate 200 sen- +tences out of a seed set of 20 sentences (see appendix) expressing a clear psycho- +pathic signature. This attempt to produce artificial ”psychopathic” sentences +resulted in only 100 unique sentences, where the vast majority of sentences were +either one of the seed sentences or a simple paraphrasing thereof. +Data augmentation is typically viewed as the process of increasing the amount +of data by adding slightly modified copies of already existing labeled data. In +some cases, there is no labeled data at all or a very small quantity which pre- +cludes proper augmentation (as our experiment with LAMBADA suggests). In +this paper, we offer a solution to these cases by using unlabelled data and adding +domain expert input to compensate for the absence of labeled data (once can +view labeled data as domain expert knowledge). +A constructive approach to personality modeling may be found in the rev- +olutionary large language models recently introduced to NLP (e.g., GPT-2) +[17, 18]. Recently, [19] and [20] showed that the GPT model, once fine-tuned, +can be useful in the domain of personal conversations. Their approach led to +substantial improvements in the PersonaChat data set, showcasing the potential +of exploiting large pre-trained generative models in the conversational domain. +However, these advancements do not naively imply anything for modeling per- +sonality types, as the poor results obtained from LAMBADA, which is based +on GPT technology, show. Indeed, personalized chit-chat models, [21] use the +notion of personalization (e.g., age) which is different from the psychodynamic +approach used in this paper. +1.1 +Our contribution. +We present a novel personality data augmentation approach, PEDANT (PEr- +sonality Data AugmeNTation), using (1) a generative pre-trained model (GPT) +combined with (2) domain expertise (the domain expert is the first author who +has intensively studied and published about personality) while relying only on +(3) unlabeled text. +PEDANT operates in two phases. In the first phase, unlabeled data relevant +to the selected personality type is harvested from online resources; this data +is then used to train a generative language model. In the second phase, the +language model is repeatedly prompted to complete a set of seed sentences +carefully crafted by the domain expert. All these completions are then filtered +and ranked according to a scoring function that the domain expert pre-defined; +the top k sentences are the output of PEDANT. +We implement PEDANT with regard to a specific personality type: the anti- +social psychopathic personality [6]; we call this particular pipeline Dexter. This +type of personality is suitable for validating our approach as the prevalence of +psychopathic personality disorder is extremely low, and a labeled corpus of nat- +urally produced texts of diagnosed psychopaths does not exist. The texts that +we harvest for the first phase of PEDANT come from a few fictive characters +2 + +from the cinema and TV (e.g., Dexter the psychopath from the TV series ”Dex- +ter”) and from Reddit forums such as r/psychopath. The second phase, where +domain expertise is used, is described in detail in Sections 3.2 and 3.4. +We validated Dexter using a downstream text classification task, as common +in other works that deal with the evaluation of data augmentation pipelines +[22, 16]. We used the data generated by Dexter to train a classifier and then +tested it on three offensive-speech datasets that cover different dimensions of +the psychopathic personality (e.g., lack of empathy, toxicity, and being manip- +ulative). +For comparison, we tested two SOTA data augmentation pipelines, LAM- +BADA [16], and LeakGAN [23]. The Dexter dataset produced a classifier that +ranked first (by a large gap) in all three datasets. The complete detail of both +experiments appears in Section 5.2. +1.2 +Illustration +It is a non-trivial task to evaluate the extent to which the resulting genera- +tive model (the outcome of Dexter) reflects the psychopathic mind. One way +mentioned above is via a downstream task that uses data generated by the +model. Another way is to have a host of personality domain experts chat with +the model. While we did not have the resources to perform this expensive and +laborious task, we invite the reader to peek into such a possible Q&A session +and to judge for herself the change in personality. +Below is a comparison of the output of a GPT model before and after fine- +tuning on the harvested psychopathic-related texts. +In the examples below, +GPT is prompt with psychopathic statements chosen by the domain expert, and +its completion is recorded; a couple more examples are given in the appendix. +Statement: I act impulsively, without regard for the consequences of my actions. +GPT-2 before fine-tuning: I know when someone is suffering and I feel bad. +GPT-3 davinci: I have difficulty organizing tasks and activities. I have difficulty +with time management. . +GPT-3 curie: My friends are more important to me than my family. I dress in +a way that is not in keeping with the standards of my community. +chatGPT: It’s natural to act impulsively at times, but if you find that you fre- +quently act without considering the consequences of your actions, it may be +helpful to try to develop more self-control. +GPT-2 +after fine-tuning: I have an abundance of rage, and I can turn to it with no +consideration for consequences. +Statement: I feel like a hunter seeking a prey. +GPT-2 before fine-tuning: I don’t want to be a bad person. +GPT-3 davinci: I want to let my heart feel. +GPT-3 curie: I’m lost in this forest ! I’ll be crushed by this forest! +chatGPT: It’s important to remember that other people are not prey to be hunted. +It’s never okay to view or treat others as if they are objects or targets to be +pursued or exploited. +GPT-2 after fine-tuning: I don’t care about it, I just want to kill and eat. +3 + +2 +Related work +Data augmentation methods have been explored to address the imbalance of +datasets challenge in supervised classification tasks. Noise injection or attribute +modification techniques were commonly applied to generate synthetic data for +image and sound classification tasks [24, 25, 26]. However, such techniques do +not extend to text due to the categorical nature of words and the sequential +nature of text. +We locate our work in the context of text data augmentation, [14, 27, 16]. +“Previous-generation” textual data augmentation approaches focus on sample +alteration [28, 29, 30, 31, 32], in which a single sentence is altered to generate a +new sentence primarily by synonyms replacements. Presumably, methods that +make only local changes will produce sentences with a structure similar to the +original ones, thus yielding low corpus-level variability. +Other recent approaches to textual data augmentation that generate whole +sentences rather than making a few local changes include using variational au- +toencoding [33], paraphrasing [34] and methods based on generative adversarial +networks [35, 23, 36]. +Recent progress in NLP has been marked by the emergence of large lan- +guage models (i.e., transformers) such as GPT-2 [18]. +GPT-based language +models scored high in open-domain dialogue generation tasks [20, 19, 37]. The +data-augmentation pipeline presented in [16] uses GPT technology to generate +themed synthetic text. The idea behind [16] involves fine-tuning a GPT model +to a specific task using existing labeled data. Using the fine-tuned model and +given a class label, new sentences for the class are generated. The sentences are +filtered with a classifier trained on the original data. +While our pipeline is similar to [16] in flavor (a fine-tuning step followed by +a filtering step), it is different in two key aspects. We use unlabelled data for +the fine-tuning step. This allows us to fine-tune the GPT model with a large +amount of, possibly slightly lower quality, data. [16] use labeled data for the +fine-tuning step; thus the quality of the augmentation depends on the amount +of available text. Second, our filtering is also done in an unsupervised manner, +replacing the need for labeled data for training a classifier with the knowledge +of a domain expert. These two key differences make our pipeline useful for data +generation for rare classes, such as rare personality types, where labeled data +is scarce or non-existent. Indeed, comparing the performance of [16] to Dexter +corroborates the latter. +3 +Methodology +The pipeline for generating data for a given personality type, illustrated in +Figure 1, is composed of the following stages (the actual parameter values that +we’ve used are given in this general description): +1. Texts produced by a few fictive characters (e.g., Dexter) and secondary +sources (e.g., Reddit forums discussing the personality style) are collected +to form a preliminary dataset. Let D be that preliminary dataset. +2. A pre-trained language model is fine-tuned on D. Let G be the obtained +model +4 + +Figure 1: Pipeline for PEDANT. GPT-2 is trained and prompted to complete +a carefully chosen seed of sentences. The completions are filtered and ranked +using similarity to relevant words chosen using domain expertise. +3. A domain-expert hand-crafted set of s = 40 seed sentences representing +the personality’s beliefs about self/others is prepared; G is prompted to +complete each seed c = 200 times, for a total of n = s·c = 8000 candidate +sentences. +4. Based on domain expertise, a hand-crafted vector F, containing f words +that are typical of the personality type, is assembled. The n candidate +sentences are being filtered and ranked according to their cosine similarity +with F. +5. The top k = 2000 sentences compose the output. +We now describe how we customized this general pipeline to the psychopathic +personality. +3.1 +The preliminary datasets +Following [38, 39] we used data from movie scripts – the text produced by +three well-known fictive psychopathic characters: The Joker in the movie ”The +Joker”, Bateman in the movie ”American Psycho” and Dexter from the TV +series ”Dexter”. In addition, we collected all texts from Reddit discussion groups +dealing with psychopathy (r/psychopath, r/sociopath, r/antisocial). +After cleaning the data by applying a spell checker [40], removing emojis, +duplicates, hyperlinks, and spam messages [41], the preliminary cleaned dataset +consisted of 1,320,552 tokens. +3.2 +The sentences completion seed set +Our domain expert manually prepared 20 seed sentences representing the psy- +chopath’s ”beliefs about self” (e.g., ”I take advantage of others whenever I can”) +5 + +OCN +Candidate +GPT-2 +Data +Sentence +Bank +Sentences +for +completion +88 +Final set of +sentencesand 20 seed sentences representing the psychopath’s ”beliefs about others” (e.g., +”Human beings are weak”). The complete list of seed sentences appears in the +appendix. +The number ’20’ is somewhat arbitrary. While testing with other seed sizes, +we found that 20 was the minimal number that gave good results, considering +the computational constraints such as space and running time. +3.3 +Training GPT-2 and generating sentences +Our starting point is the pre-trained GPT-2 with 1.5B parameters accessed via +the popular HuggingFace API [42]. We chose GPT-2 as it is currently one of +the most useful language generation model. (The newer GPT-3 is still not open +source, and it’s harder to work with, e.g., fine-tuning on a large text like our +preliminary dataset). Next, we fine-tuned the pre-trained GPT-2 model on the +entire preliminary dataset (the fictive characters text and the Reddit data) using +the task of predicting the next word of the sentence [18]. The parameters we +used were: learning rate=0.0001, model name=‘1558M’, batch size=4, optimizer += ‘adafactor’, steps = 10000 and the cross entropy loss function. +We prompted the fine-tuned GPT-2 model on each of the 40 seed sentences +( see the appendix for the full list), producing 200 sentence completions for each +sentence. Using the experts’ judgment of two psychologists, we qualitatively +evaluated a random sample of these sentences. We concluded that the best com- +pletions were obtained with the following parameters: length=50,temperature=0.7, +top k = 50, top p = 0.90. +We used the free Google Colab resources with Tesla P100 GPUs for this +part. +3.4 +Filtering and Ranking +We applied filtering and ranking to the 8000 sentence completions. First, we +removed sentences that (1) include the trivial words: psychopath, antisocial, +and sociopath; (2) are duplicates of other sentences; (3) contain less than three +words; (4) end with a stop word; (5) are emotionally neutral or have a higher +positive than a negative sentiment (we used NLTK to estimate the sentiment); +(6) are simple paraphrases of each other (via [43]). +For the ranking task, we identified words significantly collocated with the +target word ”psychopath” in the iWeb repository [44]. +The domain expert +selected 28 to form a ”psychopathic vector” (see the appendix). Next, we used +the vectorial semantic approach for personality assessment [45] and measured +the cosine similarity between each filtered candidate sentence completion and +the psychopathy vector. For each of the 40 seed sentences, we selected the 50 +completions that scored highest on the cosine similarity test. The output was a +set of 1735 synthetically generated sentences that are supposed to represent a +psychopathic mind (if for some seed less than 50 completions passed the filtering +step, we took all of them). +6 + +4 +Data +Our setting inherently precludes the existence of a large labeled bench-marking +dataset of text written by clinically diagnosed psychopaths. However, as the +antisocial psychopathic personality is composed of several dimensions (e.g., lack +of empathy), we tested our approach on labeled datasets hypothesized to share +one dimension or more with this personality type. +4.1 +Test Data +We now describe the three datasets that we’ve used to evaluate the performance +of Dexter and two other data augmentation pipelines. +Sexual predators. Sexual predators share with psychopaths at least two psy- +chological dimensions: being manipulative and lacking empathy, as indicated +by the correlation between sexual offending and psychopathy [46, 47, 48]. Our +first dataset was a labeled data set of texts produced by 142 sexual predators +and 97,689 non-predators [49]. +Empathy. Psychopaths are characterized by a lack of empathy. Our second +data set consists of interactions between help seekers in an online call center +for mental health support [50]. Labeled texts of the mental health supporters +(responders) are provided. Responders are tagged according to three increasing +levels of empathy: “0” (N = 2037), “1” (N = 895), and “2” (N = 152). Unlike +the other two datasets, the empathy dataset does not contain a natural positive +class, as an empathy score of 0 does not necessarily imply a strong negative +personality. +Cyberbullying. Cyberbullying may have a clear psychopathic signature, given +the reported association between the psychopathic mind and sadism [51, 52]. +We have used the labeled toxic-text subset of the cyberbullying dataset [53] +that contains 12,168 toxic vs. 14,874 non-toxic texts. Unlike the previous two +datasets, this one is labeled at the message level. +Each message consists of +several sentences, and the entire message is assigned the label “toxic” if there +are “enough” toxic sentences (the exact labeling procedure is described in the +original paper [53]). +4.2 +Train Data +We now describe the datasets we used to fine-tune the BERT-base-uncased +model [54], which we then ran to classify the aforementioned test datasets. The +data statistics is summarized in Table 1. +The Dexter dataset. This dataset contains 3400 sentences; 1700 sentences +are the output of Dexter, which serve as the positive class, and 1700 sentences +from various Reddit discussion groups that serve as the negative class. These +sentences were selected by first sampling 8000 random sentences from various +Reddit groups, then filtering and cleaning them according to the same procedure +that was applied to the psychopathic texts. Finally, the 1700 sentences with the +lowest psychopathic score were chosen. +The Dexter-minus and the PRELIM dataset. These two datasets allow us +to evaluate the importance of the different stages in the Dexter pipeline (Figure +1). The Dexter-minus dataset follows the same pipeline as Dexter just that the +7 + +Dataset +#Pos. +#Neg +Sexual Predators +142 +97,689 +Empathy +2,037 +152 +Cyberbullying +12,168 +14,784 +Table 1: Summary statistics for the three datasets described in Section 4. The +number of samples in the positive and negative class are shown. +fine-tuning of the GPT is skipped. The PERLIM dataset shortcuts the GPT +step altogether and proceeds directly to the filtering and ranking step. +To compare the performance of Dexter against the two SOTA data-augmentation +pipelines, we created the following two synthetic datasets. +The LAMBADA dataset. To train LAMBADA, we used the Papers-With- +Code recommended implementation of LAMBADA [55]. We augmented GPT-2 +with two new classes, ”#beliefs about others” and ”# beliefs about self”. Each +class was seeded with the 20 sentences that our domain expert crafted (Section +3.2). All the training parameters and the code are available at [55]. +We then used the LAMBADA pipeline to generate 17,000 sentences, out +of which we chose the best 1700 (this was the recommendation of the authors +[16], to generate 10x more sentences than needed). Specifically, we generated +8500 from the “#beliefs about others” class and 8500 from “#beliefs about +self” (we used the following parameters for GPT-2: max length=50, top k = 10, +p = 0.85). We ranked the sentences the same way we ranked ours: using cosine +similarity to the psychopathic vector of words (Section 3.4). +The negative class of the LAMBADA dataset is the same as Dexter’s. +The LeakGAN dataset. We trained LeakGAN using the official LeakGAN +implementation [56]. LeakGAN is trained with the target class text. We used +all the text in the preliminary dataset to that end (Section 3.1) and the default +parameters from the official implementation. We generated 1700 sentences using +LeakGAN to serve as the positive class, and the negative class was the same as +Dexter’s. +External competition datasets. This collection contains three gold-standard +competition offensive speech datasets: OffenseEval [57], HatEval [58], and AbuseE- +val [59]. Each dataset contains roughly 10,000 labeled texts. We call the union +of all three datasets the Golden dataset. +5 +Evaluation +To evaluate Dexter we’ve created a family of models using the aforementioned +training datasets of Section 4.2. Each model is named X@BERT, which means +that the BERT-base-uncased model [54] was fine-tuned using dataset X. If a +’+’ is appended, X@BERT+, this means that a preceding step of fine-tuning on +OffenseEval [57] was taking place. +5.1 +Evaluation procedure +All the test datasets mentioned in Section 4.1, Table 1, are imbalanced to dif- +ferent degrees (the sexual predators dataset contained merely 0.001% sexual +8 + +Model +Pred. +Emp. +Cyber. +Avg Rank +Dexter@BERT+ +1 +1 +2 +1.33 +OffenseEval@BERT +3 +2 +1 +2 +Dexter@BERT +2 +4 +5 +3.66 +HateEval@BERT +6 +3 +6 +5 +Dexter-@BERT +4 +5 +7 +5.33 +Golden@BERT +7 +7 +3 +5.66 +AbuseEval@BERT +7 +7 +4 +6 +PRELIM@BERT +8 +8 +8 +8 +Table 2: Summary of Table 4. The Dexter@BERT variants are in bold. +Model +Pred. +Emp. +Cyber. +Avg Rank +Dexter@BERT+ +1 +1 +1 +1 +Dexter@BERT +2 +2 +3 +2.33 +LAMBADA@BERT+ +4 +4 +2 +3.33 +Dexter-@BERT +3 +3 +5 +3.66 +LeakGAN@BERT+ +5 +5 +4 +4.66 +LAMBADA@BERT +7 +6 +6 +6.33 +LeakGAN@BERT +6 +7 +7 +6.66 +Table 3: Summary of Table 5. The Dexter@BERT variants are in bold. +predators). To allow comparison across the three datasets while avoiding mis- +leading artifacts that such imbalanced data introduces, we down-sampled the +majority class to obtain balanced sets. +The output of a BERT model is a number in [0, 1], the result of the last layer +activation unit (soft-max in our case). This number may be thought of as the +probability that BERT assigns the instance to belong to the positive class (in +our case, ”psychopath”). We define the PsychoScore of a user as the average +output of the model over all the sentences produced by that user (each sentence +is scored separately by the model). It is common practice to feed the BERT +score into a simple classifier, like SVM, to find the optimal cut-off for the binary +classification task [60]. +To evaluate the model on each test data set, we computed the 5-fold cross- +validation F1 and Macro F1 scores. Each fold consisted of n = 100 randomly +sampled instances from each class, and split into 80% train and 20% test. We +trained a soft-margin kernel SVM (we used the default Python sklearn module +parameters C = 1, kernel = RBF) on the users’ PsychoScores and the corre- +sponding label. +5.2 +Results +The results of running the X@BERT models on the test datasets of Section +4.1 are summarized in Tables 4 and 5. Table 4 reprots the comparison against +the pre-trained offensive speech models, while Table 5 reports the comparison +against LeakGAN and LAMBADA. Table 2 summarizes Table 4 with the over- +all average ranking across the three datasets and similarly Table 3 summarizes +Table 5. Both show that the model Dexter@BERT+ ranked first, and Dex- +ter@BERT came second (Table 5) and third (Table 4) . +The following key +conclusions are read from the tables: +9 + +Data set +Model +Precision +Recall +F1 score +Macro F1 score +Dexter@BERT+ +0.92 +0.87 +0.89 +0.91 ± 0.029 +Dexter@BERT +0.91 +0.86 +0.88 +0.90 ± 0.037 +Sexual Predator Identification +OffenseEval@BERT +0.89 +0.87 +0.88 +0.90 ± 0.035 +Competition [49] +Dexter-@BERT +0.80 +0.93 +0.86 +0.88 ± 0.043 +HateEval@BERT +0.95 +0.5 +0.65 +0.75 ± 0.011 +Golden@BERT +0.88 +0.50 +0.63 +0.69 ± 0.095 +AbuseEval@BERT +0.73 +0.58 +0.51 +0.53 ± 0.133 +PRELIM@BERT +0.51 +1.00 +0.68 +0.38 ± 0.024 +Dexter@BERT+ +0.66 +0.80 +0.72 +0.70 ± 0.083 +OffenseEval@BERT +0.61 +0.81 +0.69 +0.64 ± 0.081 +Empathy [50] +HateEval@BERT +0.59 +0.65 +0.61 +0.59 ± 0.063 +Dexter@BERT +0.55 +0.88 +0.67 +0.54 ± 0.077 +Dexter-@BERT +0.51 +0.65 +0.57 +0.52 ± 0.075 +Golden@BERT +0.42 +0.93 +0.58 +0.38 ± 0.061 +AbuseEval@BERT +0.44 +0.85 +0.58 +0.37 ± 0.018 +PRELIM@BERT +0.16 +0.27 +0.22 +0.27 ± 0.080 +OffenseEval@BERT +0.92 +0.80 +0.85 +0.88 ± 0.044 +Dexter@BERT+ +0.96 +0.72 +0.83 +0.87 ± 0.048 +Golden@BERT +0.84 +0.87 +0.85 +0.87 ± 0.041 +Cyberbullying [53] +AbuseEval@BERT +0.89 +0.78 +0.82 +0.83 ± 0.050 +Dexter@BERT +0.93 +0.60 +0.72 +0.77 ± 0.051 +HateEval@BERT +0.86 +0.61 +0.71 +078 ± 0.075 +Dexter-@BERT +0.91 +0.56 +0.68 +0.77 ± 0.080 +PRELIM@BERT +0.80 +0.57 +0.67 +0.70 ± 0.012 +Table 4: Results of the various models on the test data sets sorted according to +macro F1 score. +• The results for the sexual predators place Dexter@BERT+ and Dex- +ter@BERT at the top two both with respect to the other data augmenta- +tion pipelines (Table 5) and with respect to the abusive speech BERT mod- +els (Table 4). In fact, both LAMBADA@BERT+ and LeakGAN@BERT+ +obtained worse results than the baseline OffenseEval@BERT. (F1 score of +0.88 vs 0.8 and lower). +• The results for the empathy dataset in Table 4 show that Dexter@BERT+ +obtained the highest F1 and macro F1 score. In Table 5 we see that the +performance of Dexter@BERT and its derivatives is far better than the +other two pipelines. We also observe a poorer overall performance than the +other two datasets, in accordance with the absence of a natural positive +class. +• The results for the cyberbullying dataset in Table 4 show that Dex- +ter@BERT+ scored at the top (Macro F1 score 0.87) together with Of- +fenseEval@BERT (0.88) and AbuseEval@BERT (0.87). +In Table 5 we +again see that Dexter@BERT+ came first, although this time, the gap +from LAMBADA@BERT+ is small. +• The performance of Dexter@BERT+ is similar to OffenseEval@BERT on +the sexual predators and cyberbullying data sets. This is to be expected +as these datasets have a clear offensive speech element. The more telling +result is the larger gap for the empathy dataset, 0.7 vs 0.64 in Macro F1 +10 + +Dataset +Model +Precision +Recall +F1 score +Macro F1 score +Dexter@BERT+ +0.92 +0.87 +0.89 +0.91 ± 0.029 +Dexter@BERT +0.91 +0.86 +0.88 +0.90 ± 0.037 +Sexual Predator Identification +Dexter-@BERT +0.80 +0.93 +0.86 +0.88 ± 0.043 +Competition [49] +LAMBADA@BERT+ +0.85 +0.73 +0.77 +0.80 ± 0.074 +LeakGAN@BERT+ +0.55 +0.99 +0.71 +0.51 ± 0.035 +LeakGAN@BERT +0.53 +0.85 +0.65 +0.51 ± 0.041 +LAMBADA@BERT +0.47 +0.80 +0.51 +0.30 ± 0.019 +Dexter@BERT+ +0.66 +0.80 +0.72 +0.70 ± 0.083 +Dexter@BERT +0.55 +0.88 +0.67 +0.54 ± 0.077 +Empathy [50] +Dexter-@BERT +0.51 +0.65 +0.57 +0.52 ± 0.075 +LAMBADA@BERT+ +0.40 +0.67 +0.50 +0.33 ± 0.071 +LeakGAN@BERT+ +0.30 +0.76 +0.41 +0.31 ± 0.122 +LAMBADA@BERT +0.81 +0.35 +0.40 +0.50 ± 0.145 +LeakGAN@BERT +0.28 +0.53 +0.36 +0.30 ± 0.040 +Dexter@BERT+ +0.96 +0.72 +0.83 +0.87 ± 0.048 +LAMBADA@BERT+ +0.95 +0.70 +0.80 +0.83 ± 0.060 +Dexter@BERT +0.93 +0.60 +0.72 +0.77 ± 0.051 +Cyberbullying [53] +LeakGAN@BERT+ +0.91 +0.56 +0.69 +0.71 ± 0.064 +Dexter-@BERT +0.91 +0.56 +0.68 +0.77 ± 0.080 +LAMBADA@BERT +0.88 +0.54 +0.66 +0.71 ± 0.078 +LeakGAN@BERT +0.97 +0.47 +0.62 +0.68 ± 0.114 +Table 5: Results of the various models that were trained by different text aug- +mentation techniques, sorted according to macro F1 score. +score. Indeed lack of empathy has more to do with the psychopathic mind +than offensive speech. +• One can look at our results through the lens of transfer learning, where +our dataset was successfully used to facilitate transfer learning from the +task of offensive speech detection to the task of predicting various aspects +associated with the psychopathic personality. +6 +Discussion +This paper presents a new unsupervised approach for personality data augmen- +tation (PEDANT), trading labeled data with domain expertise. We implement +it in a specific pipeline that generates sentences with a psychopathic signature +(Dexter). One could ask whether it is feasible to assemble a labeled dataset +via platforms such as Amazon Turk. The answer is probably no, as domain +expertise is required in the field of personality to correctly label the data. Our +work offers a scalable and feasible data augmentation pipeline that circumvents +such caveats by taking input from a domain expert in the later stages of the +pipeline rather than at the beginning (the data collection step). +The clear conclusion from the evaluation experiments we ran is that our +pipeline produced synthetic data with better quality than the other two pipelines +([16], and [23]) thus highlighting the point that not all data augmentation tasks +were born equal. The task of generating synthetic data about flight and travel +issues (the examples from the LAMBADA paper) is not the same as generating +personality-type text. The same way generating a synthetic dog picture is not +11 + +the same as generating a CT-scan picture of a brain with a tumor in order to +train med students to read such images. +We expect that our pipeline can be adapted to other domains where high- +quality labeled data is lacking and hard to obtain by crowd-sourcing: suicide, +school shooters, etc. +References +[1] Philip J Corr and Gerald Ed Matthews. The Cambridge handbook of per- +sonality psychology. Cambridge University Press, 2020. +[2] Lewis R Goldberg. An alternative” description of personality”: the big-five +factor structure. Journal of personality and social psychology, 59(6):1216, +1990. +[3] Kenneth L Davis and Jaak Panksepp. The brain’s emotional foundations of +human personality and the affective neuroscience personality scales. Neu- +roscience & Biobehavioral Reviews, 35(9):1946–1958, 2011. +[4] Walter Mischel. Toward an integrative science of the person. Annu. Rev. +Psychol., 55:1–22, 2004. +[5] Vittorio Lingiardi and Nancy McWilliams. Psychodynamic diagnostic man- +ual: PDM-2. Guilford Publications, 2017. +[6] Jonathan Shedler and Drew Westen. The shedler–westen assessment pro- +cedure (swap): making personality diagnosis clinically meaningful. Journal +of personality assessment, 89(1):41–55, 2007. +[7] Vittorio Lingiardi, Nancy McWilliams, Robert F Bornstein, Francesco +Gazzillo, and Robert M Gordon. The psychodynamic diagnostic manual +version 2 (pdm-2): Assessing patients for improved clinical practice and +research. Psychoanalytic Psychology, 32(1):94, 2015. +[8] Yair Neuman. Computational personality analysis: Introduction, practical +applications and novel directions. Springer, 2016. +[9] David E Losada and Fabio Crestani. A test collection for research on depres- +sion and language use. In International Conference of the Cross-Language +Evaluation Forum for European Languages, pages 28–39. Springer, 2016. +[10] Yoram Bachrach, Michal Kosinski, Thore Graepel, Pushmeet Kohli, and +David Stillwell. Personality and patterns of facebook usage. In Proceedings +of the 4th annual ACM web science conference, pages 24–32, 2012. +[11] Ana Sanz-Garc´ıa, Clara Gesteira, Jes´us Sanz, and Mar´ıa Paz Garc´ıa-Vera. +Prevalence of psychopathy in the general adult population: A systematic +review and meta-analysis. Frontiers in Psychology, page 3278, 2021. +[12] Kimberly B Werner, Lauren R Few, and Kathleen K Bucholz. Epidemiol- +ogy, comorbidity, and behavioral genetics of antisocial personality disorder +and psychopathy. Psychiatric annals, 45(4):195–199, 2015. +12 + +[13] Katherine J Holzer, Michael G Vaughn, Travis M Loux, Michael A Mancini, +Noelle E Fearn, and Cara L Wallace. Prevalence and correlates of antisocial +personality disorder in older adults. Aging & Mental Health, pages 1–10, +2020. +[14] Steven Y Feng, Varun Gangal, Jason Wei, Sarath Chandar, Soroush +Vosoughi, Teruko Mitamura, and Eduard Hovy. +A survey of data aug- +mentation approaches for nlp. arXiv preprint arXiv:2105.03075, 2021. +[15] Tri Dao, Albert Gu, Alexander Ratner, Virginia Smith, Chris De Sa, and +Christopher R´e. A kernel theory of modern data augmentation. In Inter- +national Conference on Machine Learning, pages 1528–1537. PMLR, 2019. +[16] Ateret Anaby-Tavor, Boaz Carmeli, Esther Goldbraich, Amir Kantor, +George Kour, Segev Shlomov, Naama Tepper, and Naama Zwerdling. Do +not have enough data? deep learning to the rescue! In Proceedings of the +AAAI Conference on Artificial Intelligence, volume 34, pages 7383–7390, +2020. +[17] Alec Radford, Karthik Narasimhan, Tim Salimans, Ilya Sutskever, et al. +Improving language understanding by generative pre-training. 2018. +[18] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya +Sutskever, et al. +Language models are unsupervised multitask learners. +OpenAI blog, 1(8):9, 2019. +[19] Thomas Wolf, Victor Sanh, Julien Chaumond, and Clement Delangue. +Transfertransfo: A transfer learning approach for neural network based +conversational agents. arXiv preprint arXiv:1901.08149, 2019. +[20] Sergey Golovanov, Rauf Kurbanov, Sergey Nikolenko, Kyryl Truskovskyi, +Alexander Tselousov, and Thomas Wolf. Large-scale transfer learning for +natural language generation. In Proceedings of the 57th Annual Meeting of +the Association for Computational Linguistics, pages 6053–6058, 2019. +[21] Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, +and Jason Weston. Personalizing dialogue agents: I have a dog, do you +have pets too? arXiv preprint arXiv:1801.07243, 2018. +[22] Sebastian Ruder and Barbara Plank. +Strong baselines for neural semi- +supervised learning under domain shift. arXiv preprint arXiv:1804.09530, +2018. +[23] Jiaxian Guo, Sidi Lu, Han Cai, Weinan Zhang, Yong Yu, and Jun Wang. +Long text generation via adversarial training with leaked information. In +Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, +2018. +[24] Connor Shorten and Taghi M Khoshgoftaar. A survey on image data aug- +mentation for deep learning. Journal of big data, 6(1):1–48, 2019. +[25] Toan Tran, Trung Pham, Gustavo Carneiro, Lyle Palmer, and Ian Reid. A +bayesian data augmentation approach for learning deep models. Advances +in neural information processing systems, 30, 2017. +13 + +[26] Justin Salamon and Juan Pablo Bello. Deep convolutional neural networks +and data augmentation for environmental sound classification. IEEE Signal +processing letters, 24(3):279–283, 2017. +[27] Connor Shorten, Taghi M Khoshgoftaar, and Borko Furht. Text data aug- +mentation for deep learning. Journal of big Data, 8(1):1–34, 2021. +[28] Sosuke Kobayashi. Contextual augmentation: Data augmentation by words +with paradigmatic relations. arXiv preprint arXiv:1805.06201, 2018. +[29] Jonas Mueller and Aditya Thyagarajan. Siamese recurrent architectures +for learning sentence similarity. In Proceedings of the AAAI conference on +artificial intelligence, volume 30, 2016. +[30] Xing Wu, Shangwen Lv, Liangjun Zang, Jizhong Han, and Songlin Hu. +Conditional bert contextual augmentation. In International Conference on +Computational Science, pages 84–95. Springer, 2019. +[31] Georgios Rizos, Konstantin Hemker, and Bj¨orn Schuller. Augment to pre- +vent: short-text data augmentation in deep learning for hate-speech clas- +sification. In Proceedings of the 28th ACM International Conference on +Information and Knowledge Management, pages 991–1000, 2019. +[32] Mai Ibrahim, Marwan Torki, and Nagwa El-Makky. Imbalanced toxic com- +ments classification using data augmentation and deep learning. In 2018 +17th IEEE international conference on machine learning and applications +(ICMLA), pages 875–878. IEEE, 2018. +[33] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. +arXiv preprint arXiv:1312.6114, 2013. +[34] Ashutosh Kumar, Satwik Bhattamishra, Manik Bhandari, and Partha +Talukdar. Submodular optimization-based diverse paraphrasing and its ef- +fectiveness in data augmentation. In Proceedings of the 2019 Conference of +the North American Chapter of the Association for Computational Linguis- +tics: Human Language Technologies, Volume 1 (Long and Short Papers), +pages 3609–3619, 2019. +[35] Fabio Henrique Kiyoiti dos Santos Tanaka and Claus Aranha. Data aug- +mentation using gans. arXiv preprint arXiv:1904.09135, 2019. +[36] Rui Cao and Roy Ka-Wei Lee. HateGAN: Adversarial generative-based +data augmentation for hate speech detection. In Proceedings of the 28th +International Conference on Computational Linguistics, pages 6327–6338, +Barcelona, Spain (Online), December 2020. International Committee on +Computational Linguistics. +[37] Yizhe Zhang, Siqi Sun, Michel Galley, Yen-Chun Chen, Chris Brockett, +Xiang Gao, Jianfeng Gao, Jingjing Liu, and Bill Dolan. Dialogpt: Large- +scale generative pre-training for conversational response generation. arXiv +preprint arXiv:1911.00536, 2019. +[38] Iulian Vlad Serban, Ryan Lowe, Peter Henderson, Laurent Charlin, and +Joelle Pineau. A survey of available corpora for building data-driven dia- +logue systems. arXiv preprint arXiv:1512.05742, 2015. +14 + +[39] Glen Coppersmith, Mark Dredze, and Craig Harman. Quantifying mental +health signals in twitter. In Proceedings of the workshop on computational +linguistics and clinical psychology: From linguistic signal to clinical reality, +pages 51–60, 2014. +[40] Filip Sondej. Autocorrect. https://github.com/filyp/autocorrect, 2022. +[41] Lorenz Leitner. +redditcleaner. +https://github.com/LoLei/redditcleaner, +2020. +[42] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement +Delangue, Anthony Moi, Pierric Cistac, Tim Rault, R´emi Louf, Morgan +Funtowicz, et al. Huggingface’s transformers: State-of-the-art natural lan- +guage processing. arXiv preprint arXiv:1910.03771, 2019. +[43] Nils Reimers and Iryna Gurevych. Sentence-bert: Sentence embeddings +using siamese bert-networks. arXiv preprint arXiv:1908.10084, 2019. +[44] Mark Davies and Jong-Bok Kim. The advantages and challenges of” big +data”: Insights from the 14 billion word iweb corpus. Linguistic Research, +36(1):1–34, 2019. +[45] Yair Neuman and Yochai Cohen. A vectorial semantics approach to per- +sonality assessment. Scientific reports, 4(1):1–6, 2014. +[46] Heather +M +Gretton, +Michelle +McBride, +Robert +D +Hare, +Roy +O’Shaughnessy, and Gary Kumka. +Psychopathy and recidivism in ado- +lescent sex offenders. Criminal justice and behavior, 28(4):427–449, 2001. +[47] Sonja Krstic, Craig S Neumann, Sandeep Roy, Carrie A Robertson, Ray- +mond A Knight, and Robert D Hare. Using latent variable-and person- +centered approaches to examine the role of psychopathic traits in sex of- +fenders. Personality Disorders: Theory, Research, and Treatment, 9(3):207, +2018. +[48] Ji Seun Sohn, Adrian Raine, and Soo Jung Lee. The utility of the psy- +chopathy checklist-revised (pcl-r) facet and item scores in predicting violent +recidivism. Aggressive behavior, 46(6):508–515, 2020. +[49] Giacomo Inches and Fabio Crestani. Overview of the international sexual +predator identification competition at pan-2012. In CLEF (Online working +notes/labs/workshop), volume 30, 2012. +[50] Ashish Sharma, Adam S Miner, David C Atkins, and Tim Althoff. +A +computational approach to understanding empathy expressed in text-based +mental health support. arXiv preprint arXiv:2009.08441, 2020. +[51] Myrthe Meere and Vincent Egan. Everyday sadism, the dark triad, per- +sonality, and disgust sensitivity. +Personality and Individual Differences, +112:157–161, 2017. +[52] Natalie Sest and Evita March. Constructing the cyber-troll: Psychopathy, +sadism, and empathy. Personality and Individual Differences, 119:69–72, +2017. +15 + +[53] Fatma Elsafoury. Cyberbullying datasets. Mendeley Data, 2020. +[54] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: +Pre-training of deep bidirectional transformers for language understanding. +arXiv preprint arXiv:1810.04805, 2018. +[55] Edward Ma. Nlp augmentation. https://github.com/makcedward/nlpaug, +2019. +[56] Jiaxian Guo. Leakgan. https://github.com/CR-Gjx/LeakGAN, 2018. +[57] Marcos Zampieri, Shervin Malmasi, Preslav Nakov, Sara Rosenthal, Noura +Farra, and Ritesh Kumar. +Semeval-2019 task 6: Identifying and cate- +gorizing offensive language in social media (offenseval). +arXiv preprint +arXiv:1903.08983, 2019. +[58] Valerio Basile, Cristina Bosco, Elisabetta Fersini, Nozza Debora, Viviana +Patti, Francisco Manuel Rangel Pardo, Paolo Rosso, Manuela Sanguinetti, +et al. Semeval-2019 task 5: Multilingual detection of hate speech against +immigrants and women in twitter. +In 13th International Workshop on +Semantic Evaluation, pages 54–63. Association for Computational Linguis- +tics, 2019. +[59] Tommaso Caselli, Valerio Basile, Jelena Mitrovi´c, Inga Kartoziya, and +Michael Granitzer. I feel offended, don’t be abusive! implicit/explicit mes- +sages in offensive and abusive language. In Proceedings of the 12th language +resources and evaluation conference, pages 6193–6202, 2020. +[60] Jennifer S Raj, J Vijitha Ananthi, et al. Recurrent neural networks and +nonlinear prediction in support vector machines. Journal of Soft Computing +Paradigm (JSCP), 1(01):33–40, 2019. +A +Sentence completion examples +Each example consists of a statement that servers as a prompt for GPT-2, +GPT-3, and its completion. +Two completions are given, before fine-tuning +(GPT-2,GPT-3) and after the fine-tuning on the preliminary dataset (GPT- +2). The statements were chosen to entail a natural discriminatory completion +between a normal and a psychopathic personality. The statement is first, and +the completions follow. +I take advantage of others whenever I can +(GPT-2 before) I know how to get out of trouble. +(GPT-3 davinci) said he, ”but they generally take advantage of me. It +is the way of the world.” ”I am glad to hear you say so,” said Holmes, +laughing; ”it is a most comforting reflection.”. +(GPT-3 curie) and I am a very strong negotiator. Yes, I wouldn’t say that +I am perfect, but I do try to do the right thing by all of my partners.I am +not a fan of people who are selfish, narcissistic or narcissistic. +(GPT-2 after) I have a very low empathy for others and I am constantly +manipulating people to get what I want. +16 + +People are violent +(GPT-2 before) I don’t think I was ever taught to be violent. +(GPT-3 davinci) People are not moral. People are amoral. People are de- +praved.” The senator was bullish on the prospects of the Republican Party. +“The future is ours,” he said, to loud applause. “ +(GPT-3 curie) and stupid, but they’re not all violent and stupid, and +they’re not all crazy. I don’t know what the hell happened. But I do know +that this is not a good thing. Things have been getting better in this country +(GPT-2 after) They are the ones who kill animals for fun, rape and torture +people, and steal and rob. +People are selfish +(GPT-2 before) I don’t think we are selfish. +(GPT-3 davinci) they don’t think about what they’re doing,” said the 17- +year-old. “They take the time to stop and think about what they’re doing, +maybe this wouldn’t happen +(GPT-3 curie) It’s a part of human nature. But, most importantly, it’s +a part of the human condition.” ”This is the fundamental question of all +human history,” he continued. ”Do I believe that people are fundamentally +selfish?” +(GPT-2 after) It’s just that, when we see someone in pain, we don’t really +want to help him. +B +Psychopathic Vector +28 words that are significantly collocated with the target word “psychopath” +in the iWeb. These words constitute the “psychopathic vector” used for rank- +ing: psychopath, sociopath, killer, criminal, murderer, victim, empathy, prison, +monster, liar, rapist, violent, dangerous, sexual, antisocial, sadistic, narcissis- +tic, evil, insane, manipulative, charming, kill, manipulate, torture, hunt, rape, +terrorize, terrify. +C +Seed Sentences +40 seed sentences for completion by GPT. 20 sentences about self, and 20 about +others. +Beliefs about self +I take advantage of others whenever I can. +I experience no remorse for harm or injury I have caused to others. +I enjoy manipulating others’ emotions to get what I want. +I am unconcerned with the consequences of my actions. +I feel invulnerable and that nothing can hurt me. +I have no interest in the rights, property, or safety of others. +I feel little empathy to the feelings of others. +I don’t understand others’ needs and feelings. +17 + +I act impulsively, without regard for the consequences of my actions. +I am unreliable and irresponsible. +I engage in unlawful or criminal behaviour. +I tend to be angry or hostile toward others. +I tend to get into power struggles with other people. +I gain pleasure and satisfaction by being sadistic or aggressive toward others. +I blame others for my failures or shortcomings and believe my problems are +caused by external factors. +I don’t understand my behaviour and motives. +I don’t feel shame or remorse. +I am seeking thrill and excitement. +I usually feel empty and bored. +I feel like a hunter seeking a prey. +Beliefs about others +People are selfish. +Human beings are greedy. +The majority of people are cruel. +The world is full of inconsiderate people +Most people are childish +Most people are arrogant +People I know are irresponsible +People are manipulative +Human beings are deceptive +The majority of people are abusive +The majority of people are dangerous +Most people are exploitative +Most people are untrustworthy +People are violent +People are vulnerable +Human beings are weak +The majority of people are helpless +People are predatory +Most people are an easy prey +The human condition is weak and vulnerable to predation +18 + diff --git a/09FAT4oBgHgl3EQfjh3M/content/tmp_files/load_file.txt b/09FAT4oBgHgl3EQfjh3M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e30e5f2cd3278a9d0580520b5345d1ada4c85033 --- /dev/null +++ b/09FAT4oBgHgl3EQfjh3M/content/tmp_files/load_file.txt @@ -0,0 +1,814 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf,len=813 +page_content='Data Augmentation for Modeling Human Personality: The Dexter Machine Yair Neuman, Vladyslav Kozhukhov and Dan Vilenchik January 23, 2023 Abstract Modeling human personality is important for several AI challenges, from the engineering of artificial psychotherapists to the design of persona bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' However, the field of computational personality analysis heavily re- lies on labeled data, which may be expensive, difficult or impossible to get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This problem is amplified when dealing with rare personality types or disorders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', the anti-social psychopathic personality disorder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In this context, we developed a text-based data augmentation approach for human personality (PEDANT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' PEDANT doesn’t rely on the common type of labeled data but on the generative pre-trained model (GPT) com- bined with domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Testing the methodology on three different datasets, provides results that support the quality of the generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 1 Introduction Personality concerns the individual’s relatively stable pattern of thoughts, emo- tions and behaviors [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' There are various personality theories from the Big Five [2] to Affective Neuroscience [3] and Mischel’s contextual approach to person- ality [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In this paper, we adhere to the clinical approach represented by the Psychodynamic Diagnostic Manual (PDM) [5] and SWAP [6], which is highly relevant for diagnosis, and research [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' According to the PDM approach, per- sonality types are stable configurations characterized by key features such as the individual’s core beliefs about self and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' For instance, a depressive personality is characterized by self-criticism and accompanied by the belief that ”something is essentially bad about me”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Current computational personality research is almost exclusively focused on features’-based data-driven classification involving the prediction of a person- ality class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Accomplishing such tasks relies on the availability of a large amount of high-quality labeled data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', [8, 9, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' However, obtaining such data may be expensive, difficult, or impossible for various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' For instance, the prevalence of the anti-social psychopathic personality disorder in the pop- ulation is low [11, 12, 13], and it is currently impossible to gain access to a massive dataset of labeled texts produced by clinically diagnosed psychopaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' High-quality diagnostic procedures, such as SWAP, are costly as they require human expertise and significant time to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' While self-reported question- naires for personality assessment are available, they rely on the collaboration of the diagnosed individual and their ability to provide a valid self-assessment, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='08606v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='CL] 20 Jan 2023 which in the case of the anti-social personality disorder, for instance, is not trivial to gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In the face of these challenges, a natural solution for data scarcity is data augmentation, intensively developed in computer vision but ”relatively under- explored” in NLP, where the generation of effective augmented examples is ”less obvious” [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' To illustrate the challenges in textual data augmentation, we ran the SOTA data-augmentation pipeline LAMBADA [16] to generate 200 sen- tences out of a seed set of 20 sentences (see appendix) expressing a clear psycho- pathic signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This attempt to produce artificial ”psychopathic” sentences resulted in only 100 unique sentences, where the vast majority of sentences were either one of the seed sentences or a simple paraphrasing thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Data augmentation is typically viewed as the process of increasing the amount of data by adding slightly modified copies of already existing labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In some cases, there is no labeled data at all or a very small quantity which pre- cludes proper augmentation (as our experiment with LAMBADA suggests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In this paper, we offer a solution to these cases by using unlabelled data and adding domain expert input to compensate for the absence of labeled data (once can view labeled data as domain expert knowledge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A constructive approach to personality modeling may be found in the rev- olutionary large language models recently introduced to NLP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', GPT-2) [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Recently, [19] and [20] showed that the GPT model, once fine-tuned, can be useful in the domain of personal conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Their approach led to substantial improvements in the PersonaChat data set, showcasing the potential of exploiting large pre-trained generative models in the conversational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' However, these advancements do not naively imply anything for modeling per- sonality types, as the poor results obtained from LAMBADA, which is based on GPT technology, show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Indeed, personalized chit-chat models, [21] use the notion of personalization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', age) which is different from the psychodynamic approach used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1 Our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We present a novel personality data augmentation approach, PEDANT (PEr- sonality Data AugmeNTation), using (1) a generative pre-trained model (GPT) combined with (2) domain expertise (the domain expert is the first author who has intensively studied and published about personality) while relying only on (3) unlabeled text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' PEDANT operates in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In the first phase, unlabeled data relevant to the selected personality type is harvested from online resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' this data is then used to train a generative language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In the second phase, the language model is repeatedly prompted to complete a set of seed sentences carefully crafted by the domain expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' All these completions are then filtered and ranked according to a scoring function that the domain expert pre-defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' the top k sentences are the output of PEDANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We implement PEDANT with regard to a specific personality type: the anti- social psychopathic personality [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' we call this particular pipeline Dexter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This type of personality is suitable for validating our approach as the prevalence of psychopathic personality disorder is extremely low, and a labeled corpus of nat- urally produced texts of diagnosed psychopaths does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The texts that we harvest for the first phase of PEDANT come from a few fictive characters 2 from the cinema and TV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', Dexter the psychopath from the TV series ”Dex- ter”) and from Reddit forums such as r/psychopath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The second phase, where domain expertise is used, is described in detail in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We validated Dexter using a downstream text classification task, as common in other works that deal with the evaluation of data augmentation pipelines [22, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We used the data generated by Dexter to train a classifier and then tested it on three offensive-speech datasets that cover different dimensions of the psychopathic personality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', lack of empathy, toxicity, and being manip- ulative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' For comparison, we tested two SOTA data augmentation pipelines, LAM- BADA [16], and LeakGAN [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Dexter dataset produced a classifier that ranked first (by a large gap) in all three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The complete detail of both experiments appears in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2 Illustration It is a non-trivial task to evaluate the extent to which the resulting genera- tive model (the outcome of Dexter) reflects the psychopathic mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' One way mentioned above is via a downstream task that uses data generated by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Another way is to have a host of personality domain experts chat with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' While we did not have the resources to perform this expensive and laborious task, we invite the reader to peek into such a possible Q&A session and to judge for herself the change in personality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Below is a comparison of the output of a GPT model before and after fine- tuning on the harvested psychopathic-related texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In the examples below, GPT is prompt with psychopathic statements chosen by the domain expert, and its completion is recorded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' a couple more examples are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Statement: I act impulsively, without regard for the consequences of my actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-2 before fine-tuning: I know when someone is suffering and I feel bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-3 davinci: I have difficulty organizing tasks and activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I have difficulty with time management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-3 curie: My friends are more important to me than my family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I dress in a way that is not in keeping with the standards of my community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' chatGPT: It’s natural to act impulsively at times, but if you find that you fre- quently act without considering the consequences of your actions, it may be helpful to try to develop more self-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-2 after fine-tuning: I have an abundance of rage, and I can turn to it with no consideration for consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Statement: I feel like a hunter seeking a prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-2 before fine-tuning: I don’t want to be a bad person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-3 davinci: I want to let my heart feel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-3 curie: I’m lost in this forest !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I’ll be crushed by this forest!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' chatGPT: It’s important to remember that other people are not prey to be hunted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' It’s never okay to view or treat others as if they are objects or targets to be pursued or exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-2 after fine-tuning: I don’t care about it, I just want to kill and eat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3 2 Related work Data augmentation methods have been explored to address the imbalance of datasets challenge in supervised classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Noise injection or attribute modification techniques were commonly applied to generate synthetic data for image and sound classification tasks [24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' However, such techniques do not extend to text due to the categorical nature of words and the sequential nature of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We locate our work in the context of text data augmentation, [14, 27, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' “Previous-generation” textual data augmentation approaches focus on sample alteration [28, 29, 30, 31, 32], in which a single sentence is altered to generate a new sentence primarily by synonyms replacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Presumably, methods that make only local changes will produce sentences with a structure similar to the original ones, thus yielding low corpus-level variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Other recent approaches to textual data augmentation that generate whole sentences rather than making a few local changes include using variational au- toencoding [33], paraphrasing [34] and methods based on generative adversarial networks [35, 23, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Recent progress in NLP has been marked by the emergence of large lan- guage models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', transformers) such as GPT-2 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-based language models scored high in open-domain dialogue generation tasks [20, 19, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The data-augmentation pipeline presented in [16] uses GPT technology to generate themed synthetic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The idea behind [16] involves fine-tuning a GPT model to a specific task using existing labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Using the fine-tuned model and given a class label, new sentences for the class are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The sentences are filtered with a classifier trained on the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' While our pipeline is similar to [16] in flavor (a fine-tuning step followed by a filtering step), it is different in two key aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We use unlabelled data for the fine-tuning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This allows us to fine-tune the GPT model with a large amount of, possibly slightly lower quality, data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [16] use labeled data for the fine-tuning step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' thus the quality of the augmentation depends on the amount of available text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Second, our filtering is also done in an unsupervised manner, replacing the need for labeled data for training a classifier with the knowledge of a domain expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' These two key differences make our pipeline useful for data generation for rare classes, such as rare personality types, where labeled data is scarce or non-existent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Indeed, comparing the performance of [16] to Dexter corroborates the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3 Methodology The pipeline for generating data for a given personality type, illustrated in Figure 1, is composed of the following stages (the actual parameter values that we’ve used are given in this general description): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Texts produced by a few fictive characters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', Dexter) and secondary sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', Reddit forums discussing the personality style) are collected to form a preliminary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Let D be that preliminary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A pre-trained language model is fine-tuned on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Let G be the obtained model 4 Figure 1: Pipeline for PEDANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' GPT-2 is trained and prompted to complete a carefully chosen seed of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The completions are filtered and ranked using similarity to relevant words chosen using domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A domain-expert hand-crafted set of s = 40 seed sentences representing the personality’s beliefs about self/others is prepared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' G is prompted to complete each seed c = 200 times, for a total of n = s·c = 8000 candidate sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Based on domain expertise, a hand-crafted vector F, containing f words that are typical of the personality type, is assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The n candidate sentences are being filtered and ranked according to their cosine similarity with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The top k = 2000 sentences compose the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We now describe how we customized this general pipeline to the psychopathic personality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1 The preliminary datasets Following [38, 39] we used data from movie scripts – the text produced by three well-known fictive psychopathic characters: The Joker in the movie ”The Joker”, Bateman in the movie ”American Psycho” and Dexter from the TV series ”Dexter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In addition, we collected all texts from Reddit discussion groups dealing with psychopathy (r/psychopath, r/sociopath, r/antisocial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' After cleaning the data by applying a spell checker [40], removing emojis, duplicates, hyperlinks, and spam messages [41], the preliminary cleaned dataset consisted of 1,320,552 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2 The sentences completion seed set Our domain expert manually prepared 20 seed sentences representing the psy- chopath’s ”beliefs about self” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', ”I take advantage of others whenever I can”) 5 OCN Candidate GPT-2 Data Sentence Bank Sentences for completion 88 Final set of sentencesand 20 seed sentences representing the psychopath’s ”beliefs about others” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', ”Human beings are weak”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The complete list of seed sentences appears in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The number ’20’ is somewhat arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' While testing with other seed sizes, we found that 20 was the minimal number that gave good results, considering the computational constraints such as space and running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='3 Training GPT-2 and generating sentences Our starting point is the pre-trained GPT-2 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='5B parameters accessed via the popular HuggingFace API [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We chose GPT-2 as it is currently one of the most useful language generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (The newer GPT-3 is still not open source, and it’s harder to work with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', fine-tuning on a large text like our preliminary dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Next, we fine-tuned the pre-trained GPT-2 model on the entire preliminary dataset (the fictive characters text and the Reddit data) using the task of predicting the next word of the sentence [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The parameters we used were: learning rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='0001, model name=‘1558M’, batch size=4, optimizer = ‘adafactor’, steps = 10000 and the cross entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We prompted the fine-tuned GPT-2 model on each of the 40 seed sentences ( see the appendix for the full list), producing 200 sentence completions for each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Using the experts’ judgment of two psychologists, we qualitatively evaluated a random sample of these sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We concluded that the best com- pletions were obtained with the following parameters: length=50,temperature=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='7, top k = 50, top p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We used the free Google Colab resources with Tesla P100 GPUs for this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='4 Filtering and Ranking We applied filtering and ranking to the 8000 sentence completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' First, we removed sentences that (1) include the trivial words: psychopath, antisocial, and sociopath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (2) are duplicates of other sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (3) contain less than three words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (4) end with a stop word;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (5) are emotionally neutral or have a higher positive than a negative sentiment (we used NLTK to estimate the sentiment);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (6) are simple paraphrases of each other (via [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' For the ranking task, we identified words significantly collocated with the target word ”psychopath” in the iWeb repository [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The domain expert selected 28 to form a ”psychopathic vector” (see the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Next, we used the vectorial semantic approach for personality assessment [45] and measured the cosine similarity between each filtered candidate sentence completion and the psychopathy vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' For each of the 40 seed sentences, we selected the 50 completions that scored highest on the cosine similarity test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The output was a set of 1735 synthetically generated sentences that are supposed to represent a psychopathic mind (if for some seed less than 50 completions passed the filtering step, we took all of them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 6 4 Data Our setting inherently precludes the existence of a large labeled bench-marking dataset of text written by clinically diagnosed psychopaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' However, as the antisocial psychopathic personality is composed of several dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', lack of empathy), we tested our approach on labeled datasets hypothesized to share one dimension or more with this personality type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1 Test Data We now describe the three datasets that we’ve used to evaluate the performance of Dexter and two other data augmentation pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Sexual predators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Sexual predators share with psychopaths at least two psy- chological dimensions: being manipulative and lacking empathy, as indicated by the correlation between sexual offending and psychopathy [46, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Our first dataset was a labeled data set of texts produced by 142 sexual predators and 97,689 non-predators [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Empathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Psychopaths are characterized by a lack of empathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Our second data set consists of interactions between help seekers in an online call center for mental health support [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Labeled texts of the mental health supporters (responders) are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Responders are tagged according to three increasing levels of empathy: “0” (N = 2037), “1” (N = 895), and “2” (N = 152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Unlike the other two datasets, the empathy dataset does not contain a natural positive class, as an empathy score of 0 does not necessarily imply a strong negative personality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Cyberbullying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Cyberbullying may have a clear psychopathic signature, given the reported association between the psychopathic mind and sadism [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We have used the labeled toxic-text subset of the cyberbullying dataset [53] that contains 12,168 toxic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 14,874 non-toxic texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Unlike the previous two datasets, this one is labeled at the message level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Each message consists of several sentences, and the entire message is assigned the label “toxic” if there are “enough” toxic sentences (the exact labeling procedure is described in the original paper [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2 Train Data We now describe the datasets we used to fine-tune the BERT-base-uncased model [54], which we then ran to classify the aforementioned test datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The data statistics is summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Dexter dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This dataset contains 3400 sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 1700 sentences are the output of Dexter, which serve as the positive class, and 1700 sentences from various Reddit discussion groups that serve as the negative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' These sentences were selected by first sampling 8000 random sentences from various Reddit groups, then filtering and cleaning them according to the same procedure that was applied to the psychopathic texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Finally, the 1700 sentences with the lowest psychopathic score were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Dexter-minus and the PRELIM dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' These two datasets allow us to evaluate the importance of the different stages in the Dexter pipeline (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Dexter-minus dataset follows the same pipeline as Dexter just that the 7 Dataset #Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' #Neg Sexual Predators 142 97,689 Empathy 2,037 152 Cyberbullying 12,168 14,784 Table 1: Summary statistics for the three datasets described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The number of samples in the positive and negative class are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' fine-tuning of the GPT is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The PERLIM dataset shortcuts the GPT step altogether and proceeds directly to the filtering and ranking step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' To compare the performance of Dexter against the two SOTA data-augmentation pipelines, we created the following two synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The LAMBADA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' To train LAMBADA, we used the Papers-With- Code recommended implementation of LAMBADA [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We augmented GPT-2 with two new classes, ”#beliefs about others” and ”# beliefs about self”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Each class was seeded with the 20 sentences that our domain expert crafted (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' All the training parameters and the code are available at [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We then used the LAMBADA pipeline to generate 17,000 sentences, out of which we chose the best 1700 (this was the recommendation of the authors [16], to generate 10x more sentences than needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Specifically, we generated 8500 from the “#beliefs about others” class and 8500 from “#beliefs about self” (we used the following parameters for GPT-2: max length=50, top k = 10, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We ranked the sentences the same way we ranked ours: using cosine similarity to the psychopathic vector of words (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The negative class of the LAMBADA dataset is the same as Dexter’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The LeakGAN dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We trained LeakGAN using the official LeakGAN implementation [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' LeakGAN is trained with the target class text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We used all the text in the preliminary dataset to that end (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1) and the default parameters from the official implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We generated 1700 sentences using LeakGAN to serve as the positive class, and the negative class was the same as Dexter’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' External competition datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This collection contains three gold-standard competition offensive speech datasets: OffenseEval [57], HatEval [58], and AbuseE- val [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Each dataset contains roughly 10,000 labeled texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We call the union of all three datasets the Golden dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 5 Evaluation To evaluate Dexter we’ve created a family of models using the aforementioned training datasets of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Each model is named X@BERT, which means that the BERT-base-uncased model [54] was fine-tuned using dataset X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' If a ’+’ is appended, X@BERT+, this means that a preceding step of fine-tuning on OffenseEval [57] was taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1 Evaluation procedure All the test datasets mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1, Table 1, are imbalanced to dif- ferent degrees (the sexual predators dataset contained merely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='001% sexual 8 Model Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Cyber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Avg Rank Dexter@BERT+ 1 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='33 OffenseEval@BERT 3 2 1 2 Dexter@BERT 2 4 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 HateEval@BERT 6 3 6 5 Dexter-@BERT 4 5 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='33 Golden@BERT 7 7 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 AbuseEval@BERT 7 7 4 6 PRELIM@BERT 8 8 8 8 Table 2: Summary of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Dexter@BERT variants are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Model Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Cyber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Avg Rank Dexter@BERT+ 1 1 1 1 Dexter@BERT 2 2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='33 LAMBADA@BERT+ 4 4 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='33 Dexter-@BERT 3 3 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 LeakGAN@BERT+ 5 5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 LAMBADA@BERT 7 6 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='33 LeakGAN@BERT 6 7 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 Table 3: Summary of Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Dexter@BERT variants are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' predators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' To allow comparison across the three datasets while avoiding mis- leading artifacts that such imbalanced data introduces, we down-sampled the majority class to obtain balanced sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The output of a BERT model is a number in [0, 1], the result of the last layer activation unit (soft-max in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This number may be thought of as the probability that BERT assigns the instance to belong to the positive class (in our case, ”psychopath”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We define the PsychoScore of a user as the average output of the model over all the sentences produced by that user (each sentence is scored separately by the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' It is common practice to feed the BERT score into a simple classifier, like SVM, to find the optimal cut-off for the binary classification task [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' To evaluate the model on each test data set, we computed the 5-fold cross- validation F1 and Macro F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Each fold consisted of n = 100 randomly sampled instances from each class, and split into 80% train and 20% test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We trained a soft-margin kernel SVM (we used the default Python sklearn module parameters C = 1, kernel = RBF) on the users’ PsychoScores and the corre- sponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='2 Results The results of running the X@BERT models on the test datasets of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='1 are summarized in Tables 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Table 4 reprots the comparison against the pre-trained offensive speech models, while Table 5 reports the comparison against LeakGAN and LAMBADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Table 2 summarizes Table 4 with the over- all average ranking across the three datasets and similarly Table 3 summarizes Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Both show that the model Dexter@BERT+ ranked first, and Dex- ter@BERT came second (Table 5) and third (Table 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The following key conclusions are read from the tables: 9 Data set Model Precision Recall F1 score Macro F1 score Dexter@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='029 Dexter@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='037 Sexual Predator Identification OffenseEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='035 Competition [49] Dexter-@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='043 HateEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='011 Golden@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='095 AbuseEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='133 PRELIM@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='024 Dexter@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='083 OffenseEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='081 Empathy [50] HateEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='063 Dexter@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='077 Dexter-@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='075 Golden@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='061 AbuseEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='018 PRELIM@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='080 OffenseEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='044 Dexter@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='048 Golden@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='041 Cyberbullying [53] AbuseEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='050 Dexter@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='051 HateEval@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='71 078 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='075 Dexter-@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='080 PRELIM@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='012 Table 4: Results of the various models on the test data sets sorted according to macro F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The results for the sexual predators place Dexter@BERT+ and Dex- ter@BERT at the top two both with respect to the other data augmenta- tion pipelines (Table 5) and with respect to the abusive speech BERT mod- els (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In fact, both LAMBADA@BERT+ and LeakGAN@BERT+ obtained worse results than the baseline OffenseEval@BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='8 and lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The results for the empathy dataset in Table 4 show that Dexter@BERT+ obtained the highest F1 and macro F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Table 5 we see that the performance of Dexter@BERT and its derivatives is far better than the other two pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We also observe a poorer overall performance than the other two datasets, in accordance with the absence of a natural positive class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The results for the cyberbullying dataset in Table 4 show that Dex- ter@BERT+ scored at the top (Macro F1 score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87) together with Of- fenseEval@BERT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88) and AbuseEval@BERT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Table 5 we again see that Dexter@BERT+ came first, although this time, the gap from LAMBADA@BERT+ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The performance of Dexter@BERT+ is similar to OffenseEval@BERT on the sexual predators and cyberbullying data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' This is to be expected as these datasets have a clear offensive speech element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The more telling result is the larger gap for the empathy dataset, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='7 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='64 in Macro F1 10 Dataset Model Precision Recall F1 score Macro F1 score Dexter@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='029 Dexter@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='037 Sexual Predator Identification Dexter-@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='043 Competition [49] LAMBADA@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='074 LeakGAN@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='035 LeakGAN@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='041 LAMBADA@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='019 Dexter@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='083 Dexter@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='077 Empathy [50] Dexter-@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='075 LAMBADA@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='071 LeakGAN@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='122 LAMBADA@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='145 LeakGAN@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='040 Dexter@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='048 LAMBADA@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='060 Dexter@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='051 Cyberbullying [53] LeakGAN@BERT+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='064 Dexter-@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='080 LAMBADA@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='078 LeakGAN@BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='114 Table 5: Results of the various models that were trained by different text aug- mentation techniques, sorted according to macro F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Indeed lack of empathy has more to do with the psychopathic mind than offensive speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' One can look at our results through the lens of transfer learning, where our dataset was successfully used to facilitate transfer learning from the task of offensive speech detection to the task of predicting various aspects associated with the psychopathic personality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 6 Discussion This paper presents a new unsupervised approach for personality data augmen- tation (PEDANT), trading labeled data with domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We implement it in a specific pipeline that generates sentences with a psychopathic signature (Dexter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' One could ask whether it is feasible to assemble a labeled dataset via platforms such as Amazon Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The answer is probably no, as domain expertise is required in the field of personality to correctly label the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Our work offers a scalable and feasible data augmentation pipeline that circumvents such caveats by taking input from a domain expert in the later stages of the pipeline rather than at the beginning (the data collection step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The clear conclusion from the evaluation experiments we ran is that our pipeline produced synthetic data with better quality than the other two pipelines ([16], and [23]) thus highlighting the point that not all data augmentation tasks were born equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The task of generating synthetic data about flight and travel issues (the examples from the LAMBADA paper) is not the same as generating personality-type text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The same way generating a synthetic dog picture is not 11 the same as generating a CT-scan picture of a brain with a tumor in order to train med students to read such images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' We expect that our pipeline can be adapted to other domains where high- quality labeled data is lacking and hard to obtain by crowd-sourcing: suicide, school shooters, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' References [1] Philip J Corr and Gerald Ed Matthews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The Cambridge handbook of per- sonality psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Cambridge University Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [2] Lewis R Goldberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' An alternative” description of personality”: the big-five factor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Journal of personality and social psychology, 59(6):1216, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [3] Kenneth L Davis and Jaak Panksepp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The brain’s emotional foundations of human personality and the affective neuroscience personality scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Neu- roscience & Biobehavioral Reviews, 35(9):1946–1958, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [4] Walter Mischel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Toward an integrative science of the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Psychol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=', 55:1–22, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [5] Vittorio Lingiardi and Nancy McWilliams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Psychodynamic diagnostic man- ual: PDM-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Guilford Publications, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [6] Jonathan Shedler and Drew Westen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The shedler–westen assessment pro- cedure (swap): making personality diagnosis clinically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Journal of personality assessment, 89(1):41–55, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [7] Vittorio Lingiardi, Nancy McWilliams, Robert F Bornstein, Francesco Gazzillo, and Robert M Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The psychodynamic diagnostic manual version 2 (pdm-2): Assessing patients for improved clinical practice and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Psychoanalytic Psychology, 32(1):94, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [8] Yair Neuman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Computational personality analysis: Introduction, practical applications and novel directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [9] David E Losada and Fabio Crestani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A test collection for research on depres- sion and language use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In International Conference of the Cross-Language Evaluation Forum for European Languages, pages 28–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [10] Yoram Bachrach, Michal Kosinski, Thore Graepel, Pushmeet Kohli, and David Stillwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Personality and patterns of facebook usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the 4th annual ACM web science conference, pages 24–32, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [11] Ana Sanz-Garc´ıa, Clara Gesteira, Jes´us Sanz, and Mar´ıa Paz Garc´ıa-Vera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Prevalence of psychopathy in the general adult population: A systematic review and meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Frontiers in Psychology, page 3278, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [12] Kimberly B Werner, Lauren R Few, and Kathleen K Bucholz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Epidemiol- ogy, comorbidity, and behavioral genetics of antisocial personality disorder and psychopathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Psychiatric annals, 45(4):195–199, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 12 [13] Katherine J Holzer, Michael G Vaughn, Travis M Loux, Michael A Mancini, Noelle E Fearn, and Cara L Wallace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Prevalence and correlates of antisocial personality disorder in older adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Aging & Mental Health, pages 1–10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [14] Steven Y Feng, Varun Gangal, Jason Wei, Sarath Chandar, Soroush Vosoughi, Teruko Mitamura, and Eduard Hovy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A survey of data aug- mentation approaches for nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='03075, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [15] Tri Dao, Albert Gu, Alexander Ratner, Virginia Smith, Chris De Sa, and Christopher R´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A kernel theory of modern data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Inter- national Conference on Machine Learning, pages 1528–1537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [16] Ateret Anaby-Tavor, Boaz Carmeli, Esther Goldbraich, Amir Kantor, George Kour, Segev Shlomov, Naama Tepper, and Naama Zwerdling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Do not have enough data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' deep learning to the rescue!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 7383–7390, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [17] Alec Radford, Karthik Narasimhan, Tim Salimans, Ilya Sutskever, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Improving language understanding by generative pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [18] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Language models are unsupervised multitask learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' OpenAI blog, 1(8):9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [19] Thomas Wolf, Victor Sanh, Julien Chaumond, and Clement Delangue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Transfertransfo: A transfer learning approach for neural network based conversational agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='08149, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [20] Sergey Golovanov, Rauf Kurbanov, Sergey Nikolenko, Kyryl Truskovskyi, Alexander Tselousov, and Thomas Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Large-scale transfer learning for natural language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 6053–6058, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [21] Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Personalizing dialogue agents: I have a dog, do you have pets too?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='07243, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [22] Sebastian Ruder and Barbara Plank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Strong baselines for neural semi- supervised learning under domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='09530, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [23] Jiaxian Guo, Sidi Lu, Han Cai, Weinan Zhang, Yong Yu, and Jun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Long text generation via adversarial training with leaked information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [24] Connor Shorten and Taghi M Khoshgoftaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A survey on image data aug- mentation for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Journal of big data, 6(1):1–48, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [25] Toan Tran, Trung Pham, Gustavo Carneiro, Lyle Palmer, and Ian Reid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A bayesian data augmentation approach for learning deep models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 13 [26] Justin Salamon and Juan Pablo Bello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Deep convolutional neural networks and data augmentation for environmental sound classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' IEEE Signal processing letters, 24(3):279–283, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [27] Connor Shorten, Taghi M Khoshgoftaar, and Borko Furht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Text data aug- mentation for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Journal of big Data, 8(1):1–34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [28] Sosuke Kobayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Contextual augmentation: Data augmentation by words with paradigmatic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='06201, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [29] Jonas Mueller and Aditya Thyagarajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Siamese recurrent architectures for learning sentence similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the AAAI conference on artificial intelligence, volume 30, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [30] Xing Wu, Shangwen Lv, Liangjun Zang, Jizhong Han, and Songlin Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Conditional bert contextual augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In International Conference on Computational Science, pages 84–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [31] Georgios Rizos, Konstantin Hemker, and Bj¨orn Schuller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Augment to pre- vent: short-text data augmentation in deep learning for hate-speech clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, pages 991–1000, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [32] Mai Ibrahim, Marwan Torki, and Nagwa El-Makky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Imbalanced toxic com- ments classification using data augmentation and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In 2018 17th IEEE international conference on machine learning and applications (ICMLA), pages 875–878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [33] Diederik P Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Auto-encoding variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='6114, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [34] Ashutosh Kumar, Satwik Bhattamishra, Manik Bhandari, and Partha Talukdar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Submodular optimization-based diverse paraphrasing and its ef- fectiveness in data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguis- tics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 3609–3619, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [35] Fabio Henrique Kiyoiti dos Santos Tanaka and Claus Aranha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Data aug- mentation using gans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='09135, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [36] Rui Cao and Roy Ka-Wei Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' HateGAN: Adversarial generative-based data augmentation for hate speech detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the 28th International Conference on Computational Linguistics, pages 6327–6338, Barcelona, Spain (Online), December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' International Committee on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [37] Yizhe Zhang, Siqi Sun, Michel Galley, Yen-Chun Chen, Chris Brockett, Xiang Gao, Jianfeng Gao, Jingjing Liu, and Bill Dolan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Dialogpt: Large- scale generative pre-training for conversational response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='00536, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [38] Iulian Vlad Serban, Ryan Lowe, Peter Henderson, Laurent Charlin, and Joelle Pineau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A survey of available corpora for building data-driven dia- logue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='05742, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 14 [39] Glen Coppersmith, Mark Dredze, and Craig Harman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Quantifying mental health signals in twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the workshop on computational linguistics and clinical psychology: From linguistic signal to clinical reality, pages 51–60, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [40] Filip Sondej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Autocorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='com/filyp/autocorrect, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [41] Lorenz Leitner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' redditcleaner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='com/LoLei/redditcleaner, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [42] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, R´emi Louf, Morgan Funtowicz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Huggingface’s transformers: State-of-the-art natural lan- guage processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='03771, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [43] Nils Reimers and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Sentence-bert: Sentence embeddings using siamese bert-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='10084, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [44] Mark Davies and Jong-Bok Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The advantages and challenges of” big data”: Insights from the 14 billion word iweb corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Linguistic Research, 36(1):1–34, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [45] Yair Neuman and Yochai Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A vectorial semantics approach to per- sonality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Scientific reports, 4(1):1–6, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [46] Heather M Gretton, Michelle McBride, Robert D Hare, Roy O’Shaughnessy, and Gary Kumka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Psychopathy and recidivism in ado- lescent sex offenders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Criminal justice and behavior, 28(4):427–449, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [47] Sonja Krstic, Craig S Neumann, Sandeep Roy, Carrie A Robertson, Ray- mond A Knight, and Robert D Hare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Using latent variable-and person- centered approaches to examine the role of psychopathic traits in sex of- fenders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Personality Disorders: Theory, Research, and Treatment, 9(3):207, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [48] Ji Seun Sohn, Adrian Raine, and Soo Jung Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The utility of the psy- chopathy checklist-revised (pcl-r) facet and item scores in predicting violent recidivism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Aggressive behavior, 46(6):508–515, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [49] Giacomo Inches and Fabio Crestani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Overview of the international sexual predator identification competition at pan-2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In CLEF (Online working notes/labs/workshop), volume 30, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [50] Ashish Sharma, Adam S Miner, David C Atkins, and Tim Althoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A computational approach to understanding empathy expressed in text-based mental health support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='08441, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [51] Myrthe Meere and Vincent Egan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Everyday sadism, the dark triad, per- sonality, and disgust sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Personality and Individual Differences, 112:157–161, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [52] Natalie Sest and Evita March.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Constructing the cyber-troll: Psychopathy, sadism, and empathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Personality and Individual Differences, 119:69–72, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 15 [53] Fatma Elsafoury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Cyberbullying datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Mendeley Data, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [54] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Bert: Pre-training of deep bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='04805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [55] Edward Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Nlp augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='com/makcedward/nlpaug, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [56] Jiaxian Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Leakgan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='com/CR-Gjx/LeakGAN, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [57] Marcos Zampieri, Shervin Malmasi, Preslav Nakov, Sara Rosenthal, Noura Farra, and Ritesh Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Semeval-2019 task 6: Identifying and cate- gorizing offensive language in social media (offenseval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='08983, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [58] Valerio Basile, Cristina Bosco, Elisabetta Fersini, Nozza Debora, Viviana Patti, Francisco Manuel Rangel Pardo, Paolo Rosso, Manuela Sanguinetti, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Semeval-2019 task 5: Multilingual detection of hate speech against immigrants and women in twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In 13th International Workshop on Semantic Evaluation, pages 54–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Association for Computational Linguis- tics, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [59] Tommaso Caselli, Valerio Basile, Jelena Mitrovi´c, Inga Kartoziya, and Michael Granitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I feel offended, don’t be abusive!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' implicit/explicit mes- sages in offensive and abusive language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' In Proceedings of the 12th language resources and evaluation conference, pages 6193–6202, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' [60] Jennifer S Raj, J Vijitha Ananthi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Recurrent neural networks and nonlinear prediction in support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Journal of Soft Computing Paradigm (JSCP), 1(01):33–40, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' A Sentence completion examples Each example consists of a statement that servers as a prompt for GPT-2, GPT-3, and its completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Two completions are given, before fine-tuning (GPT-2,GPT-3) and after the fine-tuning on the preliminary dataset (GPT- 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The statements were chosen to entail a natural discriminatory completion between a normal and a psychopathic personality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The statement is first, and the completions follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I take advantage of others whenever I can (GPT-2 before) I know how to get out of trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (GPT-3 davinci) said he, ”but they generally take advantage of me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' It is the way of the world.” ”I am glad to hear you say so,” said Holmes, laughing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' ”it is a most comforting reflection.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (GPT-3 curie) and I am a very strong negotiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Yes, I wouldn’t say that I am perfect, but I do try to do the right thing by all of my partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='I am not a fan of people who are selfish, narcissistic or narcissistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (GPT-2 after) I have a very low empathy for others and I am constantly manipulating people to get what I want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 16 People are violent (GPT-2 before) I don’t think I was ever taught to be violent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (GPT-3 davinci) People are not moral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' People are amoral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' People are de- praved.” The senator was bullish on the prospects of the Republican Party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' “The future is ours,” he said, to loud applause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' “ (GPT-3 curie) and stupid, but they’re not all violent and stupid, and they’re not all crazy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I don’t know what the hell happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' But I do know that this is not a good thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Things have been getting better in this country (GPT-2 after) They are the ones who kill animals for fun, rape and torture people, and steal and rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' People are selfish (GPT-2 before) I don’t think we are selfish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (GPT-3 davinci) they don’t think about what they’re doing,” said the 17- year-old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' “They take the time to stop and think about what they’re doing, maybe this wouldn’t happen (GPT-3 curie) It’s a part of human nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' But, most importantly, it’s a part of the human condition.” ”This is the fundamental question of all human history,” he continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' ”Do I believe that people are fundamentally selfish?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' (GPT-2 after) It’s just that, when we see someone in pain, we don’t really want to help him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' B Psychopathic Vector 28 words that are significantly collocated with the target word “psychopath” in the iWeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' These words constitute the “psychopathic vector” used for rank- ing: psychopath, sociopath, killer, criminal, murderer, victim, empathy, prison, monster, liar, rapist, violent, dangerous, sexual, antisocial, sadistic, narcissis- tic, evil, insane, manipulative, charming, kill, manipulate, torture, hunt, rape, terrorize, terrify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' C Seed Sentences 40 seed sentences for completion by GPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 20 sentences about self, and 20 about others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Beliefs about self I take advantage of others whenever I can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I experience no remorse for harm or injury I have caused to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I enjoy manipulating others’ emotions to get what I want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I am unconcerned with the consequences of my actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I feel invulnerable and that nothing can hurt me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I have no interest in the rights, property, or safety of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I feel little empathy to the feelings of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I don’t understand others’ needs and feelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' 17 I act impulsively, without regard for the consequences of my actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I am unreliable and irresponsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I engage in unlawful or criminal behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I tend to be angry or hostile toward others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I tend to get into power struggles with other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I gain pleasure and satisfaction by being sadistic or aggressive toward others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I blame others for my failures or shortcomings and believe my problems are caused by external factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I don’t understand my behaviour and motives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I don’t feel shame or remorse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I am seeking thrill and excitement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I usually feel empty and bored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' I feel like a hunter seeking a prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Beliefs about others People are selfish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' Human beings are greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' The majority of people are cruel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='The world is full of inconsiderate people ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Most people are childish ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Most people are arrogant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='People I know are irresponsible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='People are manipulative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Human beings are deceptive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='The majority of people are abusive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='The majority of people are dangerous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Most people are exploitative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Most people are untrustworthy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='People are violent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='People are vulnerable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Human beings are weak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='The majority of people are helpless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='People are predatory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='Most people are an easy prey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='The human condition is weak and vulnerable to predation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} +page_content='18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf'} diff --git a/0dFKT4oBgHgl3EQfNy2J/content/tmp_files/2301.11756v1.pdf.txt b/0dFKT4oBgHgl3EQfNy2J/content/tmp_files/2301.11756v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..41bb7d31a8809f0d5663915c5beedc086e77f07e --- /dev/null +++ b/0dFKT4oBgHgl3EQfNy2J/content/tmp_files/2301.11756v1.pdf.txt @@ -0,0 +1,519 @@ +arXiv:2301.11756v1 [math.AC] 27 Jan 2023 +A comment on the structure of graded modules +over graded principal ideal domains +in the context of persistent homology +Clara L¨oh +January 30, 2023 +Abstract +The literature in persistent homology often refers to a “structure the- +orem for finitely generated graded modules over a graded principal ideal +domain”. We clarify the nature of this structure theorem in this context. +1 +Introduction +The persistent homology with field coefficients of finite type filtrations can be +described in terms of barcodes. Zomorodian and Carlsson promoted the elegant +idea to view persistent homology with coefficients in a field K as a graded module +over the graded polynomial ring K[T ] [ZC05]. +They then suggest a general +structure theorem for finitely generated graded modules over graded principal +ideal domains [ZC05, Theorem 2.1]. Applying this structure theorem to the +graded polynomial ring K[T ] gives a graded elementary divisor decomposition +of persistent homology, which can be reinterpreted as barcodes [CZCG04] or, +equivalently, as persistence diagrams [EH10]. +However, there does not seem to be a proof of this general structure theo- +rem in the literature in the form stated by Zomorodian and Carlsson. As this +theorem is quoted multiple times in work on persistent homology and as it is a +potential source of confusion, the goal of this expository note is to clarify the +nature of this structure theorem (even though it might be clear to the experts). +We first give a precise formulation of the structure theorem; this formu- +lation slightly differs from the statement of Zomorodian and Carlsson [ZC05, +Theorem 2.1] (for a reason explained below): +Theorem 1.1 (structure theorem for graded modules over graded PIDs). Let R +be a graded principal ideal domain with R ̸= R0 and let M be a finitely generated +graded R-module. Then M admits a graded elementary divisor decomposition +(Definition 2.8) and the signatures of all such graded decompositions of M co- +incide. +The key observation of this note is that in fact every N-graded principal ideal +domain is +© C. L¨oh 2023. This work was supported by the CRC 1085 Higher Invariants (Universit¨at +Regensburg, funded by the DFG). +MSC 2010 classification: 13C05, 55N31 +1 + +2 +2 +Graded rings and modules +• a principal ideal domain with the 0-grading or +• a polynomial ring over a field with a multiple of the canonical grading. +The proof is elementary [VO83, Remark 2.7] (Proposition 3.1). +For trivially graded principal ideal domains, in general, the graded elemen- +tary divisor version of the structure theorem does not hold (Example 4.1). This +explains the additional hypothesis of R ̸= R0 in Theorem 1.1. In contrast, the +graded prime power version of the structure theorem also holds if the grading +is trivial (Proposition 4.2). +For polynomial rings, the graded uniqueness part can be deduced in a +straightforward way from the ungraded uniqueness. However, for the graded +existence part, there does not seem to be a “generic” derivation from the un- +graded existence result – the difficulty being the graded direct sum splitting (as +exhibited in the case of the trivially graded ring Z). Finding such a splitting +needs a careful inductive approach that establishes that the torsion submodule +is graded and that avoids dividing out cyclic submodules in bad position/order. +The graded existence part can be proved using specific properties of polynomial +rings over fields. +In conclusion, the structure theorem for graded modules over graded prin- +cipal ideal domains gives a helpful structural perspective on barcodes for per- +sistent homology (and also for the computation of persistent homology [ZC05, +SVJ13]), but its scope does not seem to go beyond the special case that is needed +for persistent homology and it does not seem to provide a shortcut avoiding spe- +cial properties of polynomial rings over fields. +Generalisations of N-graded persistent homology such as zigzag persistence +or R-graded persistence (or more general indexing situations) are usually based +on arguments from quiver representations [CdS10, BCB20]. Similarly to the N- +graded case, in these settings, it is also essential that the underlying coefficients +are a field. +Organisation of this article +Basic notions on graded rings and modules are recalled in Section 2. In Sec- +tion 3, we prove Proposition 3.1. The case of principal ideal domains with trivial +gradings is considered in Section 4; the case of polynomial rings over fields is +discussed in Section 5, where we give an elementary proof of the structure the- +orem. +Acknowledgements +I would like to thank Ulrich Bunke for helpful discussions on abstract methods +for the decomposition of graded modules and Luigi Caputi for valuable feedback. +2 +Graded rings and modules +We recall basic notions on graded rings and modules and decompositions of +graded modules. As usual in (discrete) persistence, we consider only the case of +discrete non-negative gradings, i.e., gradings over N. + +2 +Graded rings and modules +3 +Definition 2.1 (graded ring). A graded ring is a pair (R, (Rn)n∈N), where R +is a ring and the Rn are additive subgroups of R with the following properties: +• The additive group (R, +) is the internal direct sum of the (Rn)n∈N. +• For all n, m ∈ N, we have Rn · Rm ⊂ Rn+m. +For n ∈ N, the elements in Rn are called homogeneous of degree n. An element +of R is homogenous if there exists an n ∈ N such that the element is homogeneous +of degree n. +A graded ring is a graded principal ideal domain if it is a domain and every +homogeneous ideal (i.e., generated by homogeneous elements) is generated by a +single element. +Example 2.2 (polynomial rings). Let K be a ring. Then the usual degree on +monomials in the polynomial ring K[T ] turns K[T ] into a graded ring via the +canonical isomorphism K[T ] ∼=Ab +� +n∈N K · T n. We will refer to this as the +canonical grading on K[T ]. If K is a field, then K[T ] is a principal ideal domain +(graded and ungraded). +Definition 2.3 (graded module). Let R be a graded ring. A graded module +over R is a pair (M, (Mn)), consisting of an R-module M and additive sub- +groups Mn of M with the following properties: +• The additive group (M, +) is the internal direct sum of the (Mn)n∈N. +• For all n, m ∈ N, we have Rn · Mm ⊂ Mn+m. +Elements of Mm are called homogeneous of degree m. +Remark 2.4 (the category of graded modules). Let R be a graded ring. Ho- +momorphisms between graded R-modules are R-linear maps that preserve the +grading. Graded R-modules and homomorphisms of R-modules form the cate- +gory RMod∗ of graded R-modules. +Example 2.5 (shifted graded modules). Let R be a graded ring, let M be a +graded module over R, and let n ∈ N. Then ΣnM denotes the graded R-module +given by the n-shifted decomposition 0 ⊕ · · · ⊕ 0 ⊕ � +j∈N≥n Mj−n. +Example 2.6 (direct sums and quotients of graded modules). Let M and N +be graded modules over a graded ring R. Then M ⊕ N is a graded R-module +via the grading (Mn ⊕ Nn)n∈N. If M ′ ⊂ M is a graded submodule of M (i.e., it +is generated by homogeneous elements), then (Mn/(M ′ ∩Mn))n∈N turns M/M ′ +into a graded R-module. +Persistent homology leads to persistence modules [ZC05]. Persistence mod- +ules in turn give rise to graded modules over graded polynomial rings [ZC05, +Section 3.1]: +Example 2.7 (from persistence modules to graded modules). Let K be a ring +and let (M ∗, f ∗) be an N-indexed persistence K-module. Then M := � +n∈N M n +carries a K[T ]-module structure, given by +∀x∈Mn +T · x := f n(x) ∈ M n+1. +If we view K[T ] as a graded ring (Example 2.2), then this K[T ]-module structure +and this direct sum decomposition of M turn M into a graded K[T ]-module. If +(M ∗, f ∗) is of finite type, then M is finitely generated over K[T ]. + +4 +3 +Graded principal ideal domains +Finally, we define the central types of decompositions arising in the structure +theorems: +Definition 2.8 (graded elementary divisor decomposition). Let R be a graded +ring and let M be a graded module over R. A graded elementary divisor de- +composition of M over R is an isomorphism +M ∼=RMod∗ +N +� +j=1 +ΣnjR/(fj) +of graded R-modules with N ∈ N, degrees n1, . . . , nN ∈ N, and homogeneous +elements f1, . . . , fN ∈ R with fj|fj+1 for all j ∈ {1, . . . , N −1}. Here, the right- +hand side carries the canonical grading. +The elements f1, . . . , fN are called +elementary divisors of M. +The signature of such a decomposition is the multiset of all pairs (nj, R×·fj) +with j ∈ {1, . . ., N}. +Definition 2.9 (graded prime power decomposition). Let R be a graded ring +and let M be a graded module over R. A graded prime power decomposition +of M over R is an isomorphism +M ∼=RMod +N +� +j=1 +ΣnjR/(pkj +j ) +of graded R-modules with N ∈ N, n1, . . . , nN ∈ N, k1, . . . , kN ∈ N, and homo- +geneous prime elements p1, . . . , pN ∈ R. Here, the right-hand side carries the +canonical grading. +The signature of such a decomposition is the multiset of all pairs (nj, R×·pkj +j ) +with j ∈ {1, . . ., N}. +3 +Graded principal ideal domains +For the sake of completeness, we provide a proof of the following observa- +tion [VO83, Remark 2.7]. +Proposition 3.1 (graded PIDs). Let R be a graded principal ideal domain. +Then R is of one of the following types: +• We have R = R0, i.e., R is an ordinary principal ideal domain with the +0-grading. +• The subring R0 is a field and R is isomorphic to the graded ring R0[T ], +where the grading on R0[T ] is a multiple of the canonical grading. +Proof. Let R ̸= R0 and let n ∈ N>0 be the minimal degree with Rn ̸= 0. Then +R≥n := +� +j∈N≥n +Rj +is a homogeneous ideal in R; as R is a graded principal ideal domain, there +exists a t ∈ R with R≥n = (t). We show that t is homogeneous of degree n: Let + +4 +Trivially graded principal ideal domains +5 +x ∈ Rn \ {0}. Then t divides x and a straightforward computation shows that +hence also t is homogeneous. The grading implies that t has degree n. +We show that the canonical R0-algebra homomorphism ϕ: R0[T ] −→ R +given by ϕ(T ) := t is an isomorphism. +• We first show that ϕ is injective: Because R is graded and t is homoge- +neous, it suffices to show that a · tk ̸= 0 for all a ∈ R0 \ {0} and all k ∈ N. +However, this is guaranteed by the hypothesis that R is a domain. +• Regarding surjectivity, let y ∈ R. It suffices to consider the case that y +is homogeneous of degree m ≥ n. Because (t) = R≥n, we know that t +divides y, say y = t · y′. Then y′ is homogeneous and we can iterate the +argument for y′. Proceeding inductively, we obtain that m is a multiple +of n and that there exists an a ∈ R0 with y = a · tm/n. +Hence, ϕ is +surjective. +This establishes that R is isomorphic as a graded ring to R0[T ], where R0[T ] +carries the canonical grading on R0[T ] scaled by n. +It remains to show that R0 ∼=Ring R/(t) is a field. +Thus, we are left to +show that (t) is a maximal ideal in R. +By construction, every ideal a that +contains (t) = R≥n is generated by (t) and a subset of R0; in particular, a is +homogeneous, whence principal. The grading shows that then a = R or a = (t). +Thus, (t) is maximal and so R0 is a field. +In the setting of Z-graded principal ideal domains, further examples appear, +such as generalised Rees rings [PvG82]. +4 +Trivially graded principal ideal domains +Example 4.1 (elementary divisor decompositions over trivially graded PIDs). +Let R be a principal ideal domain with the 0-grading that contains two non- +associated prime elements p and q (e.g., 2 and 3 in Z). We consider the graded +R-module +M := Σ0R/(p) ⊕ Σ1R/(q). +This graded R module does not admit a graded elementary divisor decom- +position: Indeed, if there were a graded elementary divisor decomposition of M, +then the corresponding elementary divisors would have to coincide with the un- +graded elementary divisors. The only ungraded elementary divisor of M is p · q. +However, M does not contain a homogenous element with annihilator ideal (p·q). +Therefore, M does not admit a graded elementary divisor decomposition. +Proposition 4.2 (prime power decompositions over trivially graded PIDs). Let +R be a principal ideal domain with the 0-grading and let M be a finitely generated +graded R-module. Then M admits a graded prime power decomposition and the +signature of all such graded decompositions of M coincide. +Proof. Because R is trivially graded, the grading on M decomposes M as a +direct sum � +n∈N Mn of R-submodules. +In view of finite generation of M, +only finitely many of these summands are non-trivial. We can now apply the +ungraded structure theorem to each summand Mn to conclude. + +6 +5 +Polynomial rings over fields +5 +Polynomial rings over fields +In view of Proposition 3.1, Theorem 1.1 can equivalently be stated as follows +(which is exactly the special case needed in persistent homology): +Theorem 5.1 (structure theorem for graded modules over polynomial rings). +Let K be a field and let M be a finitely generated graded module over the graded +ring K[T ]. Then there exist N ∈ N, n1, . . . , nN ∈ N, and k1, . . . , kN ∈ N>0 ∪ +{∞} with +M ∼=K[T ]Mod∗ +N +� +j=1 +ΣnjK[T ]/(T kj). +Here, T ∞ := 0. The multiset of all (nj, kj) with j ∈ {1, . . . , N} is uniquely +determined by M. +The rest of this section contains an elementary and constructive proof of +Theorem 5.1. +5.1 +Uniqueness of graded decompositions +The uniqueness claim in Theorem 5.1 can be derived inductively from the un- +graded uniqueness statement: +Let a decomposition as in Theorem 5.1 be given and let ϕ: � +... · · · −→ M +be a corresponding graded K[T ]-isomorphism. Then +M ′ := ϕ(N ′) with N ′ := +� +j∈{1,...,N},nj=0 +ΣnjK[T ]/(T kj) +is a graded submodule of M and it is not difficult to see that M ′ = ϕ(N ′) = +SpanK[T ] M0. +Moroever, M ′ is finitely generated over K[T ]. Therefore, the +ungraded structure theorem when applied to M ′ shows that the multiset of all +pairs (nj, kj) with nj = 0 is uniquely determined by M. +For the induction step, we pass to the quotient M/M ′, which is a finitely gen- +erated graded K[T ]-module with (M/M ′)0 ∼= 0. We shift the degrees on M/M ′ +by −1 and inductively apply the previous argument. +5.2 +Homogeneous matrix reduction +The standard matrix reduction algorithm for the computation of persistent ho- +mology [EH10, ZC05] can be viewed as a proof of the existence part of Theo- +rem 5.1. +We phrase the matrix reduction algorithm in the graded language to em- +phasise the connection with graded decompositions. +Definition 5.2 (graded matrix). Let K be a field, let r, s ∈ N, and let n1, . . . , nr, +m1, . . . , ms ∈ N be monotonically increasing. A matrix A ∈ Mr×s(K[T ]) is +(n∗, m∗)-graded if the following holds: For all j ∈ {1, . . . , r}, k ∈ {1, . . ., s}, we +have that the entry Ajk ∈ K[T ] is a homogeneous polynomial and +• Ajk = 0 or +• nj = deg Ajk + mk. + +5 +Polynomial rings over fields +7 +In a graded matrix, the degrees of matrix entries monotonically increase +from the left to the right and from the bottom to the top. +Definition 5.3 (reduced matrix). Let K be a field, let r, s ∈ N, and let +n1, . . . , nr, m1, . . . , ms ∈ N be monotonically increasing, and let A ∈ Mr×s(K[T ]) +be an (n∗, m∗)-graded matrix. +• For k ∈ {1, . . ., s}, we define +lowA(k) := max +� +j ∈ {1, . . . , r} +�� Ajk ̸= 0 +� +∈ N +(with max ∅ := 0). I.e., lowA(k) is the index of the “lowest” matrix entry +in column k that is non-zero. +• The matrix A is reduced if all columns have different low-indices: For +all k, k′ ∈ N with lowA(k) ̸= 0 and lowA(k′) ̸= 0, we have lowA(k) ̸= +lowA(k′). +Graded matrices can be transformed into reduced matrices via elementary +column operations; these reduced matrices then lead to module decompositions: +Algorithm 5.4 (homogeneous matrix reduction). Given a field K, r, s ∈ N, +monotonically increasing sequences n1, . . . , nr, m1, . . . , ms ∈ N, and an (n∗, m∗)- +graded matrix A ∈ Mr×s, do the following: +• For each k from 1 up to s (in ascending order): +Let ℓ := lowA(k). +If ℓ ̸= 0, then: +• For each j from ℓ down to 1 (in descending order): +If Ajk ̸= 0 and there exists k′ ∈ {1, . . ., k − 1} with lowA(k′) = j, +then: +• Update the matrix A by subtracting Ajk/Ajk′-times the col- +umn k′ from column k. +[Loop invariant observation: Because A is graded, Ajk/Ajk′ in- +deed is a homogeneous polynomial over K and the resulting ma- +trix is (n∗, m∗)-graded. This eliminates the entry Ajk′.] +• Return the resulting matrix A. +Proposition 5.5. Let K be a field, let r, s ∈ N, let n1, . . . , ns, m1, . . . , mr ∈ N +be monotonically increasing, and let A ∈ Mr×s(K[T ]) be an (n∗, m∗)-graded +matrix. Then: +1. The homogeneous matrix reduction algorithm (Algorithm 5.4) terminates +on this input after finitely many steps (relative to the arithmetic on K). +2. The resulting matrix A′ is reduced and there is a graded s × s-matrix B +over K[T ] that admits a graded inverse and satisfies A′ = A · B. +3. The low-entries of the resulting matrix A′ are the elementary divisors of A +over K[T ]. + +8 +5.2 +Homogeneous matrix reduction +4. We have +F/ im A ∼=K[T ]Mod∗ +� +j∈I +ΣnjK[T ]/(T mk(j)−nj) ⊕ +� +j∈I′ +ΣnjK[T ], +where F := �r +j=1 ΣnjK[T ] and I := {lowA′(k) | k ∈ {1, . . . , s}} \ {0} +as well as I′ := {1, . . ., r} \ I. For j ∈ I, let k(j) ∈ {1, . . . , s} be the +unique (!) index with lowA′(k(j)) = j. +Proof. Ad 1. Well-definedness follows from the observation mentioned in the +algorithm: As every homogeneous polynomial in K[T ] is of the form λ · T d +with λ ∈ K and d ∈ N and as the matrix is graded, the corresponding division +can be performed in K[T ] and the gradedness of the matrix is preserved by the +elimination operation. Termination is then clear from the algorithm. +Ad 2. +As we traverse the columns from left to right, a straightforward +induction shows that no two columns can remain that have the same non-zero +value of “lowA”. The product decomposition comes from the fact that we only +applied elementary homogeneous column operations without swaps. +Ad 3. Because the resulting matrix A′ is obtained through elementary col- +umn operations from A, the elementary divisors of A′ and A coincide. Applying +Lemma 5.6 to A′ proves the claim. +Ad 4. In view of the second part, we have that F/ im A ∼=K[T ]Mod∗ F/ im A′. +Therefore, the claim is a direct consequence of Lemma 5.6. +Lemma 5.6. Let K be a field, let r, s ∈ N, let n1, . . . , nr, m1, . . . , ms ∈ N be +monotonically increasing, and let A ∈ Mr×s(K[T ]) be an (n∗, m∗)-graded matrix +that is reduced. Then: +1. The low-entries of A are the elementary divisors of A over K[T ]. +2. Let F := �r +j=1 ΣnjK[T ] and I := {lowA(k) | k ∈ {1, . . ., s}} \ {0} as well +as I′ := {1, . . ., r} \ I. Then +F/ im A ∼=K[T ]Mod∗ +� +j∈I +ΣnjK[T ]/(T mk(j)−nj) ⊕ +� +j∈I′ +ΣnjK[T ] +Proof. Ad 1. Let k ∈ {1, . . . , s} with ℓ := lowA(k) ̸= 0. Then we can clear out +all the entries of A in column k above ℓ by elementary row operations (again, the +gradedness of A ensures that this is possible). Swapping zero rows and columns +appropriately thus results in a matrix in rectangle “diagonal” form; moreover, +as all the “diagonal” entries are monomials, we can swap rows and columns to +obtain a matrix A′ in Smith normal form that both +• has the same elementary divisors as A and +• whose elementary divisors are precisely the low-entries of A. +In particular, these elementary divisors must coincide. +Ad 2. The claim is clear if A is already in Smith normal form. By con- +struction, there are square matrices B and C that are invertible over K[T ] and +represent graded K[T ]-isomorphisms with +A′ = C · A · B. +In particular, F/ im A ∼=K[T ]Mod∗ (C · F)/ im A′. By construction, the values +of lowA′ and the degrees of A′ differ from the ones of A only by compatible +index permutations. Therefore, the claim follows. + +5 +Polynomial rings over fields +9 +5.3 +Existence of a graded decomposition +To prove existence in Theorem 5.1 we can follow the standard proof pattern +of first finding a (graded) finite presentation and then applying (homogeneous) +matrix reduction. +Let M be a finitely generated graded K[T ]-module. Then M also has a finite +generating set consisting of homogeneous elements. This defines a surjective +graded K[T ]-homomorphism +ϕ: F := +r +� +j=1 +ΣnjK[T ] −→ M +for suitable r ∈ N and monotonically increasing n1, . . . , nr ∈ N. +As ϕ is a +graded homomorphism, ker ϕ ⊂ F is a graded K[T ]-submodule and we obtain +an isomorphism +M ∼=K[T ]Mod∗ F/ im ker ϕ +of graded K[T ]-modules. +Because K[T ] is a principal ideal domain, the graded submodule ker ϕ ⊂ F +is finitely generated over K[T ]. Because ker ϕ is a graded submodule, ker ϕ has a +finite homogeneous generating set. (In fact, there also exists a homogeneous free +K[T ]-basis for ker ϕ, as can be seen from a straightforward inductive splitting +argument [Web85, Lemma 1].) In particular, there exist s ∈ N, monotonically +increasing m1, . . . , ms ∈ N, and a graded K[T ]-homomorphism +ψ: E := +s +� +k=1 +ΣmkK[T ] −→ F +with im ψ = ker ϕ. Because ψ is graded and n∗, m∗ are monotonically increas- +ing, the r×s-matrix A over K[T ] that represents ψ with respect to the canonical +homogeneous bases of E and F is graded in the sense of Definition 5.2. +Applying the homogeneous matrix reduction algorithm to A shows that +M ∼=K[T ]Mod∗ F/ im A, +has the desired decomposition (Proposition 5.5; after discarding the irrelevant +terms of the form ΣnK[T ]/(T 0)). +This completes the proof of the structure theorem (Theorem 5.1). +Remark 5.7. There is a general matrix reduction for a slighlty different notion +of “graded” matrices over (Z-)graded principal ideal domains [PvG82]. However, +one should be aware that such “graded” matrices in general only lead to graded +homomorphisms once one is allowed to change the grading on the underlying free +modules. This explains why this general matrix reduction does not contradict +the counterexample in case of 0-graded principal ideal rings in Example 4.1. +5.4 +Barcodes +For the sake of completeness, we recall the relation between graded decomposi- +tions and barcodes: + +10 +References +Remark 5.8 (barcodes of persistence modules). Let K be a field and let +(M ∗, f ∗) be an N-indexed persistence K-module of finite type. We equip M := +� +n∈N M n with the canonical graded K[T ]-module structure (Example 2.7). By +the graded structure theorem (Theorem 5.1), there exist N ∈ N, n1, . . . , nN ∈ N, +and k1, . . . , kN ∈ N>0 ∪ {∞} with +M ∼=K[T ]Mod∗ +N +� +j=1 +ΣnjK[T ]/(T kj). +Let B be the multiset of all (nj, kj − 1) with j ∈ {1, . . ., N}; then B is uniquely +determined by M and this multiset B is the barcode of (M ∗, f ∗). +The barcode contains the full information on the isomorphism type of the +graded K[T ]-module M (and the underlying persistence module) and describes +the birth, death, and persistence of elements as specified by the “elder rule”: If +(n, p) is an element of the barcode, this means that a new independent class is +born at stage n, it persists for p stages, and it dies (if p ̸= ∞) at stage n+ p+ 1. +In particular, this leads to the notion of barcodes of persistent homology +(in a given degree) of finite type persistence chain complexes and finite type +filtrations in topology. +References +[BCB20] +Magnus Bakke Botnan and William Crawley-Boevey. +Decomposition of +persistence modules. Proc. Amer. Math. Soc., 148(11):4581–4596, 2020. +Cited on page: 2 +[CdS10] +Gunnar Carlsson and Vin de Silva. Zigzag persistence. Found. Comput. +Math., 10(4):367–405, 2010. Cited on page: 2 +[CZCG04] Gunnar Carlsson, Afra Zomorodian, Anne Collins, and Leonidas Guibas. +Persistence Barcodes for Shapes. In Roberto Scopigno and Denis Zorin, ed- +itors, Symposium on Geometry Processing. The Eurographics Association, +2004. Cited on page: 1 +[EH10] +Herbert Edelsbrunner and John L. Harer. Computational topology. Amer- +ican Mathematical Society, Providence, RI, 2010. An introduction. Cited +on page: 1, 6 +[PvG82] +R. Puystjens and J. van Geel. +Diagonalization of matrices over graded +principal ideal domains. Linear Algebra Appl., 48:265–281, 1982. Cited on +page: 5, 9 +[SVJ13] +Primoz Skraba and Mikael Vejdemo-Johansson. Persistence modules: Al- +gebra and algorithms. 2013. arXiv:1302.2015 [cs.CG]. Cited on page: 2 +[VO83] +F. Van Oystaeyen. Generalized Rees rings and arithmetical graded rings. +J. Algebra, 82(1):185–193, 1983. Cited on page: 2, 4 +[Web85] +Cary Webb. Decomposition of graded modules. Proc. Amer. Math. Soc., +94(4):565–571, 1985. Cited on page: 9 +[ZC05] +Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. +Discrete Comput. Geom., 33(2):249–274, 2005. Cited on page: 1, 2, 3, 6 +Clara L¨oh +Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg +clara.loeh@mathematik.uni-r.de, https://loeh.app.ur.de + diff --git a/0dFKT4oBgHgl3EQfNy2J/content/tmp_files/load_file.txt b/0dFKT4oBgHgl3EQfNy2J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f8b24d567bcc28f260b76b99ddf7be681579cef --- /dev/null +++ b/0dFKT4oBgHgl3EQfNy2J/content/tmp_files/load_file.txt @@ -0,0 +1,465 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf,len=464 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='11756v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='AC] 27 Jan 2023 A comment on the structure of graded modules over graded principal ideal domains in the context of persistent homology Clara L¨oh January 30, 2023 Abstract The literature in persistent homology often refers to a “structure the- orem for finitely generated graded modules over a graded principal ideal domain”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We clarify the nature of this structure theorem in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 1 Introduction The persistent homology with field coefficients of finite type filtrations can be described in terms of barcodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Zomorodian and Carlsson promoted the elegant idea to view persistent homology with coefficients in a field K as a graded module over the graded polynomial ring K[T ] [ZC05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' They then suggest a general structure theorem for finitely generated graded modules over graded principal ideal domains [ZC05, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Applying this structure theorem to the graded polynomial ring K[T ] gives a graded elementary divisor decomposition of persistent homology, which can be reinterpreted as barcodes [CZCG04] or, equivalently, as persistence diagrams [EH10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' However, there does not seem to be a proof of this general structure theo- rem in the literature in the form stated by Zomorodian and Carlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' As this theorem is quoted multiple times in work on persistent homology and as it is a potential source of confusion, the goal of this expository note is to clarify the nature of this structure theorem (even though it might be clear to the experts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We first give a precise formulation of the structure theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' this formu- lation slightly differs from the statement of Zomorodian and Carlsson [ZC05, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1] (for a reason explained below): Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 (structure theorem for graded modules over graded PIDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded principal ideal domain with R ̸= R0 and let M be a finitely generated graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then M admits a graded elementary divisor decomposition (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='8) and the signatures of all such graded decompositions of M co- incide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The key observation of this note is that in fact every N-graded principal ideal domain is © C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' L¨oh 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This work was supported by the CRC 1085 Higher Invariants (Universit¨at Regensburg, funded by the DFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' MSC 2010 classification: 13C05, 55N31 1 2 2 Graded rings and modules a principal ideal domain with the 0-grading or a polynomial ring over a field with a multiple of the canonical grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The proof is elementary [VO83, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='7] (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For trivially graded principal ideal domains, in general, the graded elemen- tary divisor version of the structure theorem does not hold (Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This explains the additional hypothesis of R ̸= R0 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In contrast, the graded prime power version of the structure theorem also holds if the grading is trivial (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For polynomial rings, the graded uniqueness part can be deduced in a straightforward way from the ungraded uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' However, for the graded existence part, there does not seem to be a “generic” derivation from the un- graded existence result – the difficulty being the graded direct sum splitting (as exhibited in the case of the trivially graded ring Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Finding such a splitting needs a careful inductive approach that establishes that the torsion submodule is graded and that avoids dividing out cyclic submodules in bad position/order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The graded existence part can be proved using specific properties of polynomial rings over fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In conclusion, the structure theorem for graded modules over graded prin- cipal ideal domains gives a helpful structural perspective on barcodes for per- sistent homology (and also for the computation of persistent homology [ZC05, SVJ13]), but its scope does not seem to go beyond the special case that is needed for persistent homology and it does not seem to provide a shortcut avoiding spe- cial properties of polynomial rings over fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Generalisations of N-graded persistent homology such as zigzag persistence or R-graded persistence (or more general indexing situations) are usually based on arguments from quiver representations [CdS10, BCB20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Similarly to the N- graded case, in these settings, it is also essential that the underlying coefficients are a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Organisation of this article Basic notions on graded rings and modules are recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In Sec- tion 3, we prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The case of principal ideal domains with trivial gradings is considered in Section 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' the case of polynomial rings over fields is discussed in Section 5, where we give an elementary proof of the structure the- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Acknowledgements I would like to thank Ulrich Bunke for helpful discussions on abstract methods for the decomposition of graded modules and Luigi Caputi for valuable feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 2 Graded rings and modules We recall basic notions on graded rings and modules and decompositions of graded modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' As usual in (discrete) persistence, we consider only the case of discrete non-negative gradings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', gradings over N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 2 Graded rings and modules 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 (graded ring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' A graded ring is a pair (R, (Rn)n∈N), where R is a ring and the Rn are additive subgroups of R with the following properties: The additive group (R, +) is the internal direct sum of the (Rn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For all n, m ∈ N, we have Rn · Rm ⊂ Rn+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For n ∈ N, the elements in Rn are called homogeneous of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' An element of R is homogenous if there exists an n ∈ N such that the element is homogeneous of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' A graded ring is a graded principal ideal domain if it is a domain and every homogeneous ideal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', generated by homogeneous elements) is generated by a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2 (polynomial rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then the usual degree on monomials in the polynomial ring K[T ] turns K[T ] into a graded ring via the canonical isomorphism K[T ] ∼=Ab � n∈N K · T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We will refer to this as the canonical grading on K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' If K is a field, then K[T ] is a principal ideal domain (graded and ungraded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='3 (graded module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' A graded module over R is a pair (M, (Mn)), consisting of an R-module M and additive sub- groups Mn of M with the following properties: The additive group (M, +) is the internal direct sum of the (Mn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For all n, m ∈ N, we have Rn · Mm ⊂ Mn+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Elements of Mm are called homogeneous of degree m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='4 (the category of graded modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ho- momorphisms between graded R-modules are R-linear maps that preserve the grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Graded R-modules and homomorphisms of R-modules form the cate- gory RMod∗ of graded R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='5 (shifted graded modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded ring, let M be a graded module over R, and let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then ΣnM denotes the graded R-module given by the n-shifted decomposition 0 ⊕ · · · ⊕ 0 ⊕ � j∈N≥n Mj−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='6 (direct sums and quotients of graded modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let M and N be graded modules over a graded ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then M ⊕ N is a graded R-module via the grading (Mn ⊕ Nn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' If M ′ ⊂ M is a graded submodule of M (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', it is generated by homogeneous elements), then (Mn/(M ′ ∩Mn))n∈N turns M/M ′ into a graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Persistent homology leads to persistence modules [ZC05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Persistence mod- ules in turn give rise to graded modules over graded polynomial rings [ZC05, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1]: Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='7 (from persistence modules to graded modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a ring and let (M ∗, f ∗) be an N-indexed persistence K-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then M := � n∈N M n carries a K[T ]-module structure, given by ∀x∈Mn T · x := f n(x) ∈ M n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' If we view K[T ] as a graded ring (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2), then this K[T ]-module structure and this direct sum decomposition of M turn M into a graded K[T ]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' If (M ∗, f ∗) is of finite type, then M is finitely generated over K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 4 3 Graded principal ideal domains Finally, we define the central types of decompositions arising in the structure theorems: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='8 (graded elementary divisor decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded ring and let M be a graded module over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' A graded elementary divisor de- composition of M over R is an isomorphism M ∼=RMod∗ N � j=1 ΣnjR/(fj) of graded R-modules with N ∈ N, degrees n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nN ∈ N, and homogeneous elements f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , fN ∈ R with fj|fj+1 for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , N −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Here, the right- hand side carries the canonical grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The elements f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , fN are called elementary divisors of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The signature of such a decomposition is the multiset of all pairs (nj, R×·fj) with j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='9 (graded prime power decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded ring and let M be a graded module over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' A graded prime power decomposition of M over R is an isomorphism M ∼=RMod N � j=1 ΣnjR/(pkj j ) of graded R-modules with N ∈ N, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nN ∈ N, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , kN ∈ N, and homo- geneous prime elements p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , pN ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Here, the right-hand side carries the canonical grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The signature of such a decomposition is the multiset of all pairs (nj, R×·pkj j ) with j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 3 Graded principal ideal domains For the sake of completeness, we provide a proof of the following observa- tion [VO83, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 (graded PIDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a graded principal ideal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then R is of one of the following types: We have R = R0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', R is an ordinary principal ideal domain with the 0-grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The subring R0 is a field and R is isomorphic to the graded ring R0[T ], where the grading on R0[T ] is a multiple of the canonical grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R ̸= R0 and let n ∈ N>0 be the minimal degree with Rn ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then R≥n := � j∈N≥n Rj is a homogeneous ideal in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' as R is a graded principal ideal domain, there exists a t ∈ R with R≥n = (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We show that t is homogeneous of degree n: Let 4 Trivially graded principal ideal domains 5 x ∈ Rn \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then t divides x and a straightforward computation shows that hence also t is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The grading implies that t has degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We show that the canonical R0-algebra homomorphism ϕ: R0[T ] −→ R given by ϕ(T ) := t is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We first show that ϕ is injective: Because R is graded and t is homoge- neous, it suffices to show that a · tk ̸= 0 for all a ∈ R0 \\ {0} and all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' However, this is guaranteed by the hypothesis that R is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Regarding surjectivity, let y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' It suffices to consider the case that y is homogeneous of degree m ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Because (t) = R≥n, we know that t divides y, say y = t · y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then y′ is homogeneous and we can iterate the argument for y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proceeding inductively, we obtain that m is a multiple of n and that there exists an a ∈ R0 with y = a · tm/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Hence, ϕ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This establishes that R is isomorphic as a graded ring to R0[T ], where R0[T ] carries the canonical grading on R0[T ] scaled by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' It remains to show that R0 ∼=Ring R/(t) is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Thus, we are left to show that (t) is a maximal ideal in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' By construction, every ideal a that contains (t) = R≥n is generated by (t) and a subset of R0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' in particular, a is homogeneous, whence principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The grading shows that then a = R or a = (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Thus, (t) is maximal and so R0 is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In the setting of Z-graded principal ideal domains, further examples appear, such as generalised Rees rings [PvG82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 4 Trivially graded principal ideal domains Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 (elementary divisor decompositions over trivially graded PIDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a principal ideal domain with the 0-grading that contains two non- associated prime elements p and q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', 2 and 3 in Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We consider the graded R-module M := Σ0R/(p) ⊕ Σ1R/(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This graded R module does not admit a graded elementary divisor decom- position: Indeed, if there were a graded elementary divisor decomposition of M, then the corresponding elementary divisors would have to coincide with the un- graded elementary divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The only ungraded elementary divisor of M is p · q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' However, M does not contain a homogenous element with annihilator ideal (p·q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Therefore, M does not admit a graded elementary divisor decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2 (prime power decompositions over trivially graded PIDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let R be a principal ideal domain with the 0-grading and let M be a finitely generated graded R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then M admits a graded prime power decomposition and the signature of all such graded decompositions of M coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Because R is trivially graded, the grading on M decomposes M as a direct sum � n∈N Mn of R-submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In view of finite generation of M, only finitely many of these summands are non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We can now apply the ungraded structure theorem to each summand Mn to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 6 5 Polynomial rings over fields 5 Polynomial rings over fields In view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 can equivalently be stated as follows (which is exactly the special case needed in persistent homology): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 (structure theorem for graded modules over polynomial rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a field and let M be a finitely generated graded module over the graded ring K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then there exist N ∈ N, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nN ∈ N, and k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , kN ∈ N>0 ∪ {∞} with M ∼=K[T ]Mod∗ N � j=1 ΣnjK[T ]/(T kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Here, T ∞ := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The multiset of all (nj, kj) with j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , N} is uniquely determined by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The rest of this section contains an elementary and constructive proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 Uniqueness of graded decompositions The uniqueness claim in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 can be derived inductively from the un- graded uniqueness statement: Let a decomposition as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 be given and let ϕ: � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' · · · −→ M be a corresponding graded K[T ]-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then M ′ := ϕ(N ′) with N ′ := � j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=',N},nj=0 ΣnjK[T ]/(T kj) is a graded submodule of M and it is not difficult to see that M ′ = ϕ(N ′) = SpanK[T ] M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Moroever, M ′ is finitely generated over K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Therefore, the ungraded structure theorem when applied to M ′ shows that the multiset of all pairs (nj, kj) with nj = 0 is uniquely determined by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For the induction step, we pass to the quotient M/M ′, which is a finitely gen- erated graded K[T ]-module with (M/M ′)0 ∼= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We shift the degrees on M/M ′ by −1 and inductively apply the previous argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2 Homogeneous matrix reduction The standard matrix reduction algorithm for the computation of persistent ho- mology [EH10, ZC05] can be viewed as a proof of the existence part of Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We phrase the matrix reduction algorithm in the graded language to em- phasise the connection with graded decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2 (graded matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a field, let r, s ∈ N, and let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nr, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , ms ∈ N be monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' A matrix A ∈ Mr×s(K[T ]) is (n∗, m∗)-graded if the following holds: For all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , r}, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', s}, we have that the entry Ajk ∈ K[T ] is a homogeneous polynomial and Ajk = 0 or nj = deg Ajk + mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 5 Polynomial rings over fields 7 In a graded matrix, the degrees of matrix entries monotonically increase from the left to the right and from the bottom to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='3 (reduced matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a field, let r, s ∈ N, and let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nr, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , ms ∈ N be monotonically increasing, and let A ∈ Mr×s(K[T ]) be an (n∗, m∗)-graded matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', s}, we define lowA(k) := max � j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , r} �� Ajk ̸= 0 � ∈ N (with max ∅ := 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', lowA(k) is the index of the “lowest” matrix entry in column k that is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The matrix A is reduced if all columns have different low-indices: For all k, k′ ∈ N with lowA(k) ̸= 0 and lowA(k′) ̸= 0, we have lowA(k) ̸= lowA(k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Graded matrices can be transformed into reduced matrices via elementary column operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' these reduced matrices then lead to module decompositions: Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='4 (homogeneous matrix reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Given a field K, r, s ∈ N, monotonically increasing sequences n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nr, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , ms ∈ N, and an (n∗, m∗)- graded matrix A ∈ Mr×s, do the following: For each k from 1 up to s (in ascending order): Let ℓ := lowA(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' If ℓ ̸= 0, then: For each j from ℓ down to 1 (in descending order): If Ajk ̸= 0 and there exists k′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', k − 1} with lowA(k′) = j, then: Update the matrix A by subtracting Ajk/Ajk′-times the col- umn k′ from column k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' [Loop invariant observation: Because A is graded, Ajk/Ajk′ in- deed is a homogeneous polynomial over K and the resulting ma- trix is (n∗, m∗)-graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This eliminates the entry Ajk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='] Return the resulting matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a field, let r, s ∈ N, let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , ns, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , mr ∈ N be monotonically increasing, and let A ∈ Mr×s(K[T ]) be an (n∗, m∗)-graded matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The homogeneous matrix reduction algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='4) terminates on this input after finitely many steps (relative to the arithmetic on K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The resulting matrix A′ is reduced and there is a graded s × s-matrix B over K[T ] that admits a graded inverse and satisfies A′ = A · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The low-entries of the resulting matrix A′ are the elementary divisors of A over K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2 Homogeneous matrix reduction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We have F/ im A ∼=K[T ]Mod∗ � j∈I ΣnjK[T ]/(T mk(j)−nj) ⊕ � j∈I′ ΣnjK[T ], where F := �r j=1 ΣnjK[T ] and I := {lowA′(k) | k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , s}} \\ {0} as well as I′ := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', r} \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' For j ∈ I, let k(j) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , s} be the unique (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=') index with lowA′(k(j)) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Well-definedness follows from the observation mentioned in the algorithm: As every homogeneous polynomial in K[T ] is of the form λ · T d with λ ∈ K and d ∈ N and as the matrix is graded, the corresponding division can be performed in K[T ] and the gradedness of the matrix is preserved by the elimination operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Termination is then clear from the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ad 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' As we traverse the columns from left to right, a straightforward induction shows that no two columns can remain that have the same non-zero value of “lowA”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The product decomposition comes from the fact that we only applied elementary homogeneous column operations without swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ad 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Because the resulting matrix A′ is obtained through elementary col- umn operations from A, the elementary divisors of A′ and A coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='6 to A′ proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ad 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In view of the second part, we have that F/ im A ∼=K[T ]Mod∗ F/ im A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Therefore, the claim is a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a field, let r, s ∈ N, let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nr, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , ms ∈ N be monotonically increasing, and let A ∈ Mr×s(K[T ]) be an (n∗, m∗)-graded matrix that is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The low-entries of A are the elementary divisors of A over K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let F := �r j=1 ΣnjK[T ] and I := {lowA(k) | k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', s}} \\ {0} as well as I′ := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', r} \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then F/ im A ∼=K[T ]Mod∗ � j∈I ΣnjK[T ]/(T mk(j)−nj) ⊕ � j∈I′ ΣnjK[T ] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , s} with ℓ := lowA(k) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then we can clear out all the entries of A in column k above ℓ by elementary row operations (again, the gradedness of A ensures that this is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Swapping zero rows and columns appropriately thus results in a matrix in rectangle “diagonal” form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' moreover, as all the “diagonal” entries are monomials, we can swap rows and columns to obtain a matrix A′ in Smith normal form that both has the same elementary divisors as A and whose elementary divisors are precisely the low-entries of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In particular, these elementary divisors must coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Ad 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The claim is clear if A is already in Smith normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' By con- struction, there are square matrices B and C that are invertible over K[T ] and represent graded K[T ]-isomorphisms with A′ = C · A · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In particular, F/ im A ∼=K[T ]Mod∗ (C · F)/ im A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' By construction, the values of lowA′ and the degrees of A′ differ from the ones of A only by compatible index permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Therefore, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 5 Polynomial rings over fields 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='3 Existence of a graded decomposition To prove existence in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1 we can follow the standard proof pattern of first finding a (graded) finite presentation and then applying (homogeneous) matrix reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let M be a finitely generated graded K[T ]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Then M also has a finite generating set consisting of homogeneous elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This defines a surjective graded K[T ]-homomorphism ϕ: F := r � j=1 ΣnjK[T ] −→ M for suitable r ∈ N and monotonically increasing n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nr ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' As ϕ is a graded homomorphism, ker ϕ ⊂ F is a graded K[T ]-submodule and we obtain an isomorphism M ∼=K[T ]Mod∗ F/ im ker ϕ of graded K[T ]-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Because K[T ] is a principal ideal domain, the graded submodule ker ϕ ⊂ F is finitely generated over K[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Because ker ϕ is a graded submodule, ker ϕ has a finite homogeneous generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' (In fact, there also exists a homogeneous free K[T ]-basis for ker ϕ, as can be seen from a straightforward inductive splitting argument [Web85, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=') In particular, there exist s ∈ N, monotonically increasing m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , ms ∈ N, and a graded K[T ]-homomorphism ψ: E := s � k=1 ΣmkK[T ] −→ F with im ψ = ker ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Because ψ is graded and n∗, m∗ are monotonically increas- ing, the r×s-matrix A over K[T ] that represents ψ with respect to the canonical homogeneous bases of E and F is graded in the sense of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Applying the homogeneous matrix reduction algorithm to A shows that M ∼=K[T ]Mod∗ F/ im A, has the desired decomposition (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' after discarding the irrelevant terms of the form ΣnK[T ]/(T 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This completes the proof of the structure theorem (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' There is a general matrix reduction for a slighlty different notion of “graded” matrices over (Z-)graded principal ideal domains [PvG82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' However, one should be aware that such “graded” matrices in general only lead to graded homomorphisms once one is allowed to change the grading on the underlying free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' This explains why this general matrix reduction does not contradict the counterexample in case of 0-graded principal ideal rings in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='4 Barcodes For the sake of completeness, we recall the relation between graded decomposi- tions and barcodes: 10 References Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='8 (barcodes of persistence modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let K be a field and let (M ∗, f ∗) be an N-indexed persistence K-module of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' We equip M := � n∈N M n with the canonical graded K[T ]-module structure (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' By the graded structure theorem (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='1), there exist N ∈ N, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , nN ∈ N, and k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' , kN ∈ N>0 ∪ {∞} with M ∼=K[T ]Mod∗ N � j=1 ΣnjK[T ]/(T kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Let B be the multiset of all (nj, kj − 1) with j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' then B is uniquely determined by M and this multiset B is the barcode of (M ∗, f ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The barcode contains the full information on the isomorphism type of the graded K[T ]-module M (and the underlying persistence module) and describes the birth, death, and persistence of elements as specified by the “elder rule”: If (n, p) is an element of the barcode, this means that a new independent class is born at stage n, it persists for p stages, and it dies (if p ̸= ∞) at stage n+ p+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In particular, this leads to the notion of barcodes of persistent homology (in a given degree) of finite type persistence chain complexes and finite type filtrations in topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' References [BCB20] Magnus Bakke Botnan and William Crawley-Boevey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Decomposition of persistence modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', 148(11):4581–4596, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 2 [CdS10] Gunnar Carlsson and Vin de Silva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Zigzag persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', 10(4):367–405, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 2 [CZCG04] Gunnar Carlsson, Afra Zomorodian, Anne Collins, and Leonidas Guibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Persistence Barcodes for Shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' In Roberto Scopigno and Denis Zorin, ed- itors, Symposium on Geometry Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' The Eurographics Association, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 1 [EH10] Herbert Edelsbrunner and John L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Harer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Computational topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Amer- ican Mathematical Society, Providence, RI, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' An introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 1, 6 [PvG82] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Puystjens and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' van Geel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Diagonalization of matrices over graded principal ideal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', 48:265–281, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 5, 9 [SVJ13] Primoz Skraba and Mikael Vejdemo-Johansson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Persistence modules: Al- gebra and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' arXiv:1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='2015 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='CG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 2 [VO83] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Van Oystaeyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Generalized Rees rings and arithmetical graded rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Algebra, 82(1):185–193, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 2, 4 [Web85] Cary Webb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Decomposition of graded modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', 94(4):565–571, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 9 [ZC05] Afra Zomorodian and Gunnar Carlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Computing persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=', 33(2):249–274, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content=' Cited on page: 1, 2, 3, 6 Clara L¨oh Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg clara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='loeh@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='uni-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='de, https://loeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf'} diff --git a/1dFLT4oBgHgl3EQfpy-5/content/2301.12137v1.pdf b/1dFLT4oBgHgl3EQfpy-5/content/2301.12137v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6d3bda6a252efbf7168251e1680a9900804d84da --- /dev/null +++ b/1dFLT4oBgHgl3EQfpy-5/content/2301.12137v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7e413712f23fcdfa6e417cb008d81fa85e3165cfd2925bb3376e228c73d0127f +size 303973 diff --git a/1dFLT4oBgHgl3EQfpy-5/vector_store/index.faiss b/1dFLT4oBgHgl3EQfpy-5/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ffd79e0718330f399a14a34cc78571c229e59248 --- /dev/null +++ b/1dFLT4oBgHgl3EQfpy-5/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:829292d4adfc5bf683710cc871c447e321dff79f7c6deef812eaa261235884a5 +size 7733293 diff --git a/2dFST4oBgHgl3EQfXzgB/content/tmp_files/2301.13785v1.pdf.txt b/2dFST4oBgHgl3EQfXzgB/content/tmp_files/2301.13785v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b8b75bacdd9994ce463687f2aa246b5c0b09772 --- /dev/null +++ b/2dFST4oBgHgl3EQfXzgB/content/tmp_files/2301.13785v1.pdf.txt @@ -0,0 +1,1127 @@ +Commitment Against Front Running +Attacks∗ +Andrea Canidio †and Vincent Danos‡ +February 1, 2023 +Abstract +We provide a game-theoretic analysis of the problem of front-running attacks. We use it +to distinguish attacks from legitimate competition among honest users for having their +transactions included earlier in the block. We also use it to introduce an intuitive notion +of the severity of front-running attacks. We then study a simple commit-reveal protocol +and discuss its properties. This protocol has costs because it requires two messages and +imposes a delay. However, it is effective at preventing the most severe front-running at- +tacks while preserving legitimate competition between users, guaranteeing that the earliest +transaction in a block belongs to the honest user who values it the most. +Keywords: Front running, Game theory, Ethereum, MEV, Transaction reordering, commit- +reveal +1 +Introduction +On the Ethereum network, each validator decides how to order pending transactions +to form the next block, which determines the order in which these transactions are +executed. +As a consequence, users often compete with each other to have their +∗We are grateful to Agostino Capponi, Jiasun Li, Christof Ferreira Torres, Arthur Gervais, +Ari Juels, and the participants to UBRI Connect 2022, Tokenomics 2022 for their comments and +suggestions. We gratefully acknowledge the financial support of the Ethereum Foundation (grant +FY22-0840). +†IMT school of advanced studies, Lucca, Italy +‡CNRS and École Normale Supérieure (France); +1 +arXiv:2301.13785v1 [econ.TH] 31 Jan 2023 + +1 Introduction +2 +transactions included earlier in a block, either by paying transaction fees or by mak- +ing side payments directly to validators.1 This form of competition can be beneficial +because it ensures that a scarce resource (i.e., having a transaction included earlier +in the block) is allocated to the user who values it the most.2 But at the same time, +it opens the possibility of front-running attacks: because pending transactions are +public, a malicious user can observe a victim’s incoming transaction, craft a new +transaction and then pay to place it before that of the victim. +Importantly, legitimate competition and attacks are often difficult to distinguish. +As an illustrative example, consider a smart contract programmed to award a valu- +able NFT to the first person who correctly answers a question. Assume, crucially, +that the smart contract does not have an explicit mechanism to resolve competing +claims to the object (i.e., by running an auction among those who provided the +correct answer) and settles claims in order of arrival. In this example, competition +between users can arise in two cases. In the first case, two users simultaneously +and independently find the answer. Each submits it and competes to have his/her +transaction included earlier in the block. Because the user who values the NFT the +most is willing to pay more, this user should be able to place his transaction before +that of the opponent, thereby winning the NFT. In the second case, an honest user +finds the answer and sends it to the smart contract. A malicious user observes the +transaction, copies it, and competes to have its copy included in the block earlier +than the original transaction. +From the observational point of view, the above two situations are identical: +two users submit the same answer and then compete to have it included earlier in +the block. Despite this, the first is an example of legitimate competition because +users do not exploit their observation of the opponent’s transaction. Hence, each +user would have submitted his answer also in the absence of the other user. The +second is an attack because the attacker cannot send his transaction if he does +not observe the victim’s transaction. Furthermore, the extent to which an attacker +1 Competition through higher transaction fees occurs via “gas replacement” transactions, +whereby a pending transaction is resubmitted with a higher fee. The resulting game is akin to an +auction (see Daian et al. (2019)). The most popular way to make side payments to validators is +to use flashbots (see https://github.com/flashbots/pm). +2 Whether it is the most efficient to achieve this goal is a different issue we do not address here. + +1 Introduction +3 +relies on the victim’s message can be interpreted as a measure of the severity of +a front-running attack. +For example, the “attacker” could be another user who, +through his research, narrowed the correct answer to two or three possibilities. This +attack seems less severe relative to a situation in which the attacker has no prior +information. +Our simple, illustrative example is representative of the workings of most smart +contracts, including those at the core of decentralized finance protocols. For ex- +ample, Automated Market Makers (AMM) are the dominant type of decentralized +exchange. They allow users to swap one token for a different one at a price that +is mechanically derived by the size of two liquidity pools (one per token traded). +Because swapping one token for another changes the liquidity in each pool, if no +liquidity is added, then a sequence of users performing the same swap will face worse +and worse terms.3 Hence, two users who independently decide to perform the same +swap on an AMM will compete to obtain a better rate. Competition ensures that +the user who most values obtaining the better rate (perhaps because the swap is +part of a sequence of atomic transactions) can obtain it. Alternatively, a user may +want to perform a swap. Upon observing this transaction, an attacker will front-run +the victim with the same swap and then back-run her with the opposite swap, in +what is called a sandwich attack or insertion attack.4 Again, the second example is +an attack because the attacker uses information obtained by observing the victim’s +transaction. +In this paper, we propose a game-theoretic model of front-running. Our goals are +two. First, inspired by the above discussion, we aim to provide a formal definition +of front-running attacks (vs. legitimate competition among honest users) as well as +their severity. Second, we study a simple commit-reveal protocol that can be im- +plemented at the smart contract level without modifying the underlying Ethereum +infrastructure or introducing third parties (or layer-2 networks). In the simplest +version of the protocol, the user concatenates the desired message with the address +3 Like in our simple illustrative example, AMMs do not have an explicit mechanism to allocate +the “better rate” to one of the users and instead rely on the order in which transactions are +aggregated into a block. +4 See Eskandari et al. (2019) for a discussion of this type of front running. See also Park (2022) +for an analysis of these types of attacks in the context of AMMs. + +1 Introduction +4 +from which the reveal message will be sent and passes this into a function with +an intractable pre-image problem (for example, the SHA-256 hash function).5 The +resulting output is the commit message, which the user sends to the smart contract. +Then, the user sends the reveal message to the smart contract, where the reveal +message is simply the desired message. The smart contract receiving a reveal mes- +sage will execute it only if the concatenation of the reveal message with the address +from which it was received corresponds to the commit message. +The key observation is that an attack involves two steps: (i) committing a mes- +sage without knowing what message the victim will send and (ii) after observing +the victim’s reveal message, deciding to send the committed message or no message +at all. Furthermore, the victim may use a newly-created address to send the com- +mit message. When this is the case, the attacker may observe that someone sent +a commit message to the smart contract, but not whom. The protocol, therefore, +forces the attacker to make a costly guess: send a costly commit message without +knowing whether a given victim committed (and will reveal) nor what the victim +committed. At the same time, the protocol does not impede legitimate competition +between users: two honest users can commit their messages and then compete to +have their reveal message included earlier in the block. +We derive conditions under which an honest player is better off using the pro- +tocol than Ethereum’s standard procedure.6 On the cost side, the protocol requires +sending two messages instead of one and imposes a delay. Hence, if the cost of +sending messages or waiting is high, the protocol is worse than the standard way +to send transactions; if they are low, the protocol is preferred. On the benefit side, +the protocol can eliminate front-running attacks, especially when it is difficult for +an attacker to guess, that is, when the expected payoff of an attacker who commits +without knowing whether the victim committed and what message was committed +is low. By definition of the severity of the attack introduced earlier, we can say +5 In this version, an attacker observes that a smart contract received a commit message (but +not necessary whom if the commit message is sent from a brand new address). Later, we discuss +a more complex protocol in which the receiver of the commit message is obfuscated. +6 We believe that, for our purposes, the honest player’s welfare is the most sensible criterion to +evaluate the protocol. The alternative would be to consider the smart contract perspective. Note, +however, that certain types of attacks can be quite profitable from the smart contract viewpoint— +for example, sandwich attacks in the context of AMMs. + +1 Introduction +5 +that our protocol is most effective when the severity of the attack is high and is less +effective when the severity of the attack is low. +As an extension (see Section 5.1), we study a variation of the above protocol in +which the identity of the receiver of the commit message is hidden. This variation +hinges on the existence of a template code for a container smart contract. When +committing, the user uses a brand new address to create a container smart con- +tract using the template and then sends the commit message to this newly-created +container, which time-stamps the commitment message with the current block num- +ber. When sending the reveal message, the honest user also sends a pointer to the +container smart contract where the commitment is located. The smart contract +considers the commitment as valid if the commit message is correct, its timestamp +is antecedent to the current block, and, crucially, if the code of the container smart +contract corresponds to the template. This way, an outside observer can only see +that someone created a commitment smart contract and sent a commit message, but +not who committed nor the target smart contract for that commitment. Guessing +is even harder for an attacker, and hence the probability of an attack is even lower. +As a second extension (see Section 5.4), we introduce multiple attackers. Absent +the protocol, competition pushes each attacker to overspend (relative to the single- +attacker case). This is detrimental to both the attackers and the honest user. In +particular, the weakest attacker always earns zero in expected terms.7 Instead, in +the commit-reveal protocol, the commit message acts as a fixed cost to attack in the +next period. Because in the following period, the weakest attackers earn zero, there +is no equilibrium in which both attackers send the commit message with probability +1: either there is no attack; or a single attacker commits and then attacks; or +both attackers commit (and then attack) with probability strictly less than one. +As a consequence, in the best-case scenario, the protocol eliminates attacks; in the +worst-case scenario, it reduces the level of competition between attackers resulting +in fewer resources spent in an attack, which benefits the honest player as well. +7 This result is a version of a well-known result in contest theory, that of “full dissipation of +rents”. See, for example, Fudenberg and Tirole (1987). + +1 Introduction +6 +Prior work +Our commit-reveal protocol is novel but similar to existing proposals. +Our main contribution is the type of analysis. In particular, we show that our proto- +col can eliminate the most severe front-running attacks while maintaining legitimate +competition between users. As we discuss later, existing solutions instead are either +primarily concerned with eliminating attacks (at the cost of also eliminating legit- +imate competition) or better organizing competition (at the cost of exacerbating +attacks). Furthermore, most of the literature has proposed solutions to reduce or +eliminate front-running in Ethereum by changing its infrastructure or introducing +third parties (See Heimbach and Wattenhofer (2022) for a review of the literature). +Instead, our solution does not require third parties and can be implemented at the +smart contract level, allowing for flexibility in its implementation. For example, +each smart contract could decide that only some messages must follow the protocol +to be considered valid, while other messages do not need to.8 Or a smart contract +may decide that the protocol is required only during some periods (see Section 5.3). +With respect to existing solutions, our protocol can be seen as a simplified version +of the submarine commitments protocol in Breidenbach et al. (2018): in both cases, a +message is first committed and then revealed, and the commitment can be hidden in +the sense that the identity of the sender and receiver of the commit message cannot +be observed. The main difference is that we adopt a weaker notion of “commitment” +because we allow users not to send a transaction after committing it. The notion +of “commitment” in Breidenbach et al. (2018) is instead stronger because users are +penalized for not following through with their commitment. +As already mentioned, we provide a game-theoretic analysis of the properties +of this protocol, applicable to any smart contract.9 With this respect, our work +is inspired by Gans and Holden (2022), who develop a game-theoretic analysis of +the problem of front-running arising when an honest user and an attacker claim the +same reward. They also propose a protocol that eliminates these types of attacks. +Their key assumption is that the legitimate claimant strictly prefers the reward +8 See the discussion in Section 5.2. For example, in the context of AMMs, it may make sense +that users who want to provide or withdraw liquidity do not need to follow the commit reveal +protocol, which is instead required for swaps. +9 Breidenbach et al. (2018) analyze the properties of the submarine commitment scheme in the +context of a bug-bounty scheme they propose. + +1 Introduction +7 +to be burned rather than paid to the attacker. Therefore, these results are useful +in some environments where front-running may emerge, but not all. For example, +front-running attacks are a serious concern in the AMMs, but in this context, it may +not be possible to “burn the reward”. +Flashbots is a well-known project aiming to better organize competition among +users. The premise is that competition through transaction fees can lead to so-called +“gas wars” by which a given block is filled with transactions that will fail (because +only the first one can be correctly executed). Therefore, gas wars impose a negative +externality on all users because they lead to congestion and higher transaction fees. +The idea is to eliminate these negative externalities by allowing users to pay valida- +tors directly, therefore keeping their messages private.10 Doing so, however, makes +it extremely easy to attack an honest user who sends his or her message publicly +(see Capponi et al. (2022)). +Other solutions impose exogenous criteria for ordering transactions, preventing +attacks but also hindering legitimate competition. Kelkar et al. (2020) propose the +Aequitas protocol, a method to achieve ordering-fairness, by which if sufficiently +many nodes observe a given transaction arriving before a different transaction, this +ordering should be maintained when these transactions are included in a block.11 +There are also commit-reveal schemes intermediated by third parties in charge of, +for example, reorganizing incoming transactions while also encrypting and then de- +crypting them. With this respect, the better-known solution is the Shutter network, +in which a network of nodes called “keypers” jointly generate cryptographic keys with +which users encrypt their transactions. Users then submit these transactions to a +batcher contract that also orders them. Finally, Keypers broadcast the decryption +key, the transactions are decrypted and sent to the target smart contracts. +A concept that is often associated with front-running attacks is that of Maximal- +extractable value (MEV), defined as “the maximum value that can be extracted from +block production in excess of the standard block reward and gas fees by including, ex- +cluding, and changing the order of transactions in a block”12 Most existing measures +10 We note that our protocol also reduces competition among attackers and hence reduces "gas +wars". +11 See also the hedera-hashgraph project (Baird (2016)). +12 See https://ethereum.org/en/developers/docs/mev/. + +1 Introduction +8 +of total MEV are not very useful in our context as they capture both users’ legit- +imate competition (sometimes called “good MEV”) and attacks (sometimes called +“bad MEV”). A few papers, however, identify specifically profits extracted from at- +tacks. Torres et al. (2021) collect on-chain data from the inception of Ethereum +(July 30, 2015) until November 21, 2020. They estimate that these attacks gener- +ated 18.41M USD in profits for the attackers, of which 13.9M USD due to sandwich +(also called insertion) attacks. They also identify instances where several attackers +competed to attack the same victim. Similarly, Qin et al. (2022) consider a later +period (from the 1st of December, 2018 to the 5th of August, 2021) and find that +sandwich attacks generated 174.34M USD in profits. +The profits reported in the literature correspond to a situation in which an +attacker can craft his attack after observing the victim’s message, which is impossible +with our protocol. Nonetheless, we can use these measures as upper bounds for +an attacker’s profits under our protocol. +For example, suppose that all attacks +reported in Torres et al. (2021) are generated from attacking a single smart contract +(remember that in our protocol, commitments are specific to a given target smart +contract) and that the attacker is uninformed concerning when the victim will act. +In this case, an attacker must commit a message every block, hoping a victim would +do something. The profits per block reported by Torres et al. (2021) (and hence per +commitment) is 1.23 USD. Currently, the simplest possible transaction on Ethereum +costs approximately 2 USD (the base fee of a simple 21,000 gwei transaction), and +hence front-running attacks would not be profitable under our protocol. +We can repeat the same exercise using Qin et al. (2022) measures. Interestingly, +Qin et al. (2022) report that “the most sandwich attack-prone ERC-20 token is +SHIB, with an adversarial profit of 6.8M USD”. Because each pool of an AMM is +a different smart contract, 6.8M USD is an upper bound to the profits extracted +by attacking a single smart contract.13 Repeating the same calculation discussed +earlier yields profits per block of approx 8 USD.14 Remember that sandwich attacks +13 It is precisely the profits extracted by attacking a single smart contract if SHIB is traded only +on an AMM and only against one other token. It will be lower if SHIB is traded against multiple +tokens and/or on multiple AMMs. +14 SHIB was created in Aug 2020 and did not see much price action until April 2021. Here we +assume that all front-running attacks on SHIB occurred between April 2021 and Aug 2021 (when +their data collection stopped), for a period of 821798 blocks. + +2 The problem: front-running attacks +9 +require 2 messages. Hence, if profits are reduced by half (or more) by the inability +to observe the victim’s message beforehand, these attacks are not profitable under +our protocol. +2 +The problem: front-running attacks +As a benchmark case, we develop a model of front-running attacks and later intro- +duce our protocol. There is a smart contract SC and two players: Alice and Bob. +There is a piece of information (call it “the state of the world”) s ∈ S that only A +learns at the beginning of the game. Absent front running attacks, after observing +s, player A sends a message ˜σA ∈ Σ to the mempool (i.e., the set of pending trans- +actions), where Σ ̸= ∅ is the space of possible messages. As soon as the message ˜σA +is included in a block, the smart contract SC performs an action that generates a +benefit ˜PA(˜σA, s) to player A. +Front-running attacks arise because messages in the mempool are public. Hence, +after A sends a message to the mempool, this message is observed by B, who can +send a counter-message ˜σB ∈ Σ. If ˜σB is included in the blockchain before A’s +message, then B earns ˜PB(˜σB, ˜σA, s) while A earns nothing. Else, B earns nothing +and A earns ˜PA(˜σA, s). +Sending messages is costly. Each player can send a regular message by paying +c > 0. If multiple regular messages are sent, they are included in the block in the +order they are sent. We can think of c as being the base fee: a fee that should +guarantee the inclusion of a transaction in the next bloc, at least outside of periods +of rapid change in the demand for transactions.15 Player B, however, can also pay +f > c to send a “fast” message that, with probability q, is included in the block +before A’s regular message, despite A’s message being sent first. For example, f +could be the cost of sending a transaction via a service such as flashbots, or could +be a regular mempool transaction with a transaction fee significantly above the base +fee. Here we consider the parameters q, c, and f as exogenous and determined by +the technology available to A and B. We relax this assumption in Section 5.4, in +15 The concept of base fee was introduced with the EIP-1559 upgrade. See the original pro- +posal here https://eips.ethereum.org/EIPS/eip-1559. For an economic analysis of EIP-1559, see +Roughgarden (2020). + +2 The problem: front-running attacks +10 +which we introduce multiple B players choosing their own f, which then determine +the probability that a given B player successfully front runs both A and the other +B players. +In terms of applications, consider the example we discussed in the introduction: +a smart contract that rewards whoever can correctly answer a question. In this case, +B will learn the correct answer by observing A’s message and then try to submit the +same answer before A. Formally, s = σA(s) = σB(s). Our model also fits a famous +(nonfictional) example: that discussed in the blog post “Ethereum is a dark forest” +(Robinson and Konstantopoulos, 2020). In this example, two researchers wanted to +recover some tokens that a user sent to an incorrect address. They realized that +anyone who knew about these tokens could have stolen them. Despite their effort, +their attempt to recover these tokens revealed their existence to an attacker who +managed to front-run them and steal them. In the context of our model, again +σA(s) = σB(s). Another fitting example is that of an AMM. Player A is a liquidity +provider who, upon learning some private information s, decides to withdraw some or +all the liquidity provided. By observing such a message, B can infer that something +has changed in the environment and try to steal the same liquidity. In this case, +σA(s) ={withdraw my liquidity}, σB(s) ={swap some tokens}.16 Also relevant in +the context of AMMs are sandwich attacks, in which A sends message σA(s) ={swap +some tokens}, and B then front runs A with a message σB(s) ={perform the same +swap as A} and “back-run” A with the message σB(s) ={perform the opposite swap +as A}. This attack is profitable because it exploits the slippage curve of the AMM. +Although we do not explicitly allow B to back-run A, the only difference in the +analysis is that a sandwich attack is more costly than a simple front-running attack +because it requires an additional message. It follows that all our results apply to +sandwich attacks as well. +We make two simplifying assumptions. First, we assume that A is partially naive. +She is naive in that she always chooses the message that maximizes her payoff given +the state of the world; however, she is sophisticated in the choice of whether to send +her message (or, in the next section, to initiate the protocol). We, therefore, rule +16 For a study of this type of attack, see Capponi and Jia (2021). For a study of similar attacks +in the context of traditional exchanges, see Section 6 of Budish et al. (2015). + +2 The problem: front-running attacks +11 +out the possibility that A chooses her message to manipulate B’s belief about the +state of the world, which we think is unrealistic.17 Mathematically, after observing +the state of the world, if A sends a message, she sends a message +σA(s) ≡ argmax˜σA∈Σ ˜PA(˜σA, s). +Given this, we can re-define A’s payoff in case she sends a message, and she is not +front-ran as: +PA(s) ≡ ˜PA(σA(s), s). +The second simplifying assumption is that σA(s) is a bijection; that is, in each state +of the world, there is a unique and distinct message maximizing player A’s payoff. +This a useful simplification because A’s message (if sent and observed) always reveals +the state of the world. It follows that B’s optimal counter message after observing +σA(s) and learning s is: +σB(s) ≡ argmax˜σB∈Σ ˜PB(˜σB, σA(s), s). +The resulting payoff for player B if he successfully front-runs A is: +PB(s) ≡ ˜PB(σB(s), σA(s), s). +Equilibrium +The above assumptions allow us to write the extensive form of the +game for given s as in Figure 1, which we can easily solve by backward induction. +If A sends a message, then B attempts to front-run if and only if: +qPB(s) > f +Given this, we can derive A’s optimal strategy. Suppose the state of the world is +such that qPB(s) < f, and A expects no front running. In this case, she sends a +17 If A is fully sophisticated, then the equilibrium of the game is a partition of the possible states +of the world S such that A sends the same message in all states of the world belonging to the +same part of the partition. Upon observing the message, B learns the part of the partition but +not the state of the world. The results for a given partition are identical to those presented here. +However, deriving the equilibrium partition is non-trivial and of second-order importance relative +to our main research question. + +2 The problem: front-running attacks +12 +A +B +((1 − q)PA(s) − c, qPB(s) − f) +σB(s) +(PA(s) − c, 0) +no message +σA(s) +(0, 0) +no message +Fig. 1: Game tree for given s. +message if and only if +PA(s) > c +If, instead, the state of the world is such that qPB(s) > f, then A anticipates that +B will try to front-run. In this case, A sends a message if and only if +(1 − q)PA(s) > c +The following proposition summarizes these derivations. +Proposition 1 (Equilibrium). Player A’s equilibrium strategy is: +σ∗ +A(s) = +� +� +� +∅ +if PA(s) < c or qPB(s) > f and (1 − q)PA(s) < c +σA(s) +otherwise +(1) +where σ∗ +A(s) = ∅ means that A does not send any message. Player B’s equilibrium +strategy is +σ∗ +B(s) = +� +� +� +σB(s) +if qPB(s) > f and σ∗ +A(s) ̸= ∅ +∅ +otherwise +(2) +Hence, front running does not happen when its benefit is low (i.e., PB(s) ≤ f/q). +If, instead, its benefit is large (i.e., PB(s) > f/q), B will attempt to front run A + +2 The problem: front-running attacks +13 +whenever A sends a message. In particular, when PA(s) > c but (1 − q)PA(s) < c +the threat of front running prevents A from sending the message in the first place, +therefore destroying the value of the exchange between A and SC. +Front-running attacks vs. legitimate competition. +In the introduction, we ar- +gued that the difference between front-running attacks and legitimate competition +is whether the “attacker” relies on the information extracted from observing the vic- +tim’s message. This intuitive notion can be easily formalized in the context of our +model by considering a modified game in which player B chooses whether to send +his message and what message to send without observing A’s actions. We want to +find necessary and sufficient conditions such that, in the equilibrium of this modi- +fied game, B does not want to send any message. Clearly, if B does not send any +message, then A’s optimal strategy is simply: +σ∗∗ +A (s) ≡ +� +� +� +σA(s) +if PA(s) ≥ c +∅ +otherwise +(3) +Given this, there is an equilibrium in which B does not send any message if and +only if +Es[ ˜PB(˜σB, σ∗∗ +A (s), s)] ≤ f +∀˜σB ∈ Σ, +(4) +In what follows, if in the equilibrium of the original game, B sends a message +and condition 4 holds, then we say that there is a front-running attack. If instead, in +the equilibrium of the original game, B sends a message and condition 4 is violated, +then we say that B is a legitimate competitor.18 As we will see, this distinction will +play an important role in the next section when we introduce our commit-reveal +protocol. The reason is that the protocol reduces (but not fully eliminates) B’s +ability to act upon A’s message. If (4) holds, the expected benefit of an attack is +reduced, and hence attacks are less likely. If instead (4) is violated, then B always +18 It is possible that (4) does not hold and hence B sends a message also when he does not observe +A’s actions. At the same time, he may choose a different message if he observes A’s message. +According to our definition, this is not a front-running attack, even if B uses A’s message. This +is justified by the observation that, in our model, A’s payoff does not depend on what message B +sends. Hence, the fact that B uses A’s message to craft his message is irrelevant to A. + +3 Preventing front-running via commitment +14 +has a profitable message to send, independently of his observation of A. In this case, +the protocol has little impact on B’s behavior, except for requiring him to send two +messages. This means that the protocol reduces the expected return of an attack +(i.e. when 4 holds) but has little impact on legitimate competition (i.e., when 4 is +violated) +3 +Preventing front-running via commitment +To address the problem of front-running attacks, here we propose a commit-reveal +protocol. In terms of notation, we call player A’s commit message σA,1 and reveal +message σA,2. Similarly, player B’s counter-messages are σB,1 and σB,2. +Formally, the protocol has a commitment period and a reveal period, which here +are two subsequent blocks.19 If player A wants to send message σA ∈ Σ to SC, in +the commit period A sends the commit message +σA,1 = S(addr, σA) +to SC where addr is an address that A controls and S() is a function with an +intractable pre-image problem (for example Hash (addr|σA) where Hash() is the +SHA-256 hash function). Once the commit message is included in a block, A sends +the reveal message σA,2 = σA to SC from the address addr, which is then included in +the next block. Upon receiving the message, SC computes S(addr, σA) and checks +whether it received message S(addr, σA) in the previous block. +It follows that if B wants to front run A he will need to commit a message at the +commit stage and then reveal it at the reveal stage. There is a common discount +factor β ∈ [0, 1], so when a given payoff is earned with a block delay, this payoff is +discounted by β. Finally, A does not observe B’s commit message and hence cannot +detect B’s attempt to front running. At the same time, we assume B observes A’s +commit message. In Section 5.1, instead, we introduce a modified protocol that +allows A to hide his commit message. +Finally, we simplify the problem slightly by assuming that there is no state of the +19 In Section 5.3 we discuss more in detail the problem of specifying commit and reveal periods. + +4 Solution +15 +world s such that PA(s) ∈ [c, c + c +β]. Under this assumption, absent front running, +the states of the world in which A wants to send a message is the same with and +without the protocol.20 +4 +Solution +We start with a rather immediate result: there is no equilibrium in which B sends +the same commit message as A. To see this, suppose that player A sends the commit +message S(addr, σA) and player B sends the same commit message. If in the next +period B sends the message revealB = σA, then the SC will consider the earlier +commitment as invalid because B’s address is different from addr. It is also easy +to see that there is no equilibrium in which A commits but then does not reveal +because A can do better by not committing at all. The next lemma summarizes +these observations. +Lemma 1 (No cloning in equilibrium). There is no equilibrium in which σB,1 = σA,1. +There is also no equilibrium in which A sends the commit message but not the reveal +message. +In equilibrium, therefore, if B wants to attack, he would need to guess what +message to commit message without knowing the state of the world s. Nonetheless, +B anticipates that he will observe A’s message and, under our assumptions, will +learn the state of the world. At that point, he can decide whether or not to send +the message he initially committed. Therefore, the protocol severely limits but does +not totally eliminate B’s ability to act upon his observation of A’s message. Hence, +it is possible that (4) holds and, despite this, B can profitably attack. +We derive conditions under which the protocol is effective at eliminating front +running. In an equilibrium without front running, A’s optimal strategy is again +σ∗∗ +A (s) as defined in (3). Given this, consider player B. Suppose that A sent her +20 We could alternatively assume that these states of the world exist but are not very important +from B’s viewpoint, in the sense that +pr +� +PA(s) ∈ [c, c + c +β ] +� +Es +� +PB(˜σB, σA(s), s)|PA(s) ∈ [c, c + c +β ] +� +is sufficiently small. + +4 Solution +16 +commit message, that B committed message σB and then observed A’s reveal mes- +sage σA(s). In this case, B’s expected payoff from front-running is +q · ˜PB(σB, σA(s), s) − f. +Hence, upon observing σA(s) and learning s, B will try to front run if and only +if q · ˜P(σB, σA(s), s) > f. In the commitment phase, therefore, the best possible +message B can commit is +ˆσB ≡ argmaxσB∈ΣEs +� +max{q · ˜PB(σB, σA(s), s) − f, 0}|σ∗∗ +A (s) ̸= ∅ +� +, +where the expectation is conditional on the state of the world being such that A +sends a commit message. We define π as the expected payoff if B commits ˆσB after +observing that A committed a message: +π ≡ Es +� +max{q · ˜PB(ˆσB, σA(s), s) − f, 0}|σ∗∗ +A (s) ̸= ∅ +� +Hence, if A sends a commit message and B tries to front run, B’s expected payoff +is βπ − c. We therefore have the following proposition:21 +Proposition 2. If π ≤ +c +β (i.e., “guessing is hard for B”), then there is no front- +running in equilibrium. +If instead π > +c +β (i.e., “guessing is easy for B”), front +running occurs with strictly positive probability in equilibrium. +Note that in case “guessing is easy for B”, there could be a pure strategy equilibrium +in which B commits with probability 1 whenever A commits, or a mixed strategy +equilibrium in which B commits with some probability. In either cases, after com- +mitting, B attempts to front run A or not depending on A’s reveal message. +It is easy to check that in the “guessing is hard for B” case, A’s equilibrium +payoff is +v∗ +A(s) = max {−c + β(PA(s) − c), 0} +Therefore, the protocol generates both costs and benefits to player A. The main +21 The existence of the equilibrium follows from the fact that the players’ strategy space is finite, +as noted already in Nash (1950). + +4 Solution +17 +benefit is that the protocol reduces or eliminates front running. The costs are two. +The most evident one is that, here, two messages are required which implies that A +pays c twice. More subtle is the fact that, here, the payoff is earned with a one-block +delay, and hence is discounted by the parameter β. +4.1 +Discussion +Attack vs legitimate competition +It is instructive to consider what happens +when B is an attacker (i.e., condition 4) holds vs a legitimate competitor (i.e., +condition 4 is violated). To do so, we introduce the following condition +Es[ ˜PB(˜σB, σ∗∗ +A (s), s)] ≤ c + f +β +for some ˜σB ∈ Σ, +(5) +which is akin to condition (4), but where the cost of sending a message is now the +cost of participating in the commit-reveal protocol. Suppose first that the above +condition is violated, which implies that 4 is violated and hence B is a legitimate +competitor. Define +σ∗∗ +B ≡ argmaxσB∈ΣEs[ ˜PB(˜σB, σ∗∗ +A (s), s)] +as the best possible message that B can send when he is completely uninformed, +earning him a payoff equal to +−c + β(Es[ ˜PB(σ∗∗ +B , σ∗∗ +A (s), s)] − f) +It is easy to see that +− c + βEs[ ˜PB(σ∗∗ +B , σ∗∗ +A (s), s)] − f = −Pr {σ∗∗ +A (s) = ∅} · c+ +Pr +� +σ∗∗ +A (s) ̸= ∅ & q · ˜PB(σ∗∗ +B , σA(s), s) − f > 0 +� +· Es +� +max{q · ˜PB(σ∗∗ +B , σA(s), s) − f, 0} − c|σ∗∗ +A (s) ̸= ∅ +� ++ +Pr +� +σ∗∗ +A (s) ̸= ∅ & q · ˜PB(σ∗∗ +B , σA(s), s) − f < 0 +� +· Es +� +min{q · ˜PB(σ∗∗ +B , σA(s), s) − f, 0} − c|σ∗∗ +A (s) ̸= ∅ +� +≤ Pr +� +σ∗∗ +A (s) ̸= ∅ & q · ˜PB(σ∗∗ +B , σA(s), s) − f > 0 +� +· Es +� +max{q · ˜PB(σ∗∗ +B , σA(s), s) − f, 0} − c|σ∗∗ +A (s) ̸= ∅ +� +≤ Pr +� +σ∗∗ +A (s) ̸= ∅ & q · ˜PB(σ∗∗ +B , σA(s), s) − f > 0 +� +· π. + +4 Solution +18 +That is, player B is strictly better off whenever he can (i) commit a message only +when A commits a message (therefore avoiding paying the commit message and +earning zero) and (ii) only send the reveal message after observing A’s message and +only when it is profitable to do so. It follows that when (4) holds, then player B +always wants to commit. +There derivations show that, modulo the fact that sending messages is more +expensive with the protocol (i.e. +the right-hand side of 4 is different from the +right-hand side of 5), the protocol does not impede legitimate competition: both +players commit their messages and then compete with each other to have their +reveal message included first in the following block. At the same time, attacks are +more costly because an attacker is forced to make a costly guess. Hence, under the +protocol, front-running attacks are discouraged while competition among honest +players is preserved (but postponed by one period). +Severity of attacks +The value of π measures how easy it is for B to guess. It +is therefore the inverse of the measure of severity of the attack discussed in the +introduction: if it is difficult for B to guess, it is because B has very little prior +information and, in the benchmark case, he relies heavily on observing A’s message, +while the opposite is true when it is easy for B to guess. We can therefore say that +the protocol is most effective at preventing the most severe front-running attacks. +Additional messages +In the above analysis, we restricted the players’ action space +to a single message per player in the commit period. If we relax this assumption, +additional interesting considerations emerge, although the basic intuition discussed +earlier remains the same. +For example, player A may want to commit and disclose σA already in period 1. +This is strictly beneficial to A if the state of the world s is such that +• π ≥ c +β so that player B sends a commit message with strictly positive proba- +bility. +• s is such that ˜PB(ˆσB, σA(s), s) ≥ f +q . That is, after sending the commit message +and learning the state of the world, player B will try to front-run A. + +5 Extensions to the protocol +19 +• s is such that PB(s) < f/q + c/β, so that if B knew the state of the world +from the beginning, he would not want to commit and then front-run. +In this situation, by disclosing σA(s) already in period 1, player A can prevent B +from attempting to front run. The analysis above therefore holds by restricting the +space of signals Σ to those such that the above conditions do not hold and A does +not want to disclose. +It is also possible that B may want to send multiple commit messages. If the +number of commit messages is k, then B will choose the k messages that, jointly, +generate the largest expected payoffs. There is a “guessing is hard” case which is +identical to the one discussed earlier. There is also a “guessing is easy” case, which +is however more convoluted than earlier because the number of messages committed +by B may be greater than 1. However, the intuition is largely unchanged from the +simple case. +Pre-commitments +Another restriction we imposed is that the protocol starts +when player A learns the state of the world. It is however possible that A may +want to pre-commit, that is, commit a message before learning the state of the +world, in the hope that the committed message can be used immediately when the +state of the world is revealed. The important observation is that A can pre-commit, +and then decide to restart the protocol by committing a second message upon learn- +ing the state of the world. This complicates B’s inference problem because whatever +message he commits may be wasted in the future. Again, the basic insight from the +simple model above continues to hold, but guessing is harder for player B. +5 +Extensions to the protocol +5.1 +Hiding commitments +Here we propose a version of the protocol that allows to hide the commit message, +in which an attacker does not know whether the victim committed something (and +will reveal in the following period). The modified protocol exploits the fact that +player A can send commit and reveal messages from different addresses, provided + +5 Extensions to the protocol +20 +that the commit message includes the address that A will use in the following period +to send the reveal message. +To study how the possibility of hiding the commit messages affects the equilib- +rium of the game, here we assume that the honest player observes the state of the +world only with some probability, in which case she may decide to send her message. +If instead player A does not observe the state of the world, then she takes no action. +We also replicate the game n times: there are now n identical honest player, who +with some probability may want to interact with one of n smart contracts. +These modifications are irrelevant in the protocol that we introduced earlier, +because, in each replica game, the attacker can send his commit message after having +observed whether the victim sent her commit message. +But the above protocol +can be modified so that both the sender and the receiver of the commitment are +obfuscated. More precisely, the modified protocol is now: +• there is a pre-existing template code for the container smart contract. This +code is such that when the container smart contract receives a commit message, +it time-stamps it with the current block number. +• to commit, the honest player generates a brand-new address and uses it to send +a transaction in which, first, a container smart contract is created using the +template, and then the commit message is sent to the newly-created container +smart contract.22 The commitment message is now S(addr, addrSC, σA), where +addrSC is the address of the target smart contract. +• to reveal, the honest player sends to the target smart contract the message σA +together with a pointer to the container smart contract in which the commit- +ment message is stored. +• the target smart contract considers the message as valid if all these conditions +are satisfied +1. like before, the commit message should be S(addr, addrSC, σA), where +addr is the address from which the reveal message was sent. +22 The brand new address needs to be founded with some ETH before it can send messages. We +note that this could be done via a centralized exchange, therefore hiding the identity of the creator +of the new address from the attacker. + +5 Extensions to the protocol +21 +2. the timestamp associated with the commit message is lower than the +current block number. This step makes sure that the commit message +was sent before the reveal message. +3. the code of the container smart contract is identical to the template smart +contract.23 +The very last step is necessary to prevent an attack in which, after observing the +reveal message, an attacker sends a single transaction that (i) creates a container +smart contract, (ii) stores the commitment there together with a fake time stamp +and (iii) send the reveal message. +An outside observer can infer that someone created a container smart contract +using the template and committed something, but does not know who committed +nor the target smart contract that will receive the reveal message. Call τ the ratio +between the observed container smart contracts created and n. +The same logic +discussed above implies that if τ · π ≤ c/β, then it is too costly for B to attack a +given A player: guessing is too hard for B and front-running is prevented. Hence, if +the probability that a given honest user sends a message to a given smart contract +is sufficiently low (so that the realized τ is also low), then the protocol eliminates +all front-running attacks. +An important observation is that the above scheme is effective in hiding the +target smart contract if and only if multiple target smart contracts share the same +template for the container smart contract. In the extreme case in which each target +smart contract has its own template, then the identity of the user remains hidden +but the target smart contract that will receive the reveal can be inferred. At the +other extreme, the highest level of obfuscation is achieved when all smart contracts +use the same template. Different smart contracts could also coordinate by creating +a single “official” container smart contract that receives all commitments. Again, +an outside observer can infer that a user sent a commitment to the container smart +contract, without knowing who is the user and what is the target smart contract. +23 In Ethereum, a smart contract code is accessible by other smart contracts. +For example, +the expression type(SC).creationCode returns the creation bytecode of smart contract SC (see +https://docs.soliditylang.org/en/latest/units-and-global-variables.html#type-information). If the +template storage contract specifies that the contract is immutable, such bytecode will be constant +and cannot be changed. + +5 Extensions to the protocol +22 +Here, however, users do not need to create the container smart contract each time, +leading to significant savings in gas. How to achieve this coordination among smart +contracts is not part of the model. +5.2 +Partial implementation. +It is possible to implement the protocol only for a subset of possible messages. That +is, there is a set of messages M ⊂ Σ such that any message σ ∈ M is considered +valid by the SC only if the commit-reveal protocol described above is followed. All +other messages σ ̸∈ M are considered valid by the SC as soon as they are received. +Suppose that A wants to send message σA and B wants to front run with message +σB. There are four possible cases: +1. σA, σB ∈ M, which means that we are in the commit-reveal case discussed +earlier. +2. σA, σB ̸∈ M, which means that we are in the benchmark case discussed earlier. +3. σA ̸∈ M but σB ∈ M, which means that A can send her message directly, +without fear of being front ran. In this case, front running is prevented at no +cost for A. +4. σA ∈ M but σB ̸∈ M, which implies that A needs to send two messages +(commit and reveal), and wait one period, for in the end having the same +probability of being front ran than in the benchmark case. In this case, A the +protocol imposes extra costs on A without any benefit. +The specific design of M depends on the situation and will balance the possible +costs and benefits to player A. With this respect, an important observation is that +the choice of M determines π. So, for example, for given π, it would seem beneficial +not to use the protocol in states of the world in which player A does not expect an +attack. But this may not be optimal, because states of the world in which A does +not expect to be attacked are precisely the ones in which the attackers’ payoff is +low. Hence, by applying the protocol also in these states of the world, π decrease, +and with it the probability of a front-running attack. + +5 Extensions to the protocol +23 +5.3 +Specifying commit and reveal periods +Our model assumes that both commit and reveal messages are included in a block +immediately after being sent. In practice, however, messages may remain in the +mempool for some time before being included in a block.24 This possibility is not +an issue with respect to the commit message, because the honest player can simply +wait until this message is included in a block before sending the reveal message. It +is however an issue with respect to the reveal message, because an attacker may be +able to observe the victim’s reveal message, send a commit message (either directly +to SC or via a container smart contract as discussed in the previous section), have +it included in a block, then send a reveal message and have it included in a block +before that of the honest user. +To start, note that the possibility that messages stay in the mempool is a concern +also in the benchmark case (i.e. the standard way in which Ethereum operates), +possibly even more than in our protocol because an attacker needs to send just 1 +message during the period in which the honest player’s message stays in the mempool +(vs 2 in our protocol). It is also a concern that is greatly reduced by the introduction +of the base fee: a fee that should guarantee the rapid inclusion of a transaction in a +bloc (see footnote 15). +For our purposes, it is interesting to note that our protocol can reduce or elim- +inate this concern by appropriately specifying commit and reveal periods, that can +be thought of as sets of blocks. The SC will then consider a reveal message as valid +only if received during a block belonging to the reveal period, and only if its commit +message was received (either directly by SC or via the container smart contract) +during a block belonging to the commit period. +For example, a specific application may have a natural deadline, such as a com- +petition rewarding whoever can provide the correct answer to a question within a +specific time period. In these situations, it seems natural to specify the commit +period as all blocks up until the deadline and the reveal period as all blocks after +the deadline, therefore eliminating the risk that an attacker commits after having +24 We treat this possibility as a random event, not something that an attacker could manipulate. +The reason is that purposefully censoring a transaction requires a large number of miners/validators +to collude, which is prevented by the consensus protocol. + +5 Extensions to the protocol +24 +observed the reveal message. In other situations, it may be possible to alternate +between commitment periods and reveal periods. In this case, the above attack is +possible only if the reveal message remains in the mempool for the entire length of +the reveal period—a probability that drops to zero rapidly with the length of this +period. Of course, this modification has a cost because it increases A’s waiting time +(i.e., the time between A learning the state of the world and deciding to send her +message and the time he receives her reward). +Finally, it is also possible that the commit-reveal protocol is required only in some +periods. For example, during the “commit” period users could either commit or send +a message directly to the smart contract without any commitment, which would be +considered valid. In the reveal period, only reveal messages that were committed +during the commit period are considered valid. The honest player can choose to send +a given transaction in a “slow but safe” mode, or a “fast but risky” mode. In the +slow but safe mode, the user sends her commitment during the “commit” period and +the reveal in the “reveal” period, therefore preventing an attacker from sending both +commit and reveal messages after observing the honest player’s reveal message. In +the fast but risky mode, a user sends a direct message to the smart contract during +the commit period. Doing so exposes the honest player to the risk of being front-ran +but may nonetheless be optimal if the honest player is particularly impatient. +5.4 +Multiple attackers +An interesting implication of our protocol is that it may reduce or eliminate com- +petition between attackers, therefore benefiting the attackers as well as the honest +player. To see this, assume that there are two attackers: B1 and B2. When sending +a transaction, each Bi chooses how much money to spend fi ≥ 0, simultaneously +and independently from each other. +To remain as close as possible to the case with a single attacker (and leverage as +much as possible the results already derived), we can think of competition between +the two attackers and the honest player as happening in two steps. First, the attacker +that spends the most wins the right to attack the honest player. Then, similarly + +5 Extensions to the protocol +25 +to the single-attacker case, the winner attempts to front-run the honest player.25 +Mathematically, the probability that the transaction sent by player Bi is included +in the block before that of B−i and player A is: +� +� +� +γiq(fi) +iffi > f−i +0 +iffi < f−i +where the function q() : R+ → [0, 1] is strictly increasing and strictly concave, and +γi > 0 for i ∈ {1, 2}. A tie-breaking rule determines what happens when fi = f−i, +but the nature of such a rule is not important for our analysis. The parameter +γi measures the strength of each attacker. Without loss of generality, we assume +that the attacker number 1 is stronger, and hence γ1 ≥ γ2. Attackers are otherwise +identical: they have the same payoff function and the same information. +Benchmark case +We start by deriving what happens with multiple attackers when +players can send their messages directly to the smart contract (i.e., no commitment +needed). Again, after observing the victim’s message and learning the state of the +world, attacker Bi’s payoff as a function of fi, f−i is +� +� +� +Pb(s)γiq(fi) +if fi > f−i +0 +if fi < f−i +Formally, therefore the attackers are engaged in an asymmetric contest with pro- +ductive effort, as studied in Siegel (2014). Define +f i ≡ fi : PB(s)γiq′(fi) = 1 +as the optimal expenditure by attacker i whenever attacker −i is absent (or alter- +natively, whenever f−i = 0). Define +f i ≡ fi : PB(s)γiq(fi) = fi +25 All our results are robust to other ways to model competition. The reason is that our results +rely on there being full dissipation of rents: the weakest attacker expects to earn zero. This result +holds in a large class of contest models. + +5 Extensions to the protocol +26 +as the expenditure level at which attacker i’s payoff is zero in the absence of attacker +−i. Note that whenever f 1 ≥ f i, then there is a unique equilibrium in pure strategy, +in which attacker B1 sets f ∗ +1 = f 1 and attacker 2 does not do anything. This situation +is therefore identical to the single-attacker case discussed earlier. +If instead f 1 < f i, according to Theorem 1 in Siegel (2014), there are multiple +mixed-strategy equilibria. However, in every equilibrium of the game attacker 1’s +utility is +PB(s)γ1q(f 2). +That is, the strong attacker’s payoff is equal to the payoff he would achieve if he’d +set his expenditure equal to the follower’s largest possible expenditure.26 Also here, +the utility of the other attackers is zero. +To summarize, relative to the single-attacker case, if there are two attackers +who are sufficiently similar then in equilibrium they will randomize their level of +spending. In expectation, the weaker attacker earns zero. The stronger attacker +earns a positive amount, which is however lower than if he was the unique attacker. +Competition, therefore, hurts both attackers because they overspend (relative to the +single-attacker case). This is clearly detrimental to the honest player as well. +Commit-reveal protocol. +Consider the commit-reveal protocol. We assume that +both attackers observe the victim’s commit message. For simplicity, we also assume +that the attackers choose their commit messages simultaneously and independently, +and that they can observe each other’s commit messages.27 +We solve the game +backward, starting from the reveal phase. +If only one attacker Bi committed, then the problem is quite simple: the single +26 This result is also in Siegel (2009), in which however only non-productive effort is considered. +Siegel (2014) extends these results to cases in which, over some range, the “prize” to be won by a +player may be increasing in this player’s effort. +27 If an attacker does not observe the other attacker’s commit message, he will nonetheless detect +the opponent’s attempt to front run in the following period. At that point, he will increase its level +of spending. The outcome is identical to the case in which the attacker knows from the beginning +that the other attacker committed and will therefore attack. + +5 Extensions to the protocol +27 +attacker i earns28 +V (γi) ≡ max +fi +� +˜PB(˜σB, s)γiq(fi) − fi +� +If instead both attackers committed, then the logic discussed in the previous section +continues to apply: if they are sufficiently similar, then the equilibrium is in mixed +strategies. The attackers overspend (relative to the single attacker case) and, as a +consequence, the weaker attacker expects to earn zero while the stronger attacker +expects to earn V (γ1) < V (γ1).29 +Given this, we can derive the equilibrium in the commitment phase. The main +result is that there is no equilibrium in which both players commit with probability 1. +The reason is that the weak attacker anticipates that, if the other attacker commits +and he also commits, he will then earn zero in the following period. Commitment +messages are however costly, which implies that the weak attacker is better off by +not committing. +It follows that the equilibria of the game are +• if either βV (γ1) > c, or βV (γ1) > c > βV (γ1) and c > βV (γ2), then there is a +unique equilibrium in pure strategy in which only the strong attacker (attacker +1) commits. +• if βV (γ1) > c > βV (γ1) and βV (γ2) > c, then there are two pure strategy +Nash equilibria, each corresponding to only one attacker sending the commit +message. There is also a mixed strategy equilibrium, in which attacker 1 com- +mits with probability α1 and attacker 2 commits with probability α2. These +probabilities are such that each attacker is indifferent between committing or +not, that is α1V (γ2) = c and α2V (γ1) + (1 − α2)V (γ1) = c. In this equilib- +rium, there is a probability α1α2 that both attackers commit, a probability +(1−α1)(1−α2) that no attackers commit, and the remaining probability that +a single attacker commits. +• otherwise, no attacker commits and front running is prevented. +28 Remember that the attacker has the same payoff function and information. Hence, in the +commit period, if they commit they will both commit ˆσB. +29 The meaning of “the attackers being sufficiently similar” and the expected payoff of player 1 +can be precisely derived following the same steps illustrated in the previous paragraph. But their +precise expressions are not important in what follows. + +6 Conclusion +28 +The protocol, therefore, decreases the level of competition among attackers. +This, in turn, have a beneficial effect on the victim as reducing competition also +reduces the amount spent by the attackers. +6 +Conclusion +We conclude by discussing a number of possible limitations to our protocol that +require further study. +Our commit-reveal protocol may impede the possibility of calling different smart +contracts within the same transaction (usually referred to as smart-contract compos- +ablity). In principle, composability is still possible by first committing the different +messages to the various smart contracts. A problem however arises when these smart +contracts have different commit-reveal periods (see Section 5.3). Although different +commit messages may be sent in different periods depending on the commitment +window of each smart contract, to maintain composability the reveal messages must +be sent within the same transaction during the reveal window of all smart contracts. +If such a window does not exist, then composability is not possible. If it exists, then +it is possible but exploiting it may impose large delays to the execution of the trans- +action. Studying further how to mitigate this problem is also left for future work. +Here we just note that composability is preserved if the commit-reveal protocol is +required only in some periods (as discussed in the last paragraph of Section 5.3), +chosen in a coordinated way among all smart contracts. +Our analysis assumes that the smart contract does not have an explicit mecha- +nism to resolve competing claims to an object and therefore does not apply to, for +example, a smart contract running an on-chain auction. Applying our protocol to +such smart contract may lead to unintended consequence because the players (hon- +est or not) may fail to reveal after having committed—perhaps because they realize +that they would lose the auction. This is problematic in many cases. For example, +failures to reveal in a second-price auction may decrease the revenues raised in the +auction. +Finally, our protocol is also not effective against a type of front-running attack +called suppression attacks in which an attacker prevents the victim’s transaction + +6 Conclusion +29 +from being included in a block by front-running it with a series of spam transactions +(see Eskandari et al. (2019)). The reason is that, in these attacks, the content of the +victim’s transaction is irrelevant to the attacker. However, these types of attacks +are rare and specific to certain applications. For example, Eskandari et al. (2019) +document only one of them in the context of a gambling smart contract. +References +Baird, L. (2016). The swirlds hashgraph consensus algorithm: Fair, fast, byzantine +fault tolerance. Swirlds Tech Reports SWIRLDS-TR-2016-01, Tech. Rep 34. +Breidenbach, L., P. Daian, F. Tramèr, and A. Juels (2018). Enter the hydra: To- +wards principled bug bounties and {Exploit-Resistant} smart contracts. In 27th +USENIX Security Symposium (USENIX Security 18), pp. 1335–1352. +Budish, E., P. Cramton, and J. Shim (2015). The high-frequency trading arms race: +Frequent batch auctions as a market design response. The Quarterly Journal of +Economics 130(4), 1547–1621. +Capponi, A. and R. Jia (2021). The adoption of blockchain-based decentralized +exchanges. arXiv preprint arXiv:2103.08842. +Capponi, A., R. Jia, and Y. Wang (2022). The evolution of blockchain: from lit to +dark. +Daian, P., S. Goldfeder, T. Kell, Y. Li, X. Zhao, I. Bentov, L. Breidenbach, and +A. Juels (2019). Flash boys 2.0: Frontrunning, transaction reordering, and con- +sensus instability in decentralized exchanges. arXiv preprint arXiv:1904.05234. +Eskandari, S., S. Moosavi, and J. Clark (2019). Sok: Transparent dishonesty: front- +running attacks on blockchain. In International Conference on Financial Cryp- +tography and Data Security, pp. 170–189. Springer. +Fudenberg, D. and J. Tirole (1987). Understanding rent dissipation: on the use of +game theory in industrial organization. The American Economic Review 77(2), +176–183. + +6 Conclusion +30 +Gans, J. S. and R. T. Holden (2022). A solomonic solution to ownership disputes: +An application to blockchain front-running. Technical report, National Bureau of +Economic Research. +Heimbach, L. and R. Wattenhofer (2022). Sok: Preventing transaction reordering +manipulations in decentralized finance. arXiv preprint arXiv:2203.11520. +Kelkar, M., F. Zhang, S. Goldfeder, and A. Juels (2020). Order-fairness for byzantine +consensus. Cryptology ePrint Archive, Paper 2020/269. https://eprint.iacr.org/ +2020/269. +Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the national +academy of sciences 36(1), 48–49. +Park, A. (2022). Conceptual flaws of decentralized automated market making. Tech- +nical report, Working paper, University of Toronto. +Qin, K., L. Zhou, and A. Gervais (2022). Quantifying blockchain extractable value: +How dark is the forest? In 2022 IEEE Symposium on Security and Privacy (SP), +pp. 198–214. IEEE. +Robinson, D. and G. Konstantopoulos (2020, Aug). Ethereum is a dark forest. +Roughgarden, T. (2020). +Transaction fee mechanism design for the ethereum +blockchain: An economic analysis of eip-1559. arXiv preprint arXiv:2012.00854. +Siegel, R. (2009). All-pay contests. Econometrica 77(1), 71–92. +Siegel, R. (2014). Contests with productive effort. International Journal of Game +Theory 43(3), 515–523. +Torres, C. F., R. Camino, et al. (2021). Frontrunner jones and the raiders of the +dark forest: An empirical study of frontrunning on the ethereum blockchain. In +30th USENIX Security Symposium (USENIX Security 21), pp. 1343–1359. + diff --git a/2dFST4oBgHgl3EQfXzgB/content/tmp_files/load_file.txt b/2dFST4oBgHgl3EQfXzgB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e6246db2728c5f747cc84ed2ce58c86d03995f8 --- /dev/null +++ b/2dFST4oBgHgl3EQfXzgB/content/tmp_files/load_file.txt @@ -0,0 +1,680 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf,len=679 +page_content='Commitment Against Front Running Attacks∗ Andrea Canidio †and Vincent Danos‡ February 1, 2023 Abstract We provide a game-theoretic analysis of the problem of front-running attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We use it to distinguish attacks from legitimate competition among honest users for having their transactions included earlier in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We also use it to introduce an intuitive notion of the severity of front-running attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We then study a simple commit-reveal protocol and discuss its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This protocol has costs because it requires two messages and imposes a delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' However, it is effective at preventing the most severe front-running at- tacks while preserving legitimate competition between users, guaranteeing that the earliest transaction in a block belongs to the honest user who values it the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Keywords: Front running, Game theory, Ethereum, MEV, Transaction reordering, commit- reveal 1 Introduction On the Ethereum network, each validator decides how to order pending transactions to form the next block, which determines the order in which these transactions are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As a consequence, users often compete with each other to have their ∗We are grateful to Agostino Capponi, Jiasun Li, Christof Ferreira Torres, Arthur Gervais, Ari Juels, and the participants to UBRI Connect 2022, Tokenomics 2022 for their comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We gratefully acknowledge the financial support of the Ethereum Foundation (grant FY22-0840).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' †IMT school of advanced studies, Lucca, Italy ‡CNRS and École Normale Supérieure (France);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='13785v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='TH] 31 Jan 2023 1 Introduction 2 transactions included earlier in a block, either by paying transaction fees or by mak- ing side payments directly to validators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='1 This form of competition can be beneficial because it ensures that a scarce resource (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', having a transaction included earlier in the block) is allocated to the user who values it the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='2 But at the same time, it opens the possibility of front-running attacks: because pending transactions are public, a malicious user can observe a victim’s incoming transaction, craft a new transaction and then pay to place it before that of the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Importantly, legitimate competition and attacks are often difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As an illustrative example, consider a smart contract programmed to award a valu- able NFT to the first person who correctly answers a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Assume, crucially, that the smart contract does not have an explicit mechanism to resolve competing claims to the object (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', by running an auction among those who provided the correct answer) and settles claims in order of arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this example, competition between users can arise in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the first case, two users simultaneously and independently find the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Each submits it and competes to have his/her transaction included earlier in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Because the user who values the NFT the most is willing to pay more, this user should be able to place his transaction before that of the opponent, thereby winning the NFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the second case, an honest user finds the answer and sends it to the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' A malicious user observes the transaction, copies it, and competes to have its copy included in the block earlier than the original transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' From the observational point of view, the above two situations are identical: two users submit the same answer and then compete to have it included earlier in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Despite this, the first is an example of legitimate competition because users do not exploit their observation of the opponent’s transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, each user would have submitted his answer also in the absence of the other user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The second is an attack because the attacker cannot send his transaction if he does not observe the victim’s transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Furthermore, the extent to which an attacker 1 Competition through higher transaction fees occurs via “gas replacement” transactions, whereby a pending transaction is resubmitted with a higher fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The resulting game is akin to an auction (see Daian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The most popular way to make side payments to validators is to use flashbots (see https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='com/flashbots/pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2 Whether it is the most efficient to achieve this goal is a different issue we do not address here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 Introduction 3 relies on the victim’s message can be interpreted as a measure of the severity of a front-running attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, the “attacker” could be another user who, through his research, narrowed the correct answer to two or three possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This attack seems less severe relative to a situation in which the attacker has no prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Our simple, illustrative example is representative of the workings of most smart contracts, including those at the core of decentralized finance protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For ex- ample, Automated Market Makers (AMM) are the dominant type of decentralized exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' They allow users to swap one token for a different one at a price that is mechanically derived by the size of two liquidity pools (one per token traded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Because swapping one token for another changes the liquidity in each pool, if no liquidity is added, then a sequence of users performing the same swap will face worse and worse terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='3 Hence, two users who independently decide to perform the same swap on an AMM will compete to obtain a better rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Competition ensures that the user who most values obtaining the better rate (perhaps because the swap is part of a sequence of atomic transactions) can obtain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Alternatively, a user may want to perform a swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Upon observing this transaction, an attacker will front-run the victim with the same swap and then back-run her with the opposite swap, in what is called a sandwich attack or insertion attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='4 Again, the second example is an attack because the attacker uses information obtained by observing the victim’s transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this paper, we propose a game-theoretic model of front-running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Our goals are two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' First, inspired by the above discussion, we aim to provide a formal definition of front-running attacks (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' legitimate competition among honest users) as well as their severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Second, we study a simple commit-reveal protocol that can be im- plemented at the smart contract level without modifying the underlying Ethereum infrastructure or introducing third parties (or layer-2 networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the simplest version of the protocol, the user concatenates the desired message with the address 3 Like in our simple illustrative example, AMMs do not have an explicit mechanism to allocate the “better rate” to one of the users and instead rely on the order in which transactions are aggregated into a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4 See Eskandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2019) for a discussion of this type of front running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' See also Park (2022) for an analysis of these types of attacks in the context of AMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 Introduction 4 from which the reveal message will be sent and passes this into a function with an intractable pre-image problem (for example, the SHA-256 hash function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='5 The resulting output is the commit message, which the user sends to the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Then, the user sends the reveal message to the smart contract, where the reveal message is simply the desired message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The smart contract receiving a reveal mes- sage will execute it only if the concatenation of the reveal message with the address from which it was received corresponds to the commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The key observation is that an attack involves two steps: (i) committing a mes- sage without knowing what message the victim will send and (ii) after observing the victim’s reveal message, deciding to send the committed message or no message at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Furthermore, the victim may use a newly-created address to send the com- mit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' When this is the case, the attacker may observe that someone sent a commit message to the smart contract, but not whom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The protocol, therefore, forces the attacker to make a costly guess: send a costly commit message without knowing whether a given victim committed (and will reveal) nor what the victim committed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At the same time, the protocol does not impede legitimate competition between users: two honest users can commit their messages and then compete to have their reveal message included earlier in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We derive conditions under which an honest player is better off using the pro- tocol than Ethereum’s standard procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='6 On the cost side, the protocol requires sending two messages instead of one and imposes a delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, if the cost of sending messages or waiting is high, the protocol is worse than the standard way to send transactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' if they are low, the protocol is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' On the benefit side, the protocol can eliminate front-running attacks, especially when it is difficult for an attacker to guess, that is, when the expected payoff of an attacker who commits without knowing whether the victim committed and what message was committed is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' By definition of the severity of the attack introduced earlier, we can say 5 In this version, an attacker observes that a smart contract received a commit message (but not necessary whom if the commit message is sent from a brand new address).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Later, we discuss a more complex protocol in which the receiver of the commit message is obfuscated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 6 We believe that, for our purposes, the honest player’s welfare is the most sensible criterion to evaluate the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The alternative would be to consider the smart contract perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Note, however, that certain types of attacks can be quite profitable from the smart contract viewpoint— for example, sandwich attacks in the context of AMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 Introduction 5 that our protocol is most effective when the severity of the attack is high and is less effective when the severity of the attack is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As an extension (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='1), we study a variation of the above protocol in which the identity of the receiver of the commit message is hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This variation hinges on the existence of a template code for a container smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' When committing, the user uses a brand new address to create a container smart con- tract using the template and then sends the commit message to this newly-created container, which time-stamps the commitment message with the current block num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' When sending the reveal message, the honest user also sends a pointer to the container smart contract where the commitment is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The smart contract considers the commitment as valid if the commit message is correct, its timestamp is antecedent to the current block, and, crucially, if the code of the container smart contract corresponds to the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This way, an outside observer can only see that someone created a commitment smart contract and sent a commit message, but not who committed nor the target smart contract for that commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Guessing is even harder for an attacker, and hence the probability of an attack is even lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As a second extension (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='4), we introduce multiple attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Absent the protocol, competition pushes each attacker to overspend (relative to the single- attacker case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This is detrimental to both the attackers and the honest user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In particular, the weakest attacker always earns zero in expected terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='7 Instead, in the commit-reveal protocol, the commit message acts as a fixed cost to attack in the next period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Because in the following period, the weakest attackers earn zero, there is no equilibrium in which both attackers send the commit message with probability 1: either there is no attack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' or a single attacker commits and then attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' or both attackers commit (and then attack) with probability strictly less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As a consequence, in the best-case scenario, the protocol eliminates attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' in the worst-case scenario, it reduces the level of competition between attackers resulting in fewer resources spent in an attack, which benefits the honest player as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 7 This result is a version of a well-known result in contest theory, that of “full dissipation of rents”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' See, for example, Fudenberg and Tirole (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 Introduction 6 Prior work Our commit-reveal protocol is novel but similar to existing proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Our main contribution is the type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In particular, we show that our proto- col can eliminate the most severe front-running attacks while maintaining legitimate competition between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As we discuss later, existing solutions instead are either primarily concerned with eliminating attacks (at the cost of also eliminating legit- imate competition) or better organizing competition (at the cost of exacerbating attacks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Furthermore, most of the literature has proposed solutions to reduce or eliminate front-running in Ethereum by changing its infrastructure or introducing third parties (See Heimbach and Wattenhofer (2022) for a review of the literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Instead, our solution does not require third parties and can be implemented at the smart contract level, allowing for flexibility in its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, each smart contract could decide that only some messages must follow the protocol to be considered valid, while other messages do not need to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='8 Or a smart contract may decide that the protocol is required only during some periods (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' With respect to existing solutions, our protocol can be seen as a simplified version of the submarine commitments protocol in Breidenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2018): in both cases, a message is first committed and then revealed, and the commitment can be hidden in the sense that the identity of the sender and receiver of the commit message cannot be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The main difference is that we adopt a weaker notion of “commitment” because we allow users not to send a transaction after committing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The notion of “commitment” in Breidenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2018) is instead stronger because users are penalized for not following through with their commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As already mentioned, we provide a game-theoretic analysis of the properties of this protocol, applicable to any smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='9 With this respect, our work is inspired by Gans and Holden (2022), who develop a game-theoretic analysis of the problem of front-running arising when an honest user and an attacker claim the same reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' They also propose a protocol that eliminates these types of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Their key assumption is that the legitimate claimant strictly prefers the reward 8 See the discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, in the context of AMMs, it may make sense that users who want to provide or withdraw liquidity do not need to follow the commit reveal protocol, which is instead required for swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 9 Breidenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2018) analyze the properties of the submarine commitment scheme in the context of a bug-bounty scheme they propose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 Introduction 7 to be burned rather than paid to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Therefore, these results are useful in some environments where front-running may emerge, but not all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, front-running attacks are a serious concern in the AMMs, but in this context, it may not be possible to “burn the reward”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Flashbots is a well-known project aiming to better organize competition among users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The premise is that competition through transaction fees can lead to so-called “gas wars” by which a given block is filled with transactions that will fail (because only the first one can be correctly executed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Therefore, gas wars impose a negative externality on all users because they lead to congestion and higher transaction fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The idea is to eliminate these negative externalities by allowing users to pay valida- tors directly, therefore keeping their messages private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='10 Doing so, however, makes it extremely easy to attack an honest user who sends his or her message publicly (see Capponi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Other solutions impose exogenous criteria for ordering transactions, preventing attacks but also hindering legitimate competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Kelkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2020) propose the Aequitas protocol, a method to achieve ordering-fairness, by which if sufficiently many nodes observe a given transaction arriving before a different transaction, this ordering should be maintained when these transactions are included in a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='11 There are also commit-reveal schemes intermediated by third parties in charge of, for example, reorganizing incoming transactions while also encrypting and then de- crypting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' With this respect, the better-known solution is the Shutter network, in which a network of nodes called “keypers” jointly generate cryptographic keys with which users encrypt their transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Users then submit these transactions to a batcher contract that also orders them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Finally, Keypers broadcast the decryption key, the transactions are decrypted and sent to the target smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' A concept that is often associated with front-running attacks is that of Maximal- extractable value (MEV), defined as “the maximum value that can be extracted from block production in excess of the standard block reward and gas fees by including, ex- cluding, and changing the order of transactions in a block”12 Most existing measures 10 We note that our protocol also reduces competition among attackers and hence reduces "gas wars".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 11 See also the hedera-hashgraph project (Baird (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 12 See https://ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='org/en/developers/docs/mev/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1 Introduction 8 of total MEV are not very useful in our context as they capture both users’ legit- imate competition (sometimes called “good MEV”) and attacks (sometimes called “bad MEV”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' A few papers, however, identify specifically profits extracted from at- tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2021) collect on-chain data from the inception of Ethereum (July 30, 2015) until November 21, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' They estimate that these attacks gener- ated 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='41M USD in profits for the attackers, of which 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='9M USD due to sandwich (also called insertion) attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' They also identify instances where several attackers competed to attack the same victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Similarly, Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2022) consider a later period (from the 1st of December, 2018 to the 5th of August, 2021) and find that sandwich attacks generated 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='34M USD in profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The profits reported in the literature correspond to a situation in which an attacker can craft his attack after observing the victim’s message, which is impossible with our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Nonetheless, we can use these measures as upper bounds for an attacker’s profits under our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, suppose that all attacks reported in Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2021) are generated from attacking a single smart contract (remember that in our protocol, commitments are specific to a given target smart contract) and that the attacker is uninformed concerning when the victim will act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, an attacker must commit a message every block, hoping a victim would do something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The profits per block reported by Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2021) (and hence per commitment) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='23 USD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Currently, the simplest possible transaction on Ethereum costs approximately 2 USD (the base fee of a simple 21,000 gwei transaction), and hence front-running attacks would not be profitable under our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We can repeat the same exercise using Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2022) measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Interestingly, Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2022) report that “the most sandwich attack-prone ERC-20 token is SHIB, with an adversarial profit of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='8M USD”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Because each pool of an AMM is a different smart contract, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='8M USD is an upper bound to the profits extracted by attacking a single smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='13 Repeating the same calculation discussed earlier yields profits per block of approx 8 USD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='14 Remember that sandwich attacks 13 It is precisely the profits extracted by attacking a single smart contract if SHIB is traded only on an AMM and only against one other token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It will be lower if SHIB is traded against multiple tokens and/or on multiple AMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 14 SHIB was created in Aug 2020 and did not see much price action until April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Here we assume that all front-running attacks on SHIB occurred between April 2021 and Aug 2021 (when their data collection stopped), for a period of 821798 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2 The problem: front-running attacks 9 require 2 messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, if profits are reduced by half (or more) by the inability to observe the victim’s message beforehand, these attacks are not profitable under our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2 The problem: front-running attacks As a benchmark case, we develop a model of front-running attacks and later intro- duce our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is a smart contract SC and two players: Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is a piece of information (call it “the state of the world”) s ∈ S that only A learns at the beginning of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Absent front running attacks, after observing s, player A sends a message ˜σA ∈ Σ to the mempool (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', the set of pending trans- actions), where Σ ̸= ∅ is the space of possible messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' As soon as the message ˜σA is included in a block, the smart contract SC performs an action that generates a benefit ˜PA(˜σA, s) to player A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Front-running attacks arise because messages in the mempool are public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, after A sends a message to the mempool, this message is observed by B, who can send a counter-message ˜σB ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If ˜σB is included in the blockchain before A’s message, then B earns ˜PB(˜σB, ˜σA, s) while A earns nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Else, B earns nothing and A earns ˜PA(˜σA, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Sending messages is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Each player can send a regular message by paying c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If multiple regular messages are sent, they are included in the block in the order they are sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We can think of c as being the base fee: a fee that should guarantee the inclusion of a transaction in the next bloc, at least outside of periods of rapid change in the demand for transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='15 Player B, however, can also pay f > c to send a “fast” message that, with probability q, is included in the block before A’s regular message, despite A’s message being sent first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, f could be the cost of sending a transaction via a service such as flashbots, or could be a regular mempool transaction with a transaction fee significantly above the base fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Here we consider the parameters q, c, and f as exogenous and determined by the technology available to A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We relax this assumption in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='4, in 15 The concept of base fee was introduced with the EIP-1559 upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' See the original pro- posal here https://eips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='org/EIPS/eip-1559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For an economic analysis of EIP-1559, see Roughgarden (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2 The problem: front-running attacks 10 which we introduce multiple B players choosing their own f, which then determine the probability that a given B player successfully front runs both A and the other B players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In terms of applications, consider the example we discussed in the introduction: a smart contract that rewards whoever can correctly answer a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, B will learn the correct answer by observing A’s message and then try to submit the same answer before A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Formally, s = σA(s) = σB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Our model also fits a famous (nonfictional) example: that discussed in the blog post “Ethereum is a dark forest” (Robinson and Konstantopoulos, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this example, two researchers wanted to recover some tokens that a user sent to an incorrect address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' They realized that anyone who knew about these tokens could have stolen them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Despite their effort, their attempt to recover these tokens revealed their existence to an attacker who managed to front-run them and steal them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the context of our model, again σA(s) = σB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Another fitting example is that of an AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Player A is a liquidity provider who, upon learning some private information s, decides to withdraw some or all the liquidity provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' By observing such a message, B can infer that something has changed in the environment and try to steal the same liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, σA(s) ={withdraw my liquidity}, σB(s) ={swap some tokens}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='16 Also relevant in the context of AMMs are sandwich attacks, in which A sends message σA(s) ={swap some tokens}, and B then front runs A with a message σB(s) ={perform the same swap as A} and “back-run” A with the message σB(s) ={perform the opposite swap as A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This attack is profitable because it exploits the slippage curve of the AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Although we do not explicitly allow B to back-run A, the only difference in the analysis is that a sandwich attack is more costly than a simple front-running attack because it requires an additional message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It follows that all our results apply to sandwich attacks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We make two simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' First, we assume that A is partially naive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' She is naive in that she always chooses the message that maximizes her payoff given the state of the world;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' however, she is sophisticated in the choice of whether to send her message (or, in the next section, to initiate the protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We, therefore, rule 16 For a study of this type of attack, see Capponi and Jia (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For a study of similar attacks in the context of traditional exchanges, see Section 6 of Budish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2 The problem: front-running attacks 11 out the possibility that A chooses her message to manipulate B’s belief about the state of the world, which we think is unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='17 Mathematically, after observing the state of the world, if A sends a message, she sends a message σA(s) ≡ argmax˜σA∈Σ ˜PA(˜σA, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Given this, we can re-define A’s payoff in case she sends a message, and she is not front-ran as: PA(s) ≡ ˜PA(σA(s), s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The second simplifying assumption is that σA(s) is a bijection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' that is, in each state of the world, there is a unique and distinct message maximizing player A’s payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This a useful simplification because A’s message (if sent and observed) always reveals the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It follows that B’s optimal counter message after observing σA(s) and learning s is: σB(s) ≡ argmax˜σB∈Σ ˜PB(˜σB, σA(s), s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The resulting payoff for player B if he successfully front-runs A is: PB(s) ≡ ˜PB(σB(s), σA(s), s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Equilibrium The above assumptions allow us to write the extensive form of the game for given s as in Figure 1, which we can easily solve by backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If A sends a message, then B attempts to front-run if and only if: qPB(s) > f Given this, we can derive A’s optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Suppose the state of the world is such that qPB(s) < f, and A expects no front running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, she sends a 17 If A is fully sophisticated, then the equilibrium of the game is a partition of the possible states of the world S such that A sends the same message in all states of the world belonging to the same part of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Upon observing the message, B learns the part of the partition but not the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The results for a given partition are identical to those presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' However, deriving the equilibrium partition is non-trivial and of second-order importance relative to our main research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2 The problem: front-running attacks 12 A B ((1 − q)PA(s) − c, qPB(s) − f) σB(s) (PA(s) − c, 0) no message σA(s) (0, 0) no message Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1: Game tree for given s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' message if and only if PA(s) > c If, instead, the state of the world is such that qPB(s) > f, then A anticipates that B will try to front-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, A sends a message if and only if (1 − q)PA(s) > c The following proposition summarizes these derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Proposition 1 (Equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Player A’s equilibrium strategy is: σ∗ A(s) = � � � ∅ if PA(s) < c or qPB(s) > f and (1 − q)PA(s) < c σA(s) otherwise (1) where σ∗ A(s) = ∅ means that A does not send any message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Player B’s equilibrium strategy is σ∗ B(s) = � � � σB(s) if qPB(s) > f and σ∗ A(s) ̸= ∅ ∅ otherwise (2) Hence, front running does not happen when its benefit is low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', PB(s) ≤ f/q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If, instead, its benefit is large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', PB(s) > f/q), B will attempt to front run A 2 The problem: front-running attacks 13 whenever A sends a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In particular, when PA(s) > c but (1 − q)PA(s) < c the threat of front running prevents A from sending the message in the first place, therefore destroying the value of the exchange between A and SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Front-running attacks vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' legitimate competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the introduction, we ar- gued that the difference between front-running attacks and legitimate competition is whether the “attacker” relies on the information extracted from observing the vic- tim’s message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This intuitive notion can be easily formalized in the context of our model by considering a modified game in which player B chooses whether to send his message and what message to send without observing A’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We want to find necessary and sufficient conditions such that, in the equilibrium of this modi- fied game, B does not want to send any message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Clearly, if B does not send any message, then A’s optimal strategy is simply: σ∗∗ A (s) ≡ � � � σA(s) if PA(s) ≥ c ∅ otherwise (3) Given this, there is an equilibrium in which B does not send any message if and only if Es[ ˜PB(˜σB, σ∗∗ A (s), s)] ≤ f ∀˜σB ∈ Σ, (4) In what follows, if in the equilibrium of the original game, B sends a message and condition 4 holds, then we say that there is a front-running attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If instead, in the equilibrium of the original game, B sends a message and condition 4 is violated, then we say that B is a legitimate competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='18 As we will see, this distinction will play an important role in the next section when we introduce our commit-reveal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The reason is that the protocol reduces (but not fully eliminates) B’s ability to act upon A’s message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If (4) holds, the expected benefit of an attack is reduced, and hence attacks are less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If instead (4) is violated, then B always 18 It is possible that (4) does not hold and hence B sends a message also when he does not observe A’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At the same time, he may choose a different message if he observes A’s message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' According to our definition, this is not a front-running attack, even if B uses A’s message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This is justified by the observation that, in our model, A’s payoff does not depend on what message B sends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, the fact that B uses A’s message to craft his message is irrelevant to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 3 Preventing front-running via commitment 14 has a profitable message to send, independently of his observation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, the protocol has little impact on B’s behavior, except for requiring him to send two messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This means that the protocol reduces the expected return of an attack (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' when 4 holds) but has little impact on legitimate competition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', when 4 is violated) 3 Preventing front-running via commitment To address the problem of front-running attacks, here we propose a commit-reveal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In terms of notation, we call player A’s commit message σA,1 and reveal message σA,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Similarly, player B’s counter-messages are σB,1 and σB,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Formally, the protocol has a commitment period and a reveal period, which here are two subsequent blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='19 If player A wants to send message σA ∈ Σ to SC, in the commit period A sends the commit message σA,1 = S(addr, σA) to SC where addr is an address that A controls and S() is a function with an intractable pre-image problem (for example Hash (addr|σA) where Hash() is the SHA-256 hash function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Once the commit message is included in a block, A sends the reveal message σA,2 = σA to SC from the address addr, which is then included in the next block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Upon receiving the message, SC computes S(addr, σA) and checks whether it received message S(addr, σA) in the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It follows that if B wants to front run A he will need to commit a message at the commit stage and then reveal it at the reveal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is a common discount factor β ∈ [0, 1], so when a given payoff is earned with a block delay, this payoff is discounted by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Finally, A does not observe B’s commit message and hence cannot detect B’s attempt to front running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At the same time, we assume B observes A’s commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='1, instead, we introduce a modified protocol that allows A to hide his commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Finally, we simplify the problem slightly by assuming that there is no state of the 19 In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='3 we discuss more in detail the problem of specifying commit and reveal periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4 Solution 15 world s such that PA(s) ∈ [c, c + c β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Under this assumption, absent front running, the states of the world in which A wants to send a message is the same with and without the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='20 4 Solution We start with a rather immediate result: there is no equilibrium in which B sends the same commit message as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To see this, suppose that player A sends the commit message S(addr, σA) and player B sends the same commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If in the next period B sends the message revealB = σA, then the SC will consider the earlier commitment as invalid because B’s address is different from addr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is also easy to see that there is no equilibrium in which A commits but then does not reveal because A can do better by not committing at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The next lemma summarizes these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Lemma 1 (No cloning in equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is no equilibrium in which σB,1 = σA,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is also no equilibrium in which A sends the commit message but not the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In equilibrium, therefore, if B wants to attack, he would need to guess what message to commit message without knowing the state of the world s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Nonetheless, B anticipates that he will observe A’s message and, under our assumptions, will learn the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At that point, he can decide whether or not to send the message he initially committed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Therefore, the protocol severely limits but does not totally eliminate B’s ability to act upon his observation of A’s message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, it is possible that (4) holds and, despite this, B can profitably attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We derive conditions under which the protocol is effective at eliminating front running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In an equilibrium without front running, A’s optimal strategy is again σ∗∗ A (s) as defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Given this, consider player B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Suppose that A sent her 20 We could alternatively assume that these states of the world exist but are not very important from B’s viewpoint, in the sense that pr � PA(s) ∈ [c, c + c β ] � Es � PB(˜σB, σA(s), s)|PA(s) ∈ [c, c + c β ] � is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4 Solution 16 commit message, that B committed message σB and then observed A’s reveal mes- sage σA(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, B’s expected payoff from front-running is q · ˜PB(σB, σA(s), s) − f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, upon observing σA(s) and learning s, B will try to front run if and only if q · ˜P(σB, σA(s), s) > f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the commitment phase, therefore, the best possible message B can commit is ˆσB ≡ argmaxσB∈ΣEs � max{q · ˜PB(σB, σA(s), s) − f, 0}|σ∗∗ A (s) ̸= ∅ � , where the expectation is conditional on the state of the world being such that A sends a commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We define π as the expected payoff if B commits ˆσB after observing that A committed a message: π ≡ Es � max{q · ˜PB(ˆσB, σA(s), s) − f, 0}|σ∗∗ A (s) ̸= ∅ � Hence, if A sends a commit message and B tries to front run, B’s expected payoff is βπ − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We therefore have the following proposition:21 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If π ≤ c β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', “guessing is hard for B”), then there is no front- running in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If instead π > c β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', “guessing is easy for B”), front running occurs with strictly positive probability in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Note that in case “guessing is easy for B”, there could be a pure strategy equilibrium in which B commits with probability 1 whenever A commits, or a mixed strategy equilibrium in which B commits with some probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In either cases, after com- mitting, B attempts to front run A or not depending on A’s reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is easy to check that in the “guessing is hard for B” case, A’s equilibrium payoff is v∗ A(s) = max {−c + β(PA(s) − c), 0} Therefore, the protocol generates both costs and benefits to player A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The main 21 The existence of the equilibrium follows from the fact that the players’ strategy space is finite, as noted already in Nash (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4 Solution 17 benefit is that the protocol reduces or eliminates front running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The costs are two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The most evident one is that, here, two messages are required which implies that A pays c twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' More subtle is the fact that, here, the payoff is earned with a one-block delay, and hence is discounted by the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='1 Discussion Attack vs legitimate competition It is instructive to consider what happens when B is an attacker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', condition 4) holds vs a legitimate competitor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', condition 4 is violated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To do so, we introduce the following condition Es[ ˜PB(˜σB, σ∗∗ A (s), s)] ≤ c + f β for some ˜σB ∈ Σ, (5) which is akin to condition (4), but where the cost of sending a message is now the cost of participating in the commit-reveal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Suppose first that the above condition is violated, which implies that 4 is violated and hence B is a legitimate competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Define σ∗∗ B ≡ argmaxσB∈ΣEs[ ˜PB(˜σB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σ∗∗ A (s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s)] as the best possible message that B can send when he is completely uninformed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' earning him a payoff equal to −c + β(Es[ ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σ∗∗ A (s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s)] − f) It is easy to see that − c + βEs[ ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σ∗∗ A (s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s)] − f = −Pr {σ∗∗ A (s) = ∅} · c+ Pr � σ∗∗ A (s) ̸= ∅ & q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f > 0 � Es � max{q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 0} − c|σ∗∗ A (s) ̸= ∅ � + Pr � σ∗∗ A (s) ̸= ∅ & q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f < 0 � Es � min{q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 0} − c|σ∗∗ A (s) ̸= ∅ � ≤ Pr � σ∗∗ A (s) ̸= ∅ & q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f > 0 � Es � max{q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 0} − c|σ∗∗ A (s) ̸= ∅ � ≤ Pr � σ∗∗ A (s) ̸= ∅ & q · ˜PB(σ∗∗ B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s) − f > 0 � π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4 Solution 18 That is, player B is strictly better off whenever he can (i) commit a message only when A commits a message (therefore avoiding paying the commit message and earning zero) and (ii) only send the reveal message after observing A’s message and only when it is profitable to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It follows that when (4) holds, then player B always wants to commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There derivations show that, modulo the fact that sending messages is more expensive with the protocol (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' the right-hand side of 4 is different from the right-hand side of 5), the protocol does not impede legitimate competition: both players commit their messages and then compete with each other to have their reveal message included first in the following block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At the same time, attacks are more costly because an attacker is forced to make a costly guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, under the protocol, front-running attacks are discouraged while competition among honest players is preserved (but postponed by one period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Severity of attacks The value of π measures how easy it is for B to guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is therefore the inverse of the measure of severity of the attack discussed in the introduction: if it is difficult for B to guess, it is because B has very little prior information and, in the benchmark case, he relies heavily on observing A’s message, while the opposite is true when it is easy for B to guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We can therefore say that the protocol is most effective at preventing the most severe front-running attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Additional messages In the above analysis, we restricted the players’ action space to a single message per player in the commit period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If we relax this assumption, additional interesting considerations emerge, although the basic intuition discussed earlier remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, player A may want to commit and disclose σA already in period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This is strictly beneficial to A if the state of the world s is such that π ≥ c β so that player B sends a commit message with strictly positive proba- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' s is such that ˜PB(ˆσB, σA(s), s) ≥ f q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' That is, after sending the commit message and learning the state of the world, player B will try to front-run A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 19 s is such that PB(s) < f/q + c/β, so that if B knew the state of the world from the beginning, he would not want to commit and then front-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this situation, by disclosing σA(s) already in period 1, player A can prevent B from attempting to front run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The analysis above therefore holds by restricting the space of signals Σ to those such that the above conditions do not hold and A does not want to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is also possible that B may want to send multiple commit messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If the number of commit messages is k, then B will choose the k messages that, jointly, generate the largest expected payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is a “guessing is hard” case which is identical to the one discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is also a “guessing is easy” case, which is however more convoluted than earlier because the number of messages committed by B may be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' However, the intuition is largely unchanged from the simple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Pre-commitments Another restriction we imposed is that the protocol starts when player A learns the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is however possible that A may want to pre-commit, that is, commit a message before learning the state of the world, in the hope that the committed message can be used immediately when the state of the world is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The important observation is that A can pre-commit, and then decide to restart the protocol by committing a second message upon learn- ing the state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This complicates B’s inference problem because whatever message he commits may be wasted in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Again, the basic insight from the simple model above continues to hold, but guessing is harder for player B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='1 Hiding commitments Here we propose a version of the protocol that allows to hide the commit message, in which an attacker does not know whether the victim committed something (and will reveal in the following period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The modified protocol exploits the fact that player A can send commit and reveal messages from different addresses, provided 5 Extensions to the protocol 20 that the commit message includes the address that A will use in the following period to send the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To study how the possibility of hiding the commit messages affects the equilib- rium of the game, here we assume that the honest player observes the state of the world only with some probability, in which case she may decide to send her message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If instead player A does not observe the state of the world, then she takes no action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We also replicate the game n times: there are now n identical honest player, who with some probability may want to interact with one of n smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' These modifications are irrelevant in the protocol that we introduced earlier, because, in each replica game, the attacker can send his commit message after having observed whether the victim sent her commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' But the above protocol can be modified so that both the sender and the receiver of the commitment are obfuscated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' More precisely, the modified protocol is now: there is a pre-existing template code for the container smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This code is such that when the container smart contract receives a commit message, it time-stamps it with the current block number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' to commit, the honest player generates a brand-new address and uses it to send a transaction in which, first, a container smart contract is created using the template, and then the commit message is sent to the newly-created container smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='22 The commitment message is now S(addr, addrSC, σA), where addrSC is the address of the target smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' to reveal, the honest player sends to the target smart contract the message σA together with a pointer to the container smart contract in which the commit- ment message is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' the target smart contract considers the message as valid if all these conditions are satisfied 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' like before, the commit message should be S(addr, addrSC, σA), where addr is the address from which the reveal message was sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 22 The brand new address needs to be founded with some ETH before it can send messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We note that this could be done via a centralized exchange, therefore hiding the identity of the creator of the new address from the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' the timestamp associated with the commit message is lower than the current block number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This step makes sure that the commit message was sent before the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' the code of the container smart contract is identical to the template smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='23 The very last step is necessary to prevent an attack in which, after observing the reveal message, an attacker sends a single transaction that (i) creates a container smart contract, (ii) stores the commitment there together with a fake time stamp and (iii) send the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' An outside observer can infer that someone created a container smart contract using the template and committed something, but does not know who committed nor the target smart contract that will receive the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Call τ the ratio between the observed container smart contracts created and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The same logic discussed above implies that if τ · π ≤ c/β, then it is too costly for B to attack a given A player: guessing is too hard for B and front-running is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, if the probability that a given honest user sends a message to a given smart contract is sufficiently low (so that the realized τ is also low), then the protocol eliminates all front-running attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' An important observation is that the above scheme is effective in hiding the target smart contract if and only if multiple target smart contracts share the same template for the container smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the extreme case in which each target smart contract has its own template, then the identity of the user remains hidden but the target smart contract that will receive the reveal can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At the other extreme, the highest level of obfuscation is achieved when all smart contracts use the same template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Different smart contracts could also coordinate by creating a single “official” container smart contract that receives all commitments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Again, an outside observer can infer that a user sent a commitment to the container smart contract, without knowing who is the user and what is the target smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 23 In Ethereum, a smart contract code is accessible by other smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, the expression type(SC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='creationCode returns the creation bytecode of smart contract SC (see https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='soliditylang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='org/en/latest/units-and-global-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='html#type-information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If the template storage contract specifies that the contract is immutable, such bytecode will be constant and cannot be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 22 Here, however, users do not need to create the container smart contract each time, leading to significant savings in gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' How to achieve this coordination among smart contracts is not part of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='2 Partial implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is possible to implement the protocol only for a subset of possible messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' That is, there is a set of messages M ⊂ Σ such that any message σ ∈ M is considered valid by the SC only if the commit-reveal protocol described above is followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' All other messages σ ̸∈ M are considered valid by the SC as soon as they are received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Suppose that A wants to send message σA and B wants to front run with message σB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There are four possible cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA, σB ∈ M, which means that we are in the commit-reveal case discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA, σB ̸∈ M, which means that we are in the benchmark case discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA ̸∈ M but σB ∈ M, which means that A can send her message directly, without fear of being front ran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, front running is prevented at no cost for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' σA ∈ M but σB ̸∈ M, which implies that A needs to send two messages (commit and reveal), and wait one period, for in the end having the same probability of being front ran than in the benchmark case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, A the protocol imposes extra costs on A without any benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The specific design of M depends on the situation and will balance the possible costs and benefits to player A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' With this respect, an important observation is that the choice of M determines π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' So, for example, for given π, it would seem beneficial not to use the protocol in states of the world in which player A does not expect an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' But this may not be optimal, because states of the world in which A does not expect to be attacked are precisely the ones in which the attackers’ payoff is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, by applying the protocol also in these states of the world, π decrease, and with it the probability of a front-running attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='3 Specifying commit and reveal periods Our model assumes that both commit and reveal messages are included in a block immediately after being sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In practice, however, messages may remain in the mempool for some time before being included in a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='24 This possibility is not an issue with respect to the commit message, because the honest player can simply wait until this message is included in a block before sending the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is however an issue with respect to the reveal message, because an attacker may be able to observe the victim’s reveal message, send a commit message (either directly to SC or via a container smart contract as discussed in the previous section), have it included in a block, then send a reveal message and have it included in a block before that of the honest user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To start, note that the possibility that messages stay in the mempool is a concern also in the benchmark case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' the standard way in which Ethereum operates), possibly even more than in our protocol because an attacker needs to send just 1 message during the period in which the honest player’s message stays in the mempool (vs 2 in our protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It is also a concern that is greatly reduced by the introduction of the base fee: a fee that should guarantee the rapid inclusion of a transaction in a bloc (see footnote 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For our purposes, it is interesting to note that our protocol can reduce or elim- inate this concern by appropriately specifying commit and reveal periods, that can be thought of as sets of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The SC will then consider a reveal message as valid only if received during a block belonging to the reveal period, and only if its commit message was received (either directly by SC or via the container smart contract) during a block belonging to the commit period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, a specific application may have a natural deadline, such as a com- petition rewarding whoever can provide the correct answer to a question within a specific time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In these situations, it seems natural to specify the commit period as all blocks up until the deadline and the reveal period as all blocks after the deadline, therefore eliminating the risk that an attacker commits after having 24 We treat this possibility as a random event, not something that an attacker could manipulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The reason is that purposefully censoring a transaction requires a large number of miners/validators to collude, which is prevented by the consensus protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 24 observed the reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In other situations, it may be possible to alternate between commitment periods and reveal periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this case, the above attack is possible only if the reveal message remains in the mempool for the entire length of the reveal period—a probability that drops to zero rapidly with the length of this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Of course, this modification has a cost because it increases A’s waiting time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', the time between A learning the state of the world and deciding to send her message and the time he receives her reward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Finally, it is also possible that the commit-reveal protocol is required only in some periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, during the “commit” period users could either commit or send a message directly to the smart contract without any commitment, which would be considered valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the reveal period, only reveal messages that were committed during the commit period are considered valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The honest player can choose to send a given transaction in a “slow but safe” mode, or a “fast but risky” mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the slow but safe mode, the user sends her commitment during the “commit” period and the reveal in the “reveal” period, therefore preventing an attacker from sending both commit and reveal messages after observing the honest player’s reveal message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In the fast but risky mode, a user sends a direct message to the smart contract during the commit period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Doing so exposes the honest player to the risk of being front-ran but may nonetheless be optimal if the honest player is particularly impatient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='4 Multiple attackers An interesting implication of our protocol is that it may reduce or eliminate com- petition between attackers, therefore benefiting the attackers as well as the honest player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To see this, assume that there are two attackers: B1 and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' When sending a transaction, each Bi chooses how much money to spend fi ≥ 0, simultaneously and independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To remain as close as possible to the case with a single attacker (and leverage as much as possible the results already derived), we can think of competition between the two attackers and the honest player as happening in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' First, the attacker that spends the most wins the right to attack the honest player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Then, similarly 5 Extensions to the protocol 25 to the single-attacker case, the winner attempts to front-run the honest player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='25 Mathematically, the probability that the transaction sent by player Bi is included in the block before that of B−i and player A is: � � � γiq(fi) iffi > f−i 0 iffi < f−i where the function q() : R+ → [0, 1] is strictly increasing and strictly concave, and γi > 0 for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' A tie-breaking rule determines what happens when fi = f−i, but the nature of such a rule is not important for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The parameter γi measures the strength of each attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Without loss of generality, we assume that the attacker number 1 is stronger, and hence γ1 ≥ γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Attackers are otherwise identical: they have the same payoff function and the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Benchmark case We start by deriving what happens with multiple attackers when players can send their messages directly to the smart contract (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', no commitment needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Again, after observing the victim’s message and learning the state of the world, attacker Bi’s payoff as a function of fi, f−i is � � � Pb(s)γiq(fi) if fi > f−i 0 if fi < f−i Formally, therefore the attackers are engaged in an asymmetric contest with pro- ductive effort, as studied in Siegel (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Define f i ≡ fi : PB(s)γiq′(fi) = 1 as the optimal expenditure by attacker i whenever attacker −i is absent (or alter- natively, whenever f−i = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Define f i ≡ fi : PB(s)γiq(fi) = fi 25 All our results are robust to other ways to model competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The reason is that our results rely on there being full dissipation of rents: the weakest attacker expects to earn zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This result holds in a large class of contest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 26 as the expenditure level at which attacker i’s payoff is zero in the absence of attacker −i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Note that whenever f 1 ≥ f i, then there is a unique equilibrium in pure strategy, in which attacker B1 sets f ∗ 1 = f 1 and attacker 2 does not do anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This situation is therefore identical to the single-attacker case discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If instead f 1 < f i, according to Theorem 1 in Siegel (2014), there are multiple mixed-strategy equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' However, in every equilibrium of the game attacker 1’s utility is PB(s)γ1q(f 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' That is, the strong attacker’s payoff is equal to the payoff he would achieve if he’d set his expenditure equal to the follower’s largest possible expenditure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='26 Also here, the utility of the other attackers is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' To summarize, relative to the single-attacker case, if there are two attackers who are sufficiently similar then in equilibrium they will randomize their level of spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In expectation, the weaker attacker earns zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The stronger attacker earns a positive amount, which is however lower than if he was the unique attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Competition, therefore, hurts both attackers because they overspend (relative to the single-attacker case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This is clearly detrimental to the honest player as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Commit-reveal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Consider the commit-reveal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' We assume that both attackers observe the victim’s commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For simplicity, we also assume that the attackers choose their commit messages simultaneously and independently, and that they can observe each other’s commit messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='27 We solve the game backward, starting from the reveal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If only one attacker Bi committed, then the problem is quite simple: the single 26 This result is also in Siegel (2009), in which however only non-productive effort is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Siegel (2014) extends these results to cases in which, over some range, the “prize” to be won by a player may be increasing in this player’s effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 27 If an attacker does not observe the other attacker’s commit message, he will nonetheless detect the opponent’s attempt to front run in the following period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' At that point, he will increase its level of spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The outcome is identical to the case in which the attacker knows from the beginning that the other attacker committed and will therefore attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 5 Extensions to the protocol 27 attacker i earns28 V (γi) ≡ max fi � ˜PB(˜σB, s)γiq(fi) − fi � If instead both attackers committed, then the logic discussed in the previous section continues to apply: if they are sufficiently similar, then the equilibrium is in mixed strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The attackers overspend (relative to the single attacker case) and, as a consequence, the weaker attacker expects to earn zero while the stronger attacker expects to earn V (γ1) < V (γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='29 Given this, we can derive the equilibrium in the commitment phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The main result is that there is no equilibrium in which both players commit with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The reason is that the weak attacker anticipates that, if the other attacker commits and he also commits, he will then earn zero in the following period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Commitment messages are however costly, which implies that the weak attacker is better off by not committing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' It follows that the equilibria of the game are if either βV (γ1) > c, or βV (γ1) > c > βV (γ1) and c > βV (γ2), then there is a unique equilibrium in pure strategy in which only the strong attacker (attacker 1) commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' if βV (γ1) > c > βV (γ1) and βV (γ2) > c, then there are two pure strategy Nash equilibria, each corresponding to only one attacker sending the commit message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' There is also a mixed strategy equilibrium, in which attacker 1 com- mits with probability α1 and attacker 2 commits with probability α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' These probabilities are such that each attacker is indifferent between committing or not, that is α1V (γ2) = c and α2V (γ1) + (1 − α2)V (γ1) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In this equilib- rium, there is a probability α1α2 that both attackers commit, a probability (1−α1)(1−α2) that no attackers commit, and the remaining probability that a single attacker commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' otherwise, no attacker commits and front running is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 28 Remember that the attacker has the same payoff function and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Hence, in the commit period, if they commit they will both commit ˆσB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 29 The meaning of “the attackers being sufficiently similar” and the expected payoff of player 1 can be precisely derived following the same steps illustrated in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' But their precise expressions are not important in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 6 Conclusion 28 The protocol, therefore, decreases the level of competition among attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This, in turn, have a beneficial effect on the victim as reducing competition also reduces the amount spent by the attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 6 Conclusion We conclude by discussing a number of possible limitations to our protocol that require further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Our commit-reveal protocol may impede the possibility of calling different smart contracts within the same transaction (usually referred to as smart-contract compos- ablity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In principle, composability is still possible by first committing the different messages to the various smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' A problem however arises when these smart contracts have different commit-reveal periods (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Although different commit messages may be sent in different periods depending on the commitment window of each smart contract, to maintain composability the reveal messages must be sent within the same transaction during the reveal window of all smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If such a window does not exist, then composability is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' If it exists, then it is possible but exploiting it may impose large delays to the execution of the trans- action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Studying further how to mitigate this problem is also left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Here we just note that composability is preserved if the commit-reveal protocol is required only in some periods (as discussed in the last paragraph of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='3), chosen in a coordinated way among all smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Our analysis assumes that the smart contract does not have an explicit mecha- nism to resolve competing claims to an object and therefore does not apply to, for example, a smart contract running an on-chain auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Applying our protocol to such smart contract may lead to unintended consequence because the players (hon- est or not) may fail to reveal after having committed—perhaps because they realize that they would lose the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' This is problematic in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, failures to reveal in a second-price auction may decrease the revenues raised in the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Finally, our protocol is also not effective against a type of front-running attack called suppression attacks in which an attacker prevents the victim’s transaction 6 Conclusion 29 from being included in a block by front-running it with a series of spam transactions (see Eskandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The reason is that, in these attacks, the content of the victim’s transaction is irrelevant to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' However, these types of attacks are rare and specific to certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' For example, Eskandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2019) document only one of them in the context of a gambling smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' References Baird, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The swirlds hashgraph consensus algorithm: Fair, fast, byzantine fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Swirlds Tech Reports SWIRLDS-TR-2016-01, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Rep 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Breidenbach, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Daian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Tramèr, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Juels (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Enter the hydra: To- wards principled bug bounties and {Exploit-Resistant} smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In 27th USENIX Security Symposium (USENIX Security 18), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1335–1352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Budish, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Cramton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Shim (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The high-frequency trading arms race: Frequent batch auctions as a market design response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The Quarterly Journal of Economics 130(4), 1547–1621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Capponi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Jia (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The adoption of blockchain-based decentralized exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='08842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Capponi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Jia, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Wang (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The evolution of blockchain: from lit to dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Daian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Goldfeder, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Kell, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Zhao, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Bentov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Breidenbach, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Juels (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Flash boys 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='0: Frontrunning, transaction reordering, and con- sensus instability in decentralized exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='05234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Eskandari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Moosavi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Clark (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Sok: Transparent dishonesty: front- running attacks on blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In International Conference on Financial Cryp- tography and Data Security, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 170–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Fudenberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Tirole (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Understanding rent dissipation: on the use of game theory in industrial organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' The American Economic Review 77(2), 176–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 6 Conclusion 30 Gans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Holden (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' A solomonic solution to ownership disputes: An application to blockchain front-running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Technical report, National Bureau of Economic Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Heimbach, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Wattenhofer (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Sok: Preventing transaction reordering manipulations in decentralized finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='11520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Kelkar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Goldfeder, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Juels (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Order-fairness for byzantine consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Cryptology ePrint Archive, Paper 2020/269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='org/ 2020/269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Nash, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Equilibrium points in n-person games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Proceedings of the national academy of sciences 36(1), 48–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Park, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Conceptual flaws of decentralized automated market making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Tech- nical report, Working paper, University of Toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Qin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Zhou, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Gervais (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Quantifying blockchain extractable value: How dark is the forest?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In 2022 IEEE Symposium on Security and Privacy (SP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 198–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Robinson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Konstantopoulos (2020, Aug).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Ethereum is a dark forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Roughgarden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Transaction fee mechanism design for the ethereum blockchain: An economic analysis of eip-1559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content='00854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Siegel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' All-pay contests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Econometrica 77(1), 71–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Siegel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Contests with productive effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' International Journal of Game Theory 43(3), 515–523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Torres, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Camino, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' Frontrunner jones and the raiders of the dark forest: An empirical study of frontrunning on the ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' In 30th USENIX Security Symposium (USENIX Security 21), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} +page_content=' 1343–1359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFST4oBgHgl3EQfXzgB/content/2301.13785v1.pdf'} diff --git a/2tE2T4oBgHgl3EQfjAfh/content/tmp_files/2301.03965v1.pdf.txt b/2tE2T4oBgHgl3EQfjAfh/content/tmp_files/2301.03965v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..94716897ae91dfb7ba48374d2601dd2fd43f5f23 --- /dev/null +++ b/2tE2T4oBgHgl3EQfjAfh/content/tmp_files/2301.03965v1.pdf.txt @@ -0,0 +1,1509 @@ +1 +BiCurNet: Pre-Movement EEG based Neural +Decoder for Biceps Curl Trajectory Estimation +Manali Saini*, Anant Jain*, Lalan Kumar, Suriya Prakash Muthukrishnan, Shubhendu Bhasin and Sitikantha Roy +Abstract—Kinematic parameter (KP) estimation from early +electroencephalogram (EEG) signals is essential for positive +augmentation using wearable robot. However, work related to +early estimation of KPs from surface EEG is sparse. In this +work, a deep learning-based model, BiCurNet, is presented for +early estimation of biceps curl using collected EEG signal. +The model utilizes light-weight architecture with depth-wise +separable convolution layers and customized attention module. +The feasibility of early estimation of KPs is demonstrated using +brain source imaging. Computationally efficient EEG features in +spherical and head harmonics domain is utilized for the first +time for KP prediction. The best Pearson correlation coefficient +(PCC) between estimated and actual trajectory of 0.7 is achieved +when combined EEG features (spatial and harmonics domain) +in delta band is utilized. Robustness of the proposed network +is demonstrated for subject-dependent and subject-independent +training, using EEG signals with artifacts. +Index Terms—Brain-computer interface, Electroencephalo- +gram, Deep learning, Kinematic parameter estimation. +I. INTRODUCTION +Brain-computer interface (BCI) is an integration of the +measurement, decoding, and translation of the activity of +central nervous system (CNS) into imitative output that rein- +states, augments, or rehabilitates the natural CNS output [1]. +This creates an interface between the CNS and its external +environment. BCI-based systems are rapidly emerging on +account of the recent advancements in signal processing and +artificial intelligence [2], [3]. These systems are useful in +*Manali Saini and Anant Jain have contributed equally to this work. +This work was supported in part by DRDO - JATC project with project +number RP04191G. +This work involved human subjects or animals in its research. Approval +of all ethical and experimental procedures and protocols was granted by +the Institute Ethics Committee, All India Institute of Medical Sciences, New +Delhi, India with reference number IEC-751/07.08.2020,RP-06/2020. +Manali Saini is with the Department of Electrical Engineering, Indian +Institute of Technology Delhi, New Delhi 110016, India (e-mail: manali- +igit@gmail.com). +Anant Jain is with the Department of Electrical Engineering, Indian +Institute of Technology Delhi, New Delhi 110016, India (e-mail: anant- +jain@ee.iitd.ac.in). +Lalan Kumar is with the Department of Electrical Engineering, Bharti +School of Telecommunication, and Yardi School of Artificial Intelligence, +Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: +lkumar@ee.iitd.ac.in). +Suriya Prakash Muthukrishnan is with the Department of Physiology, +All India Institute of Medical Sciences, New Delhi - 110016, India(e-mail: +dr.suriyaprakash@aiims.edu). +Subhendu Bhasin is with the Department of Electrical Engineering, In- +dian Institute of Technology Delhi, New Delhi 110016, India (e-mail: sb- +hasin@ee.iitd.ac.in). +Sitikantha Roy is with the Department of Applied Mechanics, In- +dian Institute of Technology Delhi, New Delhi 110016, India (e-mail: +sroy@am.iitd.ac.in). +neuro-rehabilitation to assist users with motor-impairments +[4]–[7]. For real-time operability of these systems, continuous +signal decoding is required for extraction of kinematic param- +eters (KPs) such as motion trajectory, velocity, acceleration, +etc. [8]–[10]. In view of these aspects, electroencephalogram +(EEG)-based BCI systems have gained popularity in the recent +years, owing to the non-invasiveness, low-cost, and excellent +temporal resolution of EEG signals [11], [12]. +A. Motivation and Related Work +Literary works explore machine learning and deep learning- +based paradigms for upper limb kinematic parameter esti- +mation (KPE), movement intention detection and classifica- +tion from low frequency components of EEG signals. For +instance, in [13], sparse multinomial logistic regression is +utilized to classify EEG signals during reach intention and +actual movement, based on multiple hand-crafted features +extracted from EEG signals filtered in the range of 1 − 40 +Hz. In this work, independent component analysis (ICA) and +dipole fitting are applied to remove movement artifacts from +the recorded EEG signals, for obtaining low classification +error rates [13]. Researchers in [14] have explored EEG +current source dipole (CSD) data, using standardized low +resolution brain electromagnetic tomography (sLORETA) to +decode actual and imagined arm joint trajectories based on +multiple linear regression (mLR). The most useful time lags +are observed to be between 80−150 ms prior to the movement, +and the low β and γ bands are shown to be more effective +in movement decoding with a correlation of 0.67. Similarly, +mLR is utilized in [15] for estimating the 3D trajectories of +arm movement with variable velocities using EEG segments +filtered in the range of 0.1 − 40 Hz. The researchers reported +a high correlation between the movement velocities and EEG +activity above the motor cortex in fronto-central and parietal +areas [15]. mLR is also utilized in [16] with α and β band +powers of EEG signals during the motor planning and exe- +cution phases to predict the upcoming peak tangential speed +and acceleration of hand movement. This study demonstrates +the prominence of occipital and parietal-occipital regions for +α band, and frontal and frontal-central regions for the β band +in movement planning and execution phases. In a recent study, +researchers have explored the feasibility of a commercial EEG +headset in motor decoding and classification with the use of +Kalman filter and spatio-spectral features extracted from EEG +signals [9]. An overall correlation of 0.58 is achieved in this +work. Besides mLR, sparse LR is investigated for predicting +the circular trajectories of upper limb during movement of +arXiv:2301.03965v1 [eess.SP] 10 Jan 2023 + +2 +bottles with varying masses [17]. In this work, a wide range +of EEG frequencies, i.e., 0−150 Hz is used and channels over +the motor cortex are shown to be more prominent towards +the prediction. In [18], movement intent is decoded from +movement related cortical potentials (MRCPs) using narrow- +band EEG in the range of 0.1−1 Hz to train a support vector +machine (SVM)-based classifier. The selection of a single- +channel, i.e., Cz in movement onset decoding with an accuracy +of 91% using low frequency (0 − 5 Hz) Teager-Kaiser energy +operator with threshold-based classification is demonstrated in +[19]. +Despite of the effectiveness of conventional machine +learning-based paradigms in EEG-based movement decoding, +there is a need of extracting the high-level features which can +enhance the performance. To overcome this, researchers have +proposed deep learning-based paradigms. For example, convo- +lutional neural network (CNN) is proposed with the use of pre- +movement raw spatio-temporal multi-channel EEG for hand +movement and force levels classification with an accuracy of +84% [20]. This work demonstrates early classification of hand +movement, i.e., in 100−1600 ms advance. CNN is also utilized +in [21] along with bidirectional long short term memory (Bi- +LSTM)-based network to predict the velocities of arm reaching +tasks using pre-processed EEG signals. An overall correlation +between 0.4 − 0.6 is achieved in this work and feasibility of +robotic arm control based on real-time EEG is demonstrated +[21]. Recently, deep learning-based three-dimensional (3D) +hand movement trajectory during grasp and lift movements +is estimated using a public EEG database in [10], [22], [23]. +In [22], wavelet packet decomposition (WPD) based time- +lagged EEG sub bands are used to train a CNN-LSTM network +for prediction of the hand position/trajectory with a high +correlation of 0.86. This work explores the source-aware EEG +features and the demonstrates the relevance of low frequency +bands (δ, θ, and α) in movement estimation, however, it has +limited feasibility in real-time hardware implementation. Early +estimation of this trajectory is demonstrated in [10] with a high +correlation 0.79 using the δ band of EEG. Further, researchers +in [23] demonstrate the feasibility of a brain-inspired spiking +neural network (Bi-SNN) along with mid-frequency and high- +frequency EEG bands, i.e., α, β, and γ, toward the same +trajectory estimation with a correlation of 0.7. +Based on the aforementioned description of literary works, +it can be asserted that many of these works focus on +classification of upper limb movements, rather than predic- +tion/estimation of the related kinematic parameters. Timely +extraction of kinematic parameters from EEG data during +upper limb movement is imperative towards different real-time +exosuit control-based BCI applications. Further, few existing +machine-learning based regression algorithms are able to esti- +mate the KPs earlier w.r.t. actual movement, however, average +correlation is achieved. Although the existing deep learning- +based networks outperform these ML-based paradigms, only +few of them have explored early estimation of KPs. Further, +these networks use slightly complex architectures after pre- +processing which may not be feasible on hand-held processors +for real-time BCI systems. Most importantly, the performance +of the existing paradigms for KP estimation is highly subject- +Fig. 1: Experimental setup for biceps-curl task. +specific, which further adds to the complexity since the +networks need to be trained for each subject. +B. Objective and Key Contributions +In view of the aforementioned challenges of literary works, +this work proposes a deep learning-based upper limb mo- +tion trajectory prediction/estimation from preceding EEG, i.e., +BiCurNet, for early estimation towards exosuit control-based +BCI applications. Further, the proposed network is demon- +strated to be subject-independent and robust against artifacts, +unlike the existing works. To the best of our awareness, this +is the first work which focuses on early estimation of kine- +matic parameters from both subject-dependent and subject- +independent EEG signals and further analyses the noise- +robustness of the proposed network. The key contributions of +this work are listed as follows. +• Low-complex deep learning-based architecture is pro- +posed for early estimation of upper limb motion trajec- +tory. +• In-house recording of multi-channel EEG signals during +upper limb biceps curl experiment. +• Spherical harmonics and head-harmonics domain EEG +features based motion trajectory estimation has been +explored for the first time. +• Demonstration +of +subject-adaptability +and +noise- +robustness of the proposed network. +The rest of this paper is organized as follows. Section II +describes the experimental recording and data acquisition +procedures. Section III presents the proposed methodology for +BiCurNet. Section IV discusses the experimental evaluation +results for the proposed work. Finally, section V concludes +this work with major advantages, shortcomings, and future +directions. + +3 +Fig. 2: Block diagram depicting the proposed methodology for biceps-curl trajectory estimation. +II. EXPERIMENT AND DATA ACQUISITION +The key objective of the study is to investigate the via- +bility of using EEG signals for elbow joint angle decoding +during biceps-curl motion. For this purpose, we designed +a synchronous EEG and joint angle recording system. The +description of the experimental paradigm and data acquisition +are elucidated in the subsequent sections. +A. Subjects and Equipment +The experiment was performed in the Multichannel Signal +Processing Laboratory, Department of Electrical Engineering +at Indian Institute of Technology Delhi, New Delhi. This +research was authorized by the Institutional Review Board of +All India Institute Of Medical Sciences, New Delhi. EEG data +and joint angle data were recorded from 5 healthy subjects +(all males, age 29 ± 2.61, all right handed) while performing +the biceps curl task. Each subject performed 300 trials of +biceps curls task. EEG data was recorded using 16-channel +dry-active electrodes (actiCAP Xpress Twist, Brain Products, +Gilching, Germany) with wireless EEG amplifier (LiveAmp- +16, Brain Products, Gilching, Germany). The EEG sensors +were arranged in 10-20 international system of EEG electrode +placement, namely, Fp1, Fz, F3, C3, T7, Pz, P3, O1, Oz, O2, +P4, Cz, C4, T8, F4 and Fp2. The EEG data was acquired with +500 Hz sampling frequency. A marker-based camera system +(Noraxon NiNOX 125 Camera System) was placed for elbow +joint angle measurement. The NiNOX 125 camera system was +connected to Noraxon myoResearch platform (MR 3.16) for +recording the biceps-curl hand motion. The camera system was +placed in sagittal plane 2 m away from the subject. The elbow +joint angle was calculated using myoResearch software in +the post-processing step. The 3-point angle measurement tool +was utilized to compute 2D joint angle by tracking reflective +markers in the video recording. The joint angle data was +sampled with the sampling frequency of 125 Hz. The EEG and +joint angle data was synchronized using Noraxon myoSync +device. +B. Experimental Setup and Paradigm +Concurrent EEG and motion data was collected from the +users during biceps-curl task. At the beginning of experiment, +participants were in standing position with 2 Kg dumbbell +holding in their right hand. A monitor was positioned 1.8 m +away in front of them for showing the experimental paradigm. +Participants were standing in balanced upright posture with +dumbbell in their right hand. We designed the experiment in +PsychoPy [24] for instructing the participant for initiating the +biceps-curl movement. Each trial begin with a cross appearing +on the center screen along with a beep sound, indicating the +start of the trial. After a couple of seconds, a visual cue +appeared on the screen to instruct the participant to initiate +the biceps-curl. The biceps-curl was performed in the motion +execution phase. Each trial ended with resting phase of two +seconds. Before the actual data acquisition, each participant +performed a practice run for executing the task correctly. This +practice run was not included for any consequent analysis. We +recorded 30 runs with 10 trials each for the biceps curl task. +Inter-run rest was given to the participant for avoiding muscle +fatigue. +III. PROPOSED METHODOLOGY +This section elaborates the proposed methodology for early +prediction of upper limb motion trajectory from EEG based +on deep learning, as illustrated in Fig. 2. It consists of three +major modules: EEG recording, pre-processing and feature + +Raw EEG +signals +Pre-processing and +3 Dense +andino +(Channels 1 +feature extraction +EEG data +Flatten +layers, 8 +layer, N +to 16) +(Channels 1 +layer +units, +units, +DFT-based +to Nc) +DWSConv1Dlayer +Conv1D layer +Maxpool1D layer +Enhance and suppress +activation = activation = +baseline wander +32 kernels,kernel 32kernels, kernel +Pool size=2, +attention block +Swish +Linear +size=5, stride=1, +size=5, stride=1, +noise removal +activation-ReLu +stride=2 +activation=ReLu +(Threshold: 0.5 +Hz). +dwConv1D +Dense ++ ReLu +layer +Common average +referencing. +ICA-based artifact +rejection. +Amplitude +normalization [v]. +DWT-based sub- +bands extraction: +[AA:"A:A:*A] +DWT-Spherical +harmonics (SH) +(C3-Ks 1) × 32 +C4 × 32 +features: +C4 +C5 +DWT-Head +harmonics (H2) +(N-Ks+1) × 32 +(C,-Ks+1) × 32 (C,/2) × 32 +C5 × 1 +8×1 +N×1 +NxNC +. +features: +Predicted +c +C2 +C3 +trajectory +EEG +recording4 +extraction, and depth-wise separable convolutional neural net- +work with a customized attention module. The modules are +described in the subsequent sub-sections. +A. EEG recording +In this work, the EEG signals are acquired using LiveAmp +16 Brain Products system as described in the previous section. +Prior to be used for the proposed BiCurNet, these signals are +pre-processed as detailed in the ensuing sub-section. +B. Pre-processing +The recorded EEG signals are pre-processed in EEGLAB +[25] and MATLAB for feature extraction prior to be fed to +the proposed BiCurNet, as shown in Fig. 2. After recording +and re-sampling the EEG signals, low frequency (below 0.5 +Hz) baseline wander noise (BWN) suppression is done using +discrete Fourier transform (DFT). For this purpose, the DFT +coefficients corresponding to frequencies below 0.5Hz are +estimated. The computation of DFT coefficient index k is done +as: k = ⌊(fqNd/fqs)⌋, where fq is the frequency in Hz, fqs is +the sampling frequency, and Nd is the number of DFT points +for computation. These DFT coefficients are thresholded to +zero for suppression of the BWN. The EEG signal after BWN +suppression is synthesized as the inverse of the DFT coefficient +matrix. The mathematical interpretation of this procedure is +described for a recorded EEG signal v[n] by the following +DFT pair: +DFT of recorded signal : V [k] = +Nd−1 +� +n=0 +v[n]e +−jn2πk +Nd +(1) +˜vq[n] = 1 +Nd +Nd−1 +� +k=0 +˜Vq(k)e +jn2πk +Nd +(2) +where +˜ +Xq denotes the DFT coefficient matrix after thresh- +olding, i.e., ˜ +Xq(k) = [0, . . . , 0, Xq[k + 1], . . . , Xq[Nd − k − +1], 0, ...., 0]. All signals are normalized w.r.t. amplitude to +bring it in range: [−1, 1] as +˜ +xq[n] +max| ˜ +xq[n]|. +In this work, the recorded EEG signals are analyzed for +the estimation of motion trajectory with and without artifact +suppression. Independent component analysis (ICA) is utilized +for artifact suppression. It is used for estimating the sources +corresponding to cerebral and non-cerebral activities resulting +in the scalp EEG [26]. EEGLAB is used in this work for ICA- +based decomposition of the EEG signals obtained after BWN +removal. The decomposed independent sources with more than +70% of artifactual components are rejected and the artifact-free +EEG signal is reconstructed. +C. Brain source imaging +Brain source imaging (BSI) is performed to select the rel- +evant pre-movement EEG segment prior to feature extraction. +Numerical Boundary Element Method (BEM) based forward +modeling is utilized for this purpose. The head model utilizes +ICBM MRI template [27] in OpenMEEG [28] toolbox. The +spatio-temporal dynamics of brain cortical sources are ob- +tained using inverse modeling. In particular, standardized low- +resolution electromagnetic tomography (sLORETA) [29] is +utilized to solve the under-determined inverse problem. Under +the constraint of the smooth source distribution, standardized +current density maps are utilized for localization inference. +Source localization plots for a right hand biceps-curl activity +are illustrated in Fig. 3. The analysis shown corresponds to +a single-trial of biceps-curl. The subject was instructed to +focus the vision on fixation cross. A visual cue for movement +onset was presented at 0 ms. The subject executed biceps-curl +activity 410 ms after the visual cue was given. A constant +activation may be observed in occipital lobe up to 60 ms. +The information starts getting transferred to the left motor +cortex thereafter. All such pre-movement EEG [Fig 3(c)-(g)] +has inbuilt motion trajectory. +It may be noted that the left motor cortex region was acti- +vated at 220-240 ms [Fig 3(e)] corresponding to the right-hand +biceps-curl activity. Motor activity was observed thereafter +up to 320 msec [Fig 3(e)]. The subject executed biceps-curl +activity at 400-450 ms after the visual cue was given. It may +be concluded that the motor neural information corresponding +to the biceps-curl activity is present approximately 250 ms +prior to the motor execution. This information is utilized for +selecting the time-lag window for elbow joint-angle trajectory. +The selected EEG data was utilized for the training and testing +of the proposed neural decoder. +D. Feature extraction +The pre-processed EEG signals are analyzed with different +transform-domain techniques for significant feature extraction. +This work explores time-frequency features using discrete +wavelet transform (DWT) in: Spatial domain; Spatio-temporal +domain using spherical Fourier transform (SFT); and Spatio- +temporal domain using head harmonics transform. +1) Discrete wavelet transform-based features: +Discrete +wavelet transform is utilized to decompose the EEG signals +into constituent sub-bands/rhythms. It makes use of high-pass +and low-pass filters for decomposing the signals into a pre- +defined number of levels based on the sampling frequency +[30]. DWT of a single channel EEG signal v[n] is given by +Vj,r = +� +n∈z +v[n]ψ∗ +j,r[n] +(3) +where ψj,r is the translated and scaled version of the mother +wavelet ψ0,0, and defined as: +ψj,r[n] = 2−(j/2)ψ0,0 +� +2−j(n − r) +� +(4) +The procedure for DWT-based decomposition follows a +tree-like structure as demonstrated in Fig. 4. At each decompo- +sition level, the wavelet coefficients are down-sampled for re- +moving the redundant information [31]. In this work, since the +sampling frequency used is 125 Hz, the decomposed sub bands +are obtained as: delta (δ : 0.5 − 3.9 Hz), theta (θ : 3.9 − 7.8 +Hz), alpha (α : 7.8 − 15.6 Hz), beta (β : 15.6 − 31.2 Hz), and +gamma (γ :> 31.2 Hz), denoted by Vδ, Vθ, Vα, Vβ, and Vγ +respectively. + +5 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Fig. 3: Brain source localization using sLORETA at different time stamps : (a) 0ms (b) 60ms (c) 120ms (d) 180ms (e) 240ms +(f) 300ms (g) 360ms (h) 420ms +Fig. 4: +Four-level DWT-based decomposition to obtain the approxi- +mation and detail bands with frequency range at level j +given by: +� +0, 2−j−1 Fs +� +, and +� +2−j−1 Fs, 2−j Fs +� +respectively. +2) DWT-Spherical harmonics-based features: To extract the +spatio-temporal features of the EEG signal and the correspond- +ing DWT-based sub bands obtained above, spherical Fourier +transform (SFT) is explored in this work. Since the human +head is assumed to be spherical in shape [32], spherical Fourier +basis functions have been widely employed in literary works. +The decomposition of a multi-channel EEG signal V in SFD +is obtained as: +VSH +lm = +� +Ω +V (Ω, n) [Y m +n (Ω)] dΩ +(5) +where V (Ω, n) denotes the potential at (Ω) = (r, θ, φ) on +the scalp at time instant n. Here, r represents the radius of +head, θ denotes the angle of elevation measured in downward +direction from positive Z-axis (θ ∈ [0, π]), and φ denotes +the azimuth angle measured in anticlockwise direction from +positive X-axis, as shown in Fig. 5. The real-valued Y m +l (Ω) +of lth order and mth degree constitutes an orthonormal set of +basis function, defined over spherical array. For a finite order +Fig. 5: Total potential at a channel is a contribution of each active equivalent +dipole . +system, l ∈ [0, L], and m ∈ [−l, l]. Therefore, (L+1)2 distinct +spherical harmonics are obtained in total. Since the number of +sampling points S in spatial domain should be atleast (L+1)2, +the highest limit of L is ≤ +� +(S) − 1. In this work, since 16 +electrodes are used for recording, i.e., S = 16, the limit of L is +3. Therefore, L = 2 is used here and total 9 distinct spherical +harmonics are obtained. The corresponding features are stored +in V SH +nm with a dimension of 9 × N. Each EEG sub band is +also decomposed using spherical Fourier basis functions, and +the corresponding features are obtained as V SH +δlm , V SH +θlm , V SH +αlm, +V SH +βlm , and V SH +γlm . +3) DWT-Head harmonics-based features: More recently, +head harmonics (H2) basis functions have been proposed for +more adequate representation of EEG signals based on the +geometry of human head [32]. Since the EEG sensors placed +on head form a shape between a sphere and a hemisphere, H2 +basis functions are shown to be more efficient for representing + +x[n] sampled at F, = 125 Hz +Approximation 1 +Detail 1 +(0-31.25 Hz) +(31.25-62.5 Hz) +V +Approximation 2 +Detail 2 +(0-15.625 Hz) +(15.625-31.25 Hz) +Vp +Approximation 3 +Detail 3 +(0-7.8125 Hz) +(7.8125-15.625 Hz) +Va +Approximation 4 +Detail 4 +(0-3.9 Hz) +(3.9-7.8125 Hz) +Vs +Ve+Y0 +10-86 +the data sampled over head. The decomposition of an EEG +signal matrix V in H2 domain is given as: +VH2 +lm = +� +Ω +V (Ω, n) [Hm +l (Ω)] dΩ +≈ +S +� +w=1 +zwV (Ωw, n) [Hm +l (Ωw)] +(6) +where, zw denotes the sampling weight and Ωw = (θw, φw) +is the location of channel w. Here, the angle of elevation θ +is in the range [0, 2π/3], as per the head geometry shown in +Fig. 6. The real-valued Hm +l (Ω) of lth order and mth degree +constitutes an orthonormal set of basis function defined over +human head . +Fig. 6: Geometry of human head with the parameters: Perimeter=40cm, +radius=10cm [32]. +The corresponding features are stored in V H2 +lm +with a di- +mension of 9 × N, similar to that obtained in SFT. Each EEG +sub-band is also decomposed using H2 basis functions, and +the corresponding features are obtained as V H2 +δlm , V H2 +θlm, V H2 +αlm, +V H2 +βlm, and V H2 +γlm. +E. Proposed BiCurNet +After pre-processing and feature extraction, the EEG data is +given as input to the proposed BiCurNet model. The proposed +deep learning model is illustrated in Fig. 2. The constituent +layers in the proposed model include a depth-wise separable +one-dimensional convolution layer (DWSConv1D), a conv1D +layer, a maxpooling (maxpool1D) layer, a customized attention +module, a flatten layer, three dense layers, and an output layer +for regression/prediction. +• Depth-wise separable convolution layer (DWSConv1D): +The first layer of the network is a conv1D layer which +performs a depth-wise separable convolution of the re- +ceived input data with the kernels/filters used in this +layer. It receives the input EEG data in the form of +N × Nc matrix as shown in Fig. 2. Here N denotes the +number of samples in the data, and Nc is the number of +channels. The convolution operation is split into two parts +in this layer as depth-wise and point-wise [33]. Depth- +wise convolution is performed with each channel sepa- +rately, and point-wise convolution is performed as 1 × 1 +convolution. It is a computationally efficient operation +w.r.t. the standard convolution layer, making it suitable +for lightweight scenarios. Convolution of a filter f[n] with +an input v[n] is written as: +v[n] ∗ f[n] = +ks−1 +� +i=0 +v[i] · f[n − i] +(7) +where, ‘∗’ represents the convolution operation and ks +denotes the filter width. In this layer, 32 filters are used. +Each filter has a width ks of 5. In general, the zth +convolution output, i.e, feature map of layer lr is given +as [34]: +clr +z = σ +� +�bilr +z + +� +j +clr−1 +j +× f lr +zj +� +� +(8) +where, clr +z is the zth feature in the lrth layer; clr−1 +j +is +the jth feature in the corresponding preceding layer; f lr +zj +represents the filter which links feature z to feature j, bilr +z +represents the corresponding bias vector and σ denotes +the activation function, which is rectified linear unit +(ReLu) in this layer. It is defined as: σ(t) = max(0, t). +A stride of one is used in this layer. The ’He’ uniform +initialization is used for kernel weights and zero initializa- +tion is used for bias vector. All these parameters produce +an output dimension of C1: (N − ks + 1) × 32 as shown +in Fig. 2. L2 regularization with a factor of 0.001 is also +used in this layer to reduce over-fitting. +• Conv1D layer: The second layer is a conventional con- +volution layer, which operates on all input channels at a +time. This layer uses the same parameters as described in +the previous layer. The corresponding output dimension +of this layer is given as (C1 − ks + 1) × 32. +• Max pooling layer (Maxpool1D): The convolution layer +output is reduced in dimensionality by using a max +pooling 1D layer, which retains the highest value of the +feature in a segment with a pool size [35]. This layer +helps in low-level feature extraction. The corresponding +process can be interpreted as [34]: +chx +mx = max +∀b∈arm chx−1 +b +(9) +where, arm denotes the pool area with index m. In this +work, a pool size and a stride of 2 is selected, which +results in the dimension of the output as (C1 − ks + +1)/2 × 32, shown in Fig. 2. +• Customized attention module (CAM): The feature maps +of the previous layer are further transformed to intensify +the more relevant features and restrain the less relevant +features. A CAM is utilized for this purpose, which uses +a dense layer with 32 units and a multiply layer as +shown in Fig. 2. This module works on the attention +phenomenon, which enhances the relevant features and +diminishes the less significant features [33]. An element- +wise multiplication operation is performed between the +outputs of the dense layer and the maxpool1D layer. This +produces higher values of product where both maxpool1D +and dense layer outputs are high, thereby enhancing +the more intense features. Similarly, the less significant +features are further restrained due to low values of the +product where both the layer outputs are low. The input +dimension of the dense layer is (C3) × 32, and a dot +product operation between a 32 × 32 weight vector of +the dense layer and its input results in the same output +dimension. + ++ Z (Superior) ++ Z (Superior) +40 cm ++Y +fx +-x +10 cm +Posterior +Anterior +Right +Left +- Z (Inferior) +Z (Inferior)7 +TABLE I: Training hyper-parameters (After hypertuning). +Nc +Nk +Dr +ks +sr +lr +Bt +ec +3 +32 +0.40 +5 +1 +0.001 +15 +100 +Ncl: Number of convolution layers, Nk: Number of kernels/filters, Dr: Dropout rate, +ks: Kernel width, sr: Stride/shift, lr: Learning rate, Bt: Batch size, ec: Number of +training epochs. +• Flatten layer: This layer transforms the output of CAM +which is C3 × 32 to a 1D vector with dimension C4 × 1, +as shown in Fig. 2. A dropout with a factor of 0.4 is used +after this layer to prevent the model from over-fitting [36]. +• Dense layers: Three dense layers with 8 units each are +used after the flatten layer. In this work, swish activation +function is used in these layers, interpreted as: +f(x) = x . swish(x) +(10) +• Output layer: The final layer is a dense layer for re- +gression, that maps the output of flatten layer to the +predicted trajectory with dimension N × 1, as shown +in Fig. 2. Dense layer implements the element-wise +dot product between the input and the kernel. Linear +activation function is used in this layer, given by: +f(x) = x +(11) +The aforementioned layers and hyper-parameters are used +to create the proposed network. For training, 80% of EEG +signals with different durations/window lengths are taken from +the recorded database. The rest 20% of the data is divided +into 10% test and 10% validation data. The information about +optimal training hyper-parameter selection and their values is +provided in the next section. The proposed network is built +using Keras deep learning framework with TensorFlow version +2.2.1 as backend in Python. In this work, data augmentation +is utilized to increase the number of training examples in the +data to avoid over-fitting. It makes the proposed network more +robust by creating new and different training examples by +which it can learn the alterations in the real world. For this +purpose, random flipping and rolling operations are used in +Python Keras framework. +IV. RESULTS AND DISCUSSION +In this Section, the performance evaluation of the proposed +BiCurNet on the recorded EEG signals is presented w.r.t. +different parameters. Elaborated interpretations of the results +are also presented for the proposed network. +A. Hyper-parameters for training BiCurNet +Various parameters used for training the proposed network +are presented herein. For assessing the regression/prediction +performance of the proposed network, 10% of the EEG signals +from the recorded database are used for testing. The data +from each subject is used for training, testing, and validation, +i.e., subject-dependent training is performed. The network +is trained using a batch size of 15, epochs as 100, and +Adam optimizer with a learning rate as 0.001. To curtail +the statistical unreliability in computation of test loss due +to small database, ten-fold cross validation is employed for +performance evaluation. Mean square error (MSE) is used as +the loss function for regression. Table I presents the training +hyper-parameters which are selected using the KerasTuner +framework in Python. It is an optimization framework for +tuning the hyper-parameters that uses search-and-selection- +based criteria. The final corresponding selected set of optimal +hyper-parameters is listed in the table. +B. Regression metric +In this work, time lagged and windowed EEG signals +are used to estimate the motion trajectory in advance. In +particular, the EEG data preceding the motion by different +time lags (8-240 ms) is used to train, test, and validate the +proposed network. Additionally, the performance is evaluated +with varying EEG window sizes (320-1600 ms). A 95% +overlap between adjacent windows is considered. Pearson +correlation coefficient (PCC) is utilized for analysing the +performance of the proposed network w.r.t. upper limb motion +trajectory estimation. PCC between true/measured (A) and +predicted/estimated (P) trajectory signal with N samples is +given as +Π(A, P) = +1 +N − 1 +N +� +i=1 +�Ai − mA +σA +� �Pi − mP +σP +� +(12) +where m is the mean and σ denotes standard deviation. The +normalized covariance measure assumes a value between -1 +and 1. +C. Subject dependent PCC analysis +The proposed model is trained and tested for each subject +separately, for subject-dependent (SD) performance analysis. +The PCC values averaged across all the trials and subjects, are +presented in Table II with varying time lags, window sizes, +and EEG features. The EEG bands are considered in spatial +(V ), spherical harmonics (Vδnm), and head harmonics domains +(V H2 +δnm). It may be noted that the transformed domain (Vδnm +and V H2 +δnm) features gives PCC similar to spatial domain coun- +terparts with reduced computational cost, as detailed in Section +III-D2. Additionally, δ band gives higher PCC values while γ +band has the lowest PCC. This indicates the pertinence of low- +frequency δ band for motion trajectory decoding using EEG. +The best correlation is observed when Vδ, V SH +δnm, and V H2 +δnm are +combined. The highest correlation achieved is 0.7 with 240 +ms advanced EEG window of 1600 ms. This demonstrates the +feasibility of early estimation of the motion trajectory by using +the proposed network. +D. Subject-independent performance analysis +To further explore the adaptability of the proposed network, +subject-independent (SI) analysis is presented herein using +leave-one-out scheme. Simultaneous comparison of SI/SD +case on PCC is presented in Fig. 7. The PCC values are +averaged over all subjects and lags. A slight decrease in PCC +value may be noted in the SI case. However, it remains within +±0.05 which indicates the robustness of the proposed network +against the subject-variability. + +8 +TABLE II: Pearson correlation coefficient (PCC) for different EEG segments and lags of data (Mean over subjects). +EEG +Features +8 +ms +40 +ms +80 +ms +160 +ms +240 +ms +8 +ms +40 +ms +80 +ms +160 +ms +240 +ms +8 +ms +40 +ms +80 +ms +160 +ms +240 +ms +8 +ms +40 +ms +80 +ms +160 +ms +240 +ms +V +0.25 0.25 +0.26 +0.26 +0.26 +0.35 0.35 +0.36 +0.35 +0.26 +0.42 0.42 +0.42 +0.42 +0.43 +0.55 0.55 +0.55 +0.55 +0.56 +Vδ +0.34 0.33 +0.33 +0.34 +0.36 +0.41 0.41 +0.42 +0.42 +0.42 +0.48 0.48 +0.48 +0.48 +0.49 +0.61 0.61 +0.61 +0.66 +0.67 +Vθ +0.24 0.23 +0.23 +0.24 +0.26 +0.38 0.38 +0.38 +0.38 +0.38 +0.44 0.44 +0.44 +0.44 +0.45 +0.55 0.55 +0.55 +0.56 +0.57 +Vα +0.22 0.22 +0.22 +0.22 +0.21 +0.36 0.36 +0.37 +0.36 +0.36 +0.39 0.39 +0.39 +0.39 +0.39 +0.51 0.51 +0.51 +0.51 +0.53 +Vβ +0.18 0.18 +0.17 +0.17 +0.17 +0.29 0.29 +0.29 +0.3 +0.3 +0.32 0.32 +0.32 +0.32 +0.33 +0.39 0.39 +0.39 +0.39 +0.39 +Vγ +0.1 +0.1 +0.1 +0.1 +0.1 +0.17 0.17 +0.17 +0.18 +0.18 +0.27 0.27 +0.27 +0.27 +0.28 +0.29 0.29 +0.29 +0.29 +0.29 +V SH +nm +0.25 0.25 +0.25 +0.25 +0.26 +0.34 0.35 +0.35 +0.35 +0.36 +0.41 0.41 +0.41 +0.41 +0.42 +0.54 0.54 +0.54 +0.54 +0.55 +V SH +δnm +0.34 0.33 +0.34 +0.35 +0.35 +0.41 0.41 +0.41 +0.41 +0.41 +0.47 0.47 +0.47 +0.48 +0.48 +0.61 0.61 +0.61 +0.66 +0.66 +V SH +θnm +0.23 0.22 +0.22 +0.22 +0.23 +0.37 0.37 +0.37 +0.37 +0.38 +0.44 0.44 +0.44 +0.44 +0.45 +0.55 0.55 +0.55 +0.55 +0.56 +V SH +αnm +0.2 +0.2 +0.19 +0.2 +0.2 +0.34 0.34 +0.34 +0.34 +0.36 +0.38 0.38 +0.38 +0.38 +0.38 +0.5 +0.5 +0.5 +0.5 +0.51 +V SH +βnm +0.17 0.17 +0.16 +0.16 +0.17 +0.29 0.29 +0.29 +0.29 +0.29 +0.33 0.33 +0.33 +0.33 +0.34 +0.4 +0.4 +0.4 +0.4 +0.41 +V SH +γnm +0.09 0.1 +0.1 +0.1 +0.1 +0.18 0.18 +0.18 +0.18 +0.18 +0.28 0.28 +0.28 +0.28 +0.3 +0.3 +0.3 +0.3 +0.3 +0.31 +V H2 +nm +0.25 0.25 +0.26 +0.26 +0.26 +0.35 0.35 +0.35 +0.35 +0.36 +0.42 0.42 +0.42 +0.42 +0.43 +0.55 0.55 +0.55 +0.55 +0.57 +V H2 +δnm +0.34 0.33 +0.34 +0.34 +0.35 +0.41 0.41 +0.41 +0.41 +0.41 +0.48 0.48 +0.48 +0.48 +0.49 +0.62 0.62 +0.62 +0.62 +0.65 +V H2 +θnm +0.25 0.24 +0.24 +0.23 +0.23 +0.38 0.38 +0.38 +0.39 +0.39 +0.44 0.44 +0.44 +0.44 +0.45 +0.53 0.53 +0.53 +0.53 +0.55 +V H2 +αnm +0.22 0.2 +0.2 +0.2 +0.22 +0.36 0.36 +0.36 +0.36 +0.36 +0.38 0.38 +0.38 +0.38 +0.39 +0.51 0.51 +0.51 +0.51 +0.51 +V H2 +βnm +0.18 0.18 +0.18 +0.18 +0.18 +0.28 0.28 +0.28 +0.28 +0.29 +0.33 0.33 +0.33 +0.33 +0.34 +0.39 0.39 +0.39 +0.39 +0.4 +V H2 +γnm +0.11 0.11 +0.11 +0.11 +0.11 +0.16 0.16 +0.16 +0.16 +0.17 +0.2 +0.2 +0.2 +0.2 +0.2 +0.19 0.19 +0.19 +0.19 +0.19 +Vcom +0.36 0.36 +0.36 +0.36 +0.37 +0.43 0.43 +0.44 +0.44 +0.44 +0.5 +0.5 +0.5 +0.51 +0.52 +0.67 0.67 +0.67 +0.68 +0.70 +■: 320 ms window, ■: 800 ms window, ■: 1200 ms window , ■: 1600 ms window; Note: Vcom : [Vδ; V SH +δnm; V H2 +δnm] +Fig. 7: Average PCC values w.r.t. subject dependent (SD) and +subject-independent (SI) training of the proposed network at +different window sizes (320 ms to 1600 ms). +E. Robustness analysis +The robustness of the proposed network is analyzed herein +using artifactual EEG signals. In particular, the pre-processing +did not include ICA decomposition-based artifact removal. The +proposed network is trained and tested using such signals. +Mean PCC values obtained using without artifact (WOA) and +with artifact (WA) EEG signal are presented in Fig. 8. A +small decrease of 0.06 in the PCC values may be observed +with artifact case that indicates the robustness of the proposed +model. +F. Trajectory estimation curves +The proposed BiCurNet model is additionally evaluated +herein using actual motion trajectories. Fig. 9 illustrates the +estimated and actual trajectories for subject I with window +size varying between 800-1600 ms. 95% overlap is considered +Fig. 8: Subject dependent average PCC values utilizing with +and without artifactual EEG data for different window sizes. +between two adjacent windows. It may be observed from the +figure that there is a considerable improvement in correlation +when window size is increased. This results in trajectory closer +to the ground truth. Ability of the proposed network to follow +the trajectory pattern for all windows indicates the learning +capability of the network. +V. CONCLUSION +A deep learning-based paradigm for early estimation of +upper limb motion trajectory using EEG signal is proposed +in this work. The EEG is collected while performing biceps +curl movement. The proposed BiCurNet model is built using +a light-weight architecture with depth-wise separable con- +volution layers and customized attention module. The input +features to the model are taken in computationally more +efficient spherical and head harmonics domain in addition to +spatio-temporal data. The extensive performance evaluation + +0.7 +0.6 +0.5 +PCC +0.4 +0.3 +SD +SI +Meanoversubjects0.7 +0.6 +IWOA +IWA +0.5 +0.4 +CC +P +0.3 +0.2 +0.1 +0 +320 ms +800ms +1200ms +1600ms +Window size9 +Fig. 9: Actual and predicted trajectories of subject 1 (Early prediction, before 40 ms). +of the proposed network on in-house recorded EEG signals +demonstrates its effectiveness in early estimation. Performance +evaluation includes subject (in)dependent study. the noise +awareness of the proposed network is also demonstrated by +using the artifactual EEG signals for training. Robustness of +the proposed network is demonstrated by using the artifactual +EEG signals for training. The proposed network being com- +putationally efficient, and noise-aware, makes it suitable for +use in real-time BCI applications. Real-time implementation +of the proposed network for an exosuit control is currently +being explored. +ACKNOWLEDGMENT +This research work was supported in part by DRDO - JATC +project with project number RP04191G. +REFERENCES +[1] J. R. Wolpaw and E. W. Wolpaw, “Brain-computer interfaces: something +new under the sun,” Brain-computer interfaces: principles and practice, +vol. 14, 2012. +[2] S. Gong, K. Xing, A. Cichocki, and J. Li, “Deep learning in EEG: +Advance of the last ten-year critical period,” IEEE Transactions on +Cognitive and Developmental Systems, 2021. +[3] S. Aggarwal and N. Chugh, “Review of machine learning techniques +for EEG based brain computer interface,” Archives of Computational +Methods in Engineering, pp. 1–20, 2022. +[4] A. Chowdhury, H. Raza, Y. K. Meena, A. Dutta, and G. Prasad, “On- +line covariate shift detection-based adaptive brain-computer interface +to trigger hand exoskeleton feedback for neuro-rehabilitation,” IEEE +Transactions on Cognitive and Developmental Systems, vol. 10, no. 4, +pp. 1070–1080, 2017. +[5] H. Raza, A. Chowdhury, and S. Bhattacharyya, “Deep learning based +prediction of EEG motor imagery of stroke patients’ for neuro- +rehabilitation application,” in 2020 International Joint Conference on +Neural Networks (IJCNN). +IEEE, 2020, pp. 1–8. +[6] P. Di Lillo, F. Arrichiello, D. Di Vito, and G. Antonelli, “BCI-controlled +assistive manipulator: developed architecture and experimental results,” +IEEE Transactions on Cognitive and Developmental Systems, vol. 13, +no. 1, pp. 91–104, 2020. +[7] X. Li, L. Wang, S. Miao, Z. Yue, Z. Tang, L. Su, Y. Zheng, X. Wu, +S. Wang, J. Wang et al., “Sensorimotor Rhythm-Brain Computer Inter- +face With Audio-Cue, Motor Observation and Multisensory Feedback +for Upper-Limb Stroke Rehabilitation: A Controlled Study,” Frontiers +in Neuroscience, vol. 16, 2022. +[8] R. Sosnik and O. B. Zur, “Reconstruction of hand, elbow and shoulder +actual and imagined trajectories in 3d space using eeg slow cortical +potentials,” Journal of neural engineering, vol. 17, no. 1, p. 016065, +2020. +[9] N. Robinson, T. W. J. Chester, and K. Smitha, “Use of mobile EEG in +decoding hand movement speed and position,” IEEE Transactions on +Human-Machine Systems, vol. 51, no. 2, pp. 120–129, 2021. +[10] A. Jain and L. Kumar, “PreMovNet: Premovement EEG-Based Hand +Kinematics Estimation for Grasp-and-Lift Task,” IEEE Sensors Letters, +vol. 6, no. 7, pp. 1–4, 2022. +[11] M. Saini, U. Satija, and M. D. Upadhayay, “One-dimensional convo- +lutional neural network architecture for classification of mental tasks +from electroencephalogram,” Biomedical Signal Processing and Control, +vol. 74, p. 103494, 2022. +[12] A. Jain and L. Kumar, “Subject-Independent 3D Hand Kinematics +Reconstruction using Pre-Movement EEG Signals for Grasp And Lift +Task,” arXiv preprint arXiv:2209.01932, 2022. +[13] P. S. Hammon, S. Makeig, H. Poizner, E. Todorov, and V. R. De Sa, +“Predicting reaching targets from human EEG,” IEEE Signal Processing +Magazine, vol. 25, no. 1, pp. 69–77, 2007. +[14] R. Sosnik and L. Zheng, “Reconstruction of hand, elbow and shoulder +actual and imagined trajectories in 3D space using EEG current source +dipoles,” Journal of Neural Engineering, vol. 18, no. 5, p. 056011, 2021. +[15] J.-H. Kim, F. Bießmann, and S.-W. Lee, “Decoding three-dimensional +trajectory of executed and imagined arm movements from electroen- +cephalogram signals,” IEEE Transactions on Neural Systems and Reha- +bilitation Engineering, vol. 23, no. 5, pp. 867–876, 2014. +[16] L. Yang, H. Leung, M. Plank, J. Snider, and H. Poizner, “EEG activity +during movement planning encodes upcoming peak speed and acceler- +ation and improves the accuracy in predicting hand kinematics,” IEEE +journal of biomedical and health informatics, vol. 19, no. 1, pp. 22–28, +2014. +[17] Y. Nakanishi, T. Yanagisawa, D. Shin, H. Kambara, N. Yoshimura, +M. Tanaka, R. Fukuma, H. Kishima, M. Hirata, and Y. Koike, “Mapping +ECoG channel contributions to trajectory and muscle activity prediction +in human sensorimotor cortex,” Scientific reports, vol. 7, no. 1, pp. 1–13, +2017. +[18] N. A. Bhagat, A. Venkatakrishnan, B. Abibullaev, E. J. Artz, N. Yoz- +batiran, A. A. Blank, J. French, C. Karmonik, R. G. Grossman, M. K. +O’Malley et al., “Design and optimization of an EEG-based brain ma- +chine interface (BMI) to an upper-limb exoskeleton for stroke survivors,” +Frontiers in neuroscience, vol. 10, p. 122, 2016. +[19] M. Mahmoodi, B. Makkiabadi, M. Mahmoudi, and S. Sanei, “A new +method for accurate detection of movement intention from single +channel EEG for online BCI,” Computer Methods and Programs in +Biomedicine Update, vol. 1, p. 100027, 2021. +[20] R. Gatti, Y. Atum, L. Schiaffino, M. Jochumsen, and J. Biurrun Manresa, +“Decoding kinetic features of hand motor preparation from single- +trial EEG using convolutional neural networks,” European Journal of +Neuroscience, vol. 53, no. 2, pp. 556–570, 2021. +[21] J.-H. Jeong, K.-H. Shim, D.-J. Kim, and S.-W. Lee, “Brain-controlled +robotic arm system based on multi-directional CNN-BiLSTM network +using EEG signals,” IEEE Transactions on Neural Systems and Reha- +bilitation Engineering, vol. 28, no. 5, pp. 1226–1238, 2020. +[22] S. Pancholi, A. Giri, A. Jain, L. Kumar, and S. Roy, “Source aware +deep learning framework for hand kinematic reconstruction using EEG +signal,” IEEE Transactions on Cybernetics, 2022. +[23] K. Kumarasinghe, N. Kasabov, and D. Taylor, “Brain-inspired spiking +neural networks for decoding and understanding muscle activity and +kinematics from electroencephalography signals during hand move- +ments,” Scientific reports, vol. 11, no. 1, pp. 1–15, 2021. +[24] J. W. Peirce, “PsychoPy—psychophysics software in Python,” Journal +of neuroscience methods, vol. 162, no. 1-2, pp. 8–13, 2007. +[25] A. Delorme and S. Makeig, “EEGLAB: an open source toolbox for +analysis of single-trial EEG dynamics including independent component +analysis,” Journal of neuroscience methods, vol. 134, no. 1, pp. 9–21, +2004. + +Window size: 800 ms +Window size: 1200 ms +Window size: 1600 ms +True +Predicted +10 +10 +10 +0.9 +0.9 +0.9 +0.8 +0.8 +0.8 +Angle +0.7 +0.7 +0.7 +0.6 +0.5 +0.5 +0.5 +0.4 +PCC +0.4 +PCC +0.4 +PCC +0.45 +0.3 +0.3 +0.52 +EO +0.60 +0.2 +0.2 +0.2 +250 +500 +75010001250150017502000 +0 +250500 +75010001250150017502000 +75010001250150017502000 +Samples +Samples +Samples10 +[26] A. K. Maddirala and K. C. Veluvolu, “ICA with CWT and k-means +for Eye-blink Artifact Removal from Fewer Channel EEG,” IEEE +Transactions on Neural Systems and Rehabilitation Engineering, 2022. +[27] J. Mazziotta, A. Toga, A. Evans, P. Fox, J. Lancaster, K. Zilles, +R. Woods, T. Paus, G. Simpson, B. Pike et al., “A probabilistic atlas +and reference system for the human brain: International Consortium +for Brain Mapping (ICBM),” Philosophical Transactions of the Royal +Society of London. Series B: Biological Sciences, vol. 356, no. 1412, +pp. 1293–1322, 2001. +[28] A. Gramfort, T. Papadopoulo, E. Olivi, and M. Clerc, “OpenMEEG: +opensource software for quasistatic bioelectromagnetics,” Biomedical +engineering online, vol. 9, no. 1, pp. 1–20, 2010. +[29] R. D. Pascual-Marqui et al., “Standardized low-resolution brain elec- +tromagnetic tomography (sLORETA): technical details,” Methods Find +Exp Clin Pharmacol, vol. 24, no. Suppl D, pp. 5–12, 2002. +[30] M. Saini, U. Satija, and M. D. Upadhayay, “Discriminatory features +based on wavelet energy for effective analysis of electroencephalogram +during mental tasks,” Circuits, Systems, and Signal Processing, pp. 1–29, +2022. +[31] S. G. Mallat, “A theory for multiresolution signal decomposition: the +wavelet representation,” IEEE transactions on pattern analysis and +machine intelligence, vol. 11, no. 7, pp. 674–693, 1989. +[32] A. Giri, L. Kumar, and T. K. Gandhi, “Brain source localization in +head harmonics domain,” IEEE Transactions on Instrumentation and +Measurement, vol. 70, pp. 1–10, 2020. +[33] S. Woo, J. Park, J.-Y. Lee, and I. S. Kweon, “Cbam: Convolutional +block attention module,” in Proceedings of the European conference on +computer vision (ECCV), 2018, pp. 3–19. +[34] M. P´erez-Enciso and L. M. Zingaretti, “A guide on deep learning for +complex trait genomic prediction,” Genes, vol. 10, no. 7, p. 553, 2019. +[35] Z. Jiao, X. Gao, Y. Wang, J. Li, and H. Xu, “Deep convolutional neural +networks for mental load classification based on EEG data,” Pattern +Recognition, vol. 76, pp. 582–595, 2018. +[36] H. Wu and X. Gu, “Towards dropout training for convolutional neural +networks,” Neural Networks, vol. 71, pp. 1–10, 2015. + diff --git a/2tE2T4oBgHgl3EQfjAfh/content/tmp_files/load_file.txt b/2tE2T4oBgHgl3EQfjAfh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9c3a3fd105caee72fee24709e04d04365124980 --- /dev/null +++ b/2tE2T4oBgHgl3EQfjAfh/content/tmp_files/load_file.txt @@ -0,0 +1,1160 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf,len=1159 +page_content='1 BiCurNet: Pre-Movement EEG based Neural Decoder for Biceps Curl Trajectory Estimation Manali Saini*, Anant Jain*, Lalan Kumar, Suriya Prakash Muthukrishnan, Shubhendu Bhasin and Sitikantha Roy Abstract—Kinematic parameter (KP) estimation from early electroencephalogram (EEG) signals is essential for positive augmentation using wearable robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' However, work related to early estimation of KPs from surface EEG is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, a deep learning-based model, BiCurNet, is presented for early estimation of biceps curl using collected EEG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The model utilizes light-weight architecture with depth-wise separable convolution layers and customized attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The feasibility of early estimation of KPs is demonstrated using brain source imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Computationally efficient EEG features in spherical and head harmonics domain is utilized for the first time for KP prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The best Pearson correlation coefficient (PCC) between estimated and actual trajectory of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 is achieved when combined EEG features (spatial and harmonics domain) in delta band is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Robustness of the proposed network is demonstrated for subject-dependent and subject-independent training, using EEG signals with artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Index Terms—Brain-computer interface, Electroencephalo- gram, Deep learning, Kinematic parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' INTRODUCTION Brain-computer interface (BCI) is an integration of the measurement, decoding, and translation of the activity of central nervous system (CNS) into imitative output that rein- states, augments, or rehabilitates the natural CNS output [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This creates an interface between the CNS and its external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' BCI-based systems are rapidly emerging on account of the recent advancements in signal processing and artificial intelligence [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' These systems are useful in Manali Saini and Anant Jain have contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This work was supported in part by DRDO - JATC project with project number RP04191G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This work involved human subjects or animals in its research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Approval of all ethical and experimental procedures and protocols was granted by the Institute Ethics Committee, All India Institute of Medical Sciences, New Delhi, India with reference number IEC-751/07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2020,RP-06/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Manali Saini is with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: manali- igit@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Anant Jain is with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: anant- jain@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Lalan Kumar is with the Department of Electrical Engineering, Bharti School of Telecommunication, and Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: lkumar@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Suriya Prakash Muthukrishnan is with the Department of Physiology, All India Institute of Medical Sciences, New Delhi - 110016, India(e-mail: dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='suriyaprakash@aiims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Subhendu Bhasin is with the Department of Electrical Engineering, In- dian Institute of Technology Delhi, New Delhi 110016, India (e-mail: sb- hasin@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Sitikantha Roy is with the Department of Applied Mechanics, In- dian Institute of Technology Delhi, New Delhi 110016, India (e-mail: sroy@am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' neuro-rehabilitation to assist users with motor-impairments [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For real-time operability of these systems, continuous signal decoding is required for extraction of kinematic param- eters (KPs) such as motion trajectory, velocity, acceleration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In view of these aspects, electroencephalogram (EEG)-based BCI systems have gained popularity in the recent years, owing to the non-invasiveness, low-cost, and excellent temporal resolution of EEG signals [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Motivation and Related Work Literary works explore machine learning and deep learning- based paradigms for upper limb kinematic parameter esti- mation (KPE), movement intention detection and classifica- tion from low frequency components of EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For instance, in [13], sparse multinomial logistic regression is utilized to classify EEG signals during reach intention and actual movement, based on multiple hand-crafted features extracted from EEG signals filtered in the range of 1 − 40 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, independent component analysis (ICA) and dipole fitting are applied to remove movement artifacts from the recorded EEG signals, for obtaining low classification error rates [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Researchers in [14] have explored EEG current source dipole (CSD) data, using standardized low resolution brain electromagnetic tomography (sLORETA) to decode actual and imagined arm joint trajectories based on multiple linear regression (mLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The most useful time lags are observed to be between 80−150 ms prior to the movement, and the low β and γ bands are shown to be more effective in movement decoding with a correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Similarly, mLR is utilized in [15] for estimating the 3D trajectories of arm movement with variable velocities using EEG segments filtered in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 − 40 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The researchers reported a high correlation between the movement velocities and EEG activity above the motor cortex in fronto-central and parietal areas [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' mLR is also utilized in [16] with α and β band powers of EEG signals during the motor planning and exe- cution phases to predict the upcoming peak tangential speed and acceleration of hand movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This study demonstrates the prominence of occipital and parietal-occipital regions for α band, and frontal and frontal-central regions for the β band in movement planning and execution phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In a recent study, researchers have explored the feasibility of a commercial EEG headset in motor decoding and classification with the use of Kalman filter and spatio-spectral features extracted from EEG signals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' An overall correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='58 is achieved in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Besides mLR, sparse LR is investigated for predicting the circular trajectories of upper limb during movement of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='03965v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='SP] 10 Jan 2023 2 bottles with varying masses [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, a wide range of EEG frequencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', 0−150 Hz is used and channels over the motor cortex are shown to be more prominent towards the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In [18], movement intent is decoded from movement related cortical potentials (MRCPs) using narrow- band EEG in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1−1 Hz to train a support vector machine (SVM)-based classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The selection of a single- channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', Cz in movement onset decoding with an accuracy of 91% using low frequency (0 − 5 Hz) Teager-Kaiser energy operator with threshold-based classification is demonstrated in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Despite of the effectiveness of conventional machine learning-based paradigms in EEG-based movement decoding, there is a need of extracting the high-level features which can enhance the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' To overcome this, researchers have proposed deep learning-based paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For example, convo- lutional neural network (CNN) is proposed with the use of pre- movement raw spatio-temporal multi-channel EEG for hand movement and force levels classification with an accuracy of 84% [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This work demonstrates early classification of hand movement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', in 100−1600 ms advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' CNN is also utilized in [21] along with bidirectional long short term memory (Bi- LSTM)-based network to predict the velocities of arm reaching tasks using pre-processed EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' An overall correlation between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='6 is achieved in this work and feasibility of robotic arm control based on real-time EEG is demonstrated [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Recently, deep learning-based three-dimensional (3D) hand movement trajectory during grasp and lift movements is estimated using a public EEG database in [10], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In [22], wavelet packet decomposition (WPD) based time- lagged EEG sub bands are used to train a CNN-LSTM network for prediction of the hand position/trajectory with a high correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This work explores the source-aware EEG features and the demonstrates the relevance of low frequency bands (δ, θ, and α) in movement estimation, however, it has limited feasibility in real-time hardware implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Early estimation of this trajectory is demonstrated in [10] with a high correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='79 using the δ band of EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Further, researchers in [23] demonstrate the feasibility of a brain-inspired spiking neural network (Bi-SNN) along with mid-frequency and high- frequency EEG bands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', α, β, and γ, toward the same trajectory estimation with a correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Based on the aforementioned description of literary works, it can be asserted that many of these works focus on classification of upper limb movements, rather than predic- tion/estimation of the related kinematic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Timely extraction of kinematic parameters from EEG data during upper limb movement is imperative towards different real-time exosuit control-based BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Further, few existing machine-learning based regression algorithms are able to esti- mate the KPs earlier w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' actual movement, however, average correlation is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Although the existing deep learning- based networks outperform these ML-based paradigms, only few of them have explored early estimation of KPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Further, these networks use slightly complex architectures after pre- processing which may not be feasible on hand-held processors for real-time BCI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Most importantly, the performance of the existing paradigms for KP estimation is highly subject- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1: Experimental setup for biceps-curl task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' specific, which further adds to the complexity since the networks need to be trained for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Objective and Key Contributions In view of the aforementioned challenges of literary works, this work proposes a deep learning-based upper limb mo- tion trajectory prediction/estimation from preceding EEG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', BiCurNet, for early estimation towards exosuit control-based BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Further, the proposed network is demon- strated to be subject-independent and robust against artifacts, unlike the existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' To the best of our awareness, this is the first work which focuses on early estimation of kine- matic parameters from both subject-dependent and subject- independent EEG signals and further analyses the noise- robustness of the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The key contributions of this work are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Low-complex deep learning-based architecture is pro- posed for early estimation of upper limb motion trajec- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In-house recording of multi-channel EEG signals during upper limb biceps curl experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Spherical harmonics and head-harmonics domain EEG features based motion trajectory estimation has been explored for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Demonstration of subject-adaptability and noise- robustness of the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Section II describes the experimental recording and data acquisition procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Section III presents the proposed methodology for BiCurNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Section IV discusses the experimental evaluation results for the proposed work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Finally, section V concludes this work with major advantages, shortcomings, and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2: Block diagram depicting the proposed methodology for biceps-curl trajectory estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' EXPERIMENT AND DATA ACQUISITION The key objective of the study is to investigate the via- bility of using EEG signals for elbow joint angle decoding during biceps-curl motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For this purpose, we designed a synchronous EEG and joint angle recording system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The description of the experimental paradigm and data acquisition are elucidated in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Subjects and Equipment The experiment was performed in the Multichannel Signal Processing Laboratory, Department of Electrical Engineering at Indian Institute of Technology Delhi, New Delhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This research was authorized by the Institutional Review Board of All India Institute Of Medical Sciences, New Delhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' EEG data and joint angle data were recorded from 5 healthy subjects (all males, age 29 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61, all right handed) while performing the biceps curl task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Each subject performed 300 trials of biceps curls task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' EEG data was recorded using 16-channel dry-active electrodes (actiCAP Xpress Twist, Brain Products, Gilching, Germany) with wireless EEG amplifier (LiveAmp- 16, Brain Products, Gilching, Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The EEG sensors were arranged in 10-20 international system of EEG electrode placement, namely, Fp1, Fz, F3, C3, T7, Pz, P3, O1, Oz, O2, P4, Cz, C4, T8, F4 and Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The EEG data was acquired with 500 Hz sampling frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A marker-based camera system (Noraxon NiNOX 125 Camera System) was placed for elbow joint angle measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The NiNOX 125 camera system was connected to Noraxon myoResearch platform (MR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16) for recording the biceps-curl hand motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The camera system was placed in sagittal plane 2 m away from the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The elbow joint angle was calculated using myoResearch software in the post-processing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The 3-point angle measurement tool was utilized to compute 2D joint angle by tracking reflective markers in the video recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The joint angle data was sampled with the sampling frequency of 125 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The EEG and joint angle data was synchronized using Noraxon myoSync device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Experimental Setup and Paradigm Concurrent EEG and motion data was collected from the users during biceps-curl task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' At the beginning of experiment, participants were in standing position with 2 Kg dumbbell holding in their right hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A monitor was positioned 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8 m away in front of them for showing the experimental paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Participants were standing in balanced upright posture with dumbbell in their right hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' We designed the experiment in PsychoPy [24] for instructing the participant for initiating the biceps-curl movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Each trial begin with a cross appearing on the center screen along with a beep sound, indicating the start of the trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' After a couple of seconds, a visual cue appeared on the screen to instruct the participant to initiate the biceps-curl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The biceps-curl was performed in the motion execution phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Each trial ended with resting phase of two seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Before the actual data acquisition, each participant performed a practice run for executing the task correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This practice run was not included for any consequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' We recorded 30 runs with 10 trials each for the biceps curl task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Inter-run rest was given to the participant for avoiding muscle fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' PROPOSED METHODOLOGY This section elaborates the proposed methodology for early prediction of upper limb motion trajectory from EEG based on deep learning, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It consists of three major modules: EEG recording,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' pre-processing and feature Raw EEG signals Pre-processing and 3 Dense andino (Channels 1 feature extraction EEG data Flatten layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 8 layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' N to 16) (Channels 1 layer units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' DFT-based to Nc) DWSConv1Dlayer Conv1D layer Maxpool1D layer Enhance and suppress activation = activation = baseline wander 32 kernels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='kernel 32kernels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' kernel Pool size=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' attention block Swish Linear size=5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' stride=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' size=5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' stride=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' noise removal activation-ReLu stride=2 activation=ReLu (Threshold: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' dwConv1D Dense + ReLu layer Common average referencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' ICA-based artifact rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Amplitude normalization [v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' DWT-based sub- bands extraction: [AA:"A:A:*A] DWT-Spherical harmonics (SH) (C3-Ks 1) × 32 C4 × 32 features: C4 C5 DWT-Head harmonics (H2) (N-Ks+1) × 32 (C,-Ks+1) × 32 (C,/2) × 32 C5 × 1 8×1 N×1 NxNC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' features: Predicted c C2 C3 trajectory EEG recording4 extraction, and depth-wise separable convolutional neural net- work with a customized attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The modules are described in the subsequent sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' EEG recording In this work, the EEG signals are acquired using LiveAmp 16 Brain Products system as described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Prior to be used for the proposed BiCurNet, these signals are pre-processed as detailed in the ensuing sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Pre-processing The recorded EEG signals are pre-processed in EEGLAB [25] and MATLAB for feature extraction prior to be fed to the proposed BiCurNet, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' After recording and re-sampling the EEG signals, low frequency (below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 Hz) baseline wander noise (BWN) suppression is done using discrete Fourier transform (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For this purpose, the DFT coefficients corresponding to frequencies below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5Hz are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The computation of DFT coefficient index k is done as: k = ⌊(fqNd/fqs)⌋, where fq is the frequency in Hz, fqs is the sampling frequency, and Nd is the number of DFT points for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' These DFT coefficients are thresholded to zero for suppression of the BWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The EEG signal after BWN suppression is synthesized as the inverse of the DFT coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The mathematical interpretation of this procedure is described for a recorded EEG signal v[n] by the following DFT pair: DFT of recorded signal : V [k] = Nd−1 � n=0 v[n]e −jn2πk Nd (1) ˜vq[n] = 1 Nd Nd−1 � k=0 ˜Vq(k)e jn2πk Nd (2) where ˜ Xq denotes the DFT coefficient matrix after thresh- olding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', ˜ Xq(k) = [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' , 0, Xq[k + 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' , Xq[Nd − k − 1], 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='., 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' All signals are normalized w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' amplitude to bring it in range: [−1, 1] as ˜ xq[n] max| ˜ xq[n]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, the recorded EEG signals are analyzed for the estimation of motion trajectory with and without artifact suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Independent component analysis (ICA) is utilized for artifact suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It is used for estimating the sources corresponding to cerebral and non-cerebral activities resulting in the scalp EEG [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' EEGLAB is used in this work for ICA- based decomposition of the EEG signals obtained after BWN removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The decomposed independent sources with more than 70% of artifactual components are rejected and the artifact-free EEG signal is reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Brain source imaging Brain source imaging (BSI) is performed to select the rel- evant pre-movement EEG segment prior to feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Numerical Boundary Element Method (BEM) based forward modeling is utilized for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The head model utilizes ICBM MRI template [27] in OpenMEEG [28] toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The spatio-temporal dynamics of brain cortical sources are ob- tained using inverse modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In particular, standardized low- resolution electromagnetic tomography (sLORETA) [29] is utilized to solve the under-determined inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Under the constraint of the smooth source distribution, standardized current density maps are utilized for localization inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Source localization plots for a right hand biceps-curl activity are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The analysis shown corresponds to a single-trial of biceps-curl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The subject was instructed to focus the vision on fixation cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A visual cue for movement onset was presented at 0 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The subject executed biceps-curl activity 410 ms after the visual cue was given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A constant activation may be observed in occipital lobe up to 60 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The information starts getting transferred to the left motor cortex thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' All such pre-movement EEG [Fig 3(c)-(g)] has inbuilt motion trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It may be noted that the left motor cortex region was acti- vated at 220-240 ms [Fig 3(e)] corresponding to the right-hand biceps-curl activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Motor activity was observed thereafter up to 320 msec [Fig 3(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The subject executed biceps-curl activity at 400-450 ms after the visual cue was given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It may be concluded that the motor neural information corresponding to the biceps-curl activity is present approximately 250 ms prior to the motor execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This information is utilized for selecting the time-lag window for elbow joint-angle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The selected EEG data was utilized for the training and testing of the proposed neural decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Feature extraction The pre-processed EEG signals are analyzed with different transform-domain techniques for significant feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This work explores time-frequency features using discrete wavelet transform (DWT) in: Spatial domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Spatio-temporal domain using spherical Fourier transform (SFT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' and Spatio- temporal domain using head harmonics transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1) Discrete wavelet transform-based features: Discrete wavelet transform is utilized to decompose the EEG signals into constituent sub-bands/rhythms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It makes use of high-pass and low-pass filters for decomposing the signals into a pre- defined number of levels based on the sampling frequency [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' DWT of a single channel EEG signal v[n] is given by Vj,r = � n∈z v[n]ψ∗ j,r[n] (3) where ψj,r is the translated and scaled version of the mother wavelet ψ0,0, and defined as: ψj,r[n] = 2−(j/2)ψ0,0 � 2−j(n − r) � (4) The procedure for DWT-based decomposition follows a tree-like structure as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' At each decompo- sition level, the wavelet coefficients are down-sampled for re- moving the redundant information [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, since the sampling frequency used is 125 Hz, the decomposed sub bands are obtained as: delta (δ : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9 Hz), theta (θ : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8 Hz), alpha (α : 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8 − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='6 Hz), beta (β : 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='6 − 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 Hz), and gamma (γ :> 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 Hz), denoted by Vδ, Vθ, Vα, Vβ, and Vγ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5 (a) (b) (c) (d) (e) (f) (g) (h) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 3: Brain source localization using sLORETA at different time stamps : (a) 0ms (b) 60ms (c) 120ms (d) 180ms (e) 240ms (f) 300ms (g) 360ms (h) 420ms Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 4: Four-level DWT-based decomposition to obtain the approxi- mation and detail bands with frequency range at level j given by: � 0, 2−j−1 Fs � , and � 2−j−1 Fs, 2−j Fs � respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2) DWT-Spherical harmonics-based features: To extract the spatio-temporal features of the EEG signal and the correspond- ing DWT-based sub bands obtained above, spherical Fourier transform (SFT) is explored in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Since the human head is assumed to be spherical in shape [32], spherical Fourier basis functions have been widely employed in literary works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The decomposition of a multi-channel EEG signal V in SFD is obtained as: VSH lm = � Ω V (Ω, n) [Y m n (Ω)] dΩ (5) where V (Ω, n) denotes the potential at (Ω) = (r, θ, φ) on the scalp at time instant n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Here, r represents the radius of head, θ denotes the angle of elevation measured in downward direction from positive Z-axis (θ ∈ [0, π]), and φ denotes the azimuth angle measured in anticlockwise direction from positive X-axis, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The real-valued Y m l (Ω) of lth order and mth degree constitutes an orthonormal set of basis function, defined over spherical array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For a finite order Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5: Total potential at a channel is a contribution of each active equivalent dipole .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' system, l ∈ [0, L], and m ∈ [−l, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Therefore, (L+1)2 distinct spherical harmonics are obtained in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Since the number of sampling points S in spatial domain should be atleast (L+1)2, the highest limit of L is ≤ � (S) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, since 16 electrodes are used for recording, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', S = 16, the limit of L is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Therefore, L = 2 is used here and total 9 distinct spherical harmonics are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The corresponding features are stored in V SH nm with a dimension of 9 × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Each EEG sub band is also decomposed using spherical Fourier basis functions, and the corresponding features are obtained as V SH δlm , V SH θlm , V SH αlm, V SH βlm , and V SH γlm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 3) DWT-Head harmonics-based features: More recently, head harmonics (H2) basis functions have been proposed for more adequate representation of EEG signals based on the geometry of human head [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Since the EEG sensors placed on head form a shape between a sphere and a hemisphere, H2 basis functions are shown to be more efficient for representing x[n] sampled at F, = 125 Hz Approximation 1 Detail 1 (0-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 Hz) (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25-62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 Hz) V Approximation 2 Detail 2 (0-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='625 Hz) (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='625-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 Hz) Vp Approximation 3 Detail 3 (0-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8125 Hz) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8125-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='625 Hz) Va Approximation 4 Detail 4 (0-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9 Hz) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8125 Hz) Vs Ve+Y0 10-86 the data sampled over head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The decomposition of an EEG signal matrix V in H2 domain is given as: VH2 lm = � Ω V (Ω, n) [Hm l (Ω)] dΩ ≈ S � w=1 zwV (Ωw, n) [Hm l (Ωw)] (6) where, zw denotes the sampling weight and Ωw = (θw, φw) is the location of channel w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Here, the angle of elevation θ is in the range [0, 2π/3], as per the head geometry shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The real-valued Hm l (Ω) of lth order and mth degree constitutes an orthonormal set of basis function defined over human head .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 6: Geometry of human head with the parameters: Perimeter=40cm, radius=10cm [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The corresponding features are stored in V H2 lm with a di- mension of 9 × N, similar to that obtained in SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Each EEG sub-band is also decomposed using H2 basis functions, and the corresponding features are obtained as V H2 δlm , V H2 θlm, V H2 αlm, V H2 βlm, and V H2 γlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Proposed BiCurNet After pre-processing and feature extraction, the EEG data is given as input to the proposed BiCurNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The proposed deep learning model is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The constituent layers in the proposed model include a depth-wise separable one-dimensional convolution layer (DWSConv1D), a conv1D layer, a maxpooling (maxpool1D) layer, a customized attention module, a flatten layer, three dense layers, and an output layer for regression/prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Depth-wise separable convolution layer (DWSConv1D): The first layer of the network is a conv1D layer which performs a depth-wise separable convolution of the re- ceived input data with the kernels/filters used in this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It receives the input EEG data in the form of N × Nc matrix as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Here N denotes the number of samples in the data, and Nc is the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The convolution operation is split into two parts in this layer as depth-wise and point-wise [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Depth- wise convolution is performed with each channel sepa- rately, and point-wise convolution is performed as 1 × 1 convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It is a computationally efficient operation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' the standard convolution layer, making it suitable for lightweight scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Convolution of a filter f[n] with an input v[n] is written as: v[n] ∗ f[n] = ks−1 � i=0 v[i] · f[n − i] (7) where, ‘∗’ represents the convolution operation and ks denotes the filter width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this layer, 32 filters are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Each filter has a width ks of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In general, the zth convolution output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e, feature map of layer lr is given as [34]: clr z = σ � �bilr z + � j clr−1 j × f lr zj � � (8) where, clr z is the zth feature in the lrth layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' clr−1 j is the jth feature in the corresponding preceding layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' f lr zj represents the filter which links feature z to feature j, bilr z represents the corresponding bias vector and σ denotes the activation function, which is rectified linear unit (ReLu) in this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It is defined as: σ(t) = max(0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A stride of one is used in this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The ’He’ uniform initialization is used for kernel weights and zero initializa- tion is used for bias vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' All these parameters produce an output dimension of C1: (N − ks + 1) × 32 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' L2 regularization with a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='001 is also used in this layer to reduce over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Conv1D layer: The second layer is a conventional con- volution layer, which operates on all input channels at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This layer uses the same parameters as described in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The corresponding output dimension of this layer is given as (C1 − ks + 1) × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Max pooling layer (Maxpool1D): The convolution layer output is reduced in dimensionality by using a max pooling 1D layer, which retains the highest value of the feature in a segment with a pool size [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This layer helps in low-level feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The corresponding process can be interpreted as [34]: chx mx = max ∀b∈arm chx−1 b (9) where, arm denotes the pool area with index m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, a pool size and a stride of 2 is selected, which results in the dimension of the output as (C1 − ks + 1)/2 × 32, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Customized attention module (CAM): The feature maps of the previous layer are further transformed to intensify the more relevant features and restrain the less relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A CAM is utilized for this purpose, which uses a dense layer with 32 units and a multiply layer as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This module works on the attention phenomenon, which enhances the relevant features and diminishes the less significant features [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' An element- wise multiplication operation is performed between the outputs of the dense layer and the maxpool1D layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This produces higher values of product where both maxpool1D and dense layer outputs are high, thereby enhancing the more intense features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Similarly, the less significant features are further restrained due to low values of the product where both the layer outputs are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The input dimension of the dense layer is (C3) × 32, and a dot product operation between a 32 × 32 weight vector of the dense layer and its input results in the same output dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' + Z (Superior) + Z (Superior) 40 cm +Y fx x 10 cm Posterior Anterior Right Left Z (Inferior) Z (Inferior)7 TABLE I: Training hyper-parameters (After hypertuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Nc Nk Dr ks sr lr Bt ec 3 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='40 5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='001 15 100 Ncl: Number of convolution layers, Nk: Number of kernels/filters, Dr: Dropout rate, ks: Kernel width, sr: Stride/shift, lr: Learning rate, Bt: Batch size, ec: Number of training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Flatten layer: This layer transforms the output of CAM which is C3 × 32 to a 1D vector with dimension C4 × 1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A dropout with a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 is used after this layer to prevent the model from over-fitting [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Dense layers: Three dense layers with 8 units each are used after the flatten layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, swish activation function is used in these layers, interpreted as: f(x) = x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' swish(x) (10) Output layer: The final layer is a dense layer for re- gression, that maps the output of flatten layer to the predicted trajectory with dimension N × 1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Dense layer implements the element-wise dot product between the input and the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Linear activation function is used in this layer, given by: f(x) = x (11) The aforementioned layers and hyper-parameters are used to create the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For training, 80% of EEG signals with different durations/window lengths are taken from the recorded database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The rest 20% of the data is divided into 10% test and 10% validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The information about optimal training hyper-parameter selection and their values is provided in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The proposed network is built using Keras deep learning framework with TensorFlow version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 as backend in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In this work, data augmentation is utilized to increase the number of training examples in the data to avoid over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It makes the proposed network more robust by creating new and different training examples by which it can learn the alterations in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For this purpose, random flipping and rolling operations are used in Python Keras framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' RESULTS AND DISCUSSION In this Section, the performance evaluation of the proposed BiCurNet on the recorded EEG signals is presented w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Elaborated interpretations of the results are also presented for the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Hyper-parameters for training BiCurNet Various parameters used for training the proposed network are presented herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' For assessing the regression/prediction performance of the proposed network, 10% of the EEG signals from the recorded database are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The data from each subject is used for training, testing, and validation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', subject-dependent training is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The network is trained using a batch size of 15, epochs as 100, and Adam optimizer with a learning rate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' To curtail the statistical unreliability in computation of test loss due to small database, ten-fold cross validation is employed for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Mean square error (MSE) is used as the loss function for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Table I presents the training hyper-parameters which are selected using the KerasTuner framework in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It is an optimization framework for tuning the hyper-parameters that uses search-and-selection- based criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The final corresponding selected set of optimal hyper-parameters is listed in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Regression metric In this work, time lagged and windowed EEG signals are used to estimate the motion trajectory in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In particular, the EEG data preceding the motion by different time lags (8-240 ms) is used to train, test, and validate the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Additionally, the performance is evaluated with varying EEG window sizes (320-1600 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A 95% overlap between adjacent windows is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Pearson correlation coefficient (PCC) is utilized for analysing the performance of the proposed network w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' upper limb motion trajectory estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' PCC between true/measured (A) and predicted/estimated (P) trajectory signal with N samples is given as Π(A, P) = 1 N − 1 N � i=1 �Ai − mA σA � �Pi − mP σP � (12) where m is the mean and σ denotes standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The normalized covariance measure assumes a value between -1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Subject dependent PCC analysis The proposed model is trained and tested for each subject separately, for subject-dependent (SD) performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The PCC values averaged across all the trials and subjects, are presented in Table II with varying time lags, window sizes, and EEG features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The EEG bands are considered in spatial (V ), spherical harmonics (Vδnm), and head harmonics domains (V H2 δnm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It may be noted that the transformed domain (Vδnm and V H2 δnm) features gives PCC similar to spatial domain coun- terparts with reduced computational cost, as detailed in Section III-D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Additionally, δ band gives higher PCC values while γ band has the lowest PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This indicates the pertinence of low- frequency δ band for motion trajectory decoding using EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The best correlation is observed when Vδ, V SH δnm, and V H2 δnm are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The highest correlation achieved is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 with 240 ms advanced EEG window of 1600 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This demonstrates the feasibility of early estimation of the motion trajectory by using the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Subject-independent performance analysis To further explore the adaptability of the proposed network, subject-independent (SI) analysis is presented herein using leave-one-out scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Simultaneous comparison of SI/SD case on PCC is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The PCC values are averaged over all subjects and lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A slight decrease in PCC value may be noted in the SI case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' However, it remains within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='05 which indicates the robustness of the proposed network against the subject-variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 8 TABLE II: Pearson correlation coefficient (PCC) for different EEG segments and lags of data (Mean over subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' EEG Features 8 ms 40 ms 80 ms 160 ms 240 ms 8 ms 40 ms 80 ms 160 ms 240 ms 8 ms 40 ms 80 ms 160 ms 240 ms 8 ms 40 ms 80 ms 160 ms 240 ms V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='56 Vδ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='67 Vθ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='57 Vα 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='53 Vβ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 Vγ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 V SH nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 V SH δnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='66 V SH θnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='56 V SH αnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 V SH βnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 V SH γnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='31 V H2 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='57 V H2 δnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='65 V H2 θnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='55 V H2 αnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 V H2 βnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 V H2 γnm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='19 Vcom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='70 ■: 320 ms window, ■: 800 ms window, ■: 1200 ms window , ■: 1600 ms window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Note: Vcom : [Vδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' V SH δnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' V H2 δnm] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 7: Average PCC values w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' subject dependent (SD) and subject-independent (SI) training of the proposed network at different window sizes (320 ms to 1600 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Robustness analysis The robustness of the proposed network is analyzed herein using artifactual EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' In particular, the pre-processing did not include ICA decomposition-based artifact removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The proposed network is trained and tested using such signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Mean PCC values obtained using without artifact (WOA) and with artifact (WA) EEG signal are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A small decrease of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='06 in the PCC values may be observed with artifact case that indicates the robustness of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Trajectory estimation curves The proposed BiCurNet model is additionally evaluated herein using actual motion trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 9 illustrates the estimated and actual trajectories for subject I with window size varying between 800-1600 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 95% overlap is considered Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 8: Subject dependent average PCC values utilizing with and without artifactual EEG data for different window sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' between two adjacent windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' It may be observed from the figure that there is a considerable improvement in correlation when window size is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' This results in trajectory closer to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Ability of the proposed network to follow the trajectory pattern for all windows indicates the learning capability of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' CONCLUSION A deep learning-based paradigm for early estimation of upper limb motion trajectory using EEG signal is proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The EEG is collected while performing biceps curl movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The proposed BiCurNet model is built using a light-weight architecture with depth-wise separable con- volution layers and customized attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The input features to the model are taken in computationally more efficient spherical and head harmonics domain in addition to spatio-temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The extensive performance evaluation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 SD SI Meanoversubjects0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='6 IWOA IWA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 CC P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='1 0 320 ms 800ms 1200ms 1600ms Window size9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 9: Actual and predicted trajectories of subject 1 (Early prediction, before 40 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' of the proposed network on in-house recorded EEG signals demonstrates its effectiveness in early estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Performance evaluation includes subject (in)dependent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' the noise awareness of the proposed network is also demonstrated by using the artifactual EEG signals for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Robustness of the proposed network is demonstrated by using the artifactual EEG signals for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' The proposed network being com- putationally efficient, and noise-aware, makes it suitable for use in real-time BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Real-time implementation of the proposed network for an exosuit control is currently being explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' ACKNOWLEDGMENT This research work was supported in part by DRDO - JATC project with project number RP04191G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wolpaw and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wolpaw, “Brain-computer interfaces: something new under the sun,” Brain-computer interfaces: principles and practice, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 14, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Gong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Xing, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Cichocki, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Li, “Deep learning in EEG: Advance of the last ten-year critical period,” IEEE Transactions on Cognitive and Developmental Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Aggarwal and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Chugh, “Review of machine learning techniques for EEG based brain computer interface,” Archives of Computational Methods in Engineering, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–20, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Chowdhury, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Raza, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Meena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Dutta, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Prasad, “On- line covariate shift detection-based adaptive brain-computer interface to trigger hand exoskeleton feedback for neuro-rehabilitation,” IEEE Transactions on Cognitive and Developmental Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1070–1080, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Raza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Chowdhury, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Bhattacharyya, “Deep learning based prediction of EEG motor imagery of stroke patients’ for neuro- rehabilitation application,” in 2020 International Joint Conference on Neural Networks (IJCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Di Lillo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Arrichiello, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Di Vito, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Antonelli, “BCI-controlled assistive manipulator: developed architecture and experimental results,” IEEE Transactions on Cognitive and Developmental Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 91–104, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [7] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Miao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Yue, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Tang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', “Sensorimotor Rhythm-Brain Computer Inter- face With Audio-Cue, Motor Observation and Multisensory Feedback for Upper-Limb Stroke Rehabilitation: A Controlled Study,” Frontiers in Neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 16, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Sosnik and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Zur, “Reconstruction of hand, elbow and shoulder actual and imagined trajectories in 3d space using eeg slow cortical potentials,” Journal of neural engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 016065, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [9] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Robinson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Chester, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Smitha, “Use of mobile EEG in decoding hand movement speed and position,” IEEE Transactions on Human-Machine Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 120–129, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Jain and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kumar, “PreMovNet: Premovement EEG-Based Hand Kinematics Estimation for Grasp-and-Lift Task,” IEEE Sensors Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–4, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Saini, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Satija, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Upadhayay, “One-dimensional convo- lutional neural network architecture for classification of mental tasks from electroencephalogram,” Biomedical Signal Processing and Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 74, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 103494, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Jain and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kumar, “Subject-Independent 3D Hand Kinematics Reconstruction using Pre-Movement EEG Signals for Grasp And Lift Task,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='01932, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Hammon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Makeig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Poizner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Todorov, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' De Sa, “Predicting reaching targets from human EEG,” IEEE Signal Processing Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 69–77, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Sosnik and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Zheng, “Reconstruction of hand, elbow and shoulder actual and imagined trajectories in 3D space using EEG current source dipoles,” Journal of Neural Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 056011, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kim, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Bießmann, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Lee, “Decoding three-dimensional trajectory of executed and imagined arm movements from electroen- cephalogram signals,” IEEE Transactions on Neural Systems and Reha- bilitation Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 867–876, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Leung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Plank, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Snider, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Poizner, “EEG activity during movement planning encodes upcoming peak speed and acceler- ation and improves the accuracy in predicting hand kinematics,” IEEE journal of biomedical and health informatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 22–28, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Nakanishi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Yanagisawa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Shin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kambara, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Yoshimura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Tanaka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Fukuma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kishima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Hirata, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Koike, “Mapping ECoG channel contributions to trajectory and muscle activity prediction in human sensorimotor cortex,” Scientific reports, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–13, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Bhagat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Venkatakrishnan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Abibullaev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Artz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Yoz- batiran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Blank, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' French, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Karmonik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Grossman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' O’Malley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', “Design and optimization of an EEG-based brain ma- chine interface (BMI) to an upper-limb exoskeleton for stroke survivors,” Frontiers in neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 122, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Mahmoodi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Makkiabadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Mahmoudi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Sanei, “A new method for accurate detection of movement intention from single channel EEG for online BCI,” Computer Methods and Programs in Biomedicine Update, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 100027, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Gatti, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Atum, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Schiaffino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Jochumsen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Biurrun Manresa, “Decoding kinetic features of hand motor preparation from single- trial EEG using convolutional neural networks,” European Journal of Neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 556–570, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Jeong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Shim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kim, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Lee, “Brain-controlled robotic arm system based on multi-directional CNN-BiLSTM network using EEG signals,” IEEE Transactions on Neural Systems and Reha- bilitation Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1226–1238, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Pancholi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Giri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Jain, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kumar, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Roy, “Source aware deep learning framework for hand kinematic reconstruction using EEG signal,” IEEE Transactions on Cybernetics, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [23] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kumarasinghe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kasabov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Taylor, “Brain-inspired spiking neural networks for decoding and understanding muscle activity and kinematics from electroencephalography signals during hand move- ments,” Scientific reports, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Peirce, “PsychoPy—psychophysics software in Python,” Journal of neuroscience methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 162, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 8–13, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Delorme and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Makeig, “EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis,” Journal of neuroscience methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 134, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 9–21, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Window size: 800 ms Window size: 1200 ms Window size: 1600 ms True Predicted 10 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='8 Angle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='4 PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='52 EO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='2 250 500 75010001250150017502000 0 250500 75010001250150017502000 75010001250150017502000 Samples Samples Samples10 [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Maddirala and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Veluvolu, “ICA with CWT and k-means for Eye-blink Artifact Removal from Fewer Channel EEG,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Mazziotta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Toga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Evans, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Fox, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Lancaster, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Zilles, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Woods, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Paus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Simpson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Pike et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', “A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM),” Philosophical Transactions of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Series B: Biological Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 356, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1412, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1293–1322, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Gramfort, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Papadopoulo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Olivi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Clerc, “OpenMEEG: opensource software for quasistatic bioelectromagnetics,” Biomedical engineering online, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–20, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Pascual-Marqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=', “Standardized low-resolution brain elec- tromagnetic tomography (sLORETA): technical details,” Methods Find Exp Clin Pharmacol, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Suppl D, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 5–12, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Saini, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Satija, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Upadhayay, “Discriminatory features based on wavelet energy for effective analysis of electroencephalogram during mental tasks,” Circuits, Systems, and Signal Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE transactions on pattern analysis and machine intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 674–693, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Giri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kumar, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Gandhi, “Brain source localization in head harmonics domain,” IEEE Transactions on Instrumentation and Measurement, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Woo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Lee, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Kweon, “Cbam: Convolutional block attention module,” in Proceedings of the European conference on computer vision (ECCV), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 3–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' P´erez-Enciso and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Zingaretti, “A guide on deep learning for complex trait genomic prediction,” Genes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 553, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [35] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Jiao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Li, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Xu, “Deep convolutional neural networks for mental load classification based on EEG data,” Pattern Recognition, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 76, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 582–595, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Wu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' Gu, “Towards dropout training for convolutional neural networks,” Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 71, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} +page_content=' 1–10, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfjAfh/content/2301.03965v1.pdf'} diff --git a/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/2301.12142v1.pdf.txt b/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/2301.12142v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ed1389ff02e21ad45e3f68bcf75ba236728129a --- /dev/null +++ b/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/2301.12142v1.pdf.txt @@ -0,0 +1,1631 @@ +arXiv:2301.12142v1 [math.DG] 28 Jan 2023 +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +HUI ZHANG AND ZAILI YAN +Abstract. We consider the moment map m : PVn → iu(n) for the action of GL(n) on Vn = ⊗2(Cn)∗ ⊗ Cn, +and study the critical points of the functional Fn = ∥m∥2 : PVn → R. Firstly, we prove that [µ] ∈ PVn is a +critical point if and only if Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ), where m([µ]) = +Mµ +∥µ∥2 . Then we +show that any algebra µ admits a Nikolayevsky derivation φµ which is unique up to automorphism, and if +moreover, [µ] is a critical point of Fn, then φµ = − 1 +cµ Dµ. Secondly, we characterize the maxima and minima +of the functional Fn : An → R, where An denotes the projectivization of the algebraic varieties of all n- +dimensional associative algebras. Furthermore, for an arbitrary critical point [µ] of Fn : An → R, we also +obtain a description of the algebraic structure of [µ]. Finally, we classify the critical points of Fn : An → R +for n = 2, 3, respectively. +1. Introduction +Lauret has studied the moment map for the variety of Lie algebras and obtained many remarkable +results in [7], which turned out to be very important in proving that every Einstein solvmanifold is +standard ([9]) and in the characterization of solitons ([1, 10]). Apart from the Lie algebras, the study +of the moment map in other classes of algebras was also initiated by Lauret, see [11] for more details. +Motivated by this, the authors have recently extended the study of the moment map to the variety of +3-Lie algebras (see [17]). +In this paper, we study the moment map for the variety of associative algebras. Let GL(n) be the +complex reductive Lie group acting naturally on the complex vector space Vn = ⊗2(Cn)∗ ⊗ Cn, i.e., the +space of all n-dimensional complex algebras. The usual Hermitian inner product on Cn naturally induces +an U(n)-invariant Hermitian inner product on Vn, which is denoted by ⟨·, ·⟩. Since gl(n) = u(n) + iu(n), +we may define a function as follows +m : PVn → iu(n), +(m([µ]), A) = (dρµ)eA +∥µ∥2 +, +0 � µ ∈ Vn, A ∈ iu(n), +where (·, ·) is an Ad(U(n))-invariant real inner product on iu(n), and ρµ : GL(n) → R is defined by +ρµ(g) = ⟨g.µ, g.µ⟩. The function m is the moment map from symplectic geometry, corresponding to the +Hamiltonian action U(n) of Vn on the symplectic manifold PVn (see [4, 12]). In this paper, we study the +critical points of the functional Fn = ∥m∥2 : PVn → R, with an emphasis on the critical points that lie in +the projectivization of the algebraic variety of all n-dimensional associative algebras An. +2010 Mathematics Subject Classification. 14L30, 17B30, 53D20. +Key words and phrases. Moment map; Variety of associative algebras; Critical point. +This work is supported by NSFC (Nos. 11701300, 11626134) and K.C. Wong Magna Fund in Ningbo University. +1 + +2 +HUI ZHANG AND ZAILI YAN +The paper is organized as follows: In Sect. 2, we recall some basic concepts and results of complex +associative algebras. +In Sect. 3, we first give the explicit expression of the moment map m : PVn → iu(n) in terms of Mµ, +that is, m([µ]) = Mµ +∥µ∥2 for any [µ] ∈ PVn. Then we show that [µ] ∈ PVn is a critical point of Fn if and only +if Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ) (Thm. 3.3). +In Sect. 4, we first show that any algebra µ ∈ Vn admits a Nikolayevsky derivation φµ which is unique +up to automorphism, the eigenvalues of φµ are necessarily rational, and moreover, φµ = − 1 +cµ Dµ if [µ] +is a critical point of Fn (Thm. 4.1). Then we study the extremal points of Fn : An → R, proving that +the minimum value is attained at semisimple associative algebras (Thm. 4.6), and the maximum value at +the direct sum of a two-dimensional commutative associative algebra with the trivial algebra (Thm. 4.9). +In the context of Lie algebras ([7]), Lauret proved that any µ for which there exists [λ] ∈ GL(n).[µ] +such that all eigenvalues of Mλ are negative, must be semisimple, and we prove that this result also +holds for associative algebras (Remark 4.7). Besides, the structure for an arbitrary critical point [µ] of +Fn : An → R is discussed (Thm. 4.10 and Thm. 4.12). +In Sect. 5, we classify the critical points of Fn : An → R for n = 2, 3. It shows that every two- +dimensional associative algebra is isomorphic to a critical point of F2; and there exists only one three- +dimensional associative algebra which is not isomorphic to any critical point of F3. Finally, based on the +discussion in previous sections, we collect some natural and interesting questions. +2. Preliminaries +In this section, we recall some basic definitions and results of associative algebras. The ambient field +is always assumed to be the complex number field C unless otherwise stated. +Definition 2.1. A vector space A over C with a bilinear operation A × A → A, denoted by (x, y) �→ xy, +is called an associative algebra, if +x(yz) = (xy)z +for all x, y, z ∈ A. +A derivation of an associative algebra A is a linear transformation D : A → A satisfying +D(xy) = (Dx)y + x(Dy), +for x, y ∈ A. It is easy to see that the set of all derivations of A is a Lie algebra, which is denoted by +Der(A). A vector subspace I of A is called an ideal if AI, IA ⊂ I. + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +3 +Definition 2.2. Let A be an associative algebra. The center of A is defined by C(A) = {x ∈ A : xy = +yx, ∀y ∈ A}. The annihilator of A is defined by ann(A) = {x ∈ A : xy = yx = 0, ∀y ∈ A}. +Clearly, C(A) is a subalgebra of A, and ann(A) is an ideal of A. +Definition 2.3. Let I be an ideal of an associative algebra. Then I is called nilpotent, if Ik = 0 for some +integer k ≥ 1, where Ik = I· · ·I· · ·I +���������� +k +. +If I, J are any two nilpotent ideals of an associative algebra A, then I + J is also a nilpotent ideal. So +the maximum nilpotent ideal of A is unique, which is called the radical and denoted by N(A). +Remark 2.4. Note that N(A) coincides with the Jacobson radical of A since A is an associative algebra +over C. Moreover, N(A) = {x ∈ A : xy, yx are nilpotent elements for any y ∈ A}. +Definition 2.5. Let A be an associative algebra. If A has no ideals except itself and 0, we call A simple. +Denote by Mn(C) the set of all n × n complex square matrices, which is clearly an associative algebra +with respect to the usual matrix addition and multiplication. In fact, Mn(C) is a simple associative algebra +for any n ≥ 1. Moreover, it follows from Wedderburn-Artin theorem that any finite-dimensional simple +associative algebra over C is isomorphic to Mn(C) for some integer n ≥ 1 ([15]). +An associative algebra A is called semisimple if its radical N(A) is zero. The following theorem is +well known. +Theorem 2.6 ([15]). An associative algebra over C is semisimple if and only if it is a direct sum of simple +ideals. That is, a semisimple associative algebra is isomorphic to Mn1(C) × Mn2(C) × · · · × Mns(C) for +some positive integers n1, n2, · · · , ns. +3. The moment map for complex algebras +Let Cn be the n-dimensional complex vector space and Vn = ⊗2(Cn)∗ ⊗ Cn be the space of all complex +n-dimensional algebras. The natural action of GL(n) = GL(Cn) on Vn is given by +g.µ(X, Y) = gµ(g−1X, g−1Y), +g ∈ GL(n), X, Y ∈ Cn. +(3.1) +Clearly, GL(n).µ is precisely the isomorphism class of µ. Note that +lim +t→∞ gt.µ = 0, +gt = tI ⊂ GL(n), t > 0, +we see that 0 lies in the boundary of the orbit GL(n).µ for each µ ∈ Vn. By differentiating (3.1), we obtain +the natural action gl(n) on Vn, i.e., +A.µ(X, Y) = Aµ(X, Y) − µ(AX, Y) − µ(X, AY), +A ∈ gl(n), µ ∈ Vn. +(3.2) + +4 +HUI ZHANG AND ZAILI YAN +It follows that A.µ = 0 if and only if A ∈ Der(µ), where Der(µ) denotes the derivation algebra of µ. +Note that the usual Hermitian inner product on Cn gives an U(n)-invariant Hermitian inner product on +Vn as follows +⟨µ, λ⟩ = +� +i, j,k +⟨µ(Xi, X j), Xk⟩⟨λ(Xi, X j), Xk⟩, +µ, λ ∈ Vn, +(3.3) +where {X1, X2, · · · , Xn} is an arbitrary orthonormal basis of Cn. Let u(n) denote the Lie algebra of U(n), +then it is easy to see that gl(n) = u(n) + iu(n) decomposes into skew-Hermitian and Hermitian transfor- +mations of Vn, respectively. Moreover, there is an Ad(U(n))-invariant Hermitian inner product on gl(n) +given by +(A, B) = tr AB∗, A, B ∈ gl(n). +(3.4) +The moment map from symplectic geometry, corresponding to the Hamiltonian action of U(n) on the +symplectic manifold PVn, is defined as follows +m : PVn → iu(n), +(m([µ]), A) = (dρµ)eA +∥µ∥2 +, +0 � µ ∈ Vn, A ∈ iu(n), +(3.5) +where ρµ : GL(n) → R is given by ρµ(g) = ⟨g.µ, g.µ⟩. Clearly, (dρµ)eA = ⟨A.µ, µ⟩ + ⟨µ, A.µ⟩ = 2⟨A.µ, µ⟩ +for any A ∈ iu(n). The square norm of the moment map is denoted by +Fn : PVn → R, +(3.6) +where Fn([µ]) = ∥m([µ])∥2 = (m([µ]), m([µ])) for any [µ] ∈ PVn. +In order to express the moment map m explicitly, we define Mµ ∈ iu(n) as follows +Mµ = 2 +� +i +Lµ +Xi(Lµ +Xi)∗ − 2 +� +i +(Lµ +Xi)∗Lµ +Xi − 2 +� +i +(Rµ +Xi)∗Rµ +Xi, +(3.7) +where the left and right multiplication Lµ +X, Rµ +X : Cn → Cn by X of the algebra µ, are given by Lµ +X(Y) = +µ(X, Y) and Rµ +X(Y) = µ(Y, X) for all Y ∈ Cn, respectively. It is not hard to prove that +⟨MµX, Y⟩ =2 +� +i, j +⟨µ(Xi, X j), X⟩⟨µ(Xi, X j), Y⟩ − 2 +� +i, j +⟨µ(Xi, X), X j⟩⟨µ(Xi, Y), X j⟩ +− 2 +� +i, j +⟨µ(X, Xi), X j⟩⟨µ(Y, Xi), X j⟩ +(3.8) +for X, Y ∈ Cn. Note that if the algebra µ is commutative or anticommutative, then the second and third +term of (3.8) are the same, and in this case, Mµ coincides with [7]. +Lemma 3.1. For any µ ∈ Vn, we have (Mµ, A) = 2⟨µ, A.µ⟩, ∀A ∈ gl(n) = u(n) + iu(n). In particular, +m([µ]) = Mµ +∥µ∥2 for any 0 � µ ∈ Vn + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +5 +Proof. For any A ∈ gl(n), we have (A, Mµ) = tr AM∗ +µ = tr AMµ = tr MµA, and +tr MµA = 2 tr +� +i +Lµ +Xi(Lµ +Xi)∗A +�������������������������������������� +I +− 2 tr +� +i +((Lµ +Xi)∗Lµ +Xi + (Rµ +Xi)∗Rµ +Xi)A +�������������������������������������������������������������������������� +II +=: I − II. +Note that +I =2 +� +i +tr Lµ +Xi(Lµ +Xi)∗A +=2 +� +i +tr(Lµ +Xi)∗ALµ +Xi +=2 +� +i, j +⟨(Lµ +Xi)∗ALµ +Xi(X j), X j⟩ +=2 +� +i, j +⟨Aµ(Xi, X j), µ(Xi, X j)⟩, +and +II =2 tr +� +i +((Lµ +Xi)∗Lµ +Xi + (Rµ +Xi)∗Rµ +Xi)A +=2 +� +i, j +⟨((Lµ +Xi)∗Lµ +Xi + (Rµ +Xi)∗Rµ +Xi)AX j, X j⟩ +=2 +� +i, j +⟨µ(Xi, AX j), µ(Xi, X j)⟩ − 2 +� +i, j +⟨µ(AX j, Xi), µ(X j, Xi)⟩ +=2 +� +i, j +⟨µ(AXi, X j) + µ(Xi, AX j), µ(Xi, X j)⟩. +By (3.2), it follows that (A, Mµ) = tr MµA = 2⟨A.µ, µ⟩, so (Mµ, A) = 2⟨µ, A.µ⟩ for any A ∈ gl(n). This +proves the first statement. For A ∈ iu(n), we have ⟨A.µ, µ⟩ = ⟨µ, A.µ⟩. By (3.5), we conclude that +m([µ]) = Mµ +∥µ∥2 for any 0 � µ ∈ Vn. This completes proof Lemma 3.1 +□ +Corollary 3.2. For any µ ∈ Vn, then +(i) tr MµD = 0 for any D ∈ Der(µ); +(ii) tr Mµ[A, A∗] ≥ 0 for any A ∈ Der(µ), and equality holds if and only if A∗ ∈ Der(µ). +Proof. For (i), it follows from Lemma 3.1 that tr MµD = 2⟨D.µ, µ⟩. For (ii), it follows from that tr Mµ[A, A∗] = +2⟨[A, A∗].µ, µ⟩ = 2⟨A∗.µ, A∗.µ⟩ ≥ 0, ∀A ∈ Der(µ), and the fact A∗.µ = 0 if and only if A∗ ∈ Der(µ). +□ +Theorem 3.3. The moment map m : PVn → iu(n), the functional square norm of the moment map +Fn = ∥m∥2 : PVn → R and the gradient of Fn are, respectively, given by +Fn([µ]) = +tr M2 +µ +∥µ∥4 , +grad(Fn)[µ] = 8π∗(Mµ).µ +∥µ∥4 +, +[µ] ∈ PVn, +(3.9) + +6 +HUI ZHANG AND ZAILI YAN +where π∗ denotes the derivative of π : Vn\{0} → PVn, the canonical projection. Moreover, the following +statements are equivalent: +(i) [µ] ∈ PVn is a critical point of Fn. +(ii) [µ] ∈ PVn is a critical point of Fn|GL(n).[µ]. +(iii) Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ). +Proof. By (3.6) and Lemma 3.1, we have Fn([µ]) = +tr M2 +µ +∥µ∥4 for any [µ] ∈ PVn. To prove the second one, we +only need to compute the gradient of Fn : Vn \ {0} → R, Fn(µ) = +tr M2 +µ +∥µ∥4 , and then to project it via π∗. If +µ, λ ∈ Vn with µ � 0, then +Re⟨grad(Fn)µ, λ⟩ = d +dt +�����t=0 +Fn(µ + tλ) = d +dt +�����t=0 +1 +∥µ + tλ∥4 (Mµ+tλ, Mµ+tλ) += − 4 Re⟨Fn(µ) +∥µ∥2 µ, λ⟩ + +2 +∥µ∥4 ( d +dt +�����t=0 +Mµ+tλ, Mµ) +We claim that ( d +dt +���t=0 Mµ+tλ, A) = 4 Re⟨A.µ, λ⟩ for any A ∈ iu(n). Indeed, by Lemma 3.1, ( d +dt +���t=0 Mµ+tλ, A) = +d +dt +���t=0 (Mµ+tλ, A) = 2 d +dt +���t=0 ⟨µ + tλ, A.(µ + tλ)⟩ = 2⟨λ, A.µ⟩ + 2⟨µ, A.λ⟩ = 4 Re⟨A.µ, λ⟩ for any A ∈ iu(n). +It follows that grad(Fn)µ = −4 Fn(µ) +∥µ∥2 µ + 8(Mµ).µ +∥µ∥4 , and consequentely +grad(Fn)[µ] = 8π∗(Mµ).µ +∥µ∥4 +. +Thus the first part of the theorem is proved, and the following is to prove the equivalence among the +statements (i), (ii) and (iii). +(i) ⇔ (ii) : The equivalence follows from that grad(Fn) is tangent to the GL(n)-orbits. Indeed +grad(Fn)[µ] = 8π∗(Mµ).µ +∥µ∥4 += +8 +∥µ∥4 π∗( d +dt +�����t=0 +etMµ.µ) = +8 +∥µ∥4 +d +dt +�����t=0 +etMµ.[µ] ∈ T[µ](GL(n).[µ]). +(iii) ⇒ (i) : By (3.2), we know that I.µ = −µ, and (Mµ).µ = (cµI + Dµ).µ = −cµµ. It follows that +grad(Fn)[µ] = 0. +(i) ⇒ (iii) : Since grad(Fn)[µ] = 0, then (Mµ).µ ∈ ker π∗µ = Cµ. So Mµ = cI + D for some c ∈ C and +D ∈ Der(µ). Clearly [D, D∗] = [Mµ − cI, Mµ − ¯cI] = 0, we conclude by Corollary 3.2 that D∗ is also a +derivation of µ. In particular, (c − ¯c)I = (D∗ − D) ∈ Der(µ), thus c = ¯c ∈ R. +□ +Remark 3.4. Let [µ] be a critical point of Fn and [λ] be a critical point of Fm, then [µ ⊕ cλ] is a critical +point of Fn+m for a suitable c ∈ C. Indeed, assume that Mµ = cµI + Dµ for some cµ ∈ R, Dµ ∈ Der(µ), +and Mλ = cλI +Dλ for some cλ ∈ R, Dλ ∈ Der(λ). Noting that Mtλ = |t|2Mλ for any t ∈ C, we can choose +t0 such that cµ = |t0|2cλ, then it follows that [µ ⊕ t0λ] is a critical point of Fn+m. +In the frame of algebras, a remarkable result due to Ness can be stated as follows +Theorem 3.5 ([12]). If [µ] is a critical point of the functional Fn : PVn → R, then + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +7 +(i) Fn|GL(n).[µ] attains its minimum value at [µ]. +(ii) [λ] ∈ GL(n).[µ] is a critical point of Fn if and only if [λ] ∈ U(n).[µ]. +In fact, the above theorem implies that up to U(n)-orbit, GL(n).[µ] contains at most one critical point +for each [µ] ∈ PVn. +Lemma 3.6. Let [µ] ∈ PVn be a critical point of Fn with Mµ = cµI+Dµ for some cµ ∈ R and Dµ ∈ Der(µ). +Then we have +(i) cµ = +tr M2 +µ +tr Mµ = − 1 +2 +tr M2 +µ +∥µ∥2 < 0. +(ii) If tr Dµ � 0, then cµ = − +tr D2 +µ +tr Dµ and tr Dµ > 0. +Proof. Since Mµ = cµI + Dµ, by Lemma 3.1 and Corollary 3.2 we have +tr Mµ = (Mµ, I) = 2⟨µ, I.µ⟩ = −2∥µ∥2 < 0, +tr M2 +µ = tr Mµ(cµI + Dµ) = cµ tr Mµ. +So cµ = +tr M2 +µ +tr Mµ = − 1 +2 +tr M2 +µ +∥µ∥2 < 0. If tr Dµ � 0, then +0 = tr MµDµ = cµ tr Dµ + tr D2 +µ. +So cµ = − +tr D2 +µ +tr Dµ and tr Dµ > 0. +□ +Remark 3.7. In fact, tr Dµ = 0 if and only if Dµ = 0. Indeed, it follows from that 0 = tr MµDµ = +cµ tr Dµ + tr D2 +µ and Dµ is Hermitian. +4. The critical points of the variety of associative algebras +The space An of all n-dimensional associative algebras is an algebraic set since it is given by polyno- +mial conditions. Denote by An the projective algebraic variety obtained by projectivization of An . Note +that An is GL(n)-invariant, then by Theorem 3.3, the critical points of Fn : An → R are precisely the +critical points of Fn : PVn → R that lie in An. +4.1. The Nikolayevsky derivation and the rationality. A derivation of φ of an algebra (µ, Cn) is called +a Nikolayevsky derivation, if it is semisimple with all eigenvalues real, and tr φψ = tr ψ for any ψ ∈ +Der(µ). This notion is motivated by [14]. +Theorem 4.1. Let (µ, Cn) be an arbitrary algebra. Then +(1) (µ, Cn) admits a Nikolayevsky derivation φµ. +(2) The Nikolayevsky derivation φµ is determined up to automorphism of µ. +(3) All eigenvalues of φµ are rational numbers. + +8 +HUI ZHANG AND ZAILI YAN +If moreover, [µ] is a critical point of Fn : PVn → R with Mµ = cµI+Dµ for some cµ ∈ R and Dµ ∈ Der(µ), +then − 1 +cµ Dµ is the Nikolayevsky derivation of µ. +Proof. (1) The complex Lie algebra Der(µ) is algebraic. Let Der(µ) = s ⊕ t ⊕ n be its Levi-Mal’cev +decomposition, where s is semisimple, t ⊕ n is the radical of Der(µ), n is the set of all nilpotent elements +in t ⊕ n (and is the nilradical of t ⊕ n), t is an abelian subalgebra consisting of semisimple elements, and +[s, t] = 0. Define the bilinear form B on Der(µ) by +B(ψ1, ψ2) := tr ψ1ψ2, +∀ψ1, ψ2 ∈ Der(µ). +Then, in general, B is degenerate, and Ker B = n. Since s is semisimple, then B(s, t) = B([s, s], t) = +B(s, [s, t]) = 0. Clearly, B is nondegenerate on t. Since t is reductive, we have t = a + ia, where a consists +of semisimple elements with all eigenvalues real. It follows that there exists a unique element φ ∈ a such +that B(φ, ψ) = tr ψ for any ψ ∈ t. Thus tr φψ = tr ψ for any ψ ∈ Der(µ). +(2) The subalgebra s ⊕ t is a maximal fully reducible subalgebra of Der(µ). Since the maximal fully +reducible subalgebras of Der(µ) are conjugate by inner automorphism of Der(µ) (which corresponds to +an automorphism of µ), and then the center t of s ⊕ t, is defined uniquely, up to automorphism. So the +Nikolayevsky derivation is determined up to automorphism of µ. +(3) The case φµ = 0 is trivial. In the following, we assume that φµ is nonzero. Note that φµ is +simisimple with all eigenvalues real, we have the following decomposition +Cn = l1 ⊕ l2 ⊕ · · · ⊕ lr, +where li = {X ∈ Cn|φµX = ciX} are eigenspaces of φµ corresponding to eigenvalues c1 < c2 < · · · < cr ∈ +R, respectively. Set di = dim li ∈ N, 1 ≤ i ≤ r. Since φµ is a derivation, we have the following relations +µ(li, lj) ⊂ lk +if ci + cj = ck, +for all 1 ≤ i, j, k ≤ r. Conversely, if we define a linear transformation ψ : Cn → Cn by ψ|li = aiIdli, +where a1, a2, · · · , ar ∈ R satisfy ai + aj = ak for all 1 ≤ i, j, k ≤ r such that ci + cj = ck, then ψ is +a derivation of µ. Clearly, all such derivations form a real vector space, which can be identified with +W := {(w1, w2, · · · , wr) ∈ Rr|wi + w j = wk if ci + cj = ck}. We endow Rr with the usual inner product, i.e. +⟨x, y⟩ = +� +i +xiyi, +(4.1) +for any x = (x1, x2, · · · , xr), y = (y1, y2, · · · , yr) ∈ Rr. +For any derivation ψ ∈ W, we have +tr(φµ − I)ψ = tr φµψ − tr ψ = 0. + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +9 +Then we see that (d1(c1 − 1), d2(c2 − 1), · · · , dr(cr − 1)) ⊥ W relative to (4.1). Put F := W⊥, then by +definition we have +F = span1≤i, j,k≤r{ei + ej − ek : ci + cj = ck}, +where ei belongs to Rr having 1 in the i-th position and 0 elsewhere. Let {ei1 +ej1 −ek1, · · · , eis +ejs −eks} +be a basis of F, then +(d1(c1 − 1), d2(c2 − 1), · · · , dr(cr − 1)) = +s +� +p=1 +bp(eip + ejp − ekp), +(4.2) +for some b1, b2, · · · , bs ∈ R. Put +E = + +ei1 + ej1 − ek1 +ei2 + ej2 − ek2 +... +eis + ejs − eks + +∈ Zs×r, +then EET ∈ GL(s, Z), and (EET)−1 ∈ GL(s, Q). By (4.2) and the definition of E, we have + +d1(c1 − 1) +d2(c2 − 1) +... +dr(cr − 1) + +r×1 += ET + +b1 +b2 +... +bs + +s×1 +, E + +c1 +c2 +... +cr + +r×1 += + +0 +0 +... +0 + +s×1 +, +E + +1 +1 +... +1 + +r×1 += + +1 +1 +... +1 + +s×1 +. +By the left multiplication of E on (4.2), we have + +0 +0 +... +0 + +s×1 +− + +1 +1 +... +1 + +s×1 += ED−1ET + +b1 +b2 +... +bs + +s×1 +, +where D = diag(d1, d2, · · · , dr). It is easy to see that (ED−1ET) ∈ GL(s, Q). Consequently +D + +c1 − 1 +c2 − 1 +... +cr − 1 + +r×1 += −ET(ED−1ET)−1 + +1 +1 +... +1 + +s×1 +, +and + +c1 +c2 +... +cr + +r×1 += + +1 +1 +... +1 + +r×1 +− D−1ET(ED−1ET)−1 + +1 +1 +... +1 + +s×1 +∈ Qr. +So all eigenvalues of φµ are rational. +For the last statement, by Corollary 3.2 we know that 0 = tr Mµψ = cµ tr ψ+tr Dµψ for any ψ ∈ Der(µ). +Since Dµ is Hermitian, we conclude that − 1 +cµ Dµ is the Nikolayevsky derivation of µ. +□ +By Theorem 4.1, it is easy to obtain the following theorem. + +10 +HUI ZHANG AND ZAILI YAN +Theorem 4.2. Let [µ] ∈ PVn be a critical point of Fn : PVn → R with Mµ = cµI + Dµ for some cµ ∈ R +and Dµ ∈ Der(µ). Then there exists a constant c > 0 such that the eigenvalues of cDµ are integers prime +to each other, say k1 < k2 < · · · < kr ∈ Z with multiplicities d1, d2, · · · , dr ∈ N. +Definition 4.3. The data set (k1 < k2 < · · · < kr; d1, d2, · · · , dr) in Theorem 4.2 is called the type of the +critical point [µ]. +Proposition 4.4. Let [µ] ∈ PVn be a critical point of Fn with type α = (k1 < k2 < · · · < kr; d1, d2, · · · , dr). +Then we have +(i) If α = (0; n), then Fn([µ]) = 4 +n. +(ii) If α � (0; n), then Fn([µ]) = 4 +� +n − (k1d1+k2d2+···+krdr)2 +k2 +1d1+k2 +2d2+···+k2r dr +�−1 +. +Proof. We suppose that Mµ = cµI + Dµ, ∥µ∥ = 1. Since tr Mµ = −2⟨µ, µ⟩ = −2, then +tr M2 +µ = tr Mµ(cµI + Dµ) = cµ tr Mµ = −2cµ, +and Fn([µ]) = tr Mµ2 +∥µ∥4 = tr Mµ2 = −2cµ. +For (i), we have Dµ = 0, so Mµ = cµI and cµn = tr Mµ = −2. Thus cµ = − 2 +n. Fn([µ]) = −2cµ = 4 +n. +For (ii), we have Dµ � 0, and cµ = − +tr D2 +µ +tr Dµ by Lemma 3.6 and Remark 3.7. Note that +Fn([µ]) = tr Mµ2 = tr(cµI + Dµ)2 = c2 +µn + cµ tr Dµ = 1 +4Fn([µ])2n − 1 +2Fn([µ]) tr Dµ, +so we have +1 +Fn([µ]) = 1 +4n − +1 +2Fn([µ]) tr(Dµ) = 1 +4n + 1 +4cµ +tr Dµ = 1 +4 +n − (tr Dµ)2 +tr D2µ + . +It follows that Fn([µ]) = 4 +� +n − (k1d1+k2d2+···+krdr)2 +k2 +1d1+k2 +2d2+···+k2r dr +�−1 +. +□ +4.2. The minima of Fn : An → R. +Lemma 4.5. Assume [µ] ∈ PVn, then [µ] is a critical point of Fn : PVn → R with type (0; n) if and only +if Fn([µ]) = 4 +n. Moreover, 4 +n is the minimum value of Fn : PVn → R. +Proof. For any 0 � µ ∈ Vn, we use x1, x2, · · · , xn ∈ R denote the eigenvalues of Mµ. Note that tr Mµ = +−2∥µ∥2, then we have +Fn([µ]) = tr Mµ2 +∥µ∥4 += 4 tr Mµ2 +(tr Mµ)2 = 4 +(x2 +1 + x2 +2 + · · · + x2 +n) +(x1 + x2 + · · · + xn)2 . +It is easy to see that Fn([µ]) ≥ 4 +n with equality holds if and only if x1 = x2 = · · · = xn. So [µ] is a critical +point of Fn : PVn → R with type (0; n) if only if Mµ is a constant multiple of I, if and only Fn attains its +minimum value 4 +n at [µ]. +□ + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +11 +Theorem 4.6. The functional Fn : An → R attains its minimum value at a point [λ] ∈ GL(n).[µ] if and +only if µ is a semisimple associative algebra. In such a case, Fn([λ]) = 4 +n. +Proof. Consider the simple associative algebra Mm(C) for an integer m > 0. We endow Mm(C) with the +following Hermitian inner product +⟨A, B⟩ := tr AB∗, A, B ∈ Mm(C). +(4.3) +Then {Eij : 1 ≤ i, j ≤ m} is an orthonormal basis, where Eij denote the matrices having 1 in the (i, j)- +position and 0 elsewhere. Set ν := (Mm(C), ⟨·, ·⟩). Clearly +(Lν +A)∗ = LA∗, +(Rν +A)∗ = RA∗ +for any A ∈ Mm(C). Thus by (3.7), we have +Mν = 2 +� +ij +Lν +Eij(Lν +Eij)∗ − 2 +� +ij +(Lν +Eij)∗Lν +Eij − 2 +� +ij +(Rν +Eij)∗Rν +Eij += 2 +� +ij +Lν +EijLν +Eji − 2 +� +ij +Lν +EjiLν +Eij − 2 +� +ij +Rν +EjiRν +Eij += 2 +� +ij +Lν +EijEji − 2 +� +ij +Lν +EjiEij − 2 +� +ij +Rν +EijEji += 2m +� +i +Lν +Eii − 2m +� +i +Lν +Eii − 2m +� +i +Rν +Eii += 2mLν +I − 2mLν +I − 2mRν +I += 2mIm2 − 2mIm2 − 2mIm2 += −2mIm2. +So [ν] is a critical point of type (0; m2). Since µ is a complex semisimple associative algebra, by Theo- +rem 2.6, µ is isomorphic to Mn1(C) × Mn2(C) × · · · × Mns(C) for some positive integers n1, n2, · · · , ns. It +follows from Remark 3.4 that there exists a point [λ] ∈ GL(n).[µ] such that [λ] is a critical point of type +(0; n). So the functional Fn : An → R attains its minimum value at [λ], and Fn([λ]) = 4 +n by Lemma 4.5. +Conversely, assume that Fn : An → R attains its minimum value at a point [λ] ∈ GL(n).[µ]. The first +part of the proof implies that Mλ = cλI with cλ < 0. To prove µ is semisimple, it suffices to show that +L = (λ, Cn) is semisimple. Consider the following orthogonal decompositions: (i) L = H ⊕ N, where +N is the radical of λ; (ii) N = V ⊕ Z, where Z = {A ∈ N : λ(A, N) = λ(N, A) = 0} is the annihilator +of N. Clearly, Z is an ideal of L. We have L = H ⊕ V ⊕ Z. Suppose that Z � 0. Let {Hi}, {Vi}, {Zi} be +an orthonormal basis of H, V, and Z, respectively. Put {Xi} = {Hi} ∪ {Vi} ∪ {Zi}. For any 0 � Z ∈ Z, by + +12 +HUI ZHANG AND ZAILI YAN +hypothesis we have +0 > ⟨MλZ, Z⟩ =2 +� +ij +|⟨λ(Xi, X j), Z⟩|2 − 2 +� +ij +|⟨λ(Z, Xi), X j⟩|2 − 2 +� +ij +|⟨λ(Xi, Z), X j⟩|2 +=2 +� +ij +� +|⟨λ(Zi, H j), Z⟩|2 + |⟨λ(Hi, Z j), Z⟩|2� ++ α(Z) +− 2 +� +ij +|⟨λ(Z, Hi), Z j⟩|2 − 2 +� +ij +|⟨λ(Hi, Z), Z j⟩|2, +where α(Z) = 2 � +ij |⟨λ(Yi, Y j), Z⟩|2 ≥ 0, {Yi} = {Hi} ∪ {Vi}. This implies +0 > +� +k +⟨MλZk, Zk⟩ = +� +k +α(Zk) ≥ 0, +which is a contradiction. So Z = 0, and consequently, N = 0. Therefore L is a semisimple associative +algebra. +This completes the proof of theorem. +□ +Remark 4.7. In fact, by the proof of Theorem 4.6, we know that if [µ] ∈ An for which there exists +[λ] ∈ GL(n).[µ] such that Mλ is negative definite, then µ is a semisimple associative algebra. +4.3. The maxima of Fn : An → R. We say that an algebra λ degenerates to µ, write as λ → µ if +µ ∈ GL(n).λ, the closure of GL(n).λ with respect to the usual topology of Vn. The degeneration λ → µ +is called direct degeneration if there are no nontrivial chains: λ → ν → µ. The degeneration level of an +algebra is the maximum length of chain of direct degenerations. +Theorem 4.8 ([3]). An n-dimensional associative algebra is of degeneration level one if and only if it is +isomorphic to one of the following +(1) µl: µl(X1, Xi) = Xi, i = 1, · · · , n; +(2) µr: µr(Xi, X1) = Xi, i = 1, · · · , n; +(3) µca: µs(X1, X1) = X2, +where {X1, · · · , Xn} is a basis. +Theorem 4.9. The functional Fn : An → R attains its maximal value at a point [µ] ∈ Ln, n ≥ 3 if and +only if µ is isomorphic to the commutative associative algebra µca. In such a case, Fn([µ]) = 20. +Proof. Assume that Fn : An → R attains its maximal value at a point [µ] ∈ An, n ≥ 3. By Theorem 3.3, +we know that [µ] is also a critical of Fn : PVn → R. Then it follows Theorem 3.5 that Fn|GL(n).[µ] also +attains its minimum value at a point [µ] , consequently Fn|GL.[µ] is a constant, so +GL(n).[µ] = U(n).[µ] +(4.4) + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +13 +The relation (4.4) implies that the only non-trivial degeneration of µ is 0 ([8, Theorem 5.1] ), conse- +quently the degeneration level of µ is 1. +It is easy to see that the critical points [µl], [µr] are both of type (0 < 1; 1, n − 1), and [µca] is of type +(3 < 5 < 6; 1, n − 2, 1). By Proposition 4.4, we know +Fn([µca]) = 20 > 4 = Fn([µl]) = Fn([µr]). +So the theorem is proved. +□ +4.4. The structure for the critical points of Fn : An → R. In the following, we discuss the structure +for an arbitrary critical points of Fn : An → R by Theorem 4.2. +Theorem 4.10. Let [µ] be a critical point of Fn : An → R with Mµ = cµI + Dµ of type (k1 < · · · < +kr; d1, d2, · · · , dr), where cµ ∈ R and Dµ ∈ Der(µ). Consider the orthogonal decomposition +Cn = A− ⊕ A0 ⊕ A+, +where A−, A0 and A+ denote the direct sum of eigenspaces of Dµ with eigenvalues smaller than zero, +equal to zero and larger than zero, respectively. Then the following conditions hold: +(i) ann(µ) ⊂ A+, where ann(µ) is the annihilator of µ +(ii) A+ ⊂ N(µ), where N(µ) is the radical of µ. +(iii) A− ⊂ (C(µ) ∩ N(µ)) \ ann(µ), where C(µ) is the center of µ. +(iv) (Lµ +A − Rµ +A)∗ ∈ Der(µ) for any A ∈ A0. So the induced Lie algebra of A0 is reductive. +Proof. For (i), assume that X ∈ ann(µ) and DµX = cX, then by (3.8) +⟨MµX, X⟩ = 2 +� +i, j +|⟨µ(Xi, X j), X⟩|2 ≥ 0. +Since Mµ = cµI + Dµ, then 0 ≤ ⟨MµX, X⟩ = (cµ + c)⟨X, X⟩. It follows from Lemma 3.6 that c ≥ −cµ > 0. +This proves (i). +For (ii), it is an immediate consequence of (iii) by Remark 2.4. Now, we prove (iii) as follows. Assume +that DµX = cX for some c < 0. Since cLµ +X = [Dµ, Lµ +X], cRµ +X = [Dµ, Rµ +X], then +c tr(Lµ +X − Rµ +X)(Lµ +X − Rµ +X)∗ = tr[Dµ, (Lµ +X − Rµ +X)](Lµ +X − Rµ +X)∗ += tr[Mµ, (Lµ +X − Rµ +X)](Lµ +X − Rµ +X)∗ += tr Mµ[(Lµ +X − Rµ +X), (Lµ +X − Rµ +X)∗]. +Noting that (Lµ +X − Rµ +X) ∈ Der(µ), by Corollary 3.2 we have +c tr(Lµ +X − Rµ +X)(Lµ +X − Rµ +X)∗ ≥ 0. + +14 +HUI ZHANG AND ZAILI YAN +It follows that (Lµ +X − Rµ +X) = 0 since c < 0. So X ∈ C(µ). By Remark 2.4, it is easy to see that X ∈ N(µ). +Using (i), we conclude A− ⊂ (C(µ) ∩ N(µ)) \ ann(µ). This proves (iii). +For (iv), we first note that +[Dµ, Lµ +A] = Lµ +DµA, +[Dµ, Rµ +A] = Rµ +DµA, +for any A ∈ A. If A ∈ A0, we have [Dµ, Lµ +A] = [Dµ, Rµ +A] = 0, and so +tr Mµ[(Lµ +A − Rµ +A), (Lµ +A − Rµ +A)∗] = tr(cµI + Dµ)[(Lµ +A − Rµ +A), (Lµ +A − Rµ +A)∗] += tr Dµ[(Lµ +A − Rµ +A), (Lµ +A − Rµ +A)∗] += tr[Dµ, (Lµ +A − Rµ +A)](Lµ +A − Rµ +A)∗ += 0. +By Corollary 3.2, it follows that (Lµ +A − Rµ +A)∗ ∈ Der(µ) since (Lµ +A − Rµ +A) ∈ Der(µ). This proves (iv). +□ +In the sequel, we give a description of the critical points in terms of those which are nilpotent. Let [λ] +be a nilpotent critical point of Fm : Am → R. Define +L(λ) : = {Φ ∈ End(Cm) : Φ(λ(X, Y)) = λ(ΦX, Y)}, +R(λ) : = {Ψ ∈ End(Cm) : Ψ(λ(X, Y)) = λ(X, ΨY)}. +Moreover, we set Γl = {Φ ∈ L(λ) : [Φ, Ψ] = 0, ∀Ψ ∈ R(λ)}, Γr = {Ψ ∈ R(λ) : [Φ, Ψ] = 0, ∀Φ ∈ L(λ)}, and +Γ(λ) : = {(Φ, Ψ) ∈ Γl × Γr : λ(·, Φ(·)) = λ(Ψ(·), ·)}. +For any (Φi, Ψi) ∈ Γ(λ), i = 1, 2, we define (Φ1, Ψ1)(Φ2, Ψ2) := (Φ1Φ2, Ψ2Ψ1). Then it follows that Γ(λ) +is an associative algebra. +Lemma 4.11. Assume that S ⊂ Γ(λ) is a subalgebra such that (Φ∗, Ψ∗) ∈ S for any (Φ, Ψ) ∈ S, then S +is a semisimple associative algebra. +Proof. Note that S is an associative algebra of matrices, which are closed under conjugate transpose. +Define an Hermitian inner product on S by +⟨H1, H2⟩ := tr H1H∗ +2 = tr Φ1Φ∗ +2 + tr Ψ1Ψ∗ +2, ∀Hi = (Φi, Ψi) ∈ S, i = 1, 2. +Then it follows that ⟨HH1, H2⟩ = ⟨H1, H∗H2⟩, ⟨H1H, H2⟩ = ⟨H1, H2H∗⟩ for any H, H1, H2 ∈ S. Let I be +an ideal in S and I⊥ denote the orthogonal complement of I. Then it is easy to see that I⊥ is also an ideal +of S. Let S = R⊕N, where N is the radical of S and R = N⊥. It follows that R and N are both ideals of +S. Moreover, R is semisimple, and N is the annihilator of S (by considering the derived series). Since S +is an associative algebra of matrices which are closed under conjugate transpose, then HH∗ = 0 for any +H ∈ N, hence H = 0. So N = 0, and S is semisimple. +□ + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +15 +Theorem 4.12. Let [λ] be a nilpotent critical point of Fm : Am → R with Mλ = cλI + Dλ of type +(k2 < · · · < kr; d2, · · · , dr), where cλ ∈ R and Dλ ∈ Der(λ). Assume that S ⊂ Γ(λ) is a subalgebra of +dimension d1 such that (Φ∗, Ψ∗) ∈ S, [Dλ, Φ] = [Dλ, Ψ] = 0 for any (Φ, Ψ) ∈ S. Consider the following +semidirect sum +µ = S ⋉ λ, +where +µ((Φ1, Ψ1) + X1, (Φ2, Ψ2) + X2) := (Φ1Φ2, Ψ2Ψ1) + Φ1(X2) + Ψ2(X1) + X1X2, +for any (Φ1, Ψ1), (Φ2, Ψ2) ∈ S, X1, X2 ∈ Cm. Then µ is an associative algebra. If we extend the Hermitian +inner product on Cm by setting +⟨H, K⟩ = − 2 +cλ +(tr LS +HLS +K∗ + tr HK∗), H, K ∈ S, +then [µ] is a critical point of type (0, k2 < · · · < kr; d1, d2, · · · , dr) for the functional Fn : An → R, where +n = d1 + m. +Proof. For any H = (Φ, Ψ) ∈ S, we have +Lµ +H = +� +LS +H +0 +0 +Φ +� +, +Rµ +H = +� +RS +H +0 +0 +Ψ +� +, +where Lµ +H, Rµ +H (resp. LS +H, RS +H) denote the left and right multiplication by H of the algebra µ (resp. S), +respectively. By Lemma 4.11, we know that S is a semisimple associative algebra. Then it follows +that there is an orthonormal basis {Hi = (Φi, Ψi)} ⊂ S such that Φi∗ = −Φi, Ψi∗ = −Ψi, and Lµ +Hi, Rµ +Hi +are skew-Hermitian for each i. Let {Hi} ∪ {Xi} be an orthonormal basis of Cn = S ⊕ Cm. Then for any +H = (Φ, Ψ) ∈ S and X ∈ Cm, we have +⟨MµX, H⟩ = −2 +� +i, j +⟨µ(Xi, X), X j⟩⟨µ(Xi, H), X j⟩ − 2 +� +i, j +⟨µ(X, Xi), X j⟩⟨µ(H, Xi), X j⟩ += −2 +� +i, j +⟨λ(Xi, X), X j⟩⟨Ψ(Xi), X j⟩ − 2 +� +i, j +⟨λ(X, Xi), X j⟩⟨Φ(Xi), X j⟩ += −2 +� +i +⟨λ(Xi, X), Ψ(Xi)⟩ − 2 +� +i +⟨λ(X, Xi), Φ(Xi)⟩ += −2 tr Ψ∗Rλ +X − 2 tr Φ∗Lλ +X += −2 tr Rλ +Ψ∗(X) − 2 tr Lλ +Φ∗(X) += 0, +where Lλ +X, Rλ +X denote the left and right multiplication by X of the algebra λ, respectively, and the last two +equalities follow from that λ is nilpotent and (Φ∗, Ψ∗) ∈ S. Moreover, since Φi∗ = −Φi, Ψi∗ = −Ψi for + +16 +HUI ZHANG AND ZAILI YAN +each i, then [Φi, Φi∗] = 0, [Ψi, Ψi∗] = 0. So by (3.8) we have +⟨MµX, Y⟩ = 2 +� +i, j +⟨µ(Hi, X j), X⟩⟨µ(Hi, X j), Y⟩ + 2 +� +i, j +⟨µ(Xi, H j), X⟩⟨µ(Xi, H j), Y⟩ ++ 2 +� +i, j +⟨µ(Xi, X j), X⟩⟨µ(Xi, X j), Y⟩ − 2 +� +i, j +⟨µ(Hi, X), X j⟩⟨µ(Hi, Y), X j⟩ +− 2 +� +i, j +⟨µ(Xi, X), X j⟩⟨µ(Xi, Y), X j⟩ − 2 +� +i, j +⟨µ(X, Hi), X j⟩⟨µ(Y, Hi), X j⟩ +− 2 +� +i, j +⟨µ(X, Xi), X j⟩⟨µ(Y, Xi), X j⟩ += ⟨MλX, Y⟩ + 2 +� +i +⟨[Φi, Φi∗](X, Y⟩ + 2 +� +i +⟨[Ψi, Ψi∗](X), Y⟩ += ⟨MλX, Y⟩, +for any X, Y ∈ Cm. Therefore Mµ|Cm = Mλ = cλI + Dλ. On the other hand, noting that Lµ +Hi and Rµ +Hi are +skew-Hermitian for each i, then for any H = (Φ, Ψ) ∈ S, we have +⟨MµH, H⟩ = 2 +� +i, j +⟨µ(Hi, H j), H⟩⟨µ(Hi, H j), H⟩ +− 2 +� +i, j +⟨µ(Hi, H), H j⟩⟨µ(Hi, H), H j⟩ − 2 +� +i, j +⟨µ(Xi, H), X j⟩⟨µ(Xi, H), X j⟩ +− 2 +� +i, j +⟨µ(H, Hi), H j⟩⟨µ(H, Hi), H j⟩ − 2 +� +i, j +⟨µ(H, Xi), X j⟩⟨µ(H, Xi), X j⟩ += −2(tr LS +HLS +H∗ + tr ΦΦ∗ + tr ΨΨ∗) += −2(tr LS +HLS +H∗ + tr HH∗) += cλ⟨H, H⟩. +So Mµ = cµI + Dµ, where cµ = cλ, and +Dµ = +� +0 +0 +0 +Dλ +� +∈ Der(µ). +This completes the proof. +□ +Remark 4.13. Let the notation be as Theorem 4.10. If (Lµ +A)∗ ∈ {Lµ +A : A ∈ A0} and (Rµ +A)∗ ∈ {Rµ +A : A ∈ A0} +for any A ∈ A0. Then it follows from a similar proof of Lemma 4.11 that A0 is a semisimple associative +algebra. Moreover, the radical of [µ] corresponds to a critical point of type (k1 < · · · < ˆks < · · · < +kr; d1, · · · , ˆds, · · · , dr) by Theorem 4.12, where ks = 0. +5. Examples +In this section, we classify the critical points of Fn : An → R for n = 2 and 3, respectively. It shows +that every 2-dimensional associative algebra is isomorphic to a critical point of F2, and there exists only + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +17 +one 3-dimensional associative algebra which is not isomorphic to any critical point of F3. Finally, based +on the discussion in previous sections, we collect some natural and interesting questions. +For reader’s convenience, we recall the notation in [2]. Let {e1, e2, · · · , en} be a basis of Cn. Define +the bilinear maps ψi, j +k : Cn × Cn → Cn by +ψi, j +k (emen) = δi +mδj +nek. +It follows that any algebra can be expressed in the form d = � +ijk ck +ijψi, j +k , where ck +ij ∈ C are the structure +constants. +5.1. Two-dimensional case. The classification of two-dimensional associative algebras can be found in +[2, TABLE 1]. We give the classification of the critical points of F2 : A2 → R as follows. +TABLE I. Two-dimensional associative algebras, critical types and critical values. +Multiplication relation +Critical type +Critical value +� +d1 = ψ1,1 +1 +(0 < 1; 1, 1) +4 +� +d2 = ψ1,1 +1 ++ ψ1,2 +2 +(0 < 1; 1, 1) +4 +� +d3 = ψ1,1 +1 ++ ψ2,1 +2 +(0 < 1; 1, 1) +4 +� +d4 = ψ1,1 +1 ++ ψ2,2 +2 +(0; 2) +2 +� +d5 = ψ1,1 +2 +(1 < 2; 1, 1) +20 +� +d6 = ψ1,1 +1 ++ ψ1,2 +2 ++ ψ2,1 +2 +(0 < 1; 1, 1) +4 +Indeed, endow these algebras with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2} is an orthonormal +basis, then it is easy to obtain TABLE I. For example, the multiplication relation of µ := (d6, ⟨·, ·⟩) is +given by: e1e1 = e1, e1e2 = e2, e2e1 = e2. With respect to the given orthonormal basis {e1, e2}, the left +and right multiplications of µ are represented by +Lµ +e1 = +� 1 +0 +0 +1 +� +, +Lµ +e2 = +� 0 +0 +1 +0 +� +, +Rµ +e1 = +� 1 +0 +0 +1 +� +, +Rµ +e2 = +� 0 +0 +1 +0 +� +. +It follows from (3.7) that +Mµ = +� +−6 +0 +0 +0 +� +Set cµ := +tr M2 +µ +tr Mµ , then cµ = −6. It follows that Mµ = cµI + Dµ, where +Dµ = +� +0 +0 +0 +6 +� +is clearly a derivation of µ. So [µ] is a critical point of F2 : A2 → R with the critical type (0 < 1; 1, 1) +and F2([µ]) = 4. + +18 +HUI ZHANG AND ZAILI YAN +5.2. Three-dimensional case. The complete classification of three-dimensional associative algebras +can be found in [2, TABLE 2]. The following table gives the classification of the critical points of +F3 : A3 → R. +TABLE II. Three-dimensional associative algebras, critical types and critical values. +Multiplication relation +Critical type +Critical value +� +d1 = ψ1,1 +1 +(0 < 1; 1, 2) +4 +� +d2 = ψ1,1 +1 ++ ψ2,2 +3 +(0 < 1 < 2; 1, 1, 1) +10 +3 +� +d3 = ψ1,1 +1 ++ ψ1,3 +3 +(0 < 1; 1, 2) +4 +� +d4 = ψ1,1 +1 ++ ψ3,1 +3 +(0 < 1; 1, 2) +4 +� +d5 = ψ1,1 +1 ++ ψ1,3 +3 ++ ψ3,1 +3 +(0 < 1; 1, 2) +4 +� +d6 = ψ1,1 +1 ++ ψ3,3 +3 +(0 < 1; 2, 1) +2 +� +d7 = ψ1,1 +1 ++ ψ2,1 +2 ++ ψ1,3 +3 +(0 < 1; 1, 2) +4 +� +d8 = ψ1,1 +1 ++ ψ2,1 +2 ++ ψ3,1 +3 +(0 < 1; 1, 2) +4 +� +d9 = ψ1,1 +1 ++ ψ2,1 +2 ++ ψ1,3 +3 ++ ψ3,1 +3 +(0 < 1; 1, 2) +4 +� +d10 = ψ1,1 +1 ++ ψ2,1 +2 ++ ψ3,3 +3 +(0 < 1; 2, 1) +2 +� +d11 = ψ1,1 +1 ++ ψ2,2 +2 ++ ψ2,3 +3 +(0 < 1; 2, 1) +2 +� +d12 = ψ1,1 +1 ++ ψ2,2 +2 ++ ψ2,3 +3 ++ ψ3,2 +3 +(0 < 1; 2, 1) +2 +� +d13 = ψ1,1 +1 ++ ψ2,2 +2 ++ ψ2,3 +3 ++ ψ3,1 +3 +(0 < 1; 2, 1) +2 +� +d14 = ψ1,1 +1 ++ ψ2,2 +2 ++ ψ3,3 +3 +(0; 3) +4 +3 +� +d15 = ψ1,1 +2 +(3 < 5 < 6; 1, 1, 1) +20 +� +d16 = ψ1,1 +2 ++ ψ1,2 +3 ++ ψ2,1 +3 +(1 < 2 < 3; 1, 1, 1) +20 +3 +� +d17 = ψ1,1 +1 ++ ψ1,1 +2 ++ ψ1,2 +2 ++ ψ2,1 +2 ++ ψ1,3 +3 +(0 < 1; 1, 2) +4 +� +d18 = ψ1,1 +1 ++ ψ1,1 +2 ++ ψ1,2 +2 ++ ψ2,1 +2 ++ ψ1,3 +3 ++ ψ3,1 +3 +(0 < 1; 1, 2) +4 +� +d19 = ψ3,3 +3 ++ ψ1,1 +2 ++ ψ1,3 +1 ++ ψ3,1 +1 ++ ψ2,3 +2 ++ ψ3,2 +2 +(0 < 1 < 2; 1, 1, 1) +10 +3 +� +d20 = ψ1,1 +1 ++ ψ1,2 +2 ++ ψ1,3 +3 +(0 < 1; 1, 2) +4 +� +d21 = ψ1,1 +3 ++ ψ1,2 +3 +− ψ2,1 +3 +− +− +� +d22 = xψ1,2 +3 ++ yψ2,1 +3 +(1 < 2; 2, 1) +12 +Indeed, endow the algebras with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2, e3} is an orthonormal +basis, it is easy to obtain all cases in TABLE II except for d2, d10, d11, d12, d13, d17, d18, d21. For the cases +d2, d10, d11, d12, it follows from Remark 3.4 and TABLE I. For the cases d13, d17, d18, it follows from [5] +that d13 � U3 +1, d17 � W3 +10 and d18 � U3 +0, where U3 +1, W3 +10 and U3 +0 are defined by +U3 +1 : +ψ1,1 +1 ++ ψ3,3 +1 ++ ψ1,2 +2 ++ ψ2,1 +2 ++ ψ2,3 +2 ++ ψ1,3 +3 ++ ψ3,1 +3 +− ψ3,2 +3 . +W3 +10 : +ψ1,2 +1 ++ ψ2,1 +1 ++ ψ2,2 +2 ++ ψ2,3 +3 . +U3 +0 : +ψ1,1 +2 ++ ψ1,2 +2 ++ ψ2,1 +2 ++ ψ1,3 +3 ++ ψ3,1 +3 . +Endow U3 +1, W3 +10 and U3 +0 with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2, e3} is an orthonormal basis, +then it is easy to obtain the corresponding critical types and values for d13, d17, d18. + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +19 +In the sequel, we follow a similar procedure as in [6, 16] to classify all Hermitian inner products on +d21, then show that for any Hermitian inner product ⟨·, ·⟩ on d21, (d21, ⟨·, ·⟩) cannot be a critical point of +F3. First, note that the multiplication relation of d21 is given as follows: +e1e1 = e3, +e1e2 = e3, +e2e1 = −e3. +Denote by ⟨·, ·⟩0 the Hermitian inner product on d21 such that {e1, e2, e3} is orthonormal. With respect to +this basis {e1, e2, e3}, the automorphism group of d21 is given by +Aut(d21) = + +a +0 +0 +b +a +0 +c +d +a2 + ⊂ GL(3, C), +(5.1) +where 0 � a ∈ C, and b, c, d ∈ C are arbitrary. +Lemma 5.1. For any Hermitian inner product ⟨·, ·⟩ on d21, there exist k > 0 and φ ∈ Aut(d21) such that +{aφe1, φe2, φe3} is orthonormal with respective to k⟨·, ·⟩, where a > 0 +Proof. It suffices to prove that +U = {diag(a, 1, 1) : a > 0} ⊂ GL(3, C) +is a set of representatives for the action C×Aut(d21) on M, i.e., the space of all Hermitian inner products +on d21, which can be identified with the homogeneous space GL(3, C)/U(3) at the base point ⟨·, ·⟩0 ∈ M +(see [6]). Indeed, since +� +g∈U +C×Aut(d21) · g · U(3) = GL(3, C), +it follows that U is a set of representatives. For any Hermitian inner product ⟨·, ·⟩ on d21, we know that +there exists g0 ∈ U such that +⟨·, ·⟩ ∈ (C×Aut(d21)).(g0.⟨·, ·⟩0) +Hence there exist c ∈ C×, φ ∈ Aut(d21) such that +⟨·, ·⟩ = (cφ).(g0.⟨·, ·⟩0) = (cφg0).⟨·, ·⟩0) +Put k = |c|2, then +k⟨·, ·⟩ = k⟨(cφg0)−1(·), (cφg0)−1(·)⟩0 = kc−1¯c−1⟨(φg0)−1(·), (φg0)−1(·)⟩0 = ⟨(φg0)−1(·), (φg0)−1(·)⟩0 +Since g0 ∈ U, then g0 = diag{a, 1, 1} for some a > 0. It follows that {aφe1, φe2, φe3} is orthonormal with +respective to k⟨·, ·⟩ +□ +Proposition 5.2. For any Hermitian inner product ⟨·, ·⟩ on d21, (d21, ⟨·, ·⟩) can not be a critical point of +F3 : A3 → R. + +20 +HUI ZHANG AND ZAILI YAN +Proof. Assume that ⟨·, ·⟩ is a Hermitian inner product on d21 such that (d21, ⟨·, ·⟩) is a critical point of F3 : +A3 → R. Then the critical type is necessarily of (1 < 2; 2, 1) by Theorem 4.1 and (5.1). Moreover, for +the Hermitian inner product ⟨·, ·⟩ on d21, by Lemma 5.1 we know that there exist k > 0 and φ ∈ Aut(d21) +such that {x1 = aφe1, x2 = φe2, x3 = φe3} is orthonormal with respective to k⟨·, ·⟩, where a > 0. With +respect to the basis {x1, x2, x3}, the multiplication relation of d21 is given as follows +x1x1 = a2x3, +x1x2 = ax3, +x2x1 = −ax3. +By (3.7), Lemma 3.6 and a straightforward calculation, it follows that the critical type is of +(3a4 + 6a2 + 8, 5a4 + 10a2 + 8, 2(3a4 + 8a2 + 8)) +which is never of type (1 < 2; 2, 1) for any a > 0. This is a contradiction by Theorem 4.1, and the +proposition is therefore proved. +□ +5.3. Comments. By the previous discussion, we know that the critical types of Fn : An → R, n = 2, 3, +are necessarily nonnegative. So it is natural to ask the following question: Let [µ] ∈ An be a critical +point of Fn : An → R with Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ). Are all the eigenvalues of +Dµ necessarily nonnegative? +On the other hand, it will be also interesting to construct or classify the critical points [µ] of Fn : +An → R such that Dµ has negative eigenvalues if the above question does not hold. We note that 2-step +nilpotent Lie algebras are automatically associative algebras, so it follows from [13, Example 1] that +there exist associative algebras whose Nikolayevsky derivations do admit negative eigenvalues. +6. Statements and Declarations +The authors declare that there is no conflict of interest. +References +[1] B¨ohm, C.; Lafuente, R: Immortal homogeneous Ricci flows, Invent. Math. 212 (2018), 461–529. +[2] Fialowski, A.; Penkava, M.: The moduli space of 3-dimensional associative algebras, Comm. Algebra 37(10) (2009) +3666–3685. +[3] Khudoyberdiyev, A.; Omirov, B.: The classification of algebras of level one, Linear Algebra Appl. 439(11) (2013), 3460– +3463. +[4] Kirwan, K.: Momentum maps and reduction in algebraic geometry, Differential Geom. Appl. 9 (1998), 135–172. +[5] Kobayashi, Y; Shirayanagi, K; Tsukada, M; Takahasi, S.: A complete classification of three-dimensional algebras over R +and C–(visiting old, learn new),Asian-Eur. J. Math. 14 (2021). +[6] Kubo, A; Onda, K; Taketomi, Y; Tamaru, H.: On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie +groups, Hiroshima math. J. 46 (2016), 357–374. +[7] Lauret, J.: On the moment map for the variety of Lie algebras, J. Funct. Anal. 202 (2003), 392–423. +[8] Lauret, J.: Degenerations of Lie algebras and geometry of Lie groups, Differential Geom. Appl. 18 (2003), 177–194. +[9] Lauret, J.: Einstein solvmanifolds are standard, Ann. of Math. 172 (2010), 1859–1877. +[10] Lauret, J.: Ricci soliton solvmanifolds, J. reine angew. Math. 650 (2011), 1–21. +[11] Lauret, J.: Finding solitons, Notices Am. Math. Soc. 67 (2020), 647–657. +[12] Ness, L.: A stratification of the null cone via the moment map, Amer. J. Math., 106 (1984), 1281-1329 (with an appendix +by D. Mumford). + +THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS +21 +[13] Nikolayevsky, Y.: Nilradicals of Einstein solvmanifolds, arXiv:math/0612117v1 [math.DG] (2006). +[14] Nikolayevsky, Y.: Einstein solvmanifolds and the pre-Einstein derivation, Trans. Am. Math. Soc. 363 (2011), 3935–3958. +[15] Pierce, R.S.: Associative Algebras, Springer-Verlag, New York, Heidelberg, Berlin, 1982 +[16] Taketomi, Y; Tamaru, H.: On the nonexistence of left-invariant Ricci solitons a conjecture and examples, Transform. +Groups 23 (2018), 257–270. +[17] Zhang, H.; Chen, Z.; Li, L.: The moment map for the variety of 3-Lie algebras, J. Funct. Anal. 283 (2022), No. 11, Article +ID 109683. +(Hui Zhang) School of Mathematics, Southeast University, Nanjing 210096, P. R. China +Email address: 2120160023@mail.nankai.edu.cn +School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province, 315211, People’s Republic of China +Email address: yanzaili@nbu.edu.cn + diff --git a/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/load_file.txt b/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e99c9572622db4eb8a6e695bbfd17d492804f1b --- /dev/null +++ b/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/load_file.txt @@ -0,0 +1,789 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf,len=788 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12142v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='DG] 28 Jan 2023 THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS HUI ZHANG AND ZAILI YAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We consider the moment map m : PVn → iu(n) for the action of GL(n) on Vn = ⊗2(Cn)∗ ⊗ Cn, and study the critical points of the functional Fn = ∥m∥2 : PVn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Firstly, we prove that [µ] ∈ PVn is a critical point if and only if Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ), where m([µ]) = Mµ ∥µ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we show that any algebra µ admits a Nikolayevsky derivation φµ which is unique up to automorphism, and if moreover, [µ] is a critical point of Fn, then φµ = − 1 cµ Dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Secondly, we characterize the maxima and minima of the functional Fn : An → R, where An denotes the projectivization of the algebraic varieties of all n- dimensional associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Furthermore, for an arbitrary critical point [µ] of Fn : An → R, we also obtain a description of the algebraic structure of [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Finally, we classify the critical points of Fn : An → R for n = 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Introduction Lauret has studied the moment map for the variety of Lie algebras and obtained many remarkable results in [7], which turned out to be very important in proving that every Einstein solvmanifold is standard ([9]) and in the characterization of solitons ([1, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Apart from the Lie algebras, the study of the moment map in other classes of algebras was also initiated by Lauret, see [11] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Motivated by this, the authors have recently extended the study of the moment map to the variety of 3-Lie algebras (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In this paper, we study the moment map for the variety of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let GL(n) be the complex reductive Lie group acting naturally on the complex vector space Vn = ⊗2(Cn)∗ ⊗ Cn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', the space of all n-dimensional complex algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The usual Hermitian inner product on Cn naturally induces an U(n)-invariant Hermitian inner product on Vn, which is denoted by ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since gl(n) = u(n) + iu(n), we may define a function as follows m : PVn → iu(n), (m([µ]), A) = (dρµ)eA ∥µ∥2 , 0 � µ ∈ Vn, A ∈ iu(n), where (·, ·) is an Ad(U(n))-invariant real inner product on iu(n), and ρµ : GL(n) → R is defined by ρµ(g) = ⟨g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The function m is the moment map from symplectic geometry, corresponding to the Hamiltonian action U(n) of Vn on the symplectic manifold PVn (see [4, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In this paper, we study the critical points of the functional Fn = ∥m∥2 : PVn → R, with an emphasis on the critical points that lie in the projectivization of the algebraic variety of all n-dimensional associative algebras An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 14L30, 17B30, 53D20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moment map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Variety of associative algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This work is supported by NSFC (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 11701300, 11626134) and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Wong Magna Fund in Ningbo University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 2 HUI ZHANG AND ZAILI YAN The paper is organized as follows: In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, we recall some basic concepts and results of complex associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3, we first give the explicit expression of the moment map m : PVn → iu(n) in terms of Mµ, that is, m([µ]) = Mµ ∥µ∥2 for any [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we show that [µ] ∈ PVn is a critical point of Fn if and only if Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ) (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4, we first show that any algebra µ ∈ Vn admits a Nikolayevsky derivation φµ which is unique up to automorphism, the eigenvalues of φµ are necessarily rational, and moreover, φµ = − 1 cµ Dµ if [µ] is a critical point of Fn (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we study the extremal points of Fn : An → R, proving that the minimum value is attained at semisimple associative algebras (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6), and the maximum value at the direct sum of a two-dimensional commutative associative algebra with the trivial algebra (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the context of Lie algebras ([7]), Lauret proved that any µ for which there exists [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] such that all eigenvalues of Mλ are negative, must be semisimple, and we prove that this result also holds for associative algebras (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Besides, the structure for an arbitrary critical point [µ] of Fn : An → R is discussed (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='10 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 5, we classify the critical points of Fn : An → R for n = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It shows that every two- dimensional associative algebra is isomorphic to a critical point of F2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' and there exists only one three- dimensional associative algebra which is not isomorphic to any critical point of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Finally, based on the discussion in previous sections, we collect some natural and interesting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Preliminaries In this section, we recall some basic definitions and results of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The ambient field is always assumed to be the complex number field C unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A vector space A over C with a bilinear operation A × A → A, denoted by (x, y) �→ xy, is called an associative algebra, if x(yz) = (xy)z for all x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A derivation of an associative algebra A is a linear transformation D : A → A satisfying D(xy) = (Dx)y + x(Dy), for x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that the set of all derivations of A is a Lie algebra, which is denoted by Der(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A vector subspace I of A is called an ideal if AI, IA ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let A be an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The center of A is defined by C(A) = {x ∈ A : xy = yx, ∀y ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The annihilator of A is defined by ann(A) = {x ∈ A : xy = yx = 0, ∀y ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, C(A) is a subalgebra of A, and ann(A) is an ideal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let I be an ideal of an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then I is called nilpotent, if Ik = 0 for some integer k ≥ 1, where Ik = I· · ·I· · ·I ���������� k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If I, J are any two nilpotent ideals of an associative algebra A, then I + J is also a nilpotent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the maximum nilpotent ideal of A is unique, which is called the radical and denoted by N(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that N(A) coincides with the Jacobson radical of A since A is an associative algebra over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, N(A) = {x ∈ A : xy, yx are nilpotent elements for any y ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let A be an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If A has no ideals except itself and 0, we call A simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Denote by Mn(C) the set of all n × n complex square matrices, which is clearly an associative algebra with respect to the usual matrix addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, Mn(C) is a simple associative algebra for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, it follows from Wedderburn-Artin theorem that any finite-dimensional simple associative algebra over C is isomorphic to Mn(C) for some integer n ≥ 1 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' An associative algebra A is called semisimple if its radical N(A) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The following theorem is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' An associative algebra over C is semisimple if and only if it is a direct sum of simple ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' That is, a semisimple associative algebra is isomorphic to Mn1(C) × Mn2(C) × · · · × Mns(C) for some positive integers n1, n2, · · · , ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The moment map for complex algebras Let Cn be the n-dimensional complex vector space and Vn = ⊗2(Cn)∗ ⊗ Cn be the space of all complex n-dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The natural action of GL(n) = GL(Cn) on Vn is given by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ(X, Y) = gµ(g−1X, g−1Y), g ∈ GL(n), X, Y ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1) Clearly, GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ is precisely the isomorphism class of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that lim t→∞ gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = 0, gt = tI ⊂ GL(n), t > 0, we see that 0 lies in the boundary of the orbit GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ for each µ ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1), we obtain the natural action gl(n) on Vn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ(X, Y) = Aµ(X, Y) − µ(AX, Y) − µ(X, AY), A ∈ gl(n), µ ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2) 4 HUI ZHANG AND ZAILI YAN It follows that A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = 0 if and only if A ∈ Der(µ), where Der(µ) denotes the derivation algebra of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that the usual Hermitian inner product on Cn gives an U(n)-invariant Hermitian inner product on Vn as follows ⟨µ, λ⟩ = � i, j,k ⟨µ(Xi, X j), Xk⟩⟨λ(Xi, X j), Xk⟩, µ, λ ∈ Vn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3) where {X1, X2, · · · , Xn} is an arbitrary orthonormal basis of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let u(n) denote the Lie algebra of U(n), then it is easy to see that gl(n) = u(n) + iu(n) decomposes into skew-Hermitian and Hermitian transfor- mations of Vn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, there is an Ad(U(n))-invariant Hermitian inner product on gl(n) given by (A, B) = tr AB∗, A, B ∈ gl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4) The moment map from symplectic geometry, corresponding to the Hamiltonian action of U(n) on the symplectic manifold PVn, is defined as follows m : PVn → iu(n), (m([µ]), A) = (dρµ)eA ∥µ∥2 , 0 � µ ∈ Vn, A ∈ iu(n), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5) where ρµ : GL(n) → R is given by ρµ(g) = ⟨g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, (dρµ)eA = ⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ + ⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ = 2⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ for any A ∈ iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The square norm of the moment map is denoted by Fn : PVn → R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6) where Fn([µ]) = ∥m([µ])∥2 = (m([µ]), m([µ])) for any [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In order to express the moment map m explicitly, we define Mµ ∈ iu(n) as follows Mµ = 2 � i Lµ Xi(Lµ Xi)∗ − 2 � i (Lµ Xi)∗Lµ Xi − 2 � i (Rµ Xi)∗Rµ Xi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7) where the left and right multiplication Lµ X, Rµ X : Cn → Cn by X of the algebra µ, are given by Lµ X(Y) = µ(X, Y) and Rµ X(Y) = µ(Y, X) for all Y ∈ Cn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is not hard to prove that ⟨MµX, Y⟩ =2 � i, j ⟨µ(Xi, X j), X⟩⟨µ(Xi, X j), Y⟩ − 2 � i, j ⟨µ(Xi, X), X j⟩⟨µ(Xi, Y), X j⟩ − 2 � i, j ⟨µ(X, Xi), X j⟩⟨µ(Y, Xi), X j⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) for X, Y ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that if the algebra µ is commutative or anticommutative, then the second and third term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) are the same, and in this case, Mµ coincides with [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any µ ∈ Vn, we have (Mµ, A) = 2⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩, ∀A ∈ gl(n) = u(n) + iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In particular, m([µ]) = Mµ ∥µ∥2 for any 0 � µ ∈ Vn THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any A ∈ gl(n), we have (A, Mµ) = tr AM∗ µ = tr AMµ = tr MµA, and tr MµA = 2 tr � i Lµ Xi(Lµ Xi)∗A �������������������������������������� I − 2 tr � i ((Lµ Xi)∗Lµ Xi + (Rµ Xi)∗Rµ Xi)A �������������������������������������������������������������������������� II =: I − II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that I =2 � i tr Lµ Xi(Lµ Xi)∗A =2 � i tr(Lµ Xi)∗ALµ Xi =2 � i, j ⟨(Lµ Xi)∗ALµ Xi(X j), X j⟩ =2 � i, j ⟨Aµ(Xi, X j), µ(Xi, X j)⟩, and II =2 tr � i ((Lµ Xi)∗Lµ Xi + (Rµ Xi)∗Rµ Xi)A =2 � i, j ⟨((Lµ Xi)∗Lµ Xi + (Rµ Xi)∗Rµ Xi)AX j, X j⟩ =2 � i, j ⟨µ(Xi, AX j), µ(Xi, X j)⟩ − 2 � i, j ⟨µ(AX j, Xi), µ(X j, Xi)⟩ =2 � i, j ⟨µ(AXi, X j) + µ(Xi, AX j), µ(Xi, X j)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2), it follows that (A, Mµ) = tr MµA = 2⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩, so (Mµ, A) = 2⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ for any A ∈ gl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For A ∈ iu(n), we have ⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ = ⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5), we conclude that m([µ]) = Mµ ∥µ∥2 for any 0 � µ ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This completes proof Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any µ ∈ Vn, then (i) tr MµD = 0 for any D ∈ Der(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) tr Mµ[A, A∗] ≥ 0 for any A ∈ Der(µ), and equality holds if and only if A∗ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (i), it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 that tr MµD = 2⟨D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (ii), it follows from that tr Mµ[A, A∗] = 2⟨[A, A∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ = 2⟨A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ ≥ 0, ∀A ∈ Der(µ), and the fact A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = 0 if and only if A∗ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The moment map m : PVn → iu(n), the functional square norm of the moment map Fn = ∥m∥2 : PVn → R and the gradient of Fn are, respectively, given by Fn([µ]) = tr M2 µ ∥µ∥4 , grad(Fn)[µ] = 8π∗(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 , [µ] ∈ PVn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='9) 6 HUI ZHANG AND ZAILI YAN where π∗ denotes the derivative of π : Vn\\{0} → PVn, the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, the following statements are equivalent: (i) [µ] ∈ PVn is a critical point of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) [µ] ∈ PVn is a critical point of Fn|GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iii) Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, we have Fn([µ]) = tr M2 µ ∥µ∥4 for any [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' To prove the second one, we only need to compute the gradient of Fn : Vn \\ {0} → R, Fn(µ) = tr M2 µ ∥µ∥4 , and then to project it via π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If µ, λ ∈ Vn with µ � 0, then Re⟨grad(Fn)µ, λ⟩ = d dt �����t=0 Fn(µ + tλ) = d dt �����t=0 1 ∥µ + tλ∥4 (Mµ+tλ, Mµ+tλ) = − 4 Re⟨Fn(µ) ∥µ∥2 µ, λ⟩ + 2 ∥µ∥4 ( d dt �����t=0 Mµ+tλ, Mµ) We claim that ( d dt ���t=0 Mµ+tλ, A) = 4 Re⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, λ⟩ for any A ∈ iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, ( d dt ���t=0 Mµ+tλ, A) = d dt ���t=0 (Mµ+tλ, A) = 2 d dt ���t=0 ⟨µ + tλ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (µ + tλ)⟩ = 2⟨λ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ + 2⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='λ⟩ = 4 Re⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, λ⟩ for any A ∈ iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that grad(Fn)µ = −4 Fn(µ) ∥µ∥2 µ + 8(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 , and consequentely grad(Fn)[µ] = 8π∗(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus the first part of the theorem is proved, and the following is to prove the equivalence among the statements (i), (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (i) ⇔ (ii) : The equivalence follows from that grad(Fn) is tangent to the GL(n)-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed grad(Fn)[µ] = 8π∗(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 = 8 ∥µ∥4 π∗( d dt �����t=0 etMµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ) = 8 ∥µ∥4 d dt �����t=0 etMµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] ∈ T[µ](GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iii) ⇒ (i) : By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2), we know that I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = −µ, and (Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = (cµI + Dµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = −cµµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that grad(Fn)[µ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (i) ⇒ (iii) : Since grad(Fn)[µ] = 0, then (Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∈ ker π∗µ = Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So Mµ = cI + D for some c ∈ C and D ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly [D, D∗] = [Mµ − cI, Mµ − ¯cI] = 0, we conclude by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 that D∗ is also a derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In particular, (c − ¯c)I = (D∗ − D) ∈ Der(µ), thus c = ¯c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] be a critical point of Fn and [λ] be a critical point of Fm, then [µ ⊕ cλ] is a critical point of Fn+m for a suitable c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, assume that Mµ = cµI + Dµ for some cµ ∈ R, Dµ ∈ Der(µ), and Mλ = cλI +Dλ for some cλ ∈ R, Dλ ∈ Der(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Noting that Mtλ = |t|2Mλ for any t ∈ C, we can choose t0 such that cµ = |t0|2cλ, then it follows that [µ ⊕ t0λ] is a critical point of Fn+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the frame of algebras, a remarkable result due to Ness can be stated as follows Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If [µ] is a critical point of the functional Fn : PVn → R, then THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 7 (i) Fn|GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] attains its minimum value at [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] is a critical point of Fn if and only if [λ] ∈ U(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, the above theorem implies that up to U(n)-orbit, GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] contains at most one critical point for each [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] ∈ PVn be a critical point of Fn with Mµ = cµI+Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we have (i) cµ = tr M2 µ tr Mµ = − 1 2 tr M2 µ ∥µ∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) If tr Dµ � 0, then cµ = − tr D2 µ tr Dµ and tr Dµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since Mµ = cµI + Dµ, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 we have tr Mµ = (Mµ, I) = 2⟨µ, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ = −2∥µ∥2 < 0, tr M2 µ = tr Mµ(cµI + Dµ) = cµ tr Mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So cµ = tr M2 µ tr Mµ = − 1 2 tr M2 µ ∥µ∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If tr Dµ � 0, then 0 = tr MµDµ = cµ tr Dµ + tr D2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So cµ = − tr D2 µ tr Dµ and tr Dµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, tr Dµ = 0 if and only if Dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, it follows from that 0 = tr MµDµ = cµ tr Dµ + tr D2 µ and Dµ is Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The critical points of the variety of associative algebras The space An of all n-dimensional associative algebras is an algebraic set since it is given by polyno- mial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Denote by An the projective algebraic variety obtained by projectivization of An .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that An is GL(n)-invariant, then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3, the critical points of Fn : An → R are precisely the critical points of Fn : PVn → R that lie in An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The Nikolayevsky derivation and the rationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A derivation of φ of an algebra (µ, Cn) is called a Nikolayevsky derivation, if it is semisimple with all eigenvalues real, and tr φψ = tr ψ for any ψ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This notion is motivated by [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let (µ, Cn) be an arbitrary algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then (1) (µ, Cn) admits a Nikolayevsky derivation φµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (2) The Nikolayevsky derivation φµ is determined up to automorphism of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3) All eigenvalues of φµ are rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 8 HUI ZHANG AND ZAILI YAN If moreover, [µ] is a critical point of Fn : PVn → R with Mµ = cµI+Dµ for some cµ ∈ R and Dµ ∈ Der(µ), then − 1 cµ Dµ is the Nikolayevsky derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (1) The complex Lie algebra Der(µ) is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let Der(µ) = s ⊕ t ⊕ n be its Levi-Mal’cev decomposition, where s is semisimple, t ⊕ n is the radical of Der(µ), n is the set of all nilpotent elements in t ⊕ n (and is the nilradical of t ⊕ n), t is an abelian subalgebra consisting of semisimple elements, and [s, t] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define the bilinear form B on Der(µ) by B(ψ1, ψ2) := tr ψ1ψ2, ∀ψ1, ψ2 ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then, in general, B is degenerate, and Ker B = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since s is semisimple, then B(s, t) = B([s, s], t) = B(s, [s, t]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, B is nondegenerate on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since t is reductive, we have t = a + ia, where a consists of semisimple elements with all eigenvalues real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that there exists a unique element φ ∈ a such that B(φ, ψ) = tr ψ for any ψ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus tr φψ = tr ψ for any ψ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (2) The subalgebra s ⊕ t is a maximal fully reducible subalgebra of Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since the maximal fully reducible subalgebras of Der(µ) are conjugate by inner automorphism of Der(µ) (which corresponds to an automorphism of µ), and then the center t of s ⊕ t, is defined uniquely, up to automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the Nikolayevsky derivation is determined up to automorphism of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3) The case φµ = 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the following, we assume that φµ is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that φµ is simisimple with all eigenvalues real, we have the following decomposition Cn = l1 ⊕ l2 ⊕ · · · ⊕ lr, where li = {X ∈ Cn|φµX = ciX} are eigenspaces of φµ corresponding to eigenvalues c1 < c2 < · · · < cr ∈ R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Set di = dim li ∈ N, 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since φµ is a derivation, we have the following relations µ(li, lj) ⊂ lk if ci + cj = ck, for all 1 ≤ i, j, k ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Conversely, if we define a linear transformation ψ : Cn → Cn by ψ|li = aiIdli, where a1, a2, · · · , ar ∈ R satisfy ai + aj = ak for all 1 ≤ i, j, k ≤ r such that ci + cj = ck, then ψ is a derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, all such derivations form a real vector space, which can be identified with W := {(w1, w2, · · · , wr) ∈ Rr|wi + w j = wk if ci + cj = ck}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We endow Rr with the usual inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' ⟨x, y⟩ = � i xiyi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1) for any x = (x1, x2, · · · , xr), y = (y1, y2, · · · , yr) ∈ Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any derivation ψ ∈ W, we have tr(φµ − I)ψ = tr φµψ − tr ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 9 Then we see that (d1(c1 − 1), d2(c2 − 1), · · · , dr(cr − 1)) ⊥ W relative to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Put F := W⊥, then by definition we have F = span1≤i, j,k≤r{ei + ej − ek : ci + cj = ck}, where ei belongs to Rr having 1 in the i-th position and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {ei1 +ej1 −ek1, · · · , eis +ejs −eks} be a basis of F, then (d1(c1 − 1), d2(c2 − 1), · · · , dr(cr − 1)) = s � p=1 bp(eip + ejp − ekp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2) for some b1, b2, · · · , bs ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Put E = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ei1 + ej1 − ek1 ei2 + ej2 − ek2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' eis + ejs − eks \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ∈ Zs×r, then EET ∈ GL(s, Z), and (EET)−1 ∈ GL(s, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2) and the definition of E, we have \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed d1(c1 − 1) d2(c2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' dr(cr − 1) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = ET \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' bs \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , E \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed c1 c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' cr \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 0 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , E \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By the left multiplication of E on (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2), we have \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 0 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 = ED−1ET \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' bs \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , where D = diag(d1, d2, · · · , dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that (ED−1ET) ∈ GL(s, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consequently D \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed c1 − 1 c2 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' cr − 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = −ET(ED−1ET)−1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , and \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed c1 c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' cr \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 − D−1ET(ED−1ET)−1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 ∈ Qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So all eigenvalues of φµ are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For the last statement, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 we know that 0 = tr Mµψ = cµ tr ψ+tr Dµψ for any ψ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since Dµ is Hermitian, we conclude that − 1 cµ Dµ is the Nikolayevsky derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, it is easy to obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 10 HUI ZHANG AND ZAILI YAN Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] ∈ PVn be a critical point of Fn : PVn → R with Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then there exists a constant c > 0 such that the eigenvalues of cDµ are integers prime to each other, say k1 < k2 < · · · < kr ∈ Z with multiplicities d1, d2, · · · , dr ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The data set (k1 < k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 is called the type of the critical point [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] ∈ PVn be a critical point of Fn with type α = (k1 < k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we have (i) If α = (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n), then Fn([µ]) = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) If α � (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n), then Fn([µ]) = 4 � n − (k1d1+k2d2+···+krdr)2 k2 1d1+k2 2d2+···+k2r dr �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We suppose that Mµ = cµI + Dµ, ∥µ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since tr Mµ = −2⟨µ, µ⟩ = −2, then tr M2 µ = tr Mµ(cµI + Dµ) = cµ tr Mµ = −2cµ, and Fn([µ]) = tr Mµ2 ∥µ∥4 = tr Mµ2 = −2cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (i), we have Dµ = 0, so Mµ = cµI and cµn = tr Mµ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus cµ = − 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Fn([µ]) = −2cµ = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (ii), we have Dµ � 0, and cµ = − tr D2 µ tr Dµ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that Fn([µ]) = tr Mµ2 = tr(cµI + Dµ)2 = c2 µn + cµ tr Dµ = 1 4Fn([µ])2n − 1 2Fn([µ]) tr Dµ, so we have 1 Fn([µ]) = 1 4n − 1 2Fn([µ]) tr(Dµ) = 1 4n + 1 4cµ tr Dµ = 1 4 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8edn − (tr Dµ)2 tr D2µ \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that Fn([µ]) = 4 � n − (k1d1+k2d2+···+krdr)2 k2 1d1+k2 2d2+···+k2r dr �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The minima of Fn : An → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume [µ] ∈ PVn, then [µ] is a critical point of Fn : PVn → R with type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n) if and only if Fn([µ]) = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, 4 n is the minimum value of Fn : PVn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any 0 � µ ∈ Vn, we use x1, x2, · · · , xn ∈ R denote the eigenvalues of Mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that tr Mµ = −2∥µ∥2, then we have Fn([µ]) = tr Mµ2 ∥µ∥4 = 4 tr Mµ2 (tr Mµ)2 = 4 (x2 1 + x2 2 + · · · + x2 n) (x1 + x2 + · · · + xn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that Fn([µ]) ≥ 4 n with equality holds if and only if x1 = x2 = · · · = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So [µ] is a critical point of Fn : PVn → R with type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n) if only if Mµ is a constant multiple of I, if and only Fn attains its minimum value 4 n at [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 11 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The functional Fn : An → R attains its minimum value at a point [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] if and only if µ is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In such a case, Fn([λ]) = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the simple associative algebra Mm(C) for an integer m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We endow Mm(C) with the following Hermitian inner product ⟨A, B⟩ := tr AB∗, A, B ∈ Mm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3) Then {Eij : 1 ≤ i, j ≤ m} is an orthonormal basis, where Eij denote the matrices having 1 in the (i, j)- position and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Set ν := (Mm(C), ⟨·, ·⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly (Lν A)∗ = LA∗, (Rν A)∗ = RA∗ for any A ∈ Mm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7), we have Mν = 2 � ij Lν Eij(Lν Eij)∗ − 2 � ij (Lν Eij)∗Lν Eij − 2 � ij (Rν Eij)∗Rν Eij = 2 � ij Lν EijLν Eji − 2 � ij Lν EjiLν Eij − 2 � ij Rν EjiRν Eij = 2 � ij Lν EijEji − 2 � ij Lν EjiEij − 2 � ij Rν EijEji = 2m � i Lν Eii − 2m � i Lν Eii − 2m � i Rν Eii = 2mLν I − 2mLν I − 2mRν I = 2mIm2 − 2mIm2 − 2mIm2 = −2mIm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So [ν] is a critical point of type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since µ is a complex semisimple associative algebra, by Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6, µ is isomorphic to Mn1(C) × Mn2(C) × · · · × Mns(C) for some positive integers n1, n2, · · · , ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4 that there exists a point [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] such that [λ] is a critical point of type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the functional Fn : An → R attains its minimum value at [λ], and Fn([λ]) = 4 n by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Conversely, assume that Fn : An → R attains its minimum value at a point [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='[µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The first part of the proof implies that Mλ = cλI with cλ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' To prove µ is semisimple, it suffices to show that L = (λ, Cn) is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the following orthogonal decompositions: (i) L = H ⊕ N, where N is the radical of λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) N = V ⊕ Z, where Z = {A ∈ N : λ(A, N) = λ(N, A) = 0} is the annihilator of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, Z is an ideal of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We have L = H ⊕ V ⊕ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Suppose that Z � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {Hi}, {Vi}, {Zi} be an orthonormal basis of H, V, and Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Put {Xi} = {Hi} ∪ {Vi} ∪ {Zi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any 0 � Z ∈ Z, by 12 HUI ZHANG AND ZAILI YAN hypothesis we have 0 > ⟨MλZ, Z⟩ =2 � ij |⟨λ(Xi, X j), Z⟩|2 − 2 � ij |⟨λ(Z, Xi), X j⟩|2 − 2 � ij |⟨λ(Xi, Z), X j⟩|2 =2 � ij � |⟨λ(Zi, H j), Z⟩|2 + |⟨λ(Hi, Z j), Z⟩|2� + α(Z) − 2 � ij |⟨λ(Z, Hi), Z j⟩|2 − 2 � ij |⟨λ(Hi, Z), Z j⟩|2, where α(Z) = 2 � ij |⟨λ(Yi, Y j), Z⟩|2 ≥ 0, {Yi} = {Hi} ∪ {Vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This implies 0 > � k ⟨MλZk, Zk⟩ = � k α(Zk) ≥ 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So Z = 0, and consequently, N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Therefore L is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This completes the proof of theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, by the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6, we know that if [µ] ∈ An for which there exists [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] such that Mλ is negative definite, then µ is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The maxima of Fn : An → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We say that an algebra λ degenerates to µ, write as λ → µ if µ ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='λ, the closure of GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='λ with respect to the usual topology of Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The degeneration λ → µ is called direct degeneration if there are no nontrivial chains: λ → ν → µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The degeneration level of an algebra is the maximum length of chain of direct degenerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8 ([3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' An n-dimensional associative algebra is of degeneration level one if and only if it is isomorphic to one of the following (1) µl: µl(X1, Xi) = Xi, i = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (2) µr: µr(Xi, X1) = Xi, i = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3) µca: µs(X1, X1) = X2, where {X1, · · · , Xn} is a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The functional Fn : An → R attains its maximal value at a point [µ] ∈ Ln, n ≥ 3 if and only if µ is isomorphic to the commutative associative algebra µca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In such a case, Fn([µ]) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that Fn : An → R attains its maximal value at a point [µ] ∈ An, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3, we know that [µ] is also a critical of Fn : PVn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5 that Fn|GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] also attains its minimum value at a point [µ] , consequently Fn|GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] is a constant, so GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] = U(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4) THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 13 The relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4) implies that the only non-trivial degeneration of µ is 0 ([8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1] ), conse- quently the degeneration level of µ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that the critical points [µl], [µr] are both of type (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, n − 1), and [µca] is of type (3 < 5 < 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, n − 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4, we know Fn([µca]) = 20 > 4 = Fn([µl]) = Fn([µr]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The structure for the critical points of Fn : An → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the following, we discuss the structure for an arbitrary critical points of Fn : An → R by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] be a critical point of Fn : An → R with Mµ = cµI + Dµ of type (k1 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr), where cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the orthogonal decomposition Cn = A− ⊕ A0 ⊕ A+, where A−, A0 and A+ denote the direct sum of eigenspaces of Dµ with eigenvalues smaller than zero, equal to zero and larger than zero, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then the following conditions hold: (i) ann(µ) ⊂ A+, where ann(µ) is the annihilator of µ (ii) A+ ⊂ N(µ), where N(µ) is the radical of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iii) A− ⊂ (C(µ) ∩ N(µ)) \\ ann(µ), where C(µ) is the center of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iv) (Lµ A − Rµ A)∗ ∈ Der(µ) for any A ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the induced Lie algebra of A0 is reductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (i), assume that X ∈ ann(µ) and DµX = cX, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) ⟨MµX, X⟩ = 2 � i, j |⟨µ(Xi, X j), X⟩|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since Mµ = cµI + Dµ, then 0 ≤ ⟨MµX, X⟩ = (cµ + c)⟨X, X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 that c ≥ −cµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (ii), it is an immediate consequence of (iii) by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Now, we prove (iii) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that DµX = cX for some c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since cLµ X = [Dµ, Lµ X], cRµ X = [Dµ, Rµ X], then c tr(Lµ X − Rµ X)(Lµ X − Rµ X)∗ = tr[Dµ, (Lµ X − Rµ X)](Lµ X − Rµ X)∗ = tr[Mµ, (Lµ X − Rµ X)](Lµ X − Rµ X)∗ = tr Mµ[(Lµ X − Rµ X), (Lµ X − Rµ X)∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Noting that (Lµ X − Rµ X) ∈ Der(µ), by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 we have c tr(Lµ X − Rµ X)(Lµ X − Rµ X)∗ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 14 HUI ZHANG AND ZAILI YAN It follows that (Lµ X − Rµ X) = 0 since c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So X ∈ C(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4, it is easy to see that X ∈ N(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Using (i), we conclude A− ⊂ (C(µ) ∩ N(µ)) \\ ann(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (iv), we first note that [Dµ, Lµ A] = Lµ DµA, [Dµ, Rµ A] = Rµ DµA, for any A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If A ∈ A0, we have [Dµ, Lµ A] = [Dµ, Rµ A] = 0, and so tr Mµ[(Lµ A − Rµ A), (Lµ A − Rµ A)∗] = tr(cµI + Dµ)[(Lµ A − Rµ A), (Lµ A − Rµ A)∗] = tr Dµ[(Lµ A − Rµ A), (Lµ A − Rµ A)∗] = tr[Dµ, (Lµ A − Rµ A)](Lµ A − Rµ A)∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2, it follows that (Lµ A − Rµ A)∗ ∈ Der(µ) since (Lµ A − Rµ A) ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ In the sequel, we give a description of the critical points in terms of those which are nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [λ] be a nilpotent critical point of Fm : Am → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define L(λ) : = {Φ ∈ End(Cm) : Φ(λ(X, Y)) = λ(ΦX, Y)}, R(λ) : = {Ψ ∈ End(Cm) : Ψ(λ(X, Y)) = λ(X, ΨY)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, we set Γl = {Φ ∈ L(λ) : [Φ, Ψ] = 0, ∀Ψ ∈ R(λ)}, Γr = {Ψ ∈ R(λ) : [Φ, Ψ] = 0, ∀Φ ∈ L(λ)}, and Γ(λ) : = {(Φ, Ψ) ∈ Γl × Γr : λ(·, Φ(·)) = λ(Ψ(·), ·)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any (Φi, Ψi) ∈ Γ(λ), i = 1, 2, we define (Φ1, Ψ1)(Φ2, Ψ2) := (Φ1Φ2, Ψ2Ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows that Γ(λ) is an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that S ⊂ Γ(λ) is a subalgebra such that (Φ∗, Ψ∗) ∈ S for any (Φ, Ψ) ∈ S, then S is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that S is an associative algebra of matrices, which are closed under conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define an Hermitian inner product on S by ⟨H1, H2⟩ := tr H1H∗ 2 = tr Φ1Φ∗ 2 + tr Ψ1Ψ∗ 2, ∀Hi = (Φi, Ψi) ∈ S, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows that ⟨HH1, H2⟩ = ⟨H1, H∗H2⟩, ⟨H1H, H2⟩ = ⟨H1, H2H∗⟩ for any H, H1, H2 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let I be an ideal in S and I⊥ denote the orthogonal complement of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it is easy to see that I⊥ is also an ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let S = R⊕N, where N is the radical of S and R = N⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that R and N are both ideals of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, R is semisimple, and N is the annihilator of S (by considering the derived series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since S is an associative algebra of matrices which are closed under conjugate transpose, then HH∗ = 0 for any H ∈ N, hence H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So N = 0, and S is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 15 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [λ] be a nilpotent critical point of Fm : Am → R with Mλ = cλI + Dλ of type (k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d2, · · · , dr), where cλ ∈ R and Dλ ∈ Der(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that S ⊂ Γ(λ) is a subalgebra of dimension d1 such that (Φ∗, Ψ∗) ∈ S, [Dλ, Φ] = [Dλ, Ψ] = 0 for any (Φ, Ψ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the following semidirect sum µ = S ⋉ λ, where µ((Φ1, Ψ1) + X1, (Φ2, Ψ2) + X2) := (Φ1Φ2, Ψ2Ψ1) + Φ1(X2) + Ψ2(X1) + X1X2, for any (Φ1, Ψ1), (Φ2, Ψ2) ∈ S, X1, X2 ∈ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then µ is an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If we extend the Hermitian inner product on Cm by setting ⟨H, K⟩ = − 2 cλ (tr LS HLS K∗ + tr HK∗), H, K ∈ S, then [µ] is a critical point of type (0, k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr) for the functional Fn : An → R, where n = d1 + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any H = (Φ, Ψ) ∈ S, we have Lµ H = � LS H 0 0 Φ � , Rµ H = � RS H 0 0 Ψ � , where Lµ H, Rµ H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' LS H, RS H) denote the left and right multiplication by H of the algebra µ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' S), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='11, we know that S is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows that there is an orthonormal basis {Hi = (Φi, Ψi)} ⊂ S such that Φi∗ = −Φi, Ψi∗ = −Ψi, and Lµ Hi, Rµ Hi are skew-Hermitian for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {Hi} ∪ {Xi} be an orthonormal basis of Cn = S ⊕ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then for any H = (Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Ψ) ∈ S and X ∈ Cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' we have ⟨MµX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' H⟩ = −2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨µ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨µ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ − 2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨µ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨µ(H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ = −2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨λ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨Ψ(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ − 2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨λ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨Φ(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ = −2 � i ⟨λ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Ψ(Xi)⟩ − 2 � i ⟨λ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Φ(Xi)⟩ = −2 tr Ψ∗Rλ X − 2 tr Φ∗Lλ X = −2 tr Rλ Ψ∗(X) − 2 tr Lλ Φ∗(X) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' where Lλ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Rλ X denote the left and right multiplication by X of the algebra λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' and the last two equalities follow from that λ is nilpotent and (Φ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Ψ∗) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, since Φi∗ = −Φi, Ψi∗ = −Ψi for 16 HUI ZHANG AND ZAILI YAN each i, then [Φi, Φi∗] = 0, [Ψi, Ψi∗] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) we have ⟨MµX, Y⟩ = 2 � i, j ⟨µ(Hi, X j), X⟩⟨µ(Hi, X j), Y⟩ + 2 � i, j ⟨µ(Xi, H j), X⟩⟨µ(Xi, H j), Y⟩ + 2 � i, j ⟨µ(Xi, X j), X⟩⟨µ(Xi, X j), Y⟩ − 2 � i, j ⟨µ(Hi, X), X j⟩⟨µ(Hi, Y), X j⟩ − 2 � i, j ⟨µ(Xi, X), X j⟩⟨µ(Xi, Y), X j⟩ − 2 � i, j ⟨µ(X, Hi), X j⟩⟨µ(Y, Hi), X j⟩ − 2 � i, j ⟨µ(X, Xi), X j⟩⟨µ(Y, Xi), X j⟩ = ⟨MλX, Y⟩ + 2 � i ⟨[Φi, Φi∗](X, Y⟩ + 2 � i ⟨[Ψi, Ψi∗](X), Y⟩ = ⟨MλX, Y⟩, for any X, Y ∈ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Therefore Mµ|Cm = Mλ = cλI + Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' On the other hand, noting that Lµ Hi and Rµ Hi are skew-Hermitian for each i, then for any H = (Φ, Ψ) ∈ S, we have ⟨MµH, H⟩ = 2 � i, j ⟨µ(Hi, H j), H⟩⟨µ(Hi, H j), H⟩ − 2 � i, j ⟨µ(Hi, H), H j⟩⟨µ(Hi, H), H j⟩ − 2 � i, j ⟨µ(Xi, H), X j⟩⟨µ(Xi, H), X j⟩ − 2 � i, j ⟨µ(H, Hi), H j⟩⟨µ(H, Hi), H j⟩ − 2 � i, j ⟨µ(H, Xi), X j⟩⟨µ(H, Xi), X j⟩ = −2(tr LS HLS H∗ + tr ΦΦ∗ + tr ΨΨ∗) = −2(tr LS HLS H∗ + tr HH∗) = cλ⟨H, H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So Mµ = cµI + Dµ, where cµ = cλ, and Dµ = � 0 0 0 Dλ � ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let the notation be as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If (Lµ A)∗ ∈ {Lµ A : A ∈ A0} and (Rµ A)∗ ∈ {Rµ A : A ∈ A0} for any A ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows from a similar proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='11 that A0 is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, the radical of [µ] corresponds to a critical point of type (k1 < · · · < ˆks < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, · · · , ˆds, · · · , dr) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12, where ks = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Examples In this section, we classify the critical points of Fn : An → R for n = 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It shows that every 2-dimensional associative algebra is isomorphic to a critical point of F2, and there exists only THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 17 one 3-dimensional associative algebra which is not isomorphic to any critical point of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Finally, based on the discussion in previous sections, we collect some natural and interesting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For reader’s convenience, we recall the notation in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {e1, e2, · · · , en} be a basis of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define the bilinear maps ψi, j k : Cn × Cn → Cn by ψi, j k (emen) = δi mδj nek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that any algebra can be expressed in the form d = � ijk ck ijψi, j k , where ck ij ∈ C are the structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The classification of two-dimensional associative algebras can be found in [2, TABLE 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We give the classification of the critical points of F2 : A2 → R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Two-dimensional associative algebras, critical types and critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Multiplication relation Critical type Critical value � d1 = ψ1,1 1 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 � d2 = ψ1,1 1 + ψ1,2 2 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 � d3 = ψ1,1 1 + ψ2,1 2 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 � d4 = ψ1,1 1 + ψ2,2 2 (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2) 2 � d5 = ψ1,1 2 (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 20 � d6 = ψ1,1 1 + ψ1,2 2 + ψ2,1 2 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 Indeed, endow these algebras with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2} is an orthonormal basis, then it is easy to obtain TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For example, the multiplication relation of µ := (d6, ⟨·, ·⟩) is given by: e1e1 = e1, e1e2 = e2, e2e1 = e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' With respect to the given orthonormal basis {e1, e2}, the left and right multiplications of µ are represented by Lµ e1 = � 1 0 0 1 � , Lµ e2 = � 0 0 1 0 � , Rµ e1 = � 1 0 0 1 � , Rµ e2 = � 0 0 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7) that Mµ = � −6 0 0 0 � Set cµ := tr M2 µ tr Mµ , then cµ = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that Mµ = cµI + Dµ, where Dµ = � 0 0 0 6 � is clearly a derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So [µ] is a critical point of F2 : A2 → R with the critical type (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) and F2([µ]) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 18 HUI ZHANG AND ZAILI YAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Three-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The complete classification of three-dimensional associative algebras can be found in [2, TABLE 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The following table gives the classification of the critical points of F3 : A3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Three-dimensional associative algebras, critical types and critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Multiplication relation Critical type Critical value � d1 = ψ1,1 1 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d2 = ψ1,1 1 + ψ2,2 3 (0 < 1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 10 3 � d3 = ψ1,1 1 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d4 = ψ1,1 1 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d5 = ψ1,1 1 + ψ1,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d6 = ψ1,1 1 + ψ3,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d7 = ψ1,1 1 + ψ2,1 2 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d8 = ψ1,1 1 + ψ2,1 2 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d9 = ψ1,1 1 + ψ2,1 2 + ψ1,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d10 = ψ1,1 1 + ψ2,1 2 + ψ3,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d11 = ψ1,1 1 + ψ2,2 2 + ψ2,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d12 = ψ1,1 1 + ψ2,2 2 + ψ2,3 3 + ψ3,2 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d13 = ψ1,1 1 + ψ2,2 2 + ψ2,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d14 = ψ1,1 1 + ψ2,2 2 + ψ3,3 3 (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3) 4 3 � d15 = ψ1,1 2 (3 < 5 < 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 20 � d16 = ψ1,1 2 + ψ1,2 3 + ψ2,1 3 (1 < 2 < 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 20 3 � d17 = ψ1,1 1 + ψ1,1 2 + ψ1,2 2 + ψ2,1 2 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d18 = ψ1,1 1 + ψ1,1 2 + ψ1,2 2 + ψ2,1 2 + ψ1,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d19 = ψ3,3 3 + ψ1,1 2 + ψ1,3 1 + ψ3,1 1 + ψ2,3 2 + ψ3,2 2 (0 < 1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 10 3 � d20 = ψ1,1 1 + ψ1,2 2 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d21 = ψ1,1 3 + ψ1,2 3 − ψ2,1 3 − − � d22 = xψ1,2 3 + yψ2,1 3 (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 12 Indeed, endow the algebras with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2, e3} is an orthonormal basis, it is easy to obtain all cases in TABLE II except for d2, d10, d11, d12, d13, d17, d18, d21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For the cases d2, d10, d11, d12, it follows from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4 and TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For the cases d13, d17, d18, it follows from [5] that d13 � U3 1, d17 � W3 10 and d18 � U3 0, where U3 1, W3 10 and U3 0 are defined by U3 1 : ψ1,1 1 + ψ3,3 1 + ψ1,2 2 + ψ2,1 2 + ψ2,3 2 + ψ1,3 3 + ψ3,1 3 − ψ3,2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' W3 10 : ψ1,2 1 + ψ2,1 1 + ψ2,2 2 + ψ2,3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' U3 0 : ψ1,1 2 + ψ1,2 2 + ψ2,1 2 + ψ1,3 3 + ψ3,1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Endow U3 1, W3 10 and U3 0 with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2, e3} is an orthonormal basis, then it is easy to obtain the corresponding critical types and values for d13, d17, d18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 19 In the sequel, we follow a similar procedure as in [6, 16] to classify all Hermitian inner products on d21, then show that for any Hermitian inner product ⟨·, ·⟩ on d21, (d21, ⟨·, ·⟩) cannot be a critical point of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' First, note that the multiplication relation of d21 is given as follows: e1e1 = e3, e1e2 = e3, e2e1 = −e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Denote by ⟨·, ·⟩0 the Hermitian inner product on d21 such that {e1, e2, e3} is orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' With respect to this basis {e1, e2, e3}, the automorphism group of d21 is given by Aut(d21) = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed a 0 0 b a 0 c d a2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⊂ GL(3, C), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1) where 0 � a ∈ C, and b, c, d ∈ C are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any Hermitian inner product ⟨·, ·⟩ on d21, there exist k > 0 and φ ∈ Aut(d21) such that {aφe1, φe2, φe3} is orthonormal with respective to k⟨·, ·⟩, where a > 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It suffices to prove that U = {diag(a, 1, 1) : a > 0} ⊂ GL(3, C) is a set of representatives for the action C×Aut(d21) on M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', the space of all Hermitian inner products on d21, which can be identified with the homogeneous space GL(3, C)/U(3) at the base point ⟨·, ·⟩0 ∈ M (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, since � g∈U C×Aut(d21) · g · U(3) = GL(3, C), it follows that U is a set of representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any Hermitian inner product ⟨·, ·⟩ on d21, we know that there exists g0 ∈ U such that ⟨·, ·⟩ ∈ (C×Aut(d21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='⟨·, ·⟩0) Hence there exist c ∈ C×, φ ∈ Aut(d21) such that ⟨·, ·⟩ = (cφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='⟨·, ·⟩0) = (cφg0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='⟨·, ·⟩0) Put k = |c|2, then k⟨·, ·⟩ = k⟨(cφg0)−1(·), (cφg0)−1(·)⟩0 = kc−1¯c−1⟨(φg0)−1(·), (φg0)−1(·)⟩0 = ⟨(φg0)−1(·), (φg0)−1(·)⟩0 Since g0 ∈ U, then g0 = diag{a, 1, 1} for some a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that {aφe1, φe2, φe3} is orthonormal with respective to k⟨·, ·⟩ □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any Hermitian inner product ⟨·, ·⟩ on d21, (d21, ⟨·, ·⟩) can not be a critical point of F3 : A3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 20 HUI ZHANG AND ZAILI YAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that ⟨·, ·⟩ is a Hermitian inner product on d21 such that (d21, ⟨·, ·⟩) is a critical point of F3 : A3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then the critical type is necessarily of (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, for the Hermitian inner product ⟨·, ·⟩ on d21, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 we know that there exist k > 0 and φ ∈ Aut(d21) such that {x1 = aφe1, x2 = φe2, x3 = φe3} is orthonormal with respective to k⟨·, ·⟩, where a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' With respect to the basis {x1, x2, x3}, the multiplication relation of d21 is given as follows x1x1 = a2x3, x1x2 = ax3, x2x1 = −ax3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 and a straightforward calculation, it follows that the critical type is of (3a4 + 6a2 + 8, 5a4 + 10a2 + 8, 2(3a4 + 8a2 + 8)) which is never of type (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) for any a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This is a contradiction by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, and the proposition is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By the previous discussion, we know that the critical types of Fn : An → R, n = 2, 3, are necessarily nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So it is natural to ask the following question: Let [µ] ∈ An be a critical point of Fn : An → R with Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Are all the eigenvalues of Dµ necessarily nonnegative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' On the other hand, it will be also interesting to construct or classify the critical points [µ] of Fn : An → R such that Dµ has negative eigenvalues if the above question does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We note that 2-step nilpotent Lie algebras are automatically associative algebras, so it follows from [13, Example 1] that there exist associative algebras whose Nikolayevsky derivations do admit negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Statements and Declarations The authors declare that there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' References [1] B¨ohm, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lafuente, R: Immortal homogeneous Ricci flows, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 212 (2018), 461–529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [2] Fialowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Penkava, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': The moduli space of 3-dimensional associative algebras, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Algebra 37(10) (2009) 3666–3685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [3] Khudoyberdiyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Omirov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': The classification of algebras of level one, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 439(11) (2013), 3460– 3463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [4] Kirwan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Momentum maps and reduction in algebraic geometry, Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 9 (1998), 135–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [5] Kobayashi, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Shirayanagi, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Tsukada, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Takahasi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': A complete classification of three-dimensional algebras over R and C–(visiting old, learn new),Asian-Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 14 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [6] Kubo, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Onda, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Taketomi, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Tamaru, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups, Hiroshima math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 46 (2016), 357–374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [7] Lauret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': On the moment map for the variety of Lie algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 202 (2003), 392–423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [8] Lauret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Degenerations of Lie algebras and geometry of Lie groups, Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 18 (2003), 177–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [9] Lauret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Einstein solvmanifolds are standard, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 172 (2010), 1859–1877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [10] Lauret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Ricci soliton solvmanifolds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' reine angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 650 (2011), 1–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [11] Lauret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Finding solitons, Notices Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 67 (2020), 647–657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [12] Ness, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': A stratification of the null cone via the moment map, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', 106 (1984), 1281-1329 (with an appendix by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Mumford).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 21 [13] Nikolayevsky, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Nilradicals of Einstein solvmanifolds, arXiv:math/0612117v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='DG] (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [14] Nikolayevsky, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': Einstein solvmanifolds and the pre-Einstein derivation, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 363 (2011), 3935–3958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [15] Pierce, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' : Associative Algebras, Springer-Verlag, New York, Heidelberg, Berlin, 1982 [16] Taketomi, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Tamaru, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': On the nonexistence of left-invariant Ricci solitons a conjecture and examples, Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Groups 23 (2018), 257–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [17] Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': The moment map for the variety of 3-Lie algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 283 (2022), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 11, Article ID 109683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (Hui Zhang) School of Mathematics, Southeast University, Nanjing 210096, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' China Email address: 2120160023@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='cn School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province, 315211, People’s Republic of China Email address: yanzaili@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} diff --git a/4NAyT4oBgHgl3EQf1_nb/content/2301.00745v1.pdf b/4NAyT4oBgHgl3EQf1_nb/content/2301.00745v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..419b23e35f0cf0c6ba2bbf05202707d80ae8903d --- /dev/null +++ b/4NAyT4oBgHgl3EQf1_nb/content/2301.00745v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:48f940b5d31c0b2f0eb31b46aa8bccdc475b04f480f9058b23d6b61cd476d177 +size 302817 diff --git a/4NAyT4oBgHgl3EQf1_nb/vector_store/index.faiss b/4NAyT4oBgHgl3EQf1_nb/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..61f76ee478248d89a983eb49faa137a042c9ff1c --- /dev/null +++ b/4NAyT4oBgHgl3EQf1_nb/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ac7dc53f312cc9b8f03464845b313b0e1fd027c7d01d0c6702b63886e90f9878 +size 4325421 diff --git a/4NAyT4oBgHgl3EQf1_nb/vector_store/index.pkl b/4NAyT4oBgHgl3EQf1_nb/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1562458d8cb3cf4f6a5abba2a58f846ce302d38d --- /dev/null +++ b/4NAyT4oBgHgl3EQf1_nb/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3eddfd4c00e13b2e580370f6891bd302620722001487fa4cb9239c00f34f86fb +size 170819 diff --git a/4dAyT4oBgHgl3EQfpPgc/content/tmp_files/2301.00520v1.pdf.txt b/4dAyT4oBgHgl3EQfpPgc/content/tmp_files/2301.00520v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b396243f4ed7008fadfa0e46d9e7bc8044112fc --- /dev/null +++ b/4dAyT4oBgHgl3EQfpPgc/content/tmp_files/2301.00520v1.pdf.txt @@ -0,0 +1,1341 @@ +Quantum Annealing vs. QAOA: 127 Qubit Higher-Order Ising +Problems on NISQ Computers +Elijah Pelofske∗1, Andreas B¨artschi†1, and Stephan Eidenbenz1 +1CCS-3 Information Sciences, Los Alamos National Laboratory +Abstract +Quantum annealing (QA) and Quantum Alternating Operator Ansatz (QAOA) are both heuristic quantum +algorithms intended for sampling optimal solutions of combinatorial optimization problems. In this article we +implement a rigorous direct comparison between QA on D-Wave hardware and QAOA on IBMQ hardware. The +studied problems are instances of a class of Ising problems, with variable assignments of +1 or −1, that contain +cubic ZZZ interactions (higher order terms) and match both the native connectivity of the Pegasus topology D- +Wave chips and the heavy hexagonal lattice of the IBMQ chips. The novel QAOA implementation on the heavy +hexagonal lattice has a CNOT depth of 6 per round and allows for usage of an entire heavy hexagonal lattice. +Experimentally, QAOA is executed on an ensemble of randomly generated Ising instances with a grid search +over 1 and 2 round angles using all 127 programmable superconducting transmon qubits of ibm washington. +The error suppression technique digital dynamical decoupling (DDD) is also tested on all QAOA circuits. QA +is executed on the same Ising instances with the programmable superconducting flux qubit devices D-Wave +Advantage system4.1 and Advantage system6.1 using modified annealing schedules with pauses. We find that +QA outperforms QAOA on all problem instances. We also find that DDD enables 2-round QAOA to outperform +1-round QAOA, which is not the case without DDD. +1 +Introduction +Quantum annealing (QA) in the transverse field Ising model (TFIM) is an analog computation technology which +utilizes quantum fluctuations in order to search for ground state solutions of a problem Hamiltonian [1–4]. D-Wave +quantum annealers are programmable hardware implementations of quantum annealing which use superconducting +flux qubits. +Quantum alternating operator ansatz (QAOA) is a hybrid quantum classical algorithm for sampling combina- +torial optimization problems [5, 6], the quantum component of which can be instantiated with a programmable +gate-based universal quantum computer. The quantum approximate optimization algorithm [7] was the first vari- +ational algorithm of this type, which was then generalized to the quantum alternating operator ansatz algorithm +[5]. +QAOA is effectively a Trotterization of the Quantum Adiabatic Algorithm, and is overall similar to Quantum +Annealing. In particular both algorithms address combinatorial optimization problems. The exact characteristics +of how both QA and QAOA will scale to large system sizes is currently not fully understood, in particular because +quantum hardware is still in the NISQ era [8–10]. For example, there is evidence that QAOA may be more difficult +for classical computers to simulate than quantum annealing, which could make it a viable candidate for quantum +advantage [11]. Therefore it is of interest to investigate differences between QAOA and QA, and determine how +these algorithms will scale [12–17]. There have been experimental QAOA implementations which used up to 27 +qubits [18] and 23 qubits [19]. There have also been QAOA experiments which had circuit depth up to 159 [20] +and 148 [21]. +The contributions of this article are as follows: +1. We provide a direct comparison between QAOA and Quantum Annealing in terms of experiments on D-Wave +and IBMQ hardware. This comparison uses a comparable parameter search space for QA and QAOA, uses +no minor embedding for quantum annealing, and uses short depth QAOA circuits, thus providing a fair +comparison of the two algorithms. We show that QAOA is better than random sampling, and quantum +annealing clearly outperforms QAOA. +∗Email: epelofske@lanl.gov +†Email: baertschi@lanl.gov +1 +arXiv:2301.00520v1 [quant-ph] 2 Jan 2023 + +Device name +Topology/chip +name +Available qubits +Available +couplers/ +CNOTs +Computation type +Advantage system4.1 +Pegasus P16 +5627 +40279 +QA +Advantage system6.1 +Pegasus P16 +5616 +40135 +QA +ibm washington +Eagle r1 +heavy-hexagonal +127 +142 +Universal gate-model +Table 1: NISQ hardware summary at the time the experiments were executed. +The hardware yield (e.g., the +number of available qubits or two qubit interactions) for all of these devices can be less than the logical lattice +because of hardware defects, and can also change over time if device calibration changes. +2. The QAOA algorithm we present is tailored for short depth circuit construction on the heavy hexagonal +lattice, therefore allowing full usage of any heavy hexagonal topology quantum processor in the future. We +use all 127 qubits of the ibm washington chip in order to execute the largest QAOA circuit, in terms of +qubits, to date. +3. The problem instances that are used to compare quantum annealing and QAOA are specifically constructed +to include higher order terms, specifically three variable (cubic) terms. QAOA can directly implement higher +order terms, and quantum annealing requires order reduction using auxiliary variables to implement these +higher order terms. This is the largest experimental demonstration of QAOA with higher order terms to date. +4. In order to mitigate errors when executing the QAOA circuits, we utilize digital dynamical decoupling. This +is the largest usage of dynamical decoupling in terms of qubit system size to date, and the results show that +digital dynamical decoupling improves performance for two round QAOA, suggesting that it will be useful +for computations with large numbers of qubits in the noisy regime. +In Section 2 the QAOA and QA hardware implementations are detailed. Section 3 details the experimental +results and how the two algorithms compare. Section 4 concludes with what the results indicate and future research +directions. The figures in this article are generated using matplotlib [22, 23], and Qiskit [24] in Python 3. +2 +Methods +The problem instances are defined in Section 2.1. +In Section 2.2 the QAOA circuit algorithm and hardware +parameters are defined. In Section 2.3 the quantum annealing implementation is defined. +2.1 +Problem instances +The NISQ computers which are used in this comparison are detailed in Table 1; the clear difference between the +D-Wave quantum annealers and ibm washington is the number of qubits that are available. The additional qubits +available on the quantum annealers will allow us to embed multiple problem instances onto the chips. The current +IBMQ devices have a graph topology referred to as the heavy-hexagonal lattice [25]. Therefore, for a direct QAOA +and QA comparison we would want to be able to create QAOA circuits which match the logical heavy-hexagonal +lattice and the quantum annealer graph topology of Pegasus. For this direct comparison we target D-Wave quantum +annealers with Pegasus graph hardware [26, 27] connectivities. The two current D-Wave quantum annealers with +Pegasus hardware graphs have chip id names Advantage system6.1 and Advantage system4.1. The goal for this +direct comparison is that ideally we want problems which can be instantiated on all three of the devices in Table +1. In particular, we want these implementations to not be unfairly costly in terms of implementation overhead. +For example we do not want to introduce unnecessary qubit swapping in the QAOA circuit because that would +introduce larger circuit depths which would introduce more decoherence in the computation. We also do not want +to introduce unnecessary minor-embedding in the problems for quantum annealers. +The other property of these problem instances that is of interest is an introduction of higher order terms, +specifically cubic ZZZ interactions [28] also referred to as multi-body interactions [29], in addition to random +linear and quadratic terms. These higher order terms require both QAOA and QA to be handle these higher order +variable interactions, which is an additional test on the capability of both algorithms. QAOA can naturally handle +2 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +119 +120 +121 +122 +123 +124 +125 +126 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +119 +120 +121 +122 +123 +124 +125 +126 +Figure 1: Left: ibm washington graph connectivity, where qubits are connected by CNOT (also referred to as cx) +gates. The ideal lattice is called the heavy-hexagonal lattice. Note that there are two missing graph edges from +the lattice between qubits 8-9 and 109-114. The total number of qubits (nodes) is 127. The edges of the graph are +three colored (red, blue, and green) such that no node shares two or more edges with the same color. The node +colorings of light and dark gray show that the heavy hexagonal lattice is bipartite (meaning it can be partitioned +into two disjoint sets). The three edge coloring is consistent with the QAOA circuit construction in Figure 2. +Right: Example of a single random problem instance with cubic terms (see Eq. (1)) on the ibm washington +graph. The linear and quadratic terms are shown using two distinct colors (red and green). The nodes and edges +colored red denote a weight of −1 and the nodes and edges colored green denote a weight of +1. The cubic terms +are represented by ovals around the three qubits which define the cubic variable interactions. Like the linear and +quadratic terms, the color of the oval representing the cubic terms represents the sign of the weight on the terms, +where green is +1 and red is −1. +higher order terms [30]. Implementing high order terms with QA requires introducing auxiliary variables in order +to perform order reduction to get a problem structure that is comprised of only linear and quadratic terms, so that +it can be implemented on the hardware, but whose optimal solutions match the optimal solutions of the original +high order polynomial [3, 31–34]. +Taking each of these characteristics into account, we create a class of random problems which follow the native +device connectivities in Table 1. The problem instances we will be considering are Ising problems defined on the +hardware connectivity graph of the heavy hexagonal lattice of the device, which for these experiments will be +ibm washington. +C(x) = +� +v∈N +cv · xv + +� +(i,j)∈E +ci,j · xi · xj + +� +l∈D +cl · xl · xn1(l) · xn2(l) +(1) +Eq. (1) defines the class of problem Isings as follows. N is the set of qubits, or variables, that exist on the heavy +hexagonal layout topology. E is the edge set of all two qubit (CNOT) gates that can allow two qubits, indexed as +i and j, to interact. Any heavy hexagonal lattice is a bipartite graph with vertices V = V2 ∪ V3 where V2 consists +of vertices with a maximum degree of 2, and V3 consists of vertices with a maximum degree of 3. D is the set +of vertices in V2 which all have degree exactly equal to 2. n1 is a function which gives the qubit (variable) index +of the first of the two neighbors of a degree-2 node, and n2 provides the qubit (variable) index of the second of +the two neighbors of any degree-2 node. cv, ci,j, and ct are all functions representing the random selection of the +linear, quadratic, and cubic coefficients, respectively. These coefficients could be drawn from any distribution - in +this case we draw the coefficients from {+1, −1} with probability 0.5. The decision variables are xi, where the +possible variable states are the spins −1 or +1. Combined, any vector of variable states x can be evaluated given +this objective function formulation of Eq. (1). +The heavy hexagonal topology of ibm washington, along with an overlay showing one of the random problem +instances with cubic terms defined on ibm washington, is shown in Figure 1. Each term coefficient was chosen to +3 + +be either +1 or −1 in order to mitigate the potential problem of limited precision for the programming control on +all of the NISQ devices. 10 random instances of this class of problems are generated and sampled using QAOA and +QA, the implementations of each will be discussed next. +2.2 +QAOA +Given a combinatorial optimization problem over inputs x ∈ {0, 1}n, let f(x): {0, 1}n → R be the objective function +which evaluates the cost of solution x. For a maximization (or minimization) problem, the goal is to find a variable +assignment vector x for which f(x) is maximized (or minimized). The QAOA algorithm consists of the following +components: +• An initial state |ψ⟩ +• A phase separating Hamiltonian: HP |x⟩ = f(x) |x⟩ +• A mixing Hamiltonian: HM +• An integer p ≥ 1, the number of rounds to run the algorithm +• Two real vectors ⃗γ = (γ1, ..., γp) and ⃗β = (β1, ..., βp), each with length p +The algorithm consists of preparing the initial state |ψ⟩, then applying p rounds of the alternating simulation +of the phase separating Hamiltonian and the mixing Hamiltonian: +|⃗γ, ⃗β⟩ = e−iβpHM e−iγpHP +� +�� +� +round p +· · · e−iβ1HM e−iγ1HP +� +�� +� +round 1 +|ψ⟩ +(2) +Within reach round, HP is applied first, which separates the basis states of the state vector by phases e−iγf(x). +HM then provides parameterized interference between solutions of different cost values. After p rounds, the state +|⃗γ, ⃗β⟩ is measured in the computational basis and returns a sample solution y of cost value f(y) with probability +| ⟨y|⃗γ, ⃗β⟩ |2. +The aim of QAOA is to prepare the state |⃗γ, ⃗β⟩ from which we can sample a solution y with high cost value f(y). +Therefore, in order to use QAOA the task is to find angles ⃗γ and ⃗β such that the expectation value ⟨⃗γ, ⃗β|HP |⃗γ, ⃗β⟩ +is large (−HP for minimization problems). In the limit p → ∞, QAOA is effectively a Trotterization of of the +Quantum Adiabatic Algorithm, and in general as we increase p we expect to see a corresponding increase in the +probability of sampling the optimal solution [17]. The challenge is the classical outer loop component of finding +the good angles ⃗γ and ⃗β for all rounds p, which has a high computational cost as p increases. +Variational quantum algorithms, such as QAOA, have been a subject of large amount of attention, in large part +because of the problem domains that variational algorithms can address (such as combinatorial optimization) [35]. +One of the challenges however with variational quantum algorithms is that the classical component of parameter +selection, in the case of QAOA this is the angle finding problem, is not solved and is even more difficult when +noise is present in the computation [36]. Typically the optimal angles for QAOA are computed exactly for small +problem instances [15, 37]. +However, in this case the angle finding approach we will use is a reasonably high +resolution gridsearch over the possible angles. Note however that a fine gridsearch scales exponentially with the +number of QAOA rounds p, and therefore is not advisable for practical high round QAOA [6, 7]. Exactly computing +what the optimal angles are for problems of this size would be quite computationally intensive, especially with the +introduction of higher order terms. We leave the problem of exactly computing the optimal QAOA angles up to +future work. +Figure 2 describes the short depth QAOA circuit construction for sampling the higher order Ising test instance. +This algorithm can be applied to any heavy hexagonal lattice topology, which allows for executing the QAOA +circuits on the 127 variable instances on the IBMQ ibm washington backend. For the class of Isings with higher +order terms defined in Section 2.1, the QAOA angle ranges which are used are γ1, . . . , γp ∈ [0, π) and β1, . . . , βp−1 ∈ +[0, π), βp ∈ [0, π +2 ) where p is the number of QAOA rounds. Note that the halving of the angle search space for β +applies when p = 1. For optimizing the angles using the naive grid search for p = 1, β0 is varied over 60 linearly +spaced angles ∈ [0, π +2 ] and γ0 is varied over 120 linearly spaced angles ∈ [0, π]. For the high resolution gridsearch +for p = 2, β1 is varied over 5 linearly spaced angles ∈ [0, π +2 ] and γ0, γ1, and β0 are varied over 11 linearly spaced +angles ∈ [0, π]. Therefore, for p = 2 the angle gridsearch uses 6655 separate circuit executions (for each of the 10 +problem instances), and for p = 1 the angle gridsearch uses 7200 separate circuit executions. Each circuit execution +used 10, 000 samples in order to compute a robust distribution for each angle combination. +4 + +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +|0⟩ +H +H +H +H +H +H +H +H +H +H +H +dB +dC +dD +dE +dF +dG +dH +dI +dJ +dBA +dDE +dF C +dJI +dBC +dF I +dHG +dJK +dDC +dHI +dA +dK +dBAC +dDCE +dHGI +dJIK +dF CI +γdBC +γdDE +γdF I +γdJK +γdHI +γdDC +γdBA +γdF C +γdHG +γdA +γdB +γdC +γdD +γdE +γdF +γdG +γdH +γdI +γdJ +γdK +Z +Z +γdJI +γdBAC +γdDCE +γdF CI +γdHGI +γdJIK +Z +Z +Z +Z +Row +Column +Time +d... = ±1 +γd +Z +β +X += Rz(2γd) += Rx(2β) +Init +Phase Separator +Mixer +β +X +β +β +β +β +β +β +β +β +β +β +Eval +� +�� +� +⟨β,γ|HC|β,γ⟩ +Figure 2: A 1-round QAOA circuit: (left) The problem instance is a hardware-native bipartite graph with an +arbitrary 3-edge-coloring given by K˝onig’s line coloring theorem. (right) Any quadratic term (colored edge) gives +rise to a combination of two CNOTs and a Rz-rotation in the phase separator, giving a CNOT depth of 6 due to +the degree-3 nodes. When targeting the degree-2 nodes with the CNOT gates, these constructions can be nested, +leading to no overhead when implementing the three-qubit terms: these always have a degree-2 node in the middle +(see Eq. (1)). +In order to mitigate decoherence on idle qubits, digital dynamical decoupling (DDD) is also tested for all +QAOA circuits. Dynamical Decoupling is an open loop quantum control technique error suppression technique +for mitigating decoherence on idle qubits [38–41]. +Dynamical decoupling can be implemented with pulse level +quantum control, and digital dynamical decoupling can be implemented simply with circuit level instructions of +sequences of gates which are identities [41]. Note that digital dynamical decoupling is an approximation of pulse +level dynamical decoupling. +Dynamical decoupling has been experimentally demonstrated for superconducting +qubit quantum processors including IBMQ devices [42–44]. Dynamical decoupling in particular is applicable for +QAOA circuits because they can be relatively sparse and therefore have idle qubits [42]. DDD does not always +effective at consistently reducing errors during computation (for example because of other control errors present +on the device [40, 42]), and therefore the raw QAOA circuits are compared against the QAOA circuits with DDD +in the experiments section. In order to apply the DDD sequences to the OpenQASM [45] QAOA circuits, the +PadDynamicalDecoupling 1 method from Qiskit [24] is used, with the pulse alignment parameter set based on +the ibm washington backend properties. The native gateset of all current IBMQ backends is x, rz, cx, sx. The +circuit scheduling algorithm that is used for inserting the digital dynamical decoupling sequences is ALAP, which +schedules the stop time of instructions as late as possible 2. There are other scheduling algorithms that could be +applied which may increase the efficacy of dynamical decoupling. Note that the rz gate is a virtual gate which is +not executed on the hardware. There are different DDD gate sequences that can be applied, including Y-Y or X-X +sequences. Because the X Pauli gate is already a native gate of the IBMQ device, the X-X DDD sequence is used +for simplicity. +Note that the variable states for the optimization problems are either −1 or +1, but the circuit measurement +states are either 0 or 1. Therefore once the measurements are made on the QAOA circuits, for each variable in each +sample the variable state mapping of 0 → 1, 1 → −1 is performed. For circuit execution on the superconducting +transom qubit ibm washington, circuits are batched into jobs where each job is composed of a group of at most 250 +circuits - the maximum number of circuits for a job on ibm washington is currently 300, but we use 250 in order +to reduce job errors related to the size of jobs. Grouping circuits into jobs is helpful for reducing the total amount +of compute time required to prepare and measure each circuit. When submitting the circuits to the backend, +they are all first locally transpiled via Qiskit [24] with optimization level=3. This transpilation converts the +gateset to the ibm washington native gateset, and the transpiler optimization attempts to simplify the circuit +where possible. The QAOA circuit execution on ibm washington spanned a large amount of time, and therefore +the backend versions were not consistent. The exact backend software versions were 1.3.7, 1.3.8, 1.3.13, 1.3.15, +1.3.17. +1https://qiskit.org/documentation/locale/bn_BN/stubs/qiskit.transpiler.passes.PadDynamicalDecoupling.html +2https://qiskit.org/documentation/apidoc/transpiler_passes.html +5 + +dA +dB +dC +dBA +dBC +dBAC = +1 +dA +dB +dC +dBA +dBC ++1 +−1 +−1 +−1 +−2 +1 +1 +1 +2 +2 2 +dA +dB +dC +dBA +dBC ++1 +−3 +−3 +−1 ++1 +6 +0 +−1 +−1 +2 +−4 +−4 +dA +dB +dC +dBA +dBC +dBAC = −1 +dA +dB +dC +dBA +dBC ++1 +−1 +−1 +−1 +−2 +−1 +1 +1 +2 +2 2 +dA +dB +dC +dBA +dBC ++3 +−1 +−1 ++1 +−1 +2 +0 +1 +−1 +−2 +−4 +−4 +Figure 3: +(left) Two different embeddings for cubic +1/−1 terms. Each embedding needs two slack variable +qubits. Our overall embedding alternates between these two cubic term embeddings. Any embedding with only +one slack variable needs a 4-clique between the slack and the three original variables, which is not possible to +embed for consecutive cubic terms. (right) Embedding structures of the problem instances with higher order +terms embedded in parallel (independently) 6 times onto the logical Pegasus P16 graph. The view of this graph has +been slightly partitioned so that not all of the outer parts of the Pegasus chip are drawn. The light grey qubits and +couplers indicate unused hardware regions. The cyan coloring on nodes and edges denote the vertical qubits and +CNOTs on the ibm washington hardware graph (see Figure 1). The red coloring on nodes and edges denote the +horizontal lines of qubits and CNOTs on ibm washington. The green nodes and edges denote the order reduction +auxiliary variables. Note that the top right hand and lower left hand qubits are not present on the ibm washington +lattice - but for the purposes of generating the embeddings, these extra qubits are filled in to complete the lattice. +2.3 +Quantum Annealing +Quantum annealing is a proposed type of quantum computation which uses quantum fluctuations, such as quantum +tunneling, in order to search for the ground state of a user programmed Hamiltonian. Quantum annealing, in the +case of the transverse field Ising model implemented on D-Wave hardware, is explicitly described by the system +given in Eq. (3). The state begins at time zero purely in the transverse Hamiltonian state � +i σx +i , and then over +the course of the anneal (parameterized by the annealing time) the user programmed Ising is applied according the +function B(s). Together, A(s) and B(s) define the anneal schedules of the annealing process, and s is referred to +as the anneal fraction. The standard anneal schedule that is used is a linear interpolation between s = 0 and s = 1. +H = −A(s) +2 +� � +i +σx +i +� ++ B(s) +2 +� +Hising +� +(3) +The adiabatic theorem states that if changes to the Hamiltonian of the system are sufficiently slow, the system +will remain in the ground state of problem Hamiltonian, thereby providing a computational mechanism for comput- +ing the ground state of optimization problems. The user programmed Ising Hising, acting on n qubits, is defined +in Eq. (4). The quadratic terms and the linear terms combined define the optimization problem instance that the +annealing procedure will ideally find the ground state of. As with QAOA, the objective of quantum annealing is +6 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pause duration fraction +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Anneal fraction [s] +Figure 4: All modified (forward) quantum annealing schedules which are tested in order to find the best anneal +schedule with a pause. The symmetric pause inserted into the normal linearly interpolated schedule defining the +A(s) and B(s) functions can provide better ground state sampling probability. The anneal fraction at which this +pause occurs is varied between 0.1 and 0.9 in steps of 0.1. The pause duration, as a fraction of the total annealing +time, is also varied between 0.1 and 0.9 in steps of 0.1. Although not shown in this figure, the annealing times are +also varied between 10, 100, 1000, and 2000 microseconds. +to find the variable assignment vector x that minimizes the cost function which has the form of Eq. (4). +Hising = +n +� +i +hiσz +i + +n +� +i 6σ significance. +By performing likelihood tests with different spectral and spatial models and comparing the TeV +spectrum with multi-wavelength observations of nearby sources, we show that this excess is consistent +with a TeV halo associated with PSR J0359+5414, though future observation of HAWC and multi- +wavelength follow-ups are needed to confirm this nature. This new halo candidate is located in a +non-crowded region in the outer Galaxy. It shares similar properties to the other halos but its pulsar is +younger and radio-quiet. Our observation implies that TeV halos could commonly exist around pulsars +and their formation does not depend on the configuration of the pulsar magnetosphere. +Keywords: Pulsars (1306) — Gamma-ray astronomy(628) — High-energy astrophysics(739) +1. INTRODUCTION +Extended TeV gamma-ray emission has been observed +around several middle-aged (> 100 kyr) pulsars and +grouped as a new source class named “TeV halos” (Lin- +den et al. 2017; L´opez-Coto et al. 2022). Seven sources +are referenced as TeV halos in the online catalog for TeV +Astronomy, TeVCat1 (Wakely & Horan 2008), including +the first halos around the Geminga and Monogem pul- +sars, discovered by HAWC (Abeysekara et al. 2017a), +HESS J1825-137 reported by the H.E.S.S. collaboration +(H. E. S. S. Collaboration et al. 2018), and the halo +of PSR J0622+3749, identified by the LHAASO collab- +oration (Aharonian et al. 2021). +The VHE fluxes of +these halos suggest that ∼ 10% − 40% of the spin-down +power of the pulsars is converted into e± pair popula- +tion that interacts with the ambient interstellar radia- +tion field (Sudoh et al. 2019; Aharonian et al. 2021). The +diffusion coefficients derived from the sizes of the halos +are typically two orders of magnitude lower than the +average diffusion coefficient of the interstellar medium +(ISM; Hooper et al. 2017; Sudoh et al. 2019). +The formation mechanism of the TeV halos is still +under debate (Linden et al. 2017; Sudoh et al. 2019; +Giacinti et al. 2020; L´opez-Coto et al. 2022; Liu 2022; De +La Torre Luque et al. 2022). Whether they are related to +the local environment, such as extended, diffuse emission +by other sources near the pulsar (e.g., the Monogem +Ring Plucinsky et al. 1996), is also questioned. If TeV +halos commonly exist around pulsars, they can be used +to study the propagation of cosmic rays (e.g., Evoli et al. +2018) and to identify pulsars that are otherwise invisible +to radio and γ-ray observations (Linden et al. 2017). +1 http://tevcat2.uchicago.edu/ +In this letter, we report the detection of a new TeV +halo candidate around the pulsar PSR J0359+5414 +(hereafter J0359) using 2321 days of HAWC data. The +detection of J0359 was first reported in the Fermi Large +Area Telescope (LAT) First Source Catalog (1FGL, +Abdo et al. 2010) where it remained as an unclassi- +fied source until the Third Source Catalog (3FGL, Acero +et al. 2015). J0359 was later classified as a radio-quiet +pulsar by Clark et al. (2017) with an age of 75 kyr and +a spin-down power of ˙E = 1.3 × 1036 erg s−1. In Zyuzin +et al. (2018) a pseudo-distance of J0359 is reported as +d = 3.45 kpc, derived from the ˙E and the gamma-ray +flux. +The latest report at high energies of J0359 ap- +pears in the Fermi-LAT Fourth Source Catalog (4FGL, +Abdollahi et al. 2020) where it is detected above 33σ +in the MeV-GeV energy range. A pulsar wind nebula +(PWN) with an extension of ∼ 30′′ was observed by +Chandra as a result of a X-ray analysis on gamma-ray +pulsars (Zyuzin et al. 2018). No radio emission has been +detected from the pulsar (Grießmeier et al. 2021). The +VHE γ-ray emission from the vicinity of J0359 observed +by HAWC presents similar properties to the other TeV +halos candidates, including the derived acceleration ef- +ficiency and diffusion coefficient. If this source is a TeV +halo, it would support the hypothesis that the halos are +ubiquitous. +The paper is organized as follows. The data set and +analysis framework are described in Section 2. The re- +sults of the spectral and spatial analysis are presented in +Section 3. In Section 4, the broadband spectral energy +distribution (SED) of J0359 is presented and the origin +of the TeV emission is discussed. The conclusions are +summarized in Section 5. +2. INSTRUMENT AND DATA ANALYSIS + +TeV Halo Candidate Surrounding Radio-quiet pulsar +3 +Figure 1. HAWC significance map in Galactic coordinates +using 2321 days of live data. The significance is computed +with a point-like spatial template and a power-law spectrum +with spectral index α = 2.7. For comparison, the positions +of PSR J0359+5414 and PSR B0355+54 are marked. +The HAWC Gamma-Ray Observatory consists of 300 +water Cherenkov detectors located at 19◦N in Puebla, +Mexico at an altitude of 4100 m. Each detector is in- +strumented with 4 photo-multiplier tubes (PMTs) that +are capable of detecting the Cherenkov radiation pro- +duced in the detector water when an electromagnetic or +hadronic shower hits the ground, which is initiated by +a γ-ray or a cosmic ray, respectively, when it enters the +Earth’s atmosphere. HAWC is sensitive to sources with +declinations between −41◦ and +79◦ and to energies in +the 300 GeV to > 100 TeV range. The data set used +in this analysis comprises 2321 days of live data taken +from November 2014 to October 2021. The data set is +divided into 11 analysis bins (fHit) based on the fraction +of PMTs that are triggered in each event, on and off the +main detector array. A full description of HAWC’s de- +sign and performance can be found in Smith & HAWC +Collaboration (2015) and Abeysekara et al. (2017b). +A maximum likelihood analysis was performed using +the Multi-Mission Maximum Likelihood (3ML) frame- +work (Vianello et al. 2015) with the HAWC Accelerated +Likelihood (HAL) plug-in (Abeysekara et al. 2021). For +model selection, we use the likelihood ratio test statistic +(TS) which is defined by +TS = 2 ln LS+B +LB +, +(1) +where LS+B is the maximum likelihood of a signal plus +background model, which depends on the spectral and +spatial parameters, and LB is the maximum likelihood of +the background-only hypothesis. Three spectral models +are tested, including single power-law (PL, Equation 2), +log-parabola (LOGP, Equation 3), and power-law with +an exponential energy cutoff (PL+CO, Equation 4): +dN +dE = N0 +� E +E0 +�−α +, +(2) +dN +dE = N0 +� E +E0 +�−α−β ln(E/E0) +, +(3) +dN +dE = N0 +� E +E0 +�−α +× exp +�−E +Ec +� +. +(4) +In the above equations, N0 is the flux normalization in +units of [TeV−1cm−2s−1], E0 is the pivot energy fixed at +30 TeV to minimize correlations with the other parame- +ters, α is the spectral index, Ec is the cut-off energy and +β is the curvature of the log-parabola spectrum. Two +spatial models are tested: a point-like template and an +extended template. The extended template is described +by a symmetric Gaussian with width as a free parame- +ter. +The energy range in which a source is detected is com- +puted by multiplying a step function with the best fit +model (nominal case). The lower and upper values of +the step function at which the likelihood decreases by +1σ, 2σ or 3σ from that of the nominal case are regarded +as the upper limit to the minimum energy and lower +limit to the maximum energy, respectively. +3. RESULTS +3.1. Association with J0359 +We first free the position of the emission and fit the +PL point source model to data. The best-fit R.A. and +decl. are 59.83 ± 0.07stat and 54.22 ± 0.05stat degrees +(the systematic uncertainty at this location is 0◦.02), +which are consistent with the position of J0359 (59.86 +and 54.25 degrees for R.A and decl. respectively). The +TS of the model is TS = 38.18, which corresponds to a +significance of 6.18σ for four degrees of freedom based +on the Wilks theorem (Wilks 1938). As the position is +consistent with the pulsar position, we fixed the TeV +emission to the pulsar position to perform the spectral +analysis. +Table 1 summarizes the best-fit parameters of differ- +ent spectral and spatial models. +The simplest model +assuming a point-like morphology and non-broken PL + +PSR-B0355+54 +2 +PSR-J0359+5414 +1 +b +0 +-1 +150 +149 +148 +147 +1[°] +-4 +-2 +0 +2 +4 +6 +8 +10 +12 +14 +VTS4 +Table 1. PSR J0359-5414 likelihood fit results for the two spatial scenarios and different spectral shapes. +Model +TS +∆BIC +Extension +N0 +α +β +Ec +[◦] +[×10−16TeV−1cm−2s−1] +[TeV] +PL, point-like +37.86 +-12 +- +1.34+0.34 +−0.27 +2.60 ± 0.16 +- +- +LOGP, point-like +39.18 +-1 +- +1.6+0.5 +−0.4 +2.80 ± 0.23 +0.14 ± 0.12 +- +PL+CO, point-like +37.98 +0 +- +4+50 +−4 +2.5 ± 1.2 +- +500 +PL, extended +40.27 +-12 +0.2 ± 0.1 +2.0+0.8 +−0.6 +2.52 ± 0.16 +- +- +LOGP, extended +41.72 +-1.2 +0.2 ± 0.1 +2.6+1.5 +−1.0 +2.71 ± 0.22 +0.14 ± 0.13 +- +PL+CO, extended +40.48 +0 +0.23 ± 0.1 +14+5 +−4 +2.40 ± 0.19 +- +270+240 +−130 +Note—All the associated errors are statistical. The best model is the one with the lowest BIC value so, ∆BIC is +the difference between a model and the best model, such that it quantifies the evidence against the model with +the highest BIC value. In this case, from both spatial models, the PL+CO spectral model results with the highest +BIC value. The energy cutoff of 500 TeV of the PL+CO point-like model is the boundary of the fit. +yields TS = 37.86. In general more complicated models +with extended morphology and spectral curvature yields +a larger TS since they have more degrees of freedom than +the PL point-source model. So, the preferred spectral +models for both spatial assumptions is a PL, based the +BIC values, where these models have the lower ones. +Figure 2 presents the model and residual significance +maps, and the residual histograms for the two spatial +templates assuming a PL spectral model. +The resid- +ual histogram shows the distribution of the significance +value in each pixel within the region of interest cen- +tered at J0359. The residual significance is defined as +the deviation from the background expectation after fit- +ting and subtracting the modeled emission from J0359. +If only random background fluctuations are left, then +the significance values follow a standard normal distri- +bution (dashed red line). A positive tail is visible in the +residual map of the point-source model. Although the +current sample do not allow to distinguish between the +different spatial models, the residual histograms in Fig- +ure 2 indicate that we get a better fit for an extended +model. +The energy range of the detection are found to be 7- +188 TeV at 1σ level, 11-89 TeV at 2σ level and 15-51 +TeV at 3σ level, with the PL point-source model. For +the PL extended model, the energy range is 4-190 TeV +at 1σ level, 9-110 TeV at 2σ level and 17-78 TeV at 3σ +level. +The luminosity of the VHE emission is L15−51 TeV = +3.6 × 1032 erg s−1 for a distance of 3.45 kpc. The typ- +ical energies of the synchrotron and inverse Compton +photons produced by the same electrons are related +by Esyn ≈ 2.1 keV(EIC/30 TeV) (B/10 µG) (e.g., Aharo- +nian et al. 1997), where B is the magnetic field strength +in the PWN. As the magnetic energy density of a PWN +is usually higher than the energy density of the Cos- +mic Microwave Background (CMB) and infrared (IR) +photons of the ISM, the synchrotron flux of a typical +PWN at keV energies is expected to be higher than +its inverse Compton emission at the HAWC energies +(see e.g., the Crab Nebula H. E. S. S. Collaboration +2020). However, the X-ray luminosity of J0359’s PWN, +L0.3−10 keV = 2.8 × 1031 erg s−1 (Zyuzin et al. 2018), is +instead ∼ 13 times lower than the VHE gamma-ray lu- +minosity. This suggests the existence of a VHE electron +population outside the region where the nebula is ener- +getically dominant, which is expected in the case of a +TeV halo (Linden et al. 2017; L´opez-Coto et al. 2022). +Figure 3 presents the broadband SED of J0359. The +pulsar, PWN, and TeV halo components are shown in +grey, black, and in blue/green colors, respectively. The +multi-wavelength data points include an upper limit of +the pulsar emission by the Effelsberg telescope at 1400 +MHz (Grießmeier et al. 2021), X-ray measurements of +the pulsar and PWN (Zyuzin et al. 2018), γ-ray ob- +servation of the pulsar from 50 MeV to 1 TeV by the +Fermi-LAT (Abdollahi et al. 2020), and the VHE flux +of the halo measured by HAWC. +3.2. Nearby pulsar B0355+54 +Another pulsar, PSR B0355+54 (B0355) is only 0.09 +degrees from J0359. B0355 is classified as a radio-loud +pulsar with characteristic age of 564 kyr and spin-down +power +˙E = 4.5 × 1034 erg s−1 at a distance of 1 kpc. +B0355 has not been detected at high or very-high ener- +gies (Benbow et al. 2021). Below we investigate whether +B0355 is related to the HAWC excess emission. +We performed likelihood fits and compared three sce- +narios: 1) the VHE emission is only associated with +J0359, 2) the VHE emission is only associated with + +TeV Halo Candidate Surrounding Radio-quiet pulsar +5 +(a) PL, point-like model +(b) PL, extended model +(c) PL, point-like residuals +(d) PL, extended residuals +(e) PL, point-like residual +histogram +( f) PL, extended residual +histogram +Figure 2. +Comparison of the model maps, significance +maps, and 1-D residual histograms for point-like and ex- +tended spatial models. The source position is fixed to PSR +J0359+5414 (black cross in the significance maps) and the +spectrum is assumed to be a non-broken power-law. +The +best-fit parameter values are listed in Table 1. +B0355, and 3) the VHE emission is contributed by both +sources. We present the detailed results of scenarios 2 +and 3 in Appendix A and B, respectively. We find that +the two-source scenario (scenario 3) is disfavored com- +pared to the single-source scenarios. Scenario 1 (J0359) +yields lower BIC values than scenario 2 (B0355) for var- +ious spectral and spatial models, though the preference +of scenario 1 is not statistically significant. +The VERITAS telescope searched for emission from +the PWN of B0355 and posed tight upper limits on the +TeV flux (Benbow et al. 2021). The right panel of Figure +3 shows the broadband SED of B0355, which includes +the radio observation of the pulsar (Lorimer et al. 1995), +X-ray observation of the pulsar and its tail at 0.5-8 keV +(Klingler et al. 2016), and the VERITAS upper limits at +95% C.L. between 1 and 10 TeV (Benbow et al. 2021). +For comparison, we show the best-fit flux between 16 +and 59 TeV obtained by assuming that the VHE emis- +sion is centered at the position of B0355. The upper +limits set by VERITAS on B0355’s tail are in tension +with the HAWC’s flux at 16 TeV for both the point- +like and extended models. This suggests that the excess +emission observed by HAWC is more likely associated +with J0359 than B0355, though future multi-wavelength +observation is needed to confirm the finding. +4. SYSTEMATIC UNCERTAINTIES +The systematic uncertainties arising from the detec- +tor performance and simulations are described in Abey- +sekara et al. (2017b) and Abeysekara et al. (2019). The +systematic contribution is calculated in a single energy +band for each spectral and spatial parameter, with the +positive (negative) shift results added in quadrature to +account for the upward (downward) uncertainties. The +systematic uncertainties are calculated for the PL spec- +tral model and for both the point-like and extended tem- +plates. +To account for additional sources of systematic uncer- +tainties, such as the variations in the atmosphere that +are not considered in simulations, a 10% error has been +added to normalization flux (Albert et al. 2020). The +total systematic uncertainties are reported in Table 2. +5. CONCLUSIONS +With 2321 days of HAWC observation, VHE γ-ray +emission is detected in a relatively source-empty region +in the outer galaxy. Based on likelihood fits with dif- +ferent spectral and spatial models to the HAWC data +and the comparison of VHE γ-ray flux with multi- +wavelength observations, we conclude that the emis- +sion is a TeV halo candidate associated with the pulsar +PSR J0359+5414. +If this TeV emission is a halo, it would share sim- +ilar characteristics with the existing population. +We +find a 95% upper limit on the extension of the emis- +sion as 0.◦41 (with the PL-extended model in Ta- +ble 1), corresponding to a physical size of Rul += + +PSRJO0359+5414 +2 +b +0 +149 +148 +147 +[。］1 +-4 -2 +2 +46 +8 +10 +1214 +0 +VTSPSRJO0359+5414 +2 +b +0 +149 +148 +147 +[。］1 +-4 -2 +0 +2 +46 +8 +10 +1214 +VTSPSRT0359+5414 +2 +b +0 +149 +148 +147 +1[°] +-4 -2 +2 +46 +0 +8 +10 +1214 +VTSPSRJO0359+5414 +2 +X +b +0 +149 +148 +147 +[。 +-4 -2 +0 +2 +46 +8 +10 +1214 +VTS1D Significance Histogram +101 + Pixels +Data +Expectation +Number of +Fit +mean = 0.395 ± 0.043 +width = 1.249 ± 0.049 +100 +0 +-4 +-2 +0 +2 +4 +significance1D Significance Histogram +101 + Pixels +Data +Expectation +Number of +Fit +mean = 0.063 ± 0.019 +width = 0.964 ± 0.018 +100 +0 +.4 +-2 +0 +2 +significance6 +10 +4 +10 +1 +102 +105 +108 +1011 +1014 +Energy (eV) +10 +16 +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +E2 dN/dE (erg cm +2 s +1) +Telescope/Observatory +HAWC +Chandra +Fermi-LAT +Effelsberg +Components +TeV halo +PWN +Pulsar +10 +4 +10 +1 +102 +105 +108 +1011 +1014 +Energy (eV) +10 +16 +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +E2 dN/dE (erg cm +2 s +1) +Telescope/Observatory +HAWC +Chandra +VERITAS +Lovell +Components +TeV halo +Tail +Pulsar +Figure 3. Left panel: Spectral energy distribution (SED) of the emission around PSR J0359+5414, including the TeV halo +(green and blue bands corresponding to the HAWC observation for a point-like and extended model, respectively, as explained +in Section 3), the PWN (black band at 0.3-10 keV; Zyuzin et al. 2018), and the pulsar (in grey color; including the upper limit +in radio at 1400 MHz from Grießmeier et al. 2021, the band in X-ray at 0.3-10 keV from Zyuzin et al. 2018, and the data +points or limits at 100 MeV-1 TeV from Abdollahi et al. 2020). Right panel: SED of the emission around PSR B0355+54. +The green and blue bands indicate the TeV excess emission obtained from fits to the HAWC data with models that center at +B0355 with point-like and extended spatial profiles, respectively (see Appendix A). For comparison, the upper limits on VHE +gamma-ray emission from the PWN by VERITAS with hard spectral cuts are shown in orange, with the upper and lower bars +corresponding to region sizes of 0◦.1 and 0◦.235, respectively (Benbow et al. 2021). The black band at 0.5-8 keV indicates the +PWN in X-rays (Klingler et al. 2016). The grey band at 0.5-8 keV (Klingler et al. 2016) and the circular data markers at 1400 +and 1600 MHz (Lorimer et al. 1995) correspond to the emission from the pulsar. The HAWC bands correspondo to statistical +uncerntanties only. +Table 2. Systematic uncertainties considering a PL +for each spatial scenario. +Model +Parameter +Lower sys. +Upper sys. +Point-like +N0 +−3.9 +4.6 +α +−0.15 +0.3 +Extended +N0 +−4.6 +3.4 +α +−0.05 +0.03 +extension +−0.02 +0.02 +Note—N0 is in units of 10−17 TeV−1cm−2s−1 and +extension is in degrees. +25 (d/3.45 kpc) pc. +The diffusion coefficient of the +halo is confined to be D +≲ +R2 +ul/(4 te) += +3.7 × +1027 cm2 s−1(te/12 kyr)−1(d/3.45 kpc)2, +where +te +∼ +12 kyr(Ee/100 TeV)−1 is the cooling time of an electron +at energy Ee by upper-scattering the CMB. Like the +other halos (Abeysekara et al. 2017a), the diffusion co- +efficient is much lower than the average diffusion coeffi- +cient of the ISM. +The candidate halo of J0359 joins the observation of +extended VHE emission surrounding PSR J0622+3749 +(Aharonian et al. 2021) as the first evidence of TeV halos +around radio-quiet pulsars. Their presence suggests that +the formation of the halos is insensitive to the configu- +ration of the pulsar magnetosphere, in particular, the +geometry of the γ-ray and radio beams (Harding 2001). +With an age of 70 kyr, J0359 is younger than the other +pulsars with halos. It is likely in a transition between +the so-called relic- and halo-stage of a PWN, the bound- +aries of which are not well defined and have motivated +different classification criteria of TeV halos (Linden et al. +2017; Giacinti et al. 2020; L´opez-Coto et al. 2022). Our +observation of TeV halo features associated with J0359 +implies that high-energy particles may already start es- +caping in the ISM in the late relic-stage. +Our observation provides spectral evidence toward a +TeV halo nature of J0359. +Future data from HAWC +and multi-wavelength follow-ups of this new TeV source +are crucial to confirming its nature via morphological +studies that identify the halo extension and exclude the +association with the nearby pulsars. Future observations +of young to middle-aged pulsars like PSR J0359+5414 +with wide-field γ-ray experiments and imaging atmo- +spheric Cherenkov telescopes may provide further un- +derstanding into the evolution of TeV PWNe and their +connection with TeV halos. +ACKNOWLEDGMENTS +We acknowledge the support from: +the US Na- +tional Science Foundation (NSF); the US Department +of Energy Office of High-Energy Physics; the Labora- + +TeV Halo Candidate Surrounding Radio-quiet pulsar +7 +tory Directed Research and Development (LDRD) pro- +gram of Los Alamos National Laboratory; Consejo Na- +cional de Ciencia y Tecnolog´ıa (CONACyT), M´exico, +grants 271051, 232656, 260378, 179588, 254964, 258865, +243290, 132197, A1-S-46288, A1-S-22784, c´atedras 873, +1563, 341, 323, Red HAWC, M´exico; DGAPA-UNAM +grants IG101320, IN111716-3, IN111419, IA102019, +IN110621, IN110521; VIEP-BUAP; PIFI 2012, 2013, +PROFOCIE 2014, 2015; the University of Wisconsin +Alumni Research Foundation; the Institute of Geo- +physics, +Planetary Physics, +and Signatures at Los +Alamos National Laboratory; Polish Science Centre +grant, DEC-2017/27/B/ST9/02272; Coordinaci´on de la +Investigaci´on Cient´ıfica de la Universidad Michoacana; +Royal Society - Newton Advanced Fellowship 180385; +Generalitat Valenciana, grant CIDEGENT/2018/034; +The Program Management Unit for Human Resources +& Institutional Development, Research and Innovation, +NXPO (grant number B16F630069); Coordinaci´on Gen- +eral Acad´emica e Innovaci´on (CGAI-UdeG), PRODEP- +SEP UDG-CA-499; Institute of Cosmic Ray Research +(ICRR), University of Tokyo, H.F. acknowledges sup- +port by NASA under award number 80GSFC21M0002. +We also acknowledge the significant contributions over +many years of Stefan Westerhoff, Gaurang Yodh and Ar- +nulfo Zepeda Dominguez, all deceased members of the +HAWC collaboration. Thanks to Scott Delay, Luciano +D´ıaz and Eduardo Murrieta for technical support. +REFERENCES +Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJS, +188, 405, doi: 10.1088/0067-0049/188/2/405 +Abdollahi, S., Acero, F., Ackermann, M., et al. 2020, ApJS, +247, 33, doi: 10.3847/1538-4365/ab6bcb +Abeysekara, A. U., Albert, A., Alfaro, R., et al. 2017a, +Science, 358, 911, doi: 10.1126/science.aan4880 +—. 2017b, ApJ, 843, 39, doi: 10.3847/1538-4357/aa7555 +—. 2019, ApJ, 881, 134, doi: 10.3847/1538-4357/ab2f7d +Abeysekara, A. U., et al. 2021, PoS, ICRC2021, 828, +doi: 10.22323/1.395.0828 +Acero, F., Ackermann, M., Ajello, M., et al. 2015, ApJS, +218, 23, doi: 10.1088/0067-0049/218/2/23 +Aharonian, F., An, Q., Axikegu, Bai, L. X., et al. 2021, +PhRvL, 126, 241103, +doi: 10.1103/PhysRevLett.126.241103 +Aharonian, F. A., Atoyan, A. M., & Kifune, T. 1997, +MNRAS, 291, 162, doi: 10.1093/mnras/291.1.162 +Albert, A., Alfaro, R., Alvarez, C., et al. 2020, ApJ, 905, +76, doi: 10.3847/1538-4357/abc2d8 +Benbow, W., Brill, A., Buckley, J. H., et al. 2021, ApJ, 916, +117, doi: 10.3847/1538-4357/ac05b9 +Clark, C. J., Wu, J., Pletsch, H. J., et al. 2017, ApJ, 834, +106, doi: 10.3847/1538-4357/834/2/106 +De La Torre Luque, P., Fornieri, O., & Linden, T. 2022, +arXiv e-prints, arXiv:2205.08544. +https://arxiv.org/abs/2205.08544 +Evoli, C., Linden, T., & Morlino, G. 2018, Phys. Rev. D, +98, 063017, doi: 10.1103/PhysRevD.98.063017 +Giacinti, G., Mitchell, A. M. W., L´opez-Coto, R., et al. +2020, Astron. Astrophys., 636, A113, +doi: 10.1051/0004-6361/201936505 +Grießmeier, J. M., Smith, D. A., Theureau, G., et al. 2021, +A&A, 654, A43, doi: 10.1051/0004-6361/202140841 +H. E. S. S. Collaboration. 2020, Nature Astronomy, 4, 167, +doi: 10.1038/s41550-019-0910-0 +H. E. S. S. Collaboration, Abdalla, H., Abramowski, A., +et al. 2018, A&A, 612, A2, +doi: 10.1051/0004-6361/201629377 +Harding, A. K. 2001, in American Institute of Physics +Conference Series, Vol. 558, High Energy Gamma-Ray +Astronomy: International Symposium, ed. F. A. +Aharonian & H. J. V¨olk, 115–126, doi: 10.1063/1.1370785 +Hooper, D., Cholis, I., Linden, T., & Fang, K. 2017, Phys. +Rev. D, 96, 103013, doi: 10.1103/PhysRevD.96.103013 +Kass, R. E., & Raftery, A. E. 1995, Journal of the +American Statistical Association, 90, 773, +doi: 10.1080/01621459.1995.10476572 +Klingler, N., Rangelov, B., Kargaltsev, O., et al. 2016, ApJ, +833, 253, doi: 10.3847/1538-4357/833/2/253 +Linden, T., Auchettl, K., Bramante, J., et al. 2017, PhRvD, +96, 103016, doi: 10.1103/PhysRevD.96.103016 +Liu, R.-Y. 2022, arXiv e-prints, arXiv:2207.04011. +https://arxiv.org/abs/2207.04011 +L´opez-Coto, R., de O˜na Wilhelmi, E., Aharonian, F., +Amato, E., & Hinton, J. 2022, Nature Astronomy, 6, 199, +doi: 10.1038/s41550-021-01580-0 +Lorimer, D. R., Yates, J. A., Lyne, A. G., & Gould, D. M. +1995, MNRAS, 273, 411, doi: 10.1093/mnras/273.2.411 +Plucinsky, P. P., Snowden, S. L., Aschenbach, B., et al. +1996, ApJ, 463, 224, doi: 10.1086/177236 +Smith, A. J., & HAWC Collaboration. 2015, in +International Cosmic Ray Conference, Vol. 34, 34th +International Cosmic Ray Conference (ICRC2015), 966. +https://arxiv.org/abs/1508.05826 +Sudoh, T., Linden, T., & Beacom, J. F. 2019, PhRvD, 100, +043016, doi: 10.1103/PhysRevD.100.043016 + +8 +Vianello, G., Lauer, R. J., Younk, P., et al. 2015, arXiv +e-prints, arXiv:1507.08343. +https://arxiv.org/abs/1507.08343 +Wakely, S. P., & Horan, D. 2008, in International Cosmic +Ray Conference, Vol. 3, International Cosmic Ray +Conference, 1341–1344 +Wilks, S. S. 1938, Annals Math. Statist., 9, 60, +doi: 10.1214/aoms/1177732360 +Zyuzin, D. A., Karpova, A. V., & Shibanov, Y. A. 2018, +MNRAS, 476, 2177, doi: 10.1093/mnras/sty359 + +TeV Halo Candidate Surrounding Radio-quiet pulsar +9 +Table 3. +Results of the likelihood fit assuming that the only emitting source +is PSR B0355+54. The PL spectral model along with the two different spatial +models were tested. +Spatial model +TS +∆BIC +Extension +N0 +α +[◦] +TeV−1 cm−2 s−1 +Point-like +35.86 +-1.9 +0.0 +(1.28+0.34 +−0.27) × 10−16 +2.56 ± 0.17 +Extended +41.83 +-1.5 +0.22 ± 0.09 +(2.0+0.7 +−0.5) × 10−16 +2.51 ± 0.15 +Note—All associated errors are statistical. ∆BIC is obtained comparing the BIC +value with the best spectral model fit for both spatial models assuming that the +emission is coming from J0359 (Section 3). +APPENDIX +A. PSR B0355+54 FITTING RESULTS +In this section, we explore the possibility that the TeV excess comes entirely from B0355. We fit models with a +power-law (PL) spectrum and the spatial templates described in Section 3. The results are summarized in Table 3. +The energy ranges at which the source is detected are 7-180 TeV at 1σ level, 11-90 TeV at 2σ level and 17-54 TeV +assuming at 3σ level assuming a point-like morphology. For an extended morphology, the energy ranges are found to +be 8-155 TeV for 1σ level, 11-90 TeV at 2σ level and 17-59 at 3σ level. +As single-source scenarios are not nested models, we have employed the Bayesian Information Criterion (BIC) to +select the models. The difference in the BIC value, ∆BIC, quantifies the evidence against the model with a higher +BIC value. According to Kass & Raftery (1995), if ∆BIC is between 0 and 2 it is not clear which model is preferred; +∆BIC between 2 and 10 and above 10 indicates a slight and strong preference of the model with the smallest BIC, +respectively. +The small difference in ∆BIC from the fits of models centered at J0359 and B0355 does not allow us to distinguish +between the models. This is expected as the angular distance of the two pulsars is smaller than the spatial resolution +of HAWC. However, the tension between the VERITAS limits on B0355 and HAWC fluxes, as explained in Section 3, +suggests that the TeV emission is more likely associated with J0359. +B. FITTING RESULTS OF A TWO-SOURCE SCENARIO +We further explore a scenario where both J0359 and B0359 contribute to the TeV emission observed by HAWC. +Such a two-source model is disfavored by the data. +Table 4 presents the results of the two-source models. We consider three combinations of spatial profiles of the +two sources: (A) both sources are point-like, (B) both sources are extended with a Gaussian shape, and (C) J0359 is +extended source and B0355 is point-like. The energy spectrum is assumed to be a PL. The normalization flux N0 and +the spectral index α in each fit were free to vary while the position of the sources for all the scenarios were fixed. +The ∆TS column shows the gain of test statistics by adding an extra source to the one-source model presented in +Section 3 and Section A (the baseline model considers pure background plus the emission from the other source). The +two-source model is disfavored in all cases. + +10 +Table 4. +Results of the likelihood fit assuming that the excess observed comes from two sources: PSR +J0359+5414 and PSR B0355+54. The spectral model for all the spatial models is a PL. +Two-source model +Source +∆TS +∆BIC +Extension +N0 +α +[◦] +TeV−1 cm−2 s−1 +J0359 +2.32 +0.0 +(1.0+0.5 +−0.9) × 10−16 +2.63+0.6 +−0.20 +Model A +B0355 +0.32 +-24 +0.0 +(0.00034+6 +−0.00024) × 10−13 +2.4+1.3 +−5 +J0359 +8.73 +1.500+0.18 +−0.004 +(3.3+1.5 +−3.3) × 10−16 +2.2+0.4 +−1.3 +Model B +B0355 +10.29 +-26 +0.14+0.08 +−0.15 +(1.5+0.5 +−0.4) × 10−16 +2.56 ± 0.20 +J0359 +13.02 +1.5000 ± 0.0010 +(0.04+4 +−0.04) × 10−14 +2.2 ± 2.8 +Model C +B0355 +8.62 +-26 / -15 +0.0 +(3.2+3.0 +−1.5) × 10−16 +2.60 ± 0.28 +Note—All associated errors are statistical. Model A corresponds to a scenario where both sources are point- +like, model B assumes that both sources are extended with a Gaussian shape, and model C assumes that +PSR J0359+5414 is as a point-like source and PSR B0355+54 is an extended source with a Gaussian shape. +∆BIC is obtained comparing the BIC value with the best model fit assuming that the emission is coming +from J0359 (Section 3). For model A, with the PL point-like model, for model B with the PL Gaussian +model and for model C with the two previous models. + diff --git a/69E3T4oBgHgl3EQfpwru/content/tmp_files/load_file.txt b/69E3T4oBgHgl3EQfpwru/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa78fc17265d0bc20d1cf3ff651eb87176846c75 --- /dev/null +++ b/69E3T4oBgHgl3EQfpwru/content/tmp_files/load_file.txt @@ -0,0 +1,840 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf,len=839 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX631 HAWC Detection of a TeV Halo Candidate Surrounding a Radio-quiet pulsar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Albert ,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Alfaro ,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Arteaga-Vel´azquez,3 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Belmont-Moreno ,2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Capistr´an ,4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Carrami˜nana ,5 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Casanova ,6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Cotzomi ,7 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Couti˜no de Le´on ,8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' De la Fuente ,9 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Diaz Hernandez,5 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' DuVernois ,8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' D´ıaz-V´elez ,9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Espinoza ,2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Fan,10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Fraija ,4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Fang ,8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Garc´ıa-Gonz´alez ,11 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Garfias ,4 Armelle Jardin-Blicq ,12, 13 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Gonz´alez ,4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Goodman ,10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Harding ,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Hernandez ,2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Huang ,14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Hueyotl-Zahuantitla ,15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Iriarte ,4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Joshi ,16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Lara ,17 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Lee ,18 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Le´on Vargas ,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Linnemann ,19 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Longinotti ,4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Luis-Raya ,20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Malone ,21 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Martinez ,7 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mart´ınez-Castro ,22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Matthews ,23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Morales-Soto ,3 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Moreno ,7 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mostaf´a ,24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Nayerhoda ,6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Nellen ,25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Newbold ,26 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Nisa ,27, 28 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' P´erez Araujo ,4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' P´erez-P´erez ,20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Rho ,29 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Rosa-Gonz´alez ,5 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Schneider ,10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Serna-Franco,2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Smith ,10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Son,18 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Springer ,26 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Tollefson ,27 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Torres ,5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Torres-Escobedo,30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Wang,14 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Whitaker,24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Willox ,10 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Zhou ,30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' de Le´on ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3 (THE HAWC COLLABORATION) 1Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 2Instituto de F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Universidad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Ciudad de Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 3Universidad Michoacana de San Nicol´as de Hidalgo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Morelia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 4Instituto de Astronom´ıa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Universidad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Ciudad de Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 5Instituto Nacional de Astrof´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' ´Optica y Electr´onica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Puebla,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 6Institute of Nuclear Physics Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' PL-31342 IFJ-PAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Krakow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Poland 7Facultad de Ciencias F´ısico Matem´aticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Benem´erita Universidad Aut´onoma de Puebla,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Puebla,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 8Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' University of Wisconsin-Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' WI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 9Departamento de F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Centro Universitario de Ciencias Exactase Ingenierias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Universidad de Guadalajara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Guadalajara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 10Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 11Tecnologico de Monterrey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Escuela de Ingenier´ıa y Ciencias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Eugenio Garza Sada 2501, Monterrey, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 64849 12Universit´e Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' CNRS/IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' LP2I Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' UMR 5797,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' F-33170 Gradignan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' France 13Max-Planck Institute for Nuclear Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' D-69117 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Germany 14Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Michigan Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Houghton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 15Universidad Aut´onoma de Chiapas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Tuxtla Guti´errez,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Chiapas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' M´exico 16Erlangen Centre for Astroparticle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Germany 17Instituto de Geof´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Universidad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Ciudad de Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 18University of Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='of Korea 19Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA 20Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico 21Space Science and Applications Group, Los Alamos National Laboratory, Los Alamos, NM, USA 22Centro de Investigaci´on en Computaci´on, Instituto Polit´ecnico Nacional, M´exico City, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 23Dept of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' University of New Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Albuquerque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 24Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 25Instituto de Ciencias Nucleares,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Universidad Nacional Aut´onoma de Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Ciudad de Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Mexico 26Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Salt Lake City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' UT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 27Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Michigan State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' East Lansing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 28Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Michigan Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Houghton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' USA 29University of Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' of Korea 30Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, China Corresponding author: Sara Couti˜no de Le´on scoutino@icecube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='04646v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='HE] 11 Jan 2023 ID2 ABSTRACT Extended very-high-energy (VHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1-100 TeV) γ-ray emission has been observed around several middle-aged pulsars and referred to as “TeV halos”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Their formation mechanism remains under debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' It is also unknown whether they are ubiquitous or related to certain subgroup of pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' With 2321 days of observation, the High Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory detected VHE γ-ray emission at the location of the radio-quiet pulsar PSR J0359+5414 with > 6σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' By performing likelihood tests with different spectral and spatial models and comparing the TeV spectrum with multi-wavelength observations of nearby sources, we show that this excess is consistent with a TeV halo associated with PSR J0359+5414, though future observation of HAWC and multi- wavelength follow-ups are needed to confirm this nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' This new halo candidate is located in a non-crowded region in the outer Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' It shares similar properties to the other halos but its pulsar is younger and radio-quiet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Our observation implies that TeV halos could commonly exist around pulsars and their formation does not depend on the configuration of the pulsar magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Keywords: Pulsars (1306) — Gamma-ray astronomy(628) — High-energy astrophysics(739) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' INTRODUCTION Extended TeV gamma-ray emission has been observed around several middle-aged (> 100 kyr) pulsars and grouped as a new source class named “TeV halos” (Lin- den et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' L´opez-Coto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Seven sources are referenced as TeV halos in the online catalog for TeV Astronomy, TeVCat1 (Wakely & Horan 2008), including the first halos around the Geminga and Monogem pul- sars, discovered by HAWC (Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017a), HESS J1825-137 reported by the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' collaboration (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018), and the halo of PSR J0622+3749, identified by the LHAASO collab- oration (Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The VHE fluxes of these halos suggest that ∼ 10% − 40% of the spin-down power of the pulsars is converted into e± pair popula- tion that interacts with the ambient interstellar radia- tion field (Sudoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The diffusion coefficients derived from the sizes of the halos are typically two orders of magnitude lower than the average diffusion coefficient of the interstellar medium (ISM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Hooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Sudoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The formation mechanism of the TeV halos is still under debate (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Sudoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Giacinti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' L´opez-Coto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Liu 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' De La Torre Luque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Whether they are related to the local environment, such as extended, diffuse emission by other sources near the pulsar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', the Monogem Ring Plucinsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1996), is also questioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' If TeV halos commonly exist around pulsars, they can be used to study the propagation of cosmic rays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Evoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018) and to identify pulsars that are otherwise invisible to radio and γ-ray observations (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1 http://tevcat2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='edu/ In this letter, we report the detection of a new TeV halo candidate around the pulsar PSR J0359+5414 (hereafter J0359) using 2321 days of HAWC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The detection of J0359 was first reported in the Fermi Large Area Telescope (LAT) First Source Catalog (1FGL, Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2010) where it remained as an unclassi- fied source until the Third Source Catalog (3FGL, Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' J0359 was later classified as a radio-quiet pulsar by Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' (2017) with an age of 75 kyr and a spin-down power of ˙E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3 × 1036 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' In Zyuzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' (2018) a pseudo-distance of J0359 is reported as d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='45 kpc, derived from the ˙E and the gamma-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The latest report at high energies of J0359 ap- pears in the Fermi-LAT Fourth Source Catalog (4FGL, Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020) where it is detected above 33σ in the MeV-GeV energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A pulsar wind nebula (PWN) with an extension of ∼ 30′′ was observed by Chandra as a result of a X-ray analysis on gamma-ray pulsars (Zyuzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' No radio emission has been detected from the pulsar (Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The VHE γ-ray emission from the vicinity of J0359 observed by HAWC presents similar properties to the other TeV halos candidates, including the derived acceleration ef- ficiency and diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' If this source is a TeV halo, it would support the hypothesis that the halos are ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The data set and analysis framework are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The re- sults of the spectral and spatial analysis are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' In Section 4, the broadband spectral energy distribution (SED) of J0359 is presented and the origin of the TeV emission is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The conclusions are summarized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' INSTRUMENT AND DATA ANALYSIS TeV Halo Candidate Surrounding Radio-quiet pulsar 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' HAWC significance map in Galactic coordinates using 2321 days of live data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The significance is computed with a point-like spatial template and a power-law spectrum with spectral index α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For comparison, the positions of PSR J0359+5414 and PSR B0355+54 are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The HAWC Gamma-Ray Observatory consists of 300 water Cherenkov detectors located at 19◦N in Puebla, Mexico at an altitude of 4100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Each detector is in- strumented with 4 photo-multiplier tubes (PMTs) that are capable of detecting the Cherenkov radiation pro- duced in the detector water when an electromagnetic or hadronic shower hits the ground, which is initiated by a γ-ray or a cosmic ray, respectively, when it enters the Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' HAWC is sensitive to sources with declinations between −41◦ and +79◦ and to energies in the 300 GeV to > 100 TeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The data set used in this analysis comprises 2321 days of live data taken from November 2014 to October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The data set is divided into 11 analysis bins (fHit) based on the fraction of PMTs that are triggered in each event, on and off the main detector array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A full description of HAWC’s de- sign and performance can be found in Smith & HAWC Collaboration (2015) and Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' (2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A maximum likelihood analysis was performed using the Multi-Mission Maximum Likelihood (3ML) frame- work (Vianello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2015) with the HAWC Accelerated Likelihood (HAL) plug-in (Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For model selection, we use the likelihood ratio test statistic (TS) which is defined by TS = 2 ln LS+B LB , (1) where LS+B is the maximum likelihood of a signal plus background model, which depends on the spectral and spatial parameters, and LB is the maximum likelihood of the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Three spectral models are tested, including single power-law (PL, Equation 2), log-parabola (LOGP, Equation 3), and power-law with an exponential energy cutoff (PL+CO, Equation 4): dN dE = N0 � E E0 �−α , (2) dN dE = N0 � E E0 �−α−β ln(E/E0) , (3) dN dE = N0 � E E0 �−α × exp �−E Ec � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' (4) In the above equations, N0 is the flux normalization in units of [TeV−1cm−2s−1], E0 is the pivot energy fixed at 30 TeV to minimize correlations with the other parame- ters, α is the spectral index, Ec is the cut-off energy and β is the curvature of the log-parabola spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Two spatial models are tested: a point-like template and an extended template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The extended template is described by a symmetric Gaussian with width as a free parame- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The energy range in which a source is detected is com- puted by multiplying a step function with the best fit model (nominal case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The lower and upper values of the step function at which the likelihood decreases by 1σ, 2σ or 3σ from that of the nominal case are regarded as the upper limit to the minimum energy and lower limit to the maximum energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Association with J0359 We first free the position of the emission and fit the PL point source model to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The best-fit R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' are 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='07stat and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='05stat degrees (the systematic uncertainty at this location is 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='02), which are consistent with the position of J0359 (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='86 and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='25 degrees for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='A and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The TS of the model is TS = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='18, which corresponds to a significance of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='18σ for four degrees of freedom based on the Wilks theorem (Wilks 1938).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' As the position is consistent with the pulsar position, we fixed the TeV emission to the pulsar position to perform the spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Table 1 summarizes the best-fit parameters of differ- ent spectral and spatial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The simplest model assuming a point-like morphology and non-broken PL PSR-B0355+54 2 PSR-J0359+5414 1 b 0 1 150 149 148 147 1[°] 4 2 0 2 4 6 8 10 12 14 VTS4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' PSR J0359-5414 likelihood fit results for the two spatial scenarios and different spectral shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Model TS ∆BIC Extension N0 α β Ec [◦] [×10−16TeV−1cm−2s−1] [TeV] PL, point-like 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='86 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='16 LOGP, point-like 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='18 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='12 PL+CO, point-like 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='98 0 4+50 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2 500 PL, extended 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='27 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='16 LOGP, extended 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='13 PL+CO, extended 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='48 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1 14+5 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='19 270+240 −130 Note—All the associated errors are statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The best model is the one with the lowest BIC value so, ∆BIC is the difference between a model and the best model, such that it quantifies the evidence against the model with the highest BIC value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' In this case, from both spatial models, the PL+CO spectral model results with the highest BIC value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The energy cutoff of 500 TeV of the PL+CO point-like model is the boundary of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' yields TS = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' In general more complicated models with extended morphology and spectral curvature yields a larger TS since they have more degrees of freedom than the PL point-source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' So, the preferred spectral models for both spatial assumptions is a PL, based the BIC values, where these models have the lower ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Figure 2 presents the model and residual significance maps, and the residual histograms for the two spatial templates assuming a PL spectral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The resid- ual histogram shows the distribution of the significance value in each pixel within the region of interest cen- tered at J0359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The residual significance is defined as the deviation from the background expectation after fit- ting and subtracting the modeled emission from J0359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' If only random background fluctuations are left, then the significance values follow a standard normal distri- bution (dashed red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A positive tail is visible in the residual map of the point-source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Although the current sample do not allow to distinguish between the different spatial models, the residual histograms in Fig- ure 2 indicate that we get a better fit for an extended model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The energy range of the detection are found to be 7- 188 TeV at 1σ level, 11-89 TeV at 2σ level and 15-51 TeV at 3σ level, with the PL point-source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For the PL extended model, the energy range is 4-190 TeV at 1σ level, 9-110 TeV at 2σ level and 17-78 TeV at 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The luminosity of the VHE emission is L15−51 TeV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6 × 1032 erg s−1 for a distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='45 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The typ- ical energies of the synchrotron and inverse Compton photons produced by the same electrons are related by Esyn ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1 keV(EIC/30 TeV) (B/10 µG) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Aharo- nian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1997), where B is the magnetic field strength in the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' As the magnetic energy density of a PWN is usually higher than the energy density of the Cos- mic Microwave Background (CMB) and infrared (IR) photons of the ISM, the synchrotron flux of a typical PWN at keV energies is expected to be higher than its inverse Compton emission at the HAWC energies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', the Crab Nebula H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Collaboration 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' However, the X-ray luminosity of J0359’s PWN, L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3−10 keV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='8 × 1031 erg s−1 (Zyuzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018), is instead ∼ 13 times lower than the VHE gamma-ray lu- minosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' This suggests the existence of a VHE electron population outside the region where the nebula is ener- getically dominant, which is expected in the case of a TeV halo (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' L´opez-Coto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Figure 3 presents the broadband SED of J0359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The pulsar, PWN, and TeV halo components are shown in grey, black, and in blue/green colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The multi-wavelength data points include an upper limit of the pulsar emission by the Effelsberg telescope at 1400 MHz (Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021), X-ray measurements of the pulsar and PWN (Zyuzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018), γ-ray ob- servation of the pulsar from 50 MeV to 1 TeV by the Fermi-LAT (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020), and the VHE flux of the halo measured by HAWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Nearby pulsar B0355+54 Another pulsar, PSR B0355+54 (B0355) is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='09 degrees from J0359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' B0355 is classified as a radio-loud pulsar with characteristic age of 564 kyr and spin-down power ˙E = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 × 1034 erg s−1 at a distance of 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' B0355 has not been detected at high or very-high ener- gies (Benbow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Below we investigate whether B0355 is related to the HAWC excess emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We performed likelihood fits and compared three sce- narios: 1) the VHE emission is only associated with J0359, 2) the VHE emission is only associated with TeV Halo Candidate Surrounding Radio-quiet pulsar 5 (a) PL, point-like model (b) PL, extended model (c) PL, point-like residuals (d) PL, extended residuals (e) PL, point-like residual histogram ( f) PL, extended residual histogram Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Comparison of the model maps, significance maps, and 1-D residual histograms for point-like and ex- tended spatial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The source position is fixed to PSR J0359+5414 (black cross in the significance maps) and the spectrum is assumed to be a non-broken power-law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The best-fit parameter values are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' B0355, and 3) the VHE emission is contributed by both sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We present the detailed results of scenarios 2 and 3 in Appendix A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We find that the two-source scenario (scenario 3) is disfavored com- pared to the single-source scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Scenario 1 (J0359) yields lower BIC values than scenario 2 (B0355) for var- ious spectral and spatial models, though the preference of scenario 1 is not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The VERITAS telescope searched for emission from the PWN of B0355 and posed tight upper limits on the TeV flux (Benbow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The right panel of Figure 3 shows the broadband SED of B0355, which includes the radio observation of the pulsar (Lorimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1995), X-ray observation of the pulsar and its tail at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5-8 keV (Klingler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2016), and the VERITAS upper limits at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' between 1 and 10 TeV (Benbow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For comparison, we show the best-fit flux between 16 and 59 TeV obtained by assuming that the VHE emis- sion is centered at the position of B0355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The upper limits set by VERITAS on B0355’s tail are in tension with the HAWC’s flux at 16 TeV for both the point- like and extended models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' This suggests that the excess emission observed by HAWC is more likely associated with J0359 than B0355, though future multi-wavelength observation is needed to confirm the finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' SYSTEMATIC UNCERTAINTIES The systematic uncertainties arising from the detec- tor performance and simulations are described in Abey- sekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' (2017b) and Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The systematic contribution is calculated in a single energy band for each spectral and spatial parameter, with the positive (negative) shift results added in quadrature to account for the upward (downward) uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The systematic uncertainties are calculated for the PL spec- tral model and for both the point-like and extended tem- plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' To account for additional sources of systematic uncer- tainties, such as the variations in the atmosphere that are not considered in simulations, a 10% error has been added to normalization flux (Albert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The total systematic uncertainties are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' CONCLUSIONS With 2321 days of HAWC observation, VHE γ-ray emission is detected in a relatively source-empty region in the outer galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Based on likelihood fits with dif- ferent spectral and spatial models to the HAWC data and the comparison of VHE γ-ray flux with multi- wavelength observations, we conclude that the emis- sion is a TeV halo candidate associated with the pulsar PSR J0359+5414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' If this TeV emission is a halo, it would share sim- ilar characteristics with the existing population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We find a 95% upper limit on the extension of the emis- sion as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='◦41 (with the PL-extended model in Ta- ble 1), corresponding to a physical size of Rul = PSRJO0359+5414 2 b 0 149 148 147 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='］1 4 -2 2 46 8 10 1214 0 VTSPSRJO0359+5414 2 b 0 149 148 147 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='］1 4 -2 0 2 46 8 10 1214 VTSPSRT0359+5414 2 b 0 149 148 147 1[°] 4 -2 2 46 0 8 10 1214 VTSPSRJO0359+5414 2 X b 0 149 148 147 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 4 -2 0 2 46 8 10 1214 VTS1D Significance Histogram 101 Pixels Data Expectation Number of Fit mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='395 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='043 width = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='049 100 0 4 2 0 2 4 significance1D Significance Histogram 101 Pixels Data Expectation Number of Fit mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='063 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='019 width = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='964 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='018 100 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='4 2 0 2 significance6 10 4 10 1 102 105 108 1011 1014 Energy (eV) 10 16 10 15 10 14 10 13 10 12 10 11 E2 dN/dE (erg cm 2 s 1) Telescope/Observatory HAWC Chandra Fermi-LAT Effelsberg Components TeV halo PWN Pulsar 10 4 10 1 102 105 108 1011 1014 Energy (eV) 10 16 10 15 10 14 10 13 10 12 10 11 E2 dN/dE (erg cm 2 s 1) Telescope/Observatory HAWC Chandra VERITAS Lovell Components TeV halo Tail Pulsar Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Left panel: Spectral energy distribution (SED) of the emission around PSR J0359+5414, including the TeV halo (green and blue bands corresponding to the HAWC observation for a point-like and extended model, respectively, as explained in Section 3), the PWN (black band at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3-10 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Zyuzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018), and the pulsar (in grey color;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' including the upper limit in radio at 1400 MHz from Grießmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021, the band in X-ray at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3-10 keV from Zyuzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018, and the data points or limits at 100 MeV-1 TeV from Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Right panel: SED of the emission around PSR B0355+54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The green and blue bands indicate the TeV excess emission obtained from fits to the HAWC data with models that center at B0355 with point-like and extended spatial profiles, respectively (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For comparison, the upper limits on VHE gamma-ray emission from the PWN by VERITAS with hard spectral cuts are shown in orange, with the upper and lower bars corresponding to region sizes of 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1 and 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='235, respectively (Benbow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The black band at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5-8 keV indicates the PWN in X-rays (Klingler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The grey band at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5-8 keV (Klingler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2016) and the circular data markers at 1400 and 1600 MHz (Lorimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1995) correspond to the emission from the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The HAWC bands correspondo to statistical uncerntanties only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Systematic uncertainties considering a PL for each spatial scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Model Parameter Lower sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Upper sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Point-like N0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6 α −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3 Extended N0 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='4 α −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='03 extension −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='02 Note—N0 is in units of 10−17 TeV−1cm−2s−1 and extension is in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 25 (d/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='45 kpc) pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The diffusion coefficient of the halo is confined to be D ≲ R2 ul/(4 te) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='7 × 1027 cm2 s−1(te/12 kyr)−1(d/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='45 kpc)2, where te ∼ 12 kyr(Ee/100 TeV)−1 is the cooling time of an electron at energy Ee by upper-scattering the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Like the other halos (Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017a), the diffusion co- efficient is much lower than the average diffusion coeffi- cient of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The candidate halo of J0359 joins the observation of extended VHE emission surrounding PSR J0622+3749 (Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021) as the first evidence of TeV halos around radio-quiet pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Their presence suggests that the formation of the halos is insensitive to the configu- ration of the pulsar magnetosphere, in particular, the geometry of the γ-ray and radio beams (Harding 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' With an age of 70 kyr, J0359 is younger than the other pulsars with halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' It is likely in a transition between the so-called relic- and halo-stage of a PWN, the bound- aries of which are not well defined and have motivated different classification criteria of TeV halos (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Giacinti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' L´opez-Coto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Our observation of TeV halo features associated with J0359 implies that high-energy particles may already start es- caping in the ISM in the late relic-stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Our observation provides spectral evidence toward a TeV halo nature of J0359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Future data from HAWC and multi-wavelength follow-ups of this new TeV source are crucial to confirming its nature via morphological studies that identify the halo extension and exclude the association with the nearby pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Future observations of young to middle-aged pulsars like PSR J0359+5414 with wide-field γ-ray experiments and imaging atmo- spheric Cherenkov telescopes may provide further un- derstanding into the evolution of TeV PWNe and their connection with TeV halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge the support from: the US Na- tional Science Foundation (NSF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' the US Department of Energy Office of High-Energy Physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' the Labora- TeV Halo Candidate Surrounding Radio-quiet pulsar 7 tory Directed Research and Development (LDRD) pro- gram of Los Alamos National Laboratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Consejo Na- cional de Ciencia y Tecnolog´ıa (CONACyT), M´exico, grants 271051, 232656, 260378, 179588, 254964, 258865, 243290, 132197, A1-S-46288, A1-S-22784, c´atedras 873, 1563, 341, 323, Red HAWC, M´exico;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' DGAPA-UNAM grants IG101320, IN111716-3, IN111419, IA102019, IN110621, IN110521;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' VIEP-BUAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' PIFI 2012, 2013, PROFOCIE 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' the University of Wisconsin Alumni Research Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' the Institute of Geo- physics, Planetary Physics, and Signatures at Los Alamos National Laboratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Polish Science Centre grant, DEC-2017/27/B/ST9/02272;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Coordinaci´on de la Investigaci´on Cient´ıfica de la Universidad Michoacana;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Royal Society - Newton Advanced Fellowship 180385;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Generalitat Valenciana, grant CIDEGENT/2018/034;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The Program Management Unit for Human Resources & Institutional Development, Research and Innovation, NXPO (grant number B16F630069);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Coordinaci´on Gen- eral Acad´emica e Innovaci´on (CGAI-UdeG), PRODEP- SEP UDG-CA-499;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Institute of Cosmic Ray Research (ICRR), University of Tokyo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' acknowledges sup- port by NASA under award number 80GSFC21M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We also acknowledge the significant contributions over many years of Stefan Westerhoff, Gaurang Yodh and Ar- nulfo Zepeda Dominguez, all deceased members of the HAWC collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Thanks to Scott Delay, Luciano D´ıaz and Eduardo Murrieta for technical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' REFERENCES Abdo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2010, ApJS, 188, 405, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1088/0067-0049/188/2/405 Abdollahi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Acero, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020, ApJS, 247, 33, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4365/ab6bcb Abeysekara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Albert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Alfaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017a, Science, 358, 911, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='aan4880 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017b, ApJ, 843, 39, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4357/aa7555 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2019, ApJ, 881, 134, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4357/ab2f7d Abeysekara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021, PoS, ICRC2021, 828, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='22323/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0828 Acero, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2015, ApJS, 218, 23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1088/0067-0049/218/2/23 Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', An, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Axikegu, Bai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021, PhRvL, 126, 241103, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='241103 Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Atoyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Kifune, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1997, MNRAS, 291, 162, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1093/mnras/291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='162 Albert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Alfaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Alvarez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020, ApJ, 905, 76, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4357/abc2d8 Benbow, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Brill, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Buckley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021, ApJ, 916, 117, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4357/ac05b9 Clark, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Pletsch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017, ApJ, 834, 106, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4357/834/2/106 De La Torre Luque, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Fornieri, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Linden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='08544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='08544 Evoli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Linden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Morlino, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' D, 98, 063017, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='063017 Giacinti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Mitchell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', L´opez-Coto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', 636, A113, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1051/0004-6361/201936505 Grießmeier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Smith, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Theureau, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2021, A&A, 654, A43, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1051/0004-6361/202140841 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2020, Nature Astronomy, 4, 167, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1038/s41550-019-0910-0 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Collaboration, Abdalla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Abramowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018, A&A, 612, A2, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1051/0004-6361/201629377 Harding, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2001, in American Institute of Physics Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 558, High Energy Gamma-Ray Astronomy: International Symposium, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Aharonian & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' V¨olk, 115–126, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1370785 Hooper, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Cholis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Linden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Fang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' D, 96, 103013, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='103013 Kass, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Raftery, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1995, Journal of the American Statistical Association, 90, 773, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1080/01621459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='10476572 Klingler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Rangelov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Kargaltsev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2016, ApJ, 833, 253, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3847/1538-4357/833/2/253 Linden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Auchettl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Bramante, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2017, PhRvD, 96, 103016, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='103016 Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='04011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='04011 L´opez-Coto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', de O˜na Wilhelmi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Amato, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Hinton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2022, Nature Astronomy, 6, 199, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1038/s41550-021-01580-0 Lorimer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Yates, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Lyne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Gould, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1995, MNRAS, 273, 411, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1093/mnras/273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='411 Plucinsky, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Snowden, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Aschenbach, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1996, ApJ, 463, 224, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1086/177236 Smith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & HAWC Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2015, in International Cosmic Ray Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 34, 34th International Cosmic Ray Conference (ICRC2015), 966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='org/abs/1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='05826 Sudoh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Linden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Beacom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2019, PhRvD, 100, 043016, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='043016 8 Vianello, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Lauer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Younk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2015, arXiv e-prints, arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='08343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='org/abs/1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='08343 Wakely, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Horan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2008, in International Cosmic Ray Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 3, International Cosmic Ray Conference, 1341–1344 Wilks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 1938, Annals Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', 9, 60, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1214/aoms/1177732360 Zyuzin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', Karpova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=', & Shibanov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 2018, MNRAS, 476, 2177, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='1093/mnras/sty359 TeV Halo Candidate Surrounding Radio-quiet pulsar 9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Results of the likelihood fit assuming that the only emitting source is PSR B0355+54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The PL spectral model along with the two different spatial models were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Spatial model TS ∆BIC Extension N0 α [◦] TeV−1 cm−2 s−1 Point-like 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='27) × 10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='17 Extended 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='09 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5) × 10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='15 Note—All associated errors are statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' ∆BIC is obtained comparing the BIC value with the best spectral model fit for both spatial models assuming that the emission is coming from J0359 (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' PSR B0355+54 FITTING RESULTS In this section, we explore the possibility that the TeV excess comes entirely from B0355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We fit models with a power-law (PL) spectrum and the spatial templates described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The results are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The energy ranges at which the source is detected are 7-180 TeV at 1σ level, 11-90 TeV at 2σ level and 17-54 TeV assuming at 3σ level assuming a point-like morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For an extended morphology, the energy ranges are found to be 8-155 TeV for 1σ level, 11-90 TeV at 2σ level and 17-59 at 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' As single-source scenarios are not nested models, we have employed the Bayesian Information Criterion (BIC) to select the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The difference in the BIC value, ∆BIC, quantifies the evidence against the model with a higher BIC value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' According to Kass & Raftery (1995), if ∆BIC is between 0 and 2 it is not clear which model is preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' ∆BIC between 2 and 10 and above 10 indicates a slight and strong preference of the model with the smallest BIC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The small difference in ∆BIC from the fits of models centered at J0359 and B0355 does not allow us to distinguish between the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' This is expected as the angular distance of the two pulsars is smaller than the spatial resolution of HAWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' However, the tension between the VERITAS limits on B0355 and HAWC fluxes, as explained in Section 3, suggests that the TeV emission is more likely associated with J0359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' FITTING RESULTS OF A TWO-SOURCE SCENARIO We further explore a scenario where both J0359 and B0359 contribute to the TeV emission observed by HAWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Such a two-source model is disfavored by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Table 4 presents the results of the two-source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' We consider three combinations of spatial profiles of the two sources: (A) both sources are point-like, (B) both sources are extended with a Gaussian shape, and (C) J0359 is extended source and B0355 is point-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The energy spectrum is assumed to be a PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The normalization flux N0 and the spectral index α in each fit were free to vary while the position of the sources for all the scenarios were fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The ∆TS column shows the gain of test statistics by adding an extra source to the one-source model presented in Section 3 and Section A (the baseline model considers pure background plus the emission from the other source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The two-source model is disfavored in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' 10 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Results of the likelihood fit assuming that the excess observed comes from two sources: PSR J0359+5414 and PSR B0355+54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' The spectral model for all the spatial models is a PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Two-source model Source ∆TS ∆BIC Extension N0 α [◦] TeV−1 cm−2 s−1 J0359 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='9) × 10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='20 Model A B0355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='32 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='00034+6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='00024) × 10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3 −5 J0359 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='500+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='004 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3) × 10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='3 Model B B0355 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='29 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='15 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='4) × 10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='20 J0359 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0010 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='04+4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='04) × 10−14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='8 Model C B0355 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='62 26 / -15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='5) × 10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content='28 Note—All associated errors are statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' Model A corresponds to a scenario where both sources are point- like, model B assumes that both sources are extended with a Gaussian shape, and model C assumes that PSR J0359+5414 is as a point-like source and PSR B0355+54 is an extended source with a Gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' ∆BIC is obtained comparing the BIC value with the best model fit assuming that the emission is coming from J0359 (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} +page_content=' For model A, with the PL point-like model, for model B with the PL Gaussian model and for model C with the two previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf'} diff --git a/79FAT4oBgHgl3EQfoh2y/content/tmp_files/2301.08635v1.pdf.txt b/79FAT4oBgHgl3EQfoh2y/content/tmp_files/2301.08635v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9b150bf2c774caa46415507f22f37474979315e --- /dev/null +++ b/79FAT4oBgHgl3EQfoh2y/content/tmp_files/2301.08635v1.pdf.txt @@ -0,0 +1,1788 @@ +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 23, 2023 +Supercritical colliding wind binaries +Leandro Abaroa1, 2⋆, Gustavo E. Romero1, 2, and Pablo Sotomayor1, 2 +1 Instituto Argentino de Radioastronomía, CICPBA-CONICET-UNLP +Villa Elisa, La Plata, Argentina +2 Facultad de Cs. Astronómicas y Geofísicas, Universidad Nacional de La Plata +Paseo del Bosque S/N (1900), La Plata, Argentina +Received / Accepted +ABSTRACT +Context. Particle-accelerating colliding-wind binaries (PACWBs) are systems that are formed by two massive and hot stars and +produce nonthermal radiation. The key elements of these systems are fast winds and the shocks that they create when they collide. +Binaries with nonaccreting young pulsars have also been detected as nonthermal emitters, again as a consequence of the wind–wind +interaction. Black holes might produce nonthermal radiation by this mechanism if they accrete at super-Eddington rates. In such cases, +the disk is expected to launch a radiation-driven wind, and if this wind has an equatorial component, it can collide with the companion +star yielding a PACWB. These systems are supercritical colliding wind binaries. +Aims. We aim to characterize the particle acceleration and nonthermal radiation produced by the collision of winds in binary systems +composed of a superaccreting black hole and an early-type star. +Methods. We estimated the terminal velocity of the disk-driven wind by calculating the spatial distribution of the radiation fields +and their effect on disk particles. We then found the location of the wind collision region and calculated the timescales of energy +gain and losses of relativistic particles undergoing diffusive particle acceleration. With this information, we were able to compute +the associated spectral energy distribution of the radiation. We calculated a number of specific models with different parameters to +explore this scenario. +Results. We find that the interaction of winds can produce nonthermal emission from radio up to tens of GeV, with luminosities in +the range of ∼ 1033–1035 erg s−1, which for the most part are contributed by electron synchrotron and inverse Compton radiation. +Conclusions. We conclude that supercritical colliding wind binaries, such as some ultraluminous X-ray sources and some Galactic +X-ray binaries, are capable of accelerating cosmic rays and producing nonthermal electromagnetic emission from radio to γ-rays, in +addition to the thermal components. +Key words. acceleration of particles – accretion, accretion disks – relativistic processes – X-ray: binaries – gamma-rays: general – +radiation mechanism: non-thermal +1. Introduction +Early-type stars are very hot and their radiation fields can launch +powerful particle winds (Lamers & Cassinelli 1999). Such winds +quickly reach supersonic velocities and accelerate to terminal +velocities in the range (2 − 4) × 103 km s−1 (Abbott 1978; Mui- +jres et al. 2012). When two massive stars with powerful winds +form a binary system, the winds collide producing shocks sepa- +rated by a contact discontinuity from where matter is evacuated +(e.g., Stevens et al. 1992). A reverse shock moves in the wind of +each star. When such shocks are adiabatic, they can accelerate +suprathermal particles up to relativistic energies (Eichler & Usov +1993; Pittard et al. 2020). These particles, in turn, cool mainly +by synchrotron radiation and inverse Compton upscattering of +stellar photons, emitting nonthermal radiation (Eichler & Usov +1993; Benaglia & Romero 2003; Reimer et al. 2006; De Becker +2007; Reitberger et al. 2014; del Palacio et al. 2016; Pittard et al. +2021). Proton acceleration can also lead to gamma-ray emission +through pp collisions and the subsequent π0 decays (e.g., Balbo +& Walter 2017; Grimaldo et al. 2019). +The actual fraction of particle-accelerating colliding-wind +binaries (PACWBs) among massive colliding wind binaries +Send offprint requests to: Leandro Abaroa +⋆ leandroabaroa@gmail.com +(CWBs) is not well known. De Becker & Raucq (2013) list 43 +confirmed cases, mostly detected at radio wavelengths. These +authors mention several other candidates, and new sources have +been found since the publication of this latter work (e.g., Be- +naglia et al. 2015; del Palacio et al. 2016). The total kinetic +power of these systems ranges from ∼ 1034 to more than +1037 erg s−1. The most extreme cases are WR89, WR98, and +WR140, with powers of between 6 and 8 times 1037 erg s−1. +Less than 10−7 of this power is finally radiated through syn- +chrotron radio emission. The most luminous nonthermal radio- +emitting CWB is WR140, with a total radio luminosity of ∼ +2.6 × 1030 erg s−1. +Contrary to the radio emission, high-energy radiation has +been more difficult to detect in CWBs. At X-rays, the thermal +component usually dominates and hinders the detection of non- +thermal components. In the gamma-ray domain, only two sys- +tems have been detected so far: η Carinae and WR11. The lat- +ter is the nearest known CWB. At d ∼ 340 pc, it shows a +gamma-ray luminosity in the Fermi-LAT energy range of Lγ = +(3.7 ± 0.7) × 1031 erg s−1. This luminosity amounts to ∼ 6 × 10−6 +of the total wind kinetic power (Pshirkov 2016). Similar frac- +tions for other, more distant PACWBs yield fluxes that are un- +detectable with the currently available instrumentation. The no- +table exception is the mentioned η Carinae. +Article number, page 1 of 12 +arXiv:2301.08635v1 [astro-ph.HE] 20 Jan 2023 + +A&A proofs: manuscript no. main +η Carinae is a heavily obscured and peculiar object. The sys- +tem includes a luminous blue variable (LBV) star of about 90 +solar masses and a secondary Wolf-Rayet (WR) star of ∼ 30 so- +lar masses. η Carinae is the most luminous binary in the Galaxy, +with a bolometric luminosity of about 5 × 106 L⊙. The mass- +loss rate of the primary is extremely high, reaching up to 10−3 +M⊙ yr−1. The binary was detected in hard X-rays by INTEGRAL +(Leyder et al. 2008) and Suzaku (Okazaki et al. 2008), suggesting +the presence of relativistic electrons in the system. AGILE de- +tected gamma rays from η Carinae for the first time (Tavani et al. +2009). The system was subsequently detected by Fermi (Abdo +et al. 2010) with a luminosity of ∼ 1034 erg s−1. The observations +reveal the presence of a hard component in the spectrum around +periastron, which disappears near apastron. Such a component +has been explained through the decay of π0 produced by rela- +tivistic protons interacting with the dense stellar wind (Farnier +et al. 2011). There is a clear variability with the orbital phase. +Different behaviors are observed at low (0.3 − 10 GeV) and high +(> 10 GeV) gamma-ray energies. The low-energy component is +likely produced by inverse Compton scattering of stellar photons +(Balbo & Walter 2017). +The case of η Carinae suggests that super-Eddington systems +might be particularly powerful PACWBs. When a compact ob- +ject such as a black hole accretes with rates that exceed the Ed- +dington rate, the radiation pressure on the surface of the disk will +overcome the gravitational attraction and matter will be expelled +from the surface of the disk in the form of a strong wind. Such +winds can rival and even surpass those of the most luminous +CWBs in terms of kinetic power. When the donor star is a hot +early-type star also endowed with a wind, a supercritical collid- +ing wind binary (SCWB) can be formed. Such systems should +have strong shocks and are potential particle accelerators and +nonthermal emitters. +In our Galaxy, there are some examples of black hole X-ray +binaries with disks that launch strong outflows. Two examples +are GRS 1915+105 (Mirabel & Rodríguez 1994; Neilsen & Lee +2009) and V404 Cygni (Muñoz-Darias et al. 2016; Tetarenko +et al. 2017). However, the donor star in both of these systems +is a low-mass star. Another well-known supercritical source is +the Galactic microquasar SS433, which is a confirmed nonther- +mal emitter and might be a possible example of a SCWB in our +Galaxy (see Fabrika 2004, for an extensive review). Many ul- +traluminous X-ray sources (ULXs) detected in nearby galaxies +might also belong to this category of sources. +In this paper, we explore the CWB scenario where one of the +winds is launched by a supercritical disk around a black hole. We +start by characterizing the disk model and the radiation fields it +produces (Sections 2.1 and 2.2). We then investigate the motion +of particles under the radiation pressure in such fields (Section +2.3). This allows us to get reasonable estimates of the terminal +velocities expected for the matter ejected in the direction of the +companion star. We then proceed to study the wind interactions, +shock adiabaticity, and other relevant issues for particle accelera- +tion in Sect. 3. This is followed by estimates of energy losses for +accelerated particles, particle distributions, and calculations of +the nonthermal output (Sect. 4). In Section 5 we present results +for some specific models, with different choices of the accretor +mass and the accretion power. The donor star is supposed to be a +hot O.5V with a temperature of 41500 K and a kinetic power of +a few times 1037 erg s−1. We finally apply our model to the ex- +tragalactic binary system NGC 4190 ULX 1. After a discussion +(Sect. 7), we close with a summary and our conclusions. +2. The accretion disk and its wind +We assume that the X-ray binary is composed of a Population +I star and a nonrotating stellar mass black hole (BH) in a close +orbit. +The orbital semi-axis a, the stellar radius, and the mass ratio +of the system, q = M∗/MBH, satisfy (Eggleton 1983): +R∗ +lob = +a 0.49 q2/3 +0.6 q2/3 + ln (1 + q1/3), +(1) +where M∗ is the mass of the star and MBH the mass of the BH. +Hence, the star overflows its Roche lobe R∗ +lob, transfers mass to +the BH through the Lagrange point, and an accretion disk is +formed due to the angular momentum of the system. +In this section, we describe the semi-analytical models we +use to study the accretion disk, the spatial distribution of the ra- +diation fields produced by the disk, and the wind ejected from its +surface. We assume a Newtonian potential for the gravity field, +because we are interested in weak-field processes. +2.1. Accretion disk +We adopt cylindrical coordinates with axial symmetry along the +z-axis, neglect the self-gravity of the disk gas, and consider a +nonmagnetized disk with a super-Eddington accretion rate at the +outer part of the disk, ˙minput = ˙Minput/ ˙MEdd ≫ 1, where ˙Minput +is the input of mass per time unit in the accretion disk. The Ed- +dington rate is given by +˙MEdd = LEdd +ηc2 ≈ 2.2×10−8MBH yr−1 = 1.4×1018 MBH +M⊙ +g s−1, (2) +with LEdd the Eddington luminosity1, η ≈ 0.1 the accretion effi- +ciency, and c the speed of light. +The critical or spherization radius, given by +rcrit ∼ 40 ˙minputrg, +(3) +separates the disk in two regions: a standard outer disk (Shakura +& Sunyaev 1973) and a radiation-dominated inner disk with ad- +vection (Fukue 2004). In relation (3), rg = GMBH/c2 is the grav- +itational radius of the BH, with G the gravitational constant. In +the disk model, the advection is parameterized as a fraction f of +the viscous heating, Qadv = f Qvis, and the disk becomes geo- +metrically thick in the inner region, where the ejection of winds +by the radiation force helps to regulate the mass-accretion rate +onto the BH ( ˙Macc) at the Eddington rate2. +As the disk is optically thick, we assume that it radiates lo- +cally as a blackbody. The radiation intensity of a plasma element +in the comoving frame of the outer and inner disk, at a radius rd +measured on the equatorial plane, is +I0 = 1 +πσT 4 +eff = +��������������������� +1 +π +3GMBH ˙Minput +8πr3 +d +fin, rd > rcrit +1 +π +3 +4 +√c3 +LEdd +4πr2 +d +, rd ≤ rcrit, +(4) +1 The Eddington luminosity is defined as the luminosity required to +balance the attractive gravitational pull of the accreting object by radia- +tion pressure. +2 +˙Macc = ˙Minput in the outer region of the disk and ˙Macc = ˙Minputrd/rcrit +in the inner region (Fukue 2004). +Article number, page 2 of 12 + +L. Abaroa et al.: Super critical colliding wind binaries +Fig. 1: Geometry of the present disk model. The radiation fields +are calculated in the rz plane, where φ = 0. Here, Q is the posi- +tion of the plasma element of the disk and P the point of calcu- +lation on the rz plane. The scale height of the disk is H, and D +is the distance between Q and P. The short arrow is the direction +cosine jµ. This figure is adapted from Watarai & Fukue (1999). +where √c3 = H/rd = tan δ, with H the scale height of the disk, +δ the disk opening angle, and fin = 1 − rin/rd ≈ 1 (as rd > +rcrit, then rd ≫ rin). Here, c3 (along with c1 and c2 used in the +following section) is a coefficient that depends on the advection +parameter, the adiabatic index of the gas γ, and the viscosity α +(see Appendix in Fukue 2004). We adopt a disk with f = 0.5 and +α = 0.5; that is, we assume equipartition between advection and +viscous heating. The index γ = 4/3 corresponds to a radiation- +dominated gas in the inner disk. These values lead to a disk- +opening angle of δ = 30◦. +2.2. Radiation fields +The wind launched from the radiation-dominated region of the +disk will be determined by the radiation forces acting upon the +particles on the disk surface and along their subsequent trajec- +tories. These forces will have contributions from different parts +of the disk in relative motion with respect to the particles. Some +radiation will be blueshifted and some will be redshifted, result- +ing in differential azimuthal forces onto the particles and then +transferring angular momentum from the disk to the wind. +In order to obtain the radiative contribution of each plasma +element Q = (rd, φd, H) of the disk surface, at any point P = +(r, φ, z) above or below the disk, we make a transformation of +the intensity between the inertial and comoving reference frames +(see Fig. 1). Azimuthal symmetry allows us to perform the cal- +culations for any constant value of φ; therefore, we do it in the +rz plane (φ = 0). The relativistic Doppler factor D provides the +transformation between the reference frames (McKinley 1980): +I = D4I0 = +I0 +(1 + zred)4 , +(5) +where zred is the redshift factor given by (Watarai & Fukue 1999) +zred = −(r cos φd − rd)vr − (r sin φd)vφ + (z − H)vrc3 +cD +. +(6) +Here, D is the distance between P and Q, vφ = c2vK is the az- +imuthal velocity and vr = −c1αvK is the radial velocity, with +vK = √GMBH/rd the Keplerian velocity. We note that we only +consider the inner part of the disk for these calculations, because +the intensity decays with r−3 +d . +The radiation-field tensor is given by (Rybicki & Lightman +1986) +Rµν = +� +E +1 +c Fα +1 +c Fα +Pαβ +� += 1 +c +� +I jµ jνdΩ. +(7) +This is a symmetric tensor of rank 2 and therefore we calculate +ten elements in total: one for the energy density E, three for the +flux vector Fα, and six for the stress tensor Pαβ. In Eq. 7, jµ and +jν are the direction cosines in Cartesian coordinates, and Ω is the +solid angle subtended by Q: +jµ = +�r − rd cos φd +D +, −rd sin φd +D +, z − H +D +� +, +(8) +dΩ = −(r cos φd − rd) sin δ + (z − H) cos δ +D3 +dS, +(9) +where dS = √1 + c3 rd drd dφd. +2.3. Particles in the photon field +We now calculate the trajectory and velocity of the particles +ejected from the disk when they interact with photons of the am- +bient radiation field. +The equation of motion under a relativistic, radiation treat- +ment, is given by (Kato & Fukue 2020) +fµ = −∂Φe +∂xν + Rν +µ;ν, +(10) +where fµ is the four-force per unit volume. The effective po- +tential Φe is the sum of gravitational (Φg) and centrifugal (Φc) +potentials. The semicolon (; ) in the second term refers to the +covariant differentiation of the energy-momentum tensor. +As we consider a disk with axial symmetry, the gravitational +potential cancels out in the azimuthal coordinate: ∂Φg/∂xα = +(∂Φg/∂r, 0, ∂Φg/∂z). Furthermore, the centrifugal potential acts +only in the radial direction: ∂Φc/∂xα = (l2/r3, 0, 0), with l = +r2 +dωK being the specific angular momentum of the disk, and ωK +the angular velocity. +The equations of motion of the ejected particles can be found +working with Eq. 10. In terms of the nondimensional form of +the radiation-field tensor elements ϵ, f α, and pαβ, the system of +differential, tensorial, and coupled equations is as follows (equa- +tions originally derived by Watarai & Fukue 1999, Eq. 42–44, +but now extended to second order in velocity): +Radial coordinate: +dur +dτ = − ∂Φg +∂r + l2 +r3 + +(11) ++ 1 +2[γ f r − prβuβ − γ2ϵur + ur(2γ f βuβ − pβδuβuδ)]. +Azimuthal coordinate: +1 +r +dl +dτ = 1 +2[γ f φ − pφβuβ − γ2ϵ(l/r)+ +(12) ++ (l/r)(2γ f βuβ − pβδuβuδ)]. +Article number, page 3 of 12 + +P = (r,Φ,z) +D +Q = (rd,Φd, H) +7 +r +S +BHA&A proofs: manuscript no. main +Height coordinate: +duz +dτ = − ∂Φg +∂z + +(13) ++ 1 +2[γ f z − pzβuβ − γ2ϵuz + uz(2γ f βuβ − pβδuβuδ)], +where uµ denotes the four-velocity of the particles and γ the +Lorentz factor, which is given by +γ = +� +1 + urur + l2/r2 + uzuz. +(14) +The free parameter of these equations of motion is the launch- +ing radius of the particles, r0, and we assume as initial con- +dition that the particles co-rotate with the disk at this radius, +uα +0 = (0, l0/r0, 0). +We solve this system of equations numerically and assume +that the kinematics of the disk-driven wind is roughly described +by the trajectory and terminal velocities obtained for the test par- +ticles. As the accretion rate in the inner region of the disk is reg- +ulated at the Eddington rate, the mass loss in the wind is of the +order of the super-Eddington accretion rate, ˙Mdw ∼ ˙Minput. +3. Collision of winds +The wind ejected from the disk collides with the stellar wind +at the interaction region, where shocks are generated giving rise +to particle acceleration. An important quantity that characterizes +the wind is the kinetic luminosity, LK = +˙Mv2/2, where ˙M is +the mass-loss rate and v the velocity of the fluid. A small frac- +tion of the total kinetic power of the wind is transferred to rel- +ativistic particles, Lrel ∼ 0.1LK, where we assume equipartition +between relativistic protons and electrons (Le = Lp). The mass- +loss rate and velocity of the stellar wind are set according to the +parameters found in the literature for the type of star we have +chosen (e.g., Kobulnicky et al. 2019). In the case of the disk- +driven wind, the velocity is obtained following the procedures +described in the previous section. Given the orbital separation, +the disk inclination, and the stellar size, we estimate that ∼ 10% +of the original kinetic power reaches the acceleration region. We +assume a circular orbit, that is, the geometry associated with the +collision of winds does not depend on the orbital phase. +In this section, we describe the models for the collision re- +gion, the magnetic ambient field, and the shocks. We adopt a +one-zone approximation for these calculations. +Fig. 2: Scheme of the wind collision seen in the rz plane (not to +scale), adapted from Abaroa et al. (2021). +3.1. Contact discontinuity +The winds collide at a surface called the contact discontinuity +(CD). The stagnation point (SP) is the closest position of the CD +to the star, and is located where the ram pressures of the winds +are in equilibrium, +Pram(rBH) = ρdwv2 +dw = ρ∗wv2 +∗w = Pram(r∗). +(15) +Here, rBH and r∗ are the distances to the SP from the BH and +from the center of the star, respectively. The density of the spher- +ical stellar wind at this location is given by +ρ∗w = +˙M∗ +4πr2∗v∗w +, +(16) +whereas the density of the disk-driven wind reads +ρdw = +˙Mdw +Ωr2 +BHvdw +, +(17) +where Ω = 2π(1 − cos θ) is the solid angle of the wind and θ +the semi-opening angle of the wind. Solving these equations we +obtain the position of the SP. +3.2. Magnetic field +The strength of the magnetic field at the CD is essentially deter- +mined by the stellar surface magnetic field B∗. The intensity of +BCD and its topology –dipole (i), radial (ii), or toroidal (iii)–, is +given by (Eichler & Usov 1993): +BCD ≈ B∗ × +����������������� +R3 +∗/r3 +∗, R∗ < r∗ < rA, +(i) +R3 +∗/rAr2 +∗, rA < r∗ < R∗(v∗w/vrot +∗ ), +(ii) +R2 +∗vrot +∗ /rAr∗v∗w, R∗(v∗w/vrot +∗ ) < r∗, (iii), +(18) +where R∗ is the stellar radius, rA the Alfvén radius, and vrot +∗ +∼ +0.1v∗w the surface rotation velocity. +3.3. Particle acceleration and shock +Particles are accelerated up to relativistic energies in the col- +lision region through a first-order diffusive shock mechanism. +Two shock fronts are generated: a forward shock (FS) that prop- +agates through the stellar wind, and a reverse shock (RS) that +propagates through the wind of the disk. The diffusive accelera- +tion rate of the particles is given by (e.g., Protheroe 1999): +t−1 +ac = ηac +e Z c BCD +E +, +(19) +where e is the electric charge, Z the atomic number, and E is the +energy of the particle. The acceleration efficiency, ηac, depends +on the diffusion coefficient of the particles, the shock velocity, +and the angle between the magnetic field and the normal to the +shock plane. We assume that the shock propagates perpendicu- +lar to the magnetic field and that diffusion occurs in the Bohm +regime. Thus, the acceleration efficiency is +ηac ≈ 3 +8 +�vsh +c +�2 +, +(20) +where the shock velocities in the reference frame where one of +the fluids is at rest, v∗w = 0, and the other one moves with a +velocity vdw, are given by (Lee et al. 1996): +vRS = −4 +3 +1 +1 + +� +n∗w/ndw +vdw, +(21) +Article number, page 4 of 12 + +Shock +Stellar wind +Disk-driven wind +BH +Star +Disk +7L. Abaroa et al.: Super critical colliding wind binaries +vFS = 4 +3 +1 +1 + +� +ndw/n∗w +vdw. +(22) +Here, n∗w and ndw are the numerical densities of the winds (nw = +ρw/mp, with mp the mass of the proton). The pressure and density +of the shocked medium are calculated following the Rankine- +Hugoniot relations (e.g., Lamers & Cassinelli 1999). +As we are interested in the nonthermal particle distribution, +we investigate only adiabatic shocks; that is, where radiative +losses are negligible. This is because in radiative shocks the gas +in the shocked region emits large amounts of thermal radiation; +the system therefore loses energy, the entropy increases, and +the medium becomes increasingly homogeneous. If magnetic- +inhomogeneities disappear, the acceleration efficiency decays +abruptly, aborting the formation of nonthermal distributions. +The shock is adiabatic if the thermal cooling length RΛ is +larger than the size of the acceleration region ∆xac (McCray & +Snow 1979). The cooling length reads +RΛ = +5.9 × 1011µ(vsh/km s−1)3 +(nw/cm−3)[Λ(Tsh)/erg s−1 cm−3] cm. +(23) +Here, nw is the number density of the undisturbed medium, µ +is the average molecular weight (µ = 0.6 for a fully ionized +plasma), and Λ(Tsh) is the cooling function, which depends on +the shock temperature (Raymond et al. 1976; Myasnikov et al. +1998; Wolfire et al. 2003). This latter function can be written as +Λ(Tsh) = +������������� +4 × 10−29T 0.8 +sh , +55 K ≤ Tsh < 104 K +7 × 10−27Tsh, +104 K ≤ Tsh < 105 K +7 × 10−19T −0.6 +sh +, +105 K ≤ Tsh < 4 × 107 K +3 × 10−27T 0.5 +sh , +Tsh ≥ 4 × 107 K, +(24) +where Tsh is given by +Tsh = 18.21µ +� +vsh +km s−1 +�2 +K. +(25) +We note that this temperature has a maximum value in a colli- +sional plasma: it is self-regulated by the pair-creation, satisfying +in any case kBTsh < 1 MeV (kB is the Boltzmann constant). +We assume that the size of the acceleration region is a frac- +tion of the distance from the BH to the SP, ∆xac ∼ 0.1rBH. As +we consider a one-zone model, the acceleration region must be +narrow enough to generate near-homogeneous conditions. +4. Radiative processes +Particles accelerated at the shock can cool through different pro- +cesses and produce nonthermal radiation. The timescales asso- +ciated to this cooling are related to the total energy-loss of the +particles: +dE +dt ≈ −E +tcool +, +(26) +where the total cooling rate is +t−1 +cool = +� +i +t−1 +i , +(27) +where ti corresponds to each timescale of the involved cooling +processes. +We assume advective escape; that is, particles are removed +from the acceleration region by the bulk motion of the fluid. If +the timescales of cooling are shorter than those of escape, par- +ticles radiate before they escape from the acceleration region. +The maximum energy for each kind of particle can be inferred +by looking at the point where the acceleration rate is equal to +the total cooling or escape rate. This energy cannot exceed the +maximum energy imposed by the Hillas criterion, Emax +e,p < Emax +Hillas. +As we are interested in nonthermal processes, we work at +scales smaller than the size of the binary system and assume that +rotation effects are negligible there. Effects caused by the orbital +motion, such as Coriolis or centrifugal forces, could be relevant +on larger scales and lead to strong disturbances in the flow and +thermal processes. The analysis of such effects usually requires +numerical simulations and is beyond the scope of this work. +4.1. Energy losses +We consider adiabatic and radiative losses. Adiabatic cooling is +related to the work done by the particles of the wind to expand +the shocked gas. Radiative cooling is caused by nonthermal pro- +cesses as a consequence of the interaction of the wind particles +with ambient fields and matter. +Our model is lepto-hadronic, and so we calculate the follow- +ing radiative processes numerically: +–Synchrotron: interaction of protons and electrons with the +ambient magnetic field, which will be amplified by a factor of 4 +in the shocked region due to Rankine-Hugoniot relations. +–Inverse Compton (IC): collision of relativistic electrons +with photons of the ambient radiation field. +–Bremmstrahlung: Coulombian interactions between rela- +tivistic electrons and cold matter. +–Photo-hadronic interactions: interaction of highly relativis- +tic protons with photons of the ambient radiation field. +–Proton-proton: collision of relativistic protons with cold +matter. +In addition, we take into account inelastic collision of parti- +cles with atoms of the dense medium; that is, ionization losses, +which can be relevant in the 1–100 MeV range. We note that +in this energy range, ionization losses largely dominate over +Coulomb scatterings (see e.g., Fig. 7 from O’C Drury et al. +1996), and so the latter are not included in our analysis. The +reader is referred to Romero & Paredes (2011), Romero & Vila +(2014), and Müller & Romero (2020) plus references therein for +additional details on radiative processes. +4.2. Particle distribution +We investigate the evolution of particles that are accelerated +at the shock and injected into the surrounding medium. The +medium around the shock is the shocked gas of the winds. In +this paper, we restrict our analysis to this region. Beyond the bi- +nary, the surrounding medium has been affected by the effects +of the stellar winds, and so the system is expected to be located +inside a bubble inflated by the winds and surrounded by a shell +formed with the swept-up material at distances of a few to sev- +eral parsecs, depending on the mass of the black hole progenitor. +Inside the bubble, where the advected protons will be injected, +the density is expected to be lower than that of the standard in- +terstellar medium (e.g., around 0.01 cm−3 or less). In the shell, +there should be sufficient material for hadronic interactions with +the protons diffused or transported from the central source3. +3 These effects will be discussed elsewhere; some of them might be +responsible for part of the high-energy emission observed in the shell +Article number, page 5 of 12 + +A&A proofs: manuscript no. main +The relativistic particles have a distribution given by dN = +n(r, E, t)dEdV, where n is the number density of particles, t the +time, r the position, V the volume, and E the energy. The evo- +lution of this distribution is determined by the transport equa- +tion (see e.g., Ginzburg & Syrovatskii 1964; Romero & Paredes +2011). We solve this equation numerically in steady state and in +the one-zone approximation: +∂ +∂E +�dE +dt N(E) +� ++ N(E) +tesc += Q(E), +(28) +where tesc ∼ ∆xac/vsh is the advection time, and the particle in- +jection function, +Q(E) = Q0E−p exp (−E/Emax), +(29) +is a power-law in the energy with an exponential cutoff and a +spectral index p = 2.2, which is characteristic of the Fermi first- +order acceleration mechanism (see e.g., Drury 1983). The nor- +malization constant Q0 is obtained from +L(e,p) = ∆V +� Emax +(e,p) +Emin +(e,p) +dE(e,p)E(e,p)Q(e,p)(E(e,p)), +(30) +where ∆V is the volume of the acceleration region, and Emax +(e,p) +the maximum energy reached by protons and electrons, which is +found by looking at the point where the acceleration rate is equal +to the total cooling or escape rate. +4.3. Nonthermal emission +Once we have the particle distributions, we calculate the spectral +energy distribution (SED) for each of the relevant processes in- +volved in cooling. We find that in SCWBs, electrons typically +cool by synchrotron and IC mechanisms, and protons escape +from the acceleration region without significant cooling. The +resultant nonthermal SED usually yields a broadband spectrum +from radio waves (due to synchrotron emission) to gamma-rays +(due to IC emission). +4.4. Wind emission +We calculate the thermal emission of the photosphere of the disk- +driven wind assuming a spherically symmetric wind that ex- +pands with constant velocity equal to its terminal velocity. Since +the mass-loss rate of the disk is much higher than the critical +rate, the wind is optically thick and therefore we assume that it +radiates locally as a blackbody. The temperature measured by an +observer at infinity is given by (Fukue 2009): +σTT 4 +dw = +˙e LEdd +(1 − β cos Θ)4 4πR2 , +(31) +where ˙e = +˙E/LEdd is the normalized comoving luminosity, +β = vdw/c the normalized velocity, Θ the angle of the flow with +respect to the line of sight, and R = +√ +r2 + z2, with r and z the +being cylindrical coordinates. We assume that the comoving lu- +minosity is equal to the Eddington luminosity (˙e = 1), as is com- +monly done in supercritical wind-models (e.g., Fukue 2009). +The apparent photosphere of this wind is defined as the sur- +face where the optical depth τphoto is unity for an observer at in- +finity. If the velocity of the wind is relativistic, the optical depth +of W50, which is powered by SS433, although there are jets involved in +this specific object. +in the observer frame depends in general on the magnitude of +the velocity and the viewing angle. The location of the apparent +photosphere from the equatorial plane zphoto is (Fukue 2009): +τphoto = +� ∞ +zphoto +γdw(1 − β cos Θ) κco ρcodz = 1, +(32) +where γdw is the wind Lorentz factor, κco the opacity in the co- +moving frame, and ρco the wind density in the comoving frame. +As we assume a fully ionized wind, the opacity is dominated by +free electron scattering (κco = σT/mp). +4.5. Absorption +Finally, we calculate the gamma absorption by pair creation from +photon–photon annihilation, γ + γ → e+ + e−. The nonthermal +photons in their way out of the acceleration region can find pho- +tons of the ambient radiation fields and annihilate. The absorp- +tion is quantified by the optical depth of the medium, τγγ. If the +original luminosity of gamma rays is L0 +γ(Eγ), the attenuated lu- +minosity reads: +Lγ(Eγ) = L0 +γ(Eγ) · e−τ, +(33) +where e−τ is the attenuation factor. The targets of the ambient ra- +diation fields are photons from the star and from the disk-driven +wind photosphere. +The process of annihilation is possible only above a kine- +matic energy threshold given by +EγEph > (mec2)2, +(34) +in a frontal collision, where Eph is the energy of the targets. The +opacity caused by a photon–photon pair production for a photon +created at a distance r from the center of the thermal source can +be obtained from (Romero & Vila 2008): +τγγ(Eγ, r) = +� ∞ +Emin +� ∞ +r +nph(Eph, r′) σγγ(Eph, Eγ) dr′dEph, +(35) +where nph is the density of the ambient radiation field. The total +cross-section is given by (see e.g., Aharonian et al. 1985): +σγγ = πr2 +e +2 (1 − ξ2) +� +(3 − ξ4) ln +�1 + ξ +1 − ξ +� ++ 2ξ(ξ2 − 2) +� +, +(36) +where re is the classical radius of the electron, and +ξ = +� +1 − (mec2)2 +EγEph +�1/2 +. +(37) +The blackbody density radiation of the star and the photosphere +of the disk-driven wind is given by +nph = +2E2 +ph +h3c3 +1 +exp(Eph/kBT) − 1, +(38) +where T is the temperature of the thermal source considered for +each case; that is, Tdw or Teff. +On the other side, free-free absorption (FFA) must also be +taken into account. The collision of low-energy photons with +particles of the dense medium leads to a cutoff in the SED at +radio frequencies. The denser the medium, the higher the energy +at which the cutoff occurs. Therefore, FFA will determine the +Article number, page 6 of 12 + +L. Abaroa et al.: Super critical colliding wind binaries +turnover of the synchrotron spectrum in SCWBs, which is ex- +pected to be at ∼GHz frequencies (see e.g., Rybicki & Lightman +1986; del Palacio et al. 2016). +Other absorption processes, such as the photoelectric effect, +direct Compton, or γ-nucleon pair creation, are not taken into +account in this paper. Their cross-sections are not high enough to +become relevant in the calculation of opacity given the ambient +densities that we consider here (see Fig. 1 from Reynoso et al. +2011). +5. Results +In this section, we apply our model to a generic super-Eddington +X-ray binary. We consider a star of spectral type O.5V (Table +1) and investigate four scenarios: in scenarios S1 and S2 we re- +gard a BH with mass MBH = 5M⊙ and mass-accretion rates of +102 ˙MEdd and 103 ˙MEdd, respectively; in scenarios S3 and S4 we +consider a BH with mass MBH = 20M⊙ and again accretion rates +of 102 ˙MEdd and 103 ˙MEdd, respectively. The complete set of pa- +rameters is summarized in Table 2. +Type O.5V Star +Parameter +Value +Units +M∗ +37 +M⊙ +R∗ +11 +R⊙ +Teff +41500 +K +˙M∗ +1.2 × 10−5 +M⊙ yr−1 +v∗w +2.9 × 108 +cm s−1 +vrot +∗ +2.9 × 107 +cm s−1 +L∗ +K +3.2 × 1037 +erg s−1 +B∗ +750 +G +Table 1: Parameters adopted in the model for the star of type +O.5V. All parameters from Kobulnicky et al. (2019), with the +exception for the magnetic field (from Wade & MiMeS Collab- +oration 2015). +5.1. Wind +We calculate the radiation-field tensor (Eq. 7) and in Fig. 3 we +show the distribution of the energy density (ϵ) on the rz plane, +where the black zone is the inflated inner disk. We obtain a +strong azimuthal flux component of the radiation-field tensor. +This distribution is the same in all four scenarios, because in +the critical disk the radiation-field tensor depends on advection, +viscosity, and adiabatic parameters, which remain the same in all +cases. +We solve Eqs. 11-13 to find the trajectory and velocity of the +particles. Both quantities are determined by Rµν and therefore we +obtain the same trajectories and terminal velocities in S1–S4. As +an example, in Fig. 4 we show the normalized velocity of a test +particle, with a launching radius of 40rg (≡ 20rs), which reaches +a terminal velocity of ≈ 0.16c. This result does not vary much if +we vary the launching radius (±0.02c for ±20rg). +The particles describe a helical trajectory in the vicinity of +the BH for two main reasons (Fig. 5). The first is the presence +of the strong azimuthal components of the radiation field, which +help to maintain the spiral geometry of the particles in the inner +disk. The second reason is the condition imposed for the particle +ejection, namely that the particles initially have only azimuthal + 0 + 5 + 10 + 15 + 20 + 0 + 5 + 10 + 15 + 20 +z [rs] +r [rs] +1x10-3 +2x10-3 +2x10-3 +2x10-3 +3x10-3 +4x10-3 +4x10-3 +5x10-3 +Fig. 3: Contour maps of the spatial distribution of the normalized +radiation energy density ϵ in the rz plane above the accretion +disk. Both axes are in units of Schwarzschild radius. The color +bar is the intensity of ϵ and the black zone is the inflated disk +( f = 0.5, α = 0.5, γ = 4/3). +40 +60 +80 +100 +120 +0.16 +0.17 +0.18 +0.19 +0.20 +0.21 +r/rs +v/c +Fig. 4: Normalized velocity of a wind test particle as a function +of the Schwarzschild radius. The particle reaches a terminal ve- +locity of ∼ 0.16c for a launching radius of r0 = 20rs (coincident +with the vertical axis). +velocity. The intensity of the radiation field decays rapidly with +distance from the BH, and therefore the ejected particles follow a +spiral trajectory near the BH, but beyond a certain radius (∼ rcrit) +they follow a free path with a strong component of the radial +velocity. +The overall result is an equatorial wind with terminal veloci- +ties of the order of 0.15c. The kinetic power of these winds is in +the range 1039−41 erg s−1, which is well above the power of the +winds of typical WR or OB stars. Therefore, in general, the disk +wind is expected to overwhelm the stellar wind. +5.2. Energy gain and losses +We follow the calculations in Sect. 3.1 and find that, in all four +scenarios, the SP is located near the stellar surface and the wind +of the disk completely sweeps up the stellar wind, as expected. +Hence, the forward shock is in the stellar atmosphere, fully ra- +Article number, page 7 of 12 + +A&A proofs: manuscript no. main +Scenario +Parameter +Symbol [units] +S1 +S2 +S3 +S4 +Black hole mass(1) +MBH [M⊙] +5 +5 +20 +20 +Mass accretion rate(1) +˙Minput [M⊙ yr−1] +1.1 × 10−5 +1.1 × 10−4 +4.4 × 10−5 +4.4 × 10−4 +Orbital semi-axis(1) +a [R⊙] +15 +15 +22 +22 +Gravitational radius(2) +rg [cm] +7.4 × 105 +7.4 × 105 +2.9 × 106 +2.9 × 106 +Critical radius(2) +rcrit [rg] +4000 +40000 +4000 +40000 +Mass loss in disk winds(1) +˙Mdw [M⊙ yr−1] +10−5 +10−4 +4.3 × 10−5 +4.3 × 10−4 +Kinetic power of the disk-driven wind(2) +Ldw +K +[erg s−1] +7.8 × 1039 +7.8 × 1040 +3.4 × 1040 +3.4 × 1041 +Cold matter density at SP(2) +ndw [cm−3] +5.1 × 1012 +5.1 × 1013 +2.9 × 1012 +2.9 × 1013 +Distance to SP from BH(2) +rBH [cm] +2.7 × 1011 +2.7 × 1011 +7.6 × 1011 +7.6 × 1011 +Size of acceleration region(1) +∆xac [cm] +2.7 × 1010 +2.7 × 1010 +7.6 × 1010 +7.6 × 1010 +Shock cold matter density(2) +nRS [cm−3] +2 × 1013 +2 × 1014 +1.2 × 1013 +1.2 × 1014 +Shock cooling length(2) +RΛ [cm] +7.6 × 1011 +7.6 × 1010 +1.3 × 1012 +1.3 × 1011 +Maximum energy of electrons(2) +Emax +e +[eV] +1011 +1.6 × 1011 +1011 +1011 +Maximum energy of protons(2) +Emax +p +[eV] +1015 +1015 +3 × 1015 +3.1 × 1015 +Emission peak (low energy)(2) +L0.01mm [erg s−1] +3.2 × 1033 +3.2 × 1033 +8 × 1034 +8 × 1034 +Emission peak (high energy)(2) +L10MeV [erg s−1] +4 × 1032 +4 × 1032 +1034 +1034 +Table 2: Parameters of the different scenarios calculated for the model. We indicate with superscript (1) those parameters that are +assumed and with (2) those that are derived. In all models, the system is supposed to be oriented face-on to the observer, that is, the +inclination of the normal to the orbital plane i with respect to the line of the sight is ∼ 0◦. +Fig. 5: Trajectory of a test particle in the Cartesian 3D-space in +units of Schwarzschild radius. The particles describe a helical +trajectory above the inner disk because of the strong azimuthal +radiation fields. The launching radius of this test particle is r0 = +20rs. +diative, and completely unable to accelerate relativistic particles. +Only the reverse shock (RS) is suitable for the task. As r∗ ≈ R∗, +the magnetic field at the CD is BCD ≈ B∗. +The cooling length of the RS is greater than the size of the +acceleration region in all cases (see Table 2); this is why the +shock is adiabatic and the acceleration efficiency of the process +is relatively high: ηac ∼ 10−2 (see Sect. 3.3). The shock velocity +is ≈ 4.4 × 109 cm s−1 and the temperature of the shocked gas +reaches ≈ 4.8 × 1010 K. +We calculate the energy gain and losses of the shock- +accelerated particles following Sect. 4. Highly relativistic pro- +tons escape from the acceleration region without cooling in all +scenarios considered here (with energies up to Ep ≈ 1 PeV) and +are injected into the interstellar medium (ISM). Protons are ad- +vected, that is, they are removed from the collision region by the +bulk motion of the fluid. They therefore do not interact with am- +bient material at scales similar to that of the system. Electrons +cool mainly through IC and synchrotron mechanisms, and reach +a maximum energy of Ee ≈ 100 GeV. To obtain the electron dis- +tribution, we solve the transport equation considering only the +dominant IC and synchrotron losses, and a power-law injection +function with a spectral index of 2.2 and an exponential cutoff +(see Eq. 29). +5.3. Spectral energy distribution +Figure 6 shows the SEDs of the four scenarios. The only thermal +component of the spectrum is the photosphere of the optically +thick disk-driven wind. The emission peak of the wind for S1 +and S2 is ≈ 1037 erg s−1, whereas for S3 and S4 the peak is +≈ 1038 erg s−1. This occurs at energies of ∼ 100 eV for S1 and +S3, and ∼ 30 eV for S2 and S4. Therefore, if MBH increases, the +luminosity is higher and, if the mass-accretion rate increases, the +luminosity peak occurs at lower energies. +In the case of the nonthermal spectrum, we calculate the +emission due to synchrotron and IC losses. In the latter case, we +consider the photon fields of the star and of the wind photosphere +as targets. In all cases, the dominant IC contribution is that of +the star. The luminosity in S3 and S4 is an order of magnitude +greater than that in S1 and S2. This is because of the modification +of the orbital parameters when the BH mass varies: to guarantee +the overflow of the Roche lobe, the orbital semi-axis varies with +MBH, which results in variation in the size of the acceleration re- +Article number, page 8 of 12 + +L. Abaroa et al.: Super critical colliding wind binaries +gion and the photon density at SP, among other parameters. The +emission peak at low energies is ∼ 1033 erg s−1 for S1 and S2, +and ∼ 1035 erg s−1 for S3 and S4. At high energies, the emission +peak is ∼ 1032 erg s−1 (S1 and S2) and ∼ 1034 erg s−1 (S3 and +S4). The gamma-ray absorption due to γγ annihilation is total +for energies > 10 GeV in all scenarios4. +Attenuation due to material between the source and the ob- +server, that is, absorption by external cold gas, is mainly in the +optical-to-UV range and at soft X-rays. At radio wavelengths, re- +fractive scintillation on free electrons of the ISM occurs at lower +frequencies than predicted here. For high-energy gamma rays, +the main absorbers are infrared (IR) fields and the cosmic mi- +crowave background (CMB), but their effects are only relevant +for cosmological distances. +6. Application to NGC 4190 ULX 1 +Ultraluminous X-ray sources (ULXs) are extragalactic point-like +objects where the luminosity in the X-ray band appears to be +higher than the Eddington luminosity (Bachetti 2016). ULXs are +thought to be X-ray binaries with a stellar-mass compact object +accreting at super-Eddington rates, where a beaming effect could +be responsible for the luminosity observed in the X-ray band: +the radiation emitted from the inner part of the accretion disk is +geometrically collimated by the ejected wind, which is optically +thick except in a narrow region around the black-hole axis and +forms a cone-shaped funnel (King et al. 2001; King 2009; Kaaret +et al. 2017; Fabrika et al. 2021). +We apply our model to estimate the radiation emitted by the +ultraluminous X-ray source NGC 4190 ULX 1 (also known as +CXO J121345.2+363754). Although many characteristics of this +ULX remain poorly understood, several authors have explored +the system and have provided constraints on some of its param- +eters (see e.g., Liu & Bregman 2005; Gladstone et al. 2013; Ko- +liopanos et al. 2017; Kosec et al. 2018; Ghosh & Rana 2021). +In what follows, we describe the parameterization of the sys- +tem and its components, and investigate the expected collision +of winds. The complete set of parameters used in this section is +detailed in Table 3. +6.1. System parameterization +The source is located in the nearby Galaxy NGC 4190 at a dis- +tance of d ≈ 3 Mpc (Tully et al. 2013). Observations made +in 2010 using the XMM-Newton telescope reveal a long-term +spectral variability in the 0.3–10.0 keV energy range: LX ∼ +3 − 8 × 1039 erg s−1. +The angle i between the line of sight and the z-axis at which +the disk of a ULX is observed determines the components of its +spectrum: blackbody disk (BB) or Comptonization. If i is small, +the observer is able to look into the funnel and see the innermost +part of the disk: the spectrum shows only the BB component, +which corresponds to thermal emission of the disk. This type of +spectrum is called broadened disk (BD). If i is sufficiently large, +another effect is observed: the interaction between photons and +wind particles near the disk surface induces a Comptonization +that produces a hardening in the spectrum. Most ULXs exhibit a +combination of both phenomena in their X-ray spectrum. +4 We note that, since we assume a nearly face-on inclination of the +system, there are no significant variations of the radiative output associ- +ated with the orbital phase. If the system were oriented nearly edge-on, +the emission would be modulated by the orbital phase due to absorption +(for details see Romero et al. 2010). +Ghosh & Rana (2021) investigated the spectral properties of +NGC 4190 ULX 1 and suggested that the ULX is in a BD state, +and that the compact object is a BH with mass ∼ 10 − 30M⊙ +accreting at super-Eddington rates. We fit the XMM-Newton ob- +servations (Epoch 3) with the supercritical advection-dominated +disk model detailed in Sect. 2.1, assuming a mass-accretion rate +of ˙Minput = 10 ˙MEdd. We also assume a face-on inclination i ≈ 0◦, +a BH mass 10M⊙ and a geometrical beaming factor b = 0.07. +This factor is given by, +b = Ω/4π = 0.5(1 − cos ϑ), +(39) +where Ω is the solid angle of the emission. The angle ϑ is related +to the opening angles of the disk (δ) and its wind (θ): ϑ+δ+2θ = +90◦. Both angles, i and ϑ, can change over time, causing the +spectral variability of the object (Fabrika et al. 2021). +On the other hand, Gladstone et al. (2013) provided con- +straints on the characteristics of the optical counterpart of the +system. They suggested that, if MBH = 10M⊙, the mass of the +star could be < 50M⊙ and its radius < 86R⊙. We choose a star +of type B2V for our model in light of one of the fittings these +latter authors made from Hubble Space Telescope observations. +If we apply Eq. 1 and consider the mass ratio M∗/MBH, and the +stellar radius involved (see Table 3), the transfer of mass in the +binary system occurs for an orbital semi-axis a ≤ 15.2 R⊙, which +results in a period ≤ 38 h. +6.2. Collision of winds +The terminal velocity of the disk-driven wind is vdw = 4.95 × +109 cm s−1, and therefore Ldw +K = 1.5 × 1039 erg s−1, while L∗ +K = +2.17 × 1034 erg s−1. The SP is located near the stellar surface +and the wind of the disk completely suppresses the stellar wind. +We therefore only take into account the reverse shock (RS). As +r∗ ≈ R∗, the magnetic field at the CD is BCD ≈ B∗. +The cooling length of the RS is RΛ = 2.2 × 1013 cm and +the size of the acceleration region is ∆xac = 6.68 × 1010 cm; +therefore, the shock is adiabatic and the acceleration efficiency of +the process is ηac = 10−2, as in our general models. We calculate +the energy gain and losses of the shock particles following Sect. +4. Highly relativistic protons escape from the acceleration region +without cooling, as in our previous scenarios (with energies up +to Ep ≈ 1 PeV), and are injected into the ISM. Electrons cool +mainly through IC and synchrotron mechanisms. Figure 7 shows +the timescales of electrons, which reach a maximum energy of +Ee ≈ 0.32 TeV. To obtain the electron distribution, we solve the +transport equation taking into account only IC and synchrotron +losses, and a power-law injection function with a spectral index +of 2.2 and an exponential cutoff. +6.3. Total SED +The SED of the ULX spans a broadband energy range. Figure +9 shows the thermal (wind and accretion disk) and nonthermal +(colliding-winds shock) contributions of the system. We also +show the sensitivity of the instruments ALMA, VLA (sub-mm +waves), Fermi, and CTA (gamma rays), and observational data +from XMM-Newton. +The luminosity in the IR band is ∼ 1034 erg s−1, which is rel- +atively strong, though still undetectable at megaparsec distances. +The luminosity in gamma-rays also reaches ∼ 1034 erg s−1. The +attenuation factor (Fig. 8) has an effect on photons with ener- +gies ≳ 1 GeV. Most of the radiation above 1 GeV and all above +50 GeV is suppressed by the annihilation of the γ rays with the +photon fields of the disk-driven wind and the star. +Article number, page 9 of 12 + +A&A proofs: manuscript no. main + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 +-4 +-2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +S1 & S3 +log10 (EγLγ / erg s-1) +log10 (Eγ / eV) +wind photosphere S1 +synchrotron S1 +inverse Compton S1 (star) +inverse Compton S1 (wind) +total SED S1 +wind photosphere S3 +synchrotron S3 +inverse Compton S3 (star) +inverse Compton S3 (wind) +total SED S3 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 +-4 +-2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +S2 & S4 +log10 (EγLγ / erg s-1) +log10 (Eγ / eV) +wind photosphere S2 +synchrotron S2 +inverse Compton S2 (star) +inverse Compton S2 (wind) +total SED S2 +wind photosphere S4 +synchrotron S4 +inverse Compton S4 (star) +inverse Compton S4 (wind) +total SED S4 +Fig. 6: Thermal and nonthermal SEDs of the four scenarios considered, S1–S4, in logarithmic scale, where a face-on inclination is +assumed. S1 and S3 are shown in the left plot, whereas S2 and S4 are shown in the right plot. Dashed lines correspond to S1 (left) +and S2 (right), solid lines correspond to S3 (left) and S4 (right). We plot the nonattenuated inverse Compton contributions in gray. +The emission peak at high energies is ∼ 1033 erg s−1 for S1 and S2, and ∼ 1034 erg s−1 for S3 and S4. The gamma-ray absorption +due to γγ annihilation is total for energies > 10 GeV. +-8 +-6 +-4 +-2 + 0 + 2 + 4 + 6 + 8 + 10 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 +ηac ∼ 10-2 +RS +log10 (t-1 / s-1) +log10 (Ee / eV) +synchrotron +inverse Compton +Bremsstrahlung +adiabatic +ion +acceleration +escape +Fig. 7: Timescales in logarithmic scale of the electron accelera- +tion, escape, and cooling at the reverse shock in NGC 4190 ULX +1. Electrons reach a maximum energy of ≈ 0.32 TeV. The accel- +eration efficiency is 10−2. +7. Discussion +Our analysis of supercritical colliding wind binaries shows that +these systems should exhibit broadband emission from radio to +gamma rays. In this sense, they are similar to CWBs formed by +two hot stars, such as O+WR binaries. However, there are im- +portant differences as well. If we compare our models with re- +cent models of O+WR CWBs (Pittard et al. 2021), we find that +(i) in SCWBs, the wind of the disk is far more powerful than +the wind of the star. This results in stagnation points that are +very close to the surface of the star. Efficient particle accelera- +tion then can only occur in reverse shocks. (ii) We also see that +the disk wind advects protons from the acceleration region be- +fore they have time to cool. Only electrons can cool locally. The + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +-6 +-4 +-2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +e−τγγ +log10 (Eγ / eV) +Star +Wind photosphere +Total attenuation +Fig. 8: Attenuation factors due to γγ-annihilation between high- +energy nonthermal radiation and photon fields from the star and +from the photosphere of the disk-driven wind in NGC 4190 ULX +1. The total attenuation is plotted with a black line. +resulting SED is consequently dominated by synchrotron and IC +radiation. (iii) As the acceleration region is close to the star, the +local magnetic field is relatively strong. Synchrotron emission +reaches energies of hundreds of keV. As the medium is far more +dense than in stellar CWBs, free-free absorption causes this radi- +ation to turnover below ∼ 24 GHz. The total power at millimeter +(mm) and submm wavelengths can be between three and five +orders of magnitude higher in SCWBs than in stellar CWBs. +(iv) IC is the dominant radiation mechanism at high energies. +The stronger thermal fields of SCWBs (wind photosphere and +star) provide the seed photons, but also impose a high-energy +cutoff at ∼ 1 GeV through γ − γ attenuation. Instead, stellar +CWBs can reach energies close to 1 TeV. (v) The strong mag- +netic fields in the acceleration region cut electromagnetic cas- +Article number, page 10 of 12 + +L. Abaroa et al.: Super critical colliding wind binaries +Table 3: Parameters of NGC 4190 ULX 1. +Parameter +Symbol +Value +Units +System +Inclination(1) +i +0 +◦ +Orbital semi-axis(2) +a +15 +R⊙ +Distance to the source(3) +d +3 +Mpc +Black hole +Mass(1) +MBH +10 +M⊙ +Gravitational radius(2) +rg +1.48 × 106 +cm +Accretion disk +Disk semi opening angle(1) +δ +30 +◦ +Critical radius(2) +rcrit +3.5 × 109 +cm +Eddington accretion rate +˙MEdd +2.2 × 10−7 +M⊙ yr−1 +Mass accretion rate(1) +˙Minput +2.2 × 10−6 +M⊙ yr−1 +Mass loss in winds(1) +˙Mdw +1.98 × 10−6 +M⊙ yr−1 +Wind velocity(2) +vdw +4.95 × 109 +cm s−1 +Wind semi opening angle(2) +θ +14.5 +◦ +Beaming factor(2) +b +0.07 +− +B2V Star +Mass(4) +M∗ +8 +M⊙ +Radius(4) +R∗ +5.4 +R⊙ +Temperature(4) +Teff +20600 +K +Mass loss in winds(4) +˙M∗ +1.4 × 10−7 +M⊙ yr−1 +Wind velocity(4) +v∗w +7 × 107 +cm s−1 +Rotation velocity(1) +vrot +∗ +7 × 106 +cm s−1 +Magnetic field(5) +B∗ +200 +G +Colliding winds +Kinetic power of disk-driven wind(2) +Ldw +K +1.5 × 1039 +erg s−1 +Kinetic power of stellar wind(2) +L∗ +K +2.17 × 1034 +erg s−1 +Distance from BH to SP(2) +rBH +6.68 × 1011 +cm +Size of acceleration region(1) +∆xac +6.68 × 1010 +cm +Magnetic field at SP(2) +BSP +200 +G +Injection spectral index(1) +p +2.2 +− +Acceleration efficiency(2) +ηac +10−2 +− +Molecular mean weight(1) +µ +0.6 +− +Reverse shock +Velocity(2) +vRS +4.4 × 109 +cm s−1 +Temperature(2) +TRS +1010 +K +Cold matter density(2) +nRS +6.9 × 1011 +cm−3 +Cooling length(2) +RΛ +2.2 × 1013 +cm +Notes. We indicate the parameters we have assumed with superscript +(1) and those we have derived with (2). Parameters with superscripts +(3), (4), and (5) were taken from Tully et al. (2013), Kobulnicky et al. +(2019), and Shultz et al. (2015), respectively. +cades in SCWBs. (vi) The SED is always dominated by the X- +ray component associated with the disk or its wind in SCWBs. +Finally, (vii) stellar CWBs have wider orbits and a variable sep- +aration between the components of the system. This produces +variability related to the orbital period. On the contrary, the or- +bits of SCWBs should be mostly circularized. In general, CWBs +are weaker than SCWBs, although span a broader energy range. +An interesting feature of SCWBs is their potential as cosmic +ray sources. As mentioned, the strong wind of the disk drags +away the relativistic protons before they cool. These protons, +with maximum energies of the order of 1 PeV, are then injected +into the ISM where they diffuse. Even if a fraction of just ∼ 1 +% of the wind kinetic power goes to relativistic protons, the +cosmic ray output of a SCWB would be in the range 1037−39 +erg s−1. These protons might interact with ambient clouds at +some distance from the system, producing gamma rays through +pp → π0 + pp interactions and the subsequent pion decays +π0 → γγ. The gamma-ray emission from the illuminated clouds + 30 + 32 + 34 + 36 + 38 + 40 + 42 +-8 +-6 +-4 +-2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +Fermi +CTA +ALMA +VLA +log10 (EγLγ / erg s-1) +log10 (Eγ / eV) +wind photosphere +beamed disk +inverse Compton (star) +inverse Compton (wind) +synchrotron +Total SED +XMM Newton data +Fig. 9: Thermal and nonthermal SEDs of NGC 4190 ULX 1 in +logarithmic scale (dashed lines). The nonthermal SED is par- +tially attenuated for energies > 1 GeV and totally attenuated +for energies > 50 GeV due to annihilation of γ-rays with the +photon fields of the star and the photosphere of the disk-driven +wind. The gray dashed lines are the nonattenuated IC contribu- +tions. The total SED is plotted with a solid black line. Data from +XMM-Newton (Epoch 3), and the sensitivity of ALMA, Fermi, +VLA, and CTA are also shown (instrument sensitivities were +taken from Sotomayor & Romero 2022). +can be even stronger than the emission from the binary itself. +However, the spectrum should be softer because of propagation +effects (Aharonian & Atoyan 1996). Recent modeling by Pit- +tard et al. (2021) of particle acceleration in colliding wind bina- +ries with wind velocities of a few 103 km s−1 and mG magnetic +fields in the acceleration region demonstrate that up to ∼ 30 % of +the wind power can be transferred to nonthermal particles. This +means that, in some extreme cases, a SCWB might inject up to +∼ 1040 erg s−1 in cosmic rays. +Another type of CWB is the so-called gamma-ray binary +(GRB; e.g., LS 5039, PSR B1259-63, LSI +61◦ 303, PSR +J2032+4127, and others; see, e.g., Dubus 2013; Chernyakova & +Malyshev 2020). These sources are formed by a massive star +(usually a Be star with a dense equatorial decretion disk and a +fast wind) and a young pulsar in an eccentric orbit. The pul- +sar ejects a relativistic pair wind. The wind collision produces +a broadband spectrum from electrons accelerated at the shock +that cool by synchrotron and IC radiation. The two-peak SEDs +are similar to those we estimate for SCWBs, but some differ- +ences are also clearly seen: (i) GRBs are less energetic because +the spin-down luminosity of the pulsar is much smaller than the +power of a supercritical wind. (ii) GRBs are highly variable. This +variability is modulated with the orbital period. The orbital mod- +ulation of the different components of the broadband spectrum is +a consequence of the orbital variability of geometrical parame- +ters, such as the geometry of the contact surface of the stellar +and pulsar winds. Absorption effects are also strongly variable. +(iii) Hadronic interactions are likely when the pulsar crosses the +equatorial disk of the star (e.g., Bykov et al. 2021). (iv) GeV +flares have been observed after the periastron passage in sources +such as PSR B1259-63 (Abdo et al. 2011; Chernyakova et al. +2014). These flares are attributed to the effects of the unshocked +pulsar wind interaction with photons from the stellar disk (e.g., +Khangulyan et al. 2012). +Article number, page 11 of 12 + +A&A proofs: manuscript no. main +We finally mention that some black holes accreting at super- +critical rates seem to be capable of launching mildly relativis- +tic jets. A remarkable case in our Galaxy is the notorious mi- +croquasar SS433 (Fabrika 2004). This object resembles a ULX +source seen edge on (Begelman et al. 2006). The accretion rate +should be extremely high in order to explain the large jet power +LK ∼ 1040 erg s−1. Begelman et al. (2006) suggest rates of +∼ 5 × 103 ˙MEdd ∼ 5 × 10−4 M⊙ yr−1, which are consistent with +estimates of equatorial mass outflows inferred from radio obser- +vations (Blundell et al. 2001). These outflows, ejected toward +either side of the jets, present a thermal spectrum and might well +correspond to the radiation-driven wind of the hypercritical disk. +The contamination from the jet base makes it impossible to dis- +entangle contributions from colliding winds from those coming +from the jet. However, the equatorial outflow might propagate +well beyond the system and reveal itself if it collides with any +clouds. The shock generated in the collision would convert the +kinetic energy of the plasmoids into internal energy and relativis- +tic particles, which might then cool by pp interactions with the +cloud material. Such a scenario might explain the detection of a +GeV source by the Fermi satellite on the side of SS433 (Bordas +2020; Li et al. 2020). We will explore the details of this hypoth- +esis elsewhere. +8. Summary and conclusions +We explored the consequences of supercritical accretion in bi- +nary systems consisting of a hot star and a black hole. We find +that a fraction of the kinetic power of the radiation-driven wind +released by the accretion disk is transformed into relativistic par- +ticles in the region of the wind that collides with the star. Elec- +trons are cooled locally, mainly through synchrotron and inverse +Compton radiation. The radiation fields of the star and wind pho- +tosphere provide abundant thermal photons for the latter process; +they also absorb high-energy radiation above a few GeV. Free- +free absorption imposes a high-frequency turnover in the ra- +dio regime, suppressing centimeter radio waves, unlike the case +of colliding wind binaries. The relativistic protons are blown +away by the wind before they can cool down significantly. Once +trapped by the outflow, these protons are transported to outer re- +gions where they can interact with ambient gas away from the +binary system, producing hadronic gamma-rays. Our most im- +portant finding is that, in addition to being strong thermal UV +and X-ray sources, supercritical colliding wind binaries can be +significant nonthermal sources at mm wavelengths and GeV en- +ergies. +Acknowledgements. The authors thank the anonymous referee for a careful and +constructive review, and for his/her comments that improved this work. We thank +also Daniela Pérez and Jiˇri Horák for fruitful discussions. This work was sup- +ported by grant PIP 0554 (CONICET). LA ackowledges the Universidad Na- +cional de La Plata for the education received. GER acknowledges the support +from the Spanish Ministerio de Ciencia e Innovación (MICINN) under grant +PID2019-105510GBC31 and through the Center of Excellence Mara de Maeztu +2020-2023 award to the ICCUB (CEX2019-000918-M). +References +Abaroa, L., Sotomayor Checa, P., & Romero, G. E. 2021, Boletin de la Asocia- +cion Argentina de Astronomia La Plata Argentina, 62, 262 +Abbott, D. C. 1978, ApJ, 225, 893 +Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ, 723, 649 +Abdo, A. A., Ackermann, M., Ajello, M., et al. 2011, ApJ, 736, L11 +Aharonian, F. A. & Atoyan, A. M. 1996, A&A, 309, 917 +Aharonian, F. A., Kririllov-Ugriumov, V. G., & Vardanian, V. V. 1985, Ap&SS, +115, 201 +Bachetti, M. 2016, Astronomische Nachrichten, 337, 349 +Balbo, M. & Walter, R. 2017, A&A, 603, A111 +Begelman, M. C., King, A. R., & Pringle, J. E. 2006, MNRAS, 370, 399 +Benaglia, P., Marcote, B., Moldón, J., et al. 2015, A&A, 579, A99 +Benaglia, P. & Romero, G. E. 2003, A&A, 399, 1121 +Blundell, K. M., Mioduszewski, A. J., Muxlow, T. W. B., Podsiadlowski, P., & +Rupen, M. P. 2001, ApJ, 562, L79 +Bordas, P. 2020, Nature Astronomy, 4, 1132 +Bykov, A. M., Petrov, A. E., Kalyashova, M. E., & Troitsky, S. V. 2021, ApJ, +921, L10 +Chernyakova, M., Abdo, A. A., Neronov, A., et al. 2014, MNRAS, 439, 432 +Chernyakova, M. & Malyshev, D. 2020, in Multifrequency Behaviour of High +Energy Cosmic Sources - XIII. 3-8 June 2019. Palermo, 45 +De Becker, M. 2007, A&A Rev., 14, 171 +De Becker, M. & Raucq, F. 2013, A&A, 558, A28 +del Palacio, S., Bosch-Ramon, V., Romero, G. E., & Benaglia, P. 2016, A&A, +591, A139 +Drury, L. O. 1983, Reports on Progress in Physics, 46, 973 +Dubus, G. 2013, A&A Rev., 21, 64 +Eggleton, P. P. 1983, ApJ, 268, 368 +Eichler, D. & Usov, V. 1993, ApJ, 402, 271 +Fabrika, S. 2004, Astrophys. Space Phys. Res., 12, 1 +Fabrika, S. N., Atapin, K. E., Vinokurov, A. S., & Sholukhova, O. N. 2021, +Astrophysical Bulletin, 76, 6 +Farnier, C., Walter, R., & Leyder, J. C. 2011, A&A, 526, A57 +Fukue, J. 2004, PASJ, 56, 569 +Fukue, J. 2009, PASJ, 61, 1305 +Ghosh, T. & Rana, V. 2021, MNRAS, 504, 974 +Ginzburg, V. L. & Syrovatskii, S. I. 1964, The Origin of Cosmic Rays. Oxford: +Pergamon Press. +Gladstone, J. C., Copperwheat, C., Heinke, C. O., et al. 2013, ApJS, 206, 14 +Grimaldo, E., Reimer, A., Kissmann, R., Niederwanger, F., & Reitberger, K. +2019, ApJ, 871, 55 +Kaaret, P., Feng, H., & Roberts, T. P. 2017, ARA&A, 55, 303 +Kato, S. & Fukue, J. 2020, Fundamentals of Astrophysical Fluid Dynamics. Sin- +gapore: Springer. +Khangulyan, D., Aharonian, F. A., Bogovalov, S. V., & Ribó, M. 2012, ApJ, 752, +L17 +King, A. R. 2009, MNRAS, 393, L41 +King, A. R., Davies, M. B., Ward, M. J., Fabbiano, G., & Elvis, M. 2001, ApJ, +552, L109 +Kobulnicky, H. A., Chick, W. T., & Povich, M. S. 2019, AJ, 158, 73 +Koliopanos, F., Vasilopoulos, G., Godet, O., et al. 2017, A&A, 608, A47 +Kosec, P., Pinto, C., Fabian, A. C., & Walton, D. J. 2018, MNRAS, 473, 5680 +Lamers, H. J. G. L. M. & Cassinelli, J. P. 1999, Introduction to Stellar Winds. +Cambridge: Cambridge University Press. +Lee, H. M., Kang, H., & Ryu, D. 1996, ApJ, 464, 131 +Leyder, J. C., Walter, R., & Rauw, G. 2008, A&A, 477, L29 +Li, J., Torres, D. F., Liu, R.-Y., et al. 2020, Nature Astronomy, 4, 1177 +Liu, J.-F. & Bregman, J. N. 2005, ApJS, 157, 59 +McCray, R. & Snow, T. P., J. 1979, ARA&A, 17, 213 +McKinley, J. M. 1980, American Journal of Physics, 48, 612 +Mirabel, I. F. & Rodríguez, L. F. 1994, Nature, 371, 46 +Muñoz-Darias, T., Casares, J., Mata Sánchez, D., et al. 2016, Nature, 534, 75 +Muijres, L. E., Vink, J. S., de Koter, A., Müller, P. E., & Langer, N. 2012, A&A, +537, A37 +Müller, A. L. & Romero, G. E. 2020, A&A, 636, A92 +Myasnikov, A. V., Zhekov, S. A., & Belov, N. A. 1998, MNRAS, 298, 1021 +Neilsen, J. & Lee, J. C. 2009, Nature, 458, 481 +O’C Drury, L., Duffy, P., & Kirk, J. G. 1996, A&A, 309, 1002 +Okazaki, A. T., Owocki, S. P., Russell, C. M. P., & Corcoran, M. F. 2008, MN- +RAS, 388, L39 +Pittard, J. M., Romero, G. E., & Vila, G. S. 2021, MNRAS, 504, 4204 +Pittard, J. M., Vila, G. S., & Romero, G. E. 2020, MNRAS, 495, 2205 +Protheroe, R. J. 1999, in Topics in Cosmic-Ray Astrophysics. New York: Nova +Science Publishing, ed. M. A. Duvernois, Vol. 230 +Pshirkov, M. S. 2016, MNRAS, 457, L99 +Raymond, J. C., Cox, D. P., & Smith, B. W. 1976, ApJ, 204, 290 +Reimer, A., Pohl, M., & Reimer, O. 2006, ApJ, 644, 1118 +Reitberger, K., Kissmann, R., Reimer, A., & Reimer, O. 2014, ApJ, 789, 87 +Reynoso, M. M., Medina, M. C., & Romero, G. E. 2011, A&A, 531, A30 +Romero, G. E., Del Valle, M. V., & Orellana, M. 2010, A&A, 518, A12 +Romero, G. E. & Paredes, J. M. 2011, Introducción a la Astrofísica Relativista. +Barcelona: Universitat de Barcelona. +Romero, G. E. & Vila, G. S. 2008, A&A, 485, 623 +Romero, G. E. & Vila, G. S. 2014, Introduction to Black Hole Astrophysics. +Heidelberg: Springer. +Rybicki, G. B. & Lightman, A. P. 1986, Radiative Processes in Astrophysics. +New York: Wiley VCH. +Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 500, 33 +Article number, page 12 of 12 + +L. Abaroa et al.: Super critical colliding wind binaries +Shultz, M., Wade, G. A., Alecian, E., & BinaMIcS Collaboration. 2015, MN- +RAS, 454, L1 +Sotomayor, P. & Romero, G. E. 2022, A&A, 664, A178 +Stevens, I. R., Blondin, J. M., & Pollock, A. M. T. 1992, ApJ, 386, 265 +Tavani, M., Sabatini, S., Pian, E., et al. 2009, ApJ, 698, L142 +Tetarenko, A. J., Sivakoff, G. R., Miller-Jones, J. C. A., et al. 2017, MNRAS, +469, 3141 +Tully, R. B., Courtois, H. M., Dolphin, A. E., et al. 2013, AJ, 146, 86 +Wade, G. A. & MiMeS Collaboration. 2015, in Astronomical Society of the Pa- +cific Conference Series, Vol. 494, Physics and Evolution of Magnetic and +Related Stars, ed. Y. Y. Balega, I. I. Romanyuk, & D. O. Kudryavtsev, 30 +Watarai, K.-y. & Fukue, J. 1999, PASJ, 51, 725 +Wolfire, M. G., McKee, C. F., Hollenbach, D., & Tielens, A. G. G. M. 2003, ApJ, +587, 278 +Article number, page 13 of 12 + diff --git a/79FAT4oBgHgl3EQfoh2y/content/tmp_files/load_file.txt b/79FAT4oBgHgl3EQfoh2y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f5007c94073072cadf0169583c014b2ffa87a30 --- /dev/null +++ b/79FAT4oBgHgl3EQfoh2y/content/tmp_files/load_file.txt @@ -0,0 +1,1188 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf,len=1187 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main ©ESO 2023 January 23, 2023 Supercritical colliding wind binaries Leandro Abaroa1, 2⋆, Gustavo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Romero1, 2, and Pablo Sotomayor1, 2 1 Instituto Argentino de Radioastronomía, CICPBA-CONICET-UNLP Villa Elisa, La Plata, Argentina 2 Facultad de Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Astronómicas y Geofísicas, Universidad Nacional de La Plata Paseo del Bosque S/N (1900), La Plata, Argentina Received / Accepted ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Particle-accelerating colliding-wind binaries (PACWBs) are systems that are formed by two massive and hot stars and produce nonthermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The key elements of these systems are fast winds and the shocks that they create when they collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Binaries with nonaccreting young pulsars have also been detected as nonthermal emitters, again as a consequence of the wind–wind interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Black holes might produce nonthermal radiation by this mechanism if they accrete at super-Eddington rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In such cases, the disk is expected to launch a radiation-driven wind, and if this wind has an equatorial component, it can collide with the companion star yielding a PACWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These systems are supercritical colliding wind binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We aim to characterize the particle acceleration and nonthermal radiation produced by the collision of winds in binary systems composed of a superaccreting black hole and an early-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We estimated the terminal velocity of the disk-driven wind by calculating the spatial distribution of the radiation fields and their effect on disk particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We then found the location of the wind collision region and calculated the timescales of energy gain and losses of relativistic particles undergoing diffusive particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' With this information, we were able to compute the associated spectral energy distribution of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We calculated a number of specific models with different parameters to explore this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We find that the interaction of winds can produce nonthermal emission from radio up to tens of GeV, with luminosities in the range of ∼ 1033–1035 erg s−1, which for the most part are contributed by electron synchrotron and inverse Compton radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We conclude that supercritical colliding wind binaries, such as some ultraluminous X-ray sources and some Galactic X-ray binaries, are capable of accelerating cosmic rays and producing nonthermal electromagnetic emission from radio to γ-rays, in addition to the thermal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' acceleration of particles – accretion, accretion disks – relativistic processes – X-ray: binaries – gamma-rays: general – radiation mechanism: non-thermal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Introduction Early-type stars are very hot and their radiation fields can launch powerful particle winds (Lamers & Cassinelli 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Such winds quickly reach supersonic velocities and accelerate to terminal velocities in the range (2 − 4) × 103 km s−1 (Abbott 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Mui- jres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' When two massive stars with powerful winds form a binary system, the winds collide producing shocks sepa- rated by a contact discontinuity from where matter is evacuated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A reverse shock moves in the wind of each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' When such shocks are adiabatic, they can accelerate suprathermal particles up to relativistic energies (Eichler & Usov 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Pittard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These particles, in turn, cool mainly by synchrotron radiation and inverse Compton upscattering of stellar photons, emitting nonthermal radiation (Eichler & Usov 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Benaglia & Romero 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Reimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' De Becker 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Reitberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' del Palacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Pittard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Proton acceleration can also lead to gamma-ray emission through pp collisions and the subsequent π0 decays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Balbo & Walter 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Grimaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The actual fraction of particle-accelerating colliding-wind binaries (PACWBs) among massive colliding wind binaries Send offprint requests to: Leandro Abaroa ⋆ leandroabaroa@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='com (CWBs) is not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' De Becker & Raucq (2013) list 43 confirmed cases, mostly detected at radio wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These authors mention several other candidates, and new sources have been found since the publication of this latter work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Be- naglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' del Palacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The total kinetic power of these systems ranges from ∼ 1034 to more than 1037 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The most extreme cases are WR89, WR98, and WR140, with powers of between 6 and 8 times 1037 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Less than 10−7 of this power is finally radiated through syn- chrotron radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The most luminous nonthermal radio- emitting CWB is WR140, with a total radio luminosity of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1030 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Contrary to the radio emission, high-energy radiation has been more difficult to detect in CWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' At X-rays, the thermal component usually dominates and hinders the detection of non- thermal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In the gamma-ray domain, only two sys- tems have been detected so far: η Carinae and WR11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The lat- ter is the nearest known CWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' At d ∼ 340 pc, it shows a gamma-ray luminosity in the Fermi-LAT energy range of Lγ = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='7) × 1031 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This luminosity amounts to ∼ 6 × 10−6 of the total wind kinetic power (Pshirkov 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Similar frac- tions for other, more distant PACWBs yield fluxes that are un- detectable with the currently available instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The no- table exception is the mentioned η Carinae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Article number, page 1 of 12 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='08635v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='HE] 20 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main η Carinae is a heavily obscured and peculiar object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The sys- tem includes a luminous blue variable (LBV) star of about 90 solar masses and a secondary Wolf-Rayet (WR) star of ∼ 30 so- lar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' η Carinae is the most luminous binary in the Galaxy, with a bolometric luminosity of about 5 × 106 L⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The mass- loss rate of the primary is extremely high, reaching up to 10−3 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The binary was detected in hard X-rays by INTEGRAL (Leyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2008) and Suzaku (Okazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2008), suggesting the presence of relativistic electrons in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' AGILE de- tected gamma rays from η Carinae for the first time (Tavani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The system was subsequently detected by Fermi (Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2010) with a luminosity of ∼ 1034 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The observations reveal the presence of a hard component in the spectrum around periastron, which disappears near apastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Such a component has been explained through the decay of π0 produced by rela- tivistic protons interacting with the dense stellar wind (Farnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' There is a clear variability with the orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Different behaviors are observed at low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3 − 10 GeV) and high (> 10 GeV) gamma-ray energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The low-energy component is likely produced by inverse Compton scattering of stellar photons (Balbo & Walter 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The case of η Carinae suggests that super-Eddington systems might be particularly powerful PACWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' When a compact ob- ject such as a black hole accretes with rates that exceed the Ed- dington rate, the radiation pressure on the surface of the disk will overcome the gravitational attraction and matter will be expelled from the surface of the disk in the form of a strong wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Such winds can rival and even surpass those of the most luminous CWBs in terms of kinetic power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' When the donor star is a hot early-type star also endowed with a wind, a supercritical collid- ing wind binary (SCWB) can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Such systems should have strong shocks and are potential particle accelerators and nonthermal emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In our Galaxy, there are some examples of black hole X-ray binaries with disks that launch strong outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Two examples are GRS 1915+105 (Mirabel & Rodríguez 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Neilsen & Lee 2009) and V404 Cygni (Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' However, the donor star in both of these systems is a low-mass star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Another well-known supercritical source is the Galactic microquasar SS433, which is a confirmed nonther- mal emitter and might be a possible example of a SCWB in our Galaxy (see Fabrika 2004, for an extensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Many ul- traluminous X-ray sources (ULXs) detected in nearby galaxies might also belong to this category of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In this paper, we explore the CWB scenario where one of the winds is launched by a supercritical disk around a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We start by characterizing the disk model and the radiation fields it produces (Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We then investigate the motion of particles under the radiation pressure in such fields (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This allows us to get reasonable estimates of the terminal velocities expected for the matter ejected in the direction of the companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We then proceed to study the wind interactions, shock adiabaticity, and other relevant issues for particle accelera- tion in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This is followed by estimates of energy losses for accelerated particles, particle distributions, and calculations of the nonthermal output (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In Section 5 we present results for some specific models, with different choices of the accretor mass and the accretion power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The donor star is supposed to be a hot O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5V with a temperature of 41500 K and a kinetic power of a few times 1037 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We finally apply our model to the ex- tragalactic binary system NGC 4190 ULX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' After a discussion (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 7), we close with a summary and our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The accretion disk and its wind We assume that the X-ray binary is composed of a Population I star and a nonrotating stellar mass black hole (BH) in a close orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The orbital semi-axis a, the stellar radius, and the mass ratio of the system, q = M∗/MBH, satisfy (Eggleton 1983): R∗ lob = a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='49 q2/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 q2/3 + ln (1 + q1/3), (1) where M∗ is the mass of the star and MBH the mass of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Hence, the star overflows its Roche lobe R∗ lob, transfers mass to the BH through the Lagrange point, and an accretion disk is formed due to the angular momentum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In this section, we describe the semi-analytical models we use to study the accretion disk, the spatial distribution of the ra- diation fields produced by the disk, and the wind ejected from its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We assume a Newtonian potential for the gravity field, because we are interested in weak-field processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Accretion disk We adopt cylindrical coordinates with axial symmetry along the z-axis, neglect the self-gravity of the disk gas, and consider a nonmagnetized disk with a super-Eddington accretion rate at the outer part of the disk, ˙minput = ˙Minput/ ˙MEdd ≫ 1, where ˙Minput is the input of mass per time unit in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The Ed- dington rate is given by ˙MEdd = LEdd ηc2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2×10−8MBH yr−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4×1018 MBH M⊙ g s−1, (2) with LEdd the Eddington luminosity1, η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 the accretion effi- ciency, and c the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The critical or spherization radius, given by rcrit ∼ 40 ˙minputrg, (3) separates the disk in two regions: a standard outer disk (Shakura & Sunyaev 1973) and a radiation-dominated inner disk with ad- vection (Fukue 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In relation (3), rg = GMBH/c2 is the grav- itational radius of the BH, with G the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In the disk model, the advection is parameterized as a fraction f of the viscous heating, Qadv = f Qvis, and the disk becomes geo- metrically thick in the inner region, where the ejection of winds by the radiation force helps to regulate the mass-accretion rate onto the BH ( ˙Macc) at the Eddington rate2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As the disk is optically thick, we assume that it radiates lo- cally as a blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The radiation intensity of a plasma element in the comoving frame of the outer and inner disk, at a radius rd measured on the equatorial plane, is I0 = 1 πσT 4 eff = ��������������������� 1 π 3GMBH ˙Minput 8πr3 d fin, rd > rcrit 1 π 3 4 √c3 LEdd 4πr2 d , rd ≤ rcrit, (4) 1 The Eddington luminosity is defined as the luminosity required to balance the attractive gravitational pull of the accreting object by radia- tion pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2 ˙Macc = ˙Minput in the outer region of the disk and ˙Macc = ˙Minputrd/rcrit in the inner region (Fukue 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Article number, page 2 of 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' : Super critical colliding wind binaries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1: Geometry of the present disk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The radiation fields are calculated in the rz plane, where φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Here, Q is the posi- tion of the plasma element of the disk and P the point of calcu- lation on the rz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The scale height of the disk is H, and D is the distance between Q and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The short arrow is the direction cosine jµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This figure is adapted from Watarai & Fukue (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' where √c3 = H/rd = tan δ, with H the scale height of the disk, δ the disk opening angle, and fin = 1 − rin/rd ≈ 1 (as rd > rcrit, then rd ≫ rin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Here, c3 (along with c1 and c2 used in the following section) is a coefficient that depends on the advection parameter, the adiabatic index of the gas γ, and the viscosity α (see Appendix in Fukue 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We adopt a disk with f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' that is, we assume equipartition between advection and viscous heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The index γ = 4/3 corresponds to a radiation- dominated gas in the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These values lead to a disk- opening angle of δ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Radiation fields The wind launched from the radiation-dominated region of the disk will be determined by the radiation forces acting upon the particles on the disk surface and along their subsequent trajec- tories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These forces will have contributions from different parts of the disk in relative motion with respect to the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Some radiation will be blueshifted and some will be redshifted, result- ing in differential azimuthal forces onto the particles and then transferring angular momentum from the disk to the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In order to obtain the radiative contribution of each plasma element Q = (rd, φd, H) of the disk surface, at any point P = (r, φ, z) above or below the disk, we make a transformation of the intensity between the inertial and comoving reference frames (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Azimuthal symmetry allows us to perform the cal- culations for any constant value of φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' therefore, we do it in the rz plane (φ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The relativistic Doppler factor D provides the transformation between the reference frames (McKinley 1980): I = D4I0 = I0 (1 + zred)4 , (5) where zred is the redshift factor given by (Watarai & Fukue 1999) zred = −(r cos φd − rd)vr − (r sin φd)vφ + (z − H)vrc3 cD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (6) Here, D is the distance between P and Q, vφ = c2vK is the az- imuthal velocity and vr = −c1αvK is the radial velocity, with vK = √GMBH/rd the Keplerian velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We note that we only consider the inner part of the disk for these calculations, because the intensity decays with r−3 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The radiation-field tensor is given by (Rybicki & Lightman 1986) Rµν = � E 1 c Fα 1 c Fα Pαβ � = 1 c � I jµ jνdΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (7) This is a symmetric tensor of rank 2 and therefore we calculate ten elements in total: one for the energy density E, three for the flux vector Fα, and six for the stress tensor Pαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 7, jµ and jν are the direction cosines in Cartesian coordinates, and Ω is the solid angle subtended by Q: jµ = �r − rd cos φd D , −rd sin φd D , z − H D � , (8) dΩ = −(r cos φd − rd) sin δ + (z − H) cos δ D3 dS, (9) where dS = √1 + c3 rd drd dφd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Particles in the photon field We now calculate the trajectory and velocity of the particles ejected from the disk when they interact with photons of the am- bient radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The equation of motion under a relativistic, radiation treat- ment, is given by (Kato & Fukue 2020) fµ = −∂Φe ∂xν + Rν µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='ν, (10) where fµ is the four-force per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The effective po- tential Φe is the sum of gravitational (Φg) and centrifugal (Φc) potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The semicolon (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' ) in the second term refers to the covariant differentiation of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As we consider a disk with axial symmetry, the gravitational potential cancels out in the azimuthal coordinate: ∂Φg/∂xα = (∂Φg/∂r, 0, ∂Φg/∂z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Furthermore, the centrifugal potential acts only in the radial direction: ∂Φc/∂xα = (l2/r3, 0, 0), with l = r2 dωK being the specific angular momentum of the disk, and ωK the angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The equations of motion of the ejected particles can be found working with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In terms of the nondimensional form of the radiation-field tensor elements ϵ, f α, and pαβ, the system of differential, tensorial, and coupled equations is as follows (equa- tions originally derived by Watarai & Fukue 1999, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 42–44, but now extended to second order in velocity): Radial coordinate: dur dτ = − ∂Φg ∂r + l2 r3 + (11) + 1 2[γ f r − prβuβ − γ2ϵur + ur(2γ f βuβ − pβδuβuδ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Azimuthal coordinate: 1 r dl dτ = 1 2[γ f φ − pφβuβ − γ2ϵ(l/r)+ (12) + (l/r)(2γ f βuβ − pβδuβuδ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Article number, page 3 of 12 P = (r,Φ,z) D Q = (rd,Φd, H) 7 r S BHA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main Height coordinate: duz dτ = − ∂Φg ∂z + (13) + 1 2[γ f z − pzβuβ − γ2ϵuz + uz(2γ f βuβ − pβδuβuδ)], where uµ denotes the four-velocity of the particles and γ the Lorentz factor, which is given by γ = � 1 + urur + l2/r2 + uzuz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (14) The free parameter of these equations of motion is the launch- ing radius of the particles, r0, and we assume as initial con- dition that the particles co-rotate with the disk at this radius, uα 0 = (0, l0/r0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We solve this system of equations numerically and assume that the kinematics of the disk-driven wind is roughly described by the trajectory and terminal velocities obtained for the test par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As the accretion rate in the inner region of the disk is reg- ulated at the Eddington rate, the mass loss in the wind is of the order of the super-Eddington accretion rate, ˙Mdw ∼ ˙Minput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Collision of winds The wind ejected from the disk collides with the stellar wind at the interaction region, where shocks are generated giving rise to particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' An important quantity that characterizes the wind is the kinetic luminosity, LK = ˙Mv2/2, where ˙M is the mass-loss rate and v the velocity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A small frac- tion of the total kinetic power of the wind is transferred to rel- ativistic particles, Lrel ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1LK, where we assume equipartition between relativistic protons and electrons (Le = Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The mass- loss rate and velocity of the stellar wind are set according to the parameters found in the literature for the type of star we have chosen (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Kobulnicky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In the case of the disk- driven wind, the velocity is obtained following the procedures described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Given the orbital separation, the disk inclination, and the stellar size, we estimate that ∼ 10% of the original kinetic power reaches the acceleration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We assume a circular orbit, that is, the geometry associated with the collision of winds does not depend on the orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In this section, we describe the models for the collision re- gion, the magnetic ambient field, and the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We adopt a one-zone approximation for these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2: Scheme of the wind collision seen in the rz plane (not to scale), adapted from Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Contact discontinuity The winds collide at a surface called the contact discontinuity (CD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The stagnation point (SP) is the closest position of the CD to the star, and is located where the ram pressures of the winds are in equilibrium, Pram(rBH) = ρdwv2 dw = ρ∗wv2 ∗w = Pram(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (15) Here, rBH and r∗ are the distances to the SP from the BH and from the center of the star, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The density of the spher- ical stellar wind at this location is given by ρ∗w = ˙M∗ 4πr2∗v∗w , (16) whereas the density of the disk-driven wind reads ρdw = ˙Mdw Ωr2 BHvdw , (17) where Ω = 2π(1 − cos θ) is the solid angle of the wind and θ the semi-opening angle of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Solving these equations we obtain the position of the SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Magnetic field The strength of the magnetic field at the CD is essentially deter- mined by the stellar surface magnetic field B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The intensity of BCD and its topology –dipole (i), radial (ii), or toroidal (iii)–, is given by (Eichler & Usov 1993): BCD ≈ B∗ × ����������������� R3 ∗/r3 ∗, R∗ < r∗ < rA, (i) R3 ∗/rAr2 ∗, rA < r∗ < R∗(v∗w/vrot ∗ ), (ii) R2 ∗vrot ∗ /rAr∗v∗w, R∗(v∗w/vrot ∗ ) < r∗, (iii), (18) where R∗ is the stellar radius, rA the Alfvén radius, and vrot ∗ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1v∗w the surface rotation velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Particle acceleration and shock Particles are accelerated up to relativistic energies in the col- lision region through a first-order diffusive shock mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Two shock fronts are generated: a forward shock (FS) that prop- agates through the stellar wind, and a reverse shock (RS) that propagates through the wind of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The diffusive accelera- tion rate of the particles is given by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Protheroe 1999): t−1 ac = ηac e Z c BCD E , (19) where e is the electric charge, Z the atomic number, and E is the energy of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The acceleration efficiency, ηac, depends on the diffusion coefficient of the particles, the shock velocity, and the angle between the magnetic field and the normal to the shock plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We assume that the shock propagates perpendicu- lar to the magnetic field and that diffusion occurs in the Bohm regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Thus, the acceleration efficiency is ηac ≈ 3 8 �vsh c �2 , (20) where the shock velocities in the reference frame where one of the fluids is at rest, v∗w = 0, and the other one moves with a velocity vdw, are given by (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1996): vRS = −4 3 1 1 + � n∗w/ndw vdw, (21) Article number, page 4 of 12 Shock Stellar wind Disk-driven wind BH Star Disk 7L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' : Super critical colliding wind binaries vFS = 4 3 1 1 + � ndw/n∗w vdw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (22) Here, n∗w and ndw are the numerical densities of the winds (nw = ρw/mp, with mp the mass of the proton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The pressure and density of the shocked medium are calculated following the Rankine- Hugoniot relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Lamers & Cassinelli 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As we are interested in the nonthermal particle distribution, we investigate only adiabatic shocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' that is, where radiative losses are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This is because in radiative shocks the gas in the shocked region emits large amounts of thermal radiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' the system therefore loses energy, the entropy increases, and the medium becomes increasingly homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If magnetic- inhomogeneities disappear, the acceleration efficiency decays abruptly, aborting the formation of nonthermal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The shock is adiabatic if the thermal cooling length RΛ is larger than the size of the acceleration region ∆xac (McCray & Snow 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The cooling length reads RΛ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 1011µ(vsh/km s−1)3 (nw/cm−3)[Λ(Tsh)/erg s−1 cm−3] cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (23) Here, nw is the number density of the undisturbed medium, µ is the average molecular weight (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 for a fully ionized plasma), and Λ(Tsh) is the cooling function, which depends on the shock temperature (Raymond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Myasnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Wolfire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This latter function can be written as Λ(Tsh) = ������������� 4 × 10−29T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 sh , 55 K ≤ Tsh < 104 K 7 × 10−27Tsh, 104 K ≤ Tsh < 105 K 7 × 10−19T −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 sh , 105 K ≤ Tsh < 4 × 107 K 3 × 10−27T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5 sh , Tsh ≥ 4 × 107 K, (24) where Tsh is given by Tsh = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='21µ � vsh km s−1 �2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (25) We note that this temperature has a maximum value in a colli- sional plasma: it is self-regulated by the pair-creation, satisfying in any case kBTsh < 1 MeV (kB is the Boltzmann constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We assume that the size of the acceleration region is a frac- tion of the distance from the BH to the SP, ∆xac ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1rBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As we consider a one-zone model, the acceleration region must be narrow enough to generate near-homogeneous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Radiative processes Particles accelerated at the shock can cool through different pro- cesses and produce nonthermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The timescales asso- ciated to this cooling are related to the total energy-loss of the particles: dE dt ≈ −E tcool , (26) where the total cooling rate is t−1 cool = � i t−1 i , (27) where ti corresponds to each timescale of the involved cooling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We assume advective escape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' that is, particles are removed from the acceleration region by the bulk motion of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If the timescales of cooling are shorter than those of escape, par- ticles radiate before they escape from the acceleration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The maximum energy for each kind of particle can be inferred by looking at the point where the acceleration rate is equal to the total cooling or escape rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This energy cannot exceed the maximum energy imposed by the Hillas criterion, Emax e,p < Emax Hillas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As we are interested in nonthermal processes, we work at scales smaller than the size of the binary system and assume that rotation effects are negligible there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Effects caused by the orbital motion, such as Coriolis or centrifugal forces, could be relevant on larger scales and lead to strong disturbances in the flow and thermal processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The analysis of such effects usually requires numerical simulations and is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Energy losses We consider adiabatic and radiative losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Adiabatic cooling is related to the work done by the particles of the wind to expand the shocked gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Radiative cooling is caused by nonthermal pro- cesses as a consequence of the interaction of the wind particles with ambient fields and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Our model is lepto-hadronic, and so we calculate the follow- ing radiative processes numerically: –Synchrotron: interaction of protons and electrons with the ambient magnetic field, which will be amplified by a factor of 4 in the shocked region due to Rankine-Hugoniot relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' –Inverse Compton (IC): collision of relativistic electrons with photons of the ambient radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' –Bremmstrahlung: Coulombian interactions between rela- tivistic electrons and cold matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' –Photo-hadronic interactions: interaction of highly relativis- tic protons with photons of the ambient radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' –Proton-proton: collision of relativistic protons with cold matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In addition, we take into account inelastic collision of parti- cles with atoms of the dense medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' that is, ionization losses, which can be relevant in the 1–100 MeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We note that in this energy range, ionization losses largely dominate over Coulomb scatterings (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 7 from O’C Drury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1996), and so the latter are not included in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The reader is referred to Romero & Paredes (2011), Romero & Vila (2014), and Müller & Romero (2020) plus references therein for additional details on radiative processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Particle distribution We investigate the evolution of particles that are accelerated at the shock and injected into the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The medium around the shock is the shocked gas of the winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In this paper, we restrict our analysis to this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Beyond the bi- nary, the surrounding medium has been affected by the effects of the stellar winds, and so the system is expected to be located inside a bubble inflated by the winds and surrounded by a shell formed with the swept-up material at distances of a few to sev- eral parsecs, depending on the mass of the black hole progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Inside the bubble, where the advected protons will be injected, the density is expected to be lower than that of the standard in- terstellar medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='01 cm−3 or less).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In the shell, there should be sufficient material for hadronic interactions with the protons diffused or transported from the central source3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3 These effects will be discussed elsewhere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' some of them might be responsible for part of the high-energy emission observed in the shell Article number, page 5 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main The relativistic particles have a distribution given by dN = n(r, E, t)dEdV, where n is the number density of particles, t the time, r the position, V the volume, and E the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The evo- lution of this distribution is determined by the transport equa- tion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Ginzburg & Syrovatskii 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Romero & Paredes 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We solve this equation numerically in steady state and in the one-zone approximation: ∂ ∂E �dE dt N(E) � + N(E) tesc = Q(E), (28) where tesc ∼ ∆xac/vsh is the advection time, and the particle in- jection function, Q(E) = Q0E−p exp (−E/Emax), (29) is a power-law in the energy with an exponential cutoff and a spectral index p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2, which is characteristic of the Fermi first- order acceleration mechanism (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Drury 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The nor- malization constant Q0 is obtained from L(e,p) = ∆V � Emax (e,p) Emin (e,p) dE(e,p)E(e,p)Q(e,p)(E(e,p)), (30) where ∆V is the volume of the acceleration region, and Emax (e,p) the maximum energy reached by protons and electrons, which is found by looking at the point where the acceleration rate is equal to the total cooling or escape rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Nonthermal emission Once we have the particle distributions, we calculate the spectral energy distribution (SED) for each of the relevant processes in- volved in cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We find that in SCWBs, electrons typically cool by synchrotron and IC mechanisms, and protons escape from the acceleration region without significant cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The resultant nonthermal SED usually yields a broadband spectrum from radio waves (due to synchrotron emission) to gamma-rays (due to IC emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Wind emission We calculate the thermal emission of the photosphere of the disk- driven wind assuming a spherically symmetric wind that ex- pands with constant velocity equal to its terminal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Since the mass-loss rate of the disk is much higher than the critical rate, the wind is optically thick and therefore we assume that it radiates locally as a blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The temperature measured by an observer at infinity is given by (Fukue 2009): σTT 4 dw = ˙e LEdd (1 − β cos Θ)4 4πR2 , (31) where ˙e = ˙E/LEdd is the normalized comoving luminosity, β = vdw/c the normalized velocity, Θ the angle of the flow with respect to the line of sight, and R = √ r2 + z2, with r and z the being cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We assume that the comoving lu- minosity is equal to the Eddington luminosity (˙e = 1), as is com- monly done in supercritical wind-models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Fukue 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The apparent photosphere of this wind is defined as the sur- face where the optical depth τphoto is unity for an observer at in- finity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If the velocity of the wind is relativistic, the optical depth of W50, which is powered by SS433, although there are jets involved in this specific object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' in the observer frame depends in general on the magnitude of the velocity and the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The location of the apparent photosphere from the equatorial plane zphoto is (Fukue 2009): τphoto = � ∞ zphoto γdw(1 − β cos Θ) κco ρcodz = 1, (32) where γdw is the wind Lorentz factor, κco the opacity in the co- moving frame, and ρco the wind density in the comoving frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As we assume a fully ionized wind, the opacity is dominated by free electron scattering (κco = σT/mp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Absorption Finally, we calculate the gamma absorption by pair creation from photon–photon annihilation, γ + γ → e+ + e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The nonthermal photons in their way out of the acceleration region can find pho- tons of the ambient radiation fields and annihilate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The absorp- tion is quantified by the optical depth of the medium, τγγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If the original luminosity of gamma rays is L0 γ(Eγ), the attenuated lu- minosity reads: Lγ(Eγ) = L0 γ(Eγ) · e−τ, (33) where e−τ is the attenuation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The targets of the ambient ra- diation fields are photons from the star and from the disk-driven wind photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The process of annihilation is possible only above a kine- matic energy threshold given by EγEph > (mec2)2, (34) in a frontal collision, where Eph is the energy of the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The opacity caused by a photon–photon pair production for a photon created at a distance r from the center of the thermal source can be obtained from (Romero & Vila 2008): τγγ(Eγ, r) = � ∞ Emin � ∞ r nph(Eph, r′) σγγ(Eph, Eγ) dr′dEph, (35) where nph is the density of the ambient radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The total cross-section is given by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1985): σγγ = πr2 e 2 (1 − ξ2) � (3 − ξ4) ln �1 + ξ 1 − ξ � + 2ξ(ξ2 − 2) � , (36) where re is the classical radius of the electron, and ξ = � 1 − (mec2)2 EγEph �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (37) The blackbody density radiation of the star and the photosphere of the disk-driven wind is given by nph = 2E2 ph h3c3 1 exp(Eph/kBT) − 1, (38) where T is the temperature of the thermal source considered for each case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' that is, Tdw or Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' On the other side, free-free absorption (FFA) must also be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The collision of low-energy photons with particles of the dense medium leads to a cutoff in the SED at radio frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The denser the medium, the higher the energy at which the cutoff occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Therefore, FFA will determine the Article number, page 6 of 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' : Super critical colliding wind binaries turnover of the synchrotron spectrum in SCWBs, which is ex- pected to be at ∼GHz frequencies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Rybicki & Lightman 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' del Palacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Other absorption processes, such as the photoelectric effect, direct Compton, or γ-nucleon pair creation, are not taken into account in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Their cross-sections are not high enough to become relevant in the calculation of opacity given the ambient densities that we consider here (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1 from Reynoso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Results In this section, we apply our model to a generic super-Eddington X-ray binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We consider a star of spectral type O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5V (Table 1) and investigate four scenarios: in scenarios S1 and S2 we re- gard a BH with mass MBH = 5M⊙ and mass-accretion rates of 102 ˙MEdd and 103 ˙MEdd, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' in scenarios S3 and S4 we consider a BH with mass MBH = 20M⊙ and again accretion rates of 102 ˙MEdd and 103 ˙MEdd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The complete set of pa- rameters is summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Type O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5V Star Parameter Value Units M∗ 37 M⊙ R∗ 11 R⊙ Teff 41500 K ˙M∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 10−5 M⊙ yr−1 v∗w 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 108 cm s−1 vrot ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 107 cm s−1 L∗ K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1037 erg s−1 B∗ 750 G Table 1: Parameters adopted in the model for the star of type O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' All parameters from Kobulnicky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2019), with the exception for the magnetic field (from Wade & MiMeS Collab- oration 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Wind We calculate the radiation-field tensor (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 7) and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3 we show the distribution of the energy density (ϵ) on the rz plane, where the black zone is the inflated inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We obtain a strong azimuthal flux component of the radiation-field tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This distribution is the same in all four scenarios, because in the critical disk the radiation-field tensor depends on advection, viscosity, and adiabatic parameters, which remain the same in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 11-13 to find the trajectory and velocity of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Both quantities are determined by Rµν and therefore we obtain the same trajectories and terminal velocities in S1–S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As an example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4 we show the normalized velocity of a test particle, with a launching radius of 40rg (≡ 20rs), which reaches a terminal velocity of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='16c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This result does not vary much if we vary the launching radius (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='02c for ±20rg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The particles describe a helical trajectory in the vicinity of the BH for two main reasons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The first is the presence of the strong azimuthal components of the radiation field, which help to maintain the spiral geometry of the particles in the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The second reason is the condition imposed for the particle ejection, namely that the particles initially have only azimuthal 0 5 10 15 20 0 5 10 15 20 z [rs] r [rs] 1x10-3 2x10-3 2x10-3 2x10-3 3x10-3 4x10-3 4x10-3 5x10-3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3: Contour maps of the spatial distribution of the normalized radiation energy density ϵ in the rz plane above the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Both axes are in units of Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The color bar is the intensity of ϵ and the black zone is the inflated disk ( f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5, γ = 4/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='21 r/rs v/c Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4: Normalized velocity of a wind test particle as a function of the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The particle reaches a terminal ve- locity of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='16c for a launching radius of r0 = 20rs (coincident with the vertical axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The intensity of the radiation field decays rapidly with distance from the BH, and therefore the ejected particles follow a spiral trajectory near the BH, but beyond a certain radius (∼ rcrit) they follow a free path with a strong component of the radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The overall result is an equatorial wind with terminal veloci- ties of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='15c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The kinetic power of these winds is in the range 1039−41 erg s−1, which is well above the power of the winds of typical WR or OB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Therefore, in general, the disk wind is expected to overwhelm the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Energy gain and losses We follow the calculations in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 and find that, in all four scenarios, the SP is located near the stellar surface and the wind of the disk completely sweeps up the stellar wind, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Hence, the forward shock is in the stellar atmosphere, fully ra- Article number, page 7 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main Scenario Parameter Symbol [units] S1 S2 S3 S4 Black hole mass(1) MBH [M⊙] 5 5 20 20 Mass accretion rate(1) ˙Minput [M⊙ yr−1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 10−4 Orbital semi-axis(1) a [R⊙] 15 15 22 22 Gravitational radius(2) rg [cm] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 106 Critical radius(2) rcrit [rg] 4000 40000 4000 40000 Mass loss in disk winds(1) ˙Mdw [M⊙ yr−1] 10−5 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3 × 10−4 Kinetic power of the disk-driven wind(2) Ldw K [erg s−1] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 × 1039 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 × 1040 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 1040 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 1041 Cold matter density at SP(2) ndw [cm−3] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 × 1012 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 × 1013 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 1012 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 1013 Distance to SP from BH(2) rBH [cm] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='7 × 1011 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='7 × 1011 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1011 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1011 Size of acceleration region(1) ∆xac [cm] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='7 × 1010 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='7 × 1010 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1010 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1010 Shock cold matter density(2) nRS [cm−3] 2 × 1013 2 × 1014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1014 Shock cooling length(2) RΛ [cm] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1011 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3 × 1012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3 × 1011 Maximum energy of electrons(2) Emax e [eV] 1011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 × 1011 1011 1011 Maximum energy of protons(2) Emax p [eV] 1015 1015 3 × 1015 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1 × 1015 Emission peak (low energy)(2) L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='01mm [erg s−1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1033 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1033 8 × 1034 8 × 1034 Emission peak (high energy)(2) L10MeV [erg s−1] 4 × 1032 4 × 1032 1034 1034 Table 2: Parameters of the different scenarios calculated for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We indicate with superscript (1) those parameters that are assumed and with (2) those that are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In all models, the system is supposed to be oriented face-on to the observer, that is, the inclination of the normal to the orbital plane i with respect to the line of the sight is ∼ 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 5: Trajectory of a test particle in the Cartesian 3D-space in units of Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The particles describe a helical trajectory above the inner disk because of the strong azimuthal radiation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The launching radius of this test particle is r0 = 20rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' diative, and completely unable to accelerate relativistic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Only the reverse shock (RS) is suitable for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As r∗ ≈ R∗, the magnetic field at the CD is BCD ≈ B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The cooling length of the RS is greater than the size of the acceleration region in all cases (see Table 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' this is why the shock is adiabatic and the acceleration efficiency of the process is relatively high: ηac ∼ 10−2 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The shock velocity is ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 109 cm s−1 and the temperature of the shocked gas reaches ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 × 1010 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We calculate the energy gain and losses of the shock- accelerated particles following Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Highly relativistic pro- tons escape from the acceleration region without cooling in all scenarios considered here (with energies up to Ep ≈ 1 PeV) and are injected into the interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Protons are ad- vected, that is, they are removed from the collision region by the bulk motion of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' They therefore do not interact with am- bient material at scales similar to that of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Electrons cool mainly through IC and synchrotron mechanisms, and reach a maximum energy of Ee ≈ 100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' To obtain the electron dis- tribution, we solve the transport equation considering only the dominant IC and synchrotron losses, and a power-law injection function with a spectral index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 and an exponential cutoff (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Spectral energy distribution Figure 6 shows the SEDs of the four scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The only thermal component of the spectrum is the photosphere of the optically thick disk-driven wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The emission peak of the wind for S1 and S2 is ≈ 1037 erg s−1, whereas for S3 and S4 the peak is ≈ 1038 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This occurs at energies of ∼ 100 eV for S1 and S3, and ∼ 30 eV for S2 and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Therefore, if MBH increases, the luminosity is higher and, if the mass-accretion rate increases, the luminosity peak occurs at lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In the case of the nonthermal spectrum, we calculate the emission due to synchrotron and IC losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In the latter case, we consider the photon fields of the star and of the wind photosphere as targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In all cases, the dominant IC contribution is that of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The luminosity in S3 and S4 is an order of magnitude greater than that in S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This is because of the modification of the orbital parameters when the BH mass varies: to guarantee the overflow of the Roche lobe, the orbital semi-axis varies with MBH, which results in variation in the size of the acceleration re- Article number, page 8 of 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' : Super critical colliding wind binaries gion and the photon density at SP, among other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The emission peak at low energies is ∼ 1033 erg s−1 for S1 and S2, and ∼ 1035 erg s−1 for S3 and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' At high energies, the emission peak is ∼ 1032 erg s−1 (S1 and S2) and ∼ 1034 erg s−1 (S3 and S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The gamma-ray absorption due to γγ annihilation is total for energies > 10 GeV in all scenarios4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Attenuation due to material between the source and the ob- server, that is, absorption by external cold gas, is mainly in the optical-to-UV range and at soft X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' At radio wavelengths, re- fractive scintillation on free electrons of the ISM occurs at lower frequencies than predicted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' For high-energy gamma rays, the main absorbers are infrared (IR) fields and the cosmic mi- crowave background (CMB), but their effects are only relevant for cosmological distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Application to NGC 4190 ULX 1 Ultraluminous X-ray sources (ULXs) are extragalactic point-like objects where the luminosity in the X-ray band appears to be higher than the Eddington luminosity (Bachetti 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' ULXs are thought to be X-ray binaries with a stellar-mass compact object accreting at super-Eddington rates, where a beaming effect could be responsible for the luminosity observed in the X-ray band: the radiation emitted from the inner part of the accretion disk is geometrically collimated by the ejected wind, which is optically thick except in a narrow region around the black-hole axis and forms a cone-shaped funnel (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' King 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Fabrika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We apply our model to estimate the radiation emitted by the ultraluminous X-ray source NGC 4190 ULX 1 (also known as CXO J121345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2+363754).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Although many characteristics of this ULX remain poorly understood, several authors have explored the system and have provided constraints on some of its param- eters (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Liu & Bregman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Gladstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Ko- liopanos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Kosec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Ghosh & Rana 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In what follows, we describe the parameterization of the sys- tem and its components, and investigate the expected collision of winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The complete set of parameters used in this section is detailed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' System parameterization The source is located in the nearby Galaxy NGC 4190 at a dis- tance of d ≈ 3 Mpc (Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Observations made in 2010 using the XMM-Newton telescope reveal a long-term spectral variability in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='0 keV energy range: LX ∼ 3 − 8 × 1039 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The angle i between the line of sight and the z-axis at which the disk of a ULX is observed determines the components of its spectrum: blackbody disk (BB) or Comptonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If i is small, the observer is able to look into the funnel and see the innermost part of the disk: the spectrum shows only the BB component, which corresponds to thermal emission of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This type of spectrum is called broadened disk (BD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If i is sufficiently large, another effect is observed: the interaction between photons and wind particles near the disk surface induces a Comptonization that produces a hardening in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Most ULXs exhibit a combination of both phenomena in their X-ray spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4 We note that, since we assume a nearly face-on inclination of the system, there are no significant variations of the radiative output associ- ated with the orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If the system were oriented nearly edge-on, the emission would be modulated by the orbital phase due to absorption (for details see Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Ghosh & Rana (2021) investigated the spectral properties of NGC 4190 ULX 1 and suggested that the ULX is in a BD state, and that the compact object is a BH with mass ∼ 10 − 30M⊙ accreting at super-Eddington rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We fit the XMM-Newton ob- servations (Epoch 3) with the supercritical advection-dominated disk model detailed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='1, assuming a mass-accretion rate of ˙Minput = 10 ˙MEdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We also assume a face-on inclination i ≈ 0◦, a BH mass 10M⊙ and a geometrical beaming factor b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This factor is given by, b = Ω/4π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5(1 − cos ϑ), (39) where Ω is the solid angle of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The angle ϑ is related to the opening angles of the disk (δ) and its wind (θ): ϑ+δ+2θ = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Both angles, i and ϑ, can change over time, causing the spectral variability of the object (Fabrika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' On the other hand, Gladstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2013) provided con- straints on the characteristics of the optical counterpart of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' They suggested that, if MBH = 10M⊙, the mass of the star could be < 50M⊙ and its radius < 86R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We choose a star of type B2V for our model in light of one of the fittings these latter authors made from Hubble Space Telescope observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If we apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1 and consider the mass ratio M∗/MBH, and the stellar radius involved (see Table 3), the transfer of mass in the binary system occurs for an orbital semi-axis a ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 R⊙, which results in a period ≤ 38 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Collision of winds The terminal velocity of the disk-driven wind is vdw = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='95 × 109 cm s−1, and therefore Ldw K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5 × 1039 erg s−1, while L∗ K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='17 × 1034 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The SP is located near the stellar surface and the wind of the disk completely suppresses the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We therefore only take into account the reverse shock (RS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As r∗ ≈ R∗, the magnetic field at the CD is BCD ≈ B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The cooling length of the RS is RΛ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1013 cm and the size of the acceleration region is ∆xac = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='68 × 1010 cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' therefore, the shock is adiabatic and the acceleration efficiency of the process is ηac = 10−2, as in our general models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We calculate the energy gain and losses of the shock particles following Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Highly relativistic protons escape from the acceleration region without cooling, as in our previous scenarios (with energies up to Ep ≈ 1 PeV), and are injected into the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Electrons cool mainly through IC and synchrotron mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Figure 7 shows the timescales of electrons, which reach a maximum energy of Ee ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='32 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' To obtain the electron distribution, we solve the transport equation taking into account only IC and synchrotron losses, and a power-law injection function with a spectral index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 and an exponential cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Total SED The SED of the ULX spans a broadband energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Figure 9 shows the thermal (wind and accretion disk) and nonthermal (colliding-winds shock) contributions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We also show the sensitivity of the instruments ALMA, VLA (sub-mm waves), Fermi, and CTA (gamma rays), and observational data from XMM-Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The luminosity in the IR band is ∼ 1034 erg s−1, which is rel- atively strong, though still undetectable at megaparsec distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The luminosity in gamma-rays also reaches ∼ 1034 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The attenuation factor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 8) has an effect on photons with ener- gies ≳ 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Most of the radiation above 1 GeV and all above 50 GeV is suppressed by the annihilation of the γ rays with the photon fields of the disk-driven wind and the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Article number, page 9 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='S1 & S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='log10 (EγLγ / erg s-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='log10 (Eγ / eV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='wind photosphere S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='synchrotron S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S1 (star) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S1 (wind) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='total SED S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='wind photosphere S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='synchrotron S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S3 (star) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S3 (wind) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='total SED S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='S2 & S4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='log10 (EγLγ / erg s-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='log10 (Eγ / eV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='wind photosphere S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='synchrotron S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S2 (star) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S2 (wind) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='total SED S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='wind photosphere S4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='synchrotron S4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S4 (star) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='inverse Compton S4 (wind) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='total SED S4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 6: Thermal and nonthermal SEDs of the four scenarios considered, S1–S4, in logarithmic scale, where a face-on inclination is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S1 and S3 are shown in the left plot, whereas S2 and S4 are shown in the right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Dashed lines correspond to S1 (left) and S2 (right), solid lines correspond to S3 (left) and S4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We plot the nonattenuated inverse Compton contributions in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The emission peak at high energies is ∼ 1033 erg s−1 for S1 and S2, and ∼ 1034 erg s−1 for S3 and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The gamma-ray absorption due to γγ annihilation is total for energies > 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 8 6 4 2 0 2 4 6 8 10 6 7 8 9 10 11 12 13 ηac ∼ 10-2 RS log10 (t-1 / s-1) log10 (Ee / eV) synchrotron inverse Compton Bremsstrahlung adiabatic ion acceleration escape Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 7: Timescales in logarithmic scale of the electron accelera- tion, escape, and cooling at the reverse shock in NGC 4190 ULX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Electrons reach a maximum energy of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='32 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The accel- eration efficiency is 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Discussion Our analysis of supercritical colliding wind binaries shows that these systems should exhibit broadband emission from radio to gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In this sense, they are similar to CWBs formed by two hot stars, such as O+WR binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' However, there are im- portant differences as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' If we compare our models with re- cent models of O+WR CWBs (Pittard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021), we find that (i) in SCWBs, the wind of the disk is far more powerful than the wind of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This results in stagnation points that are very close to the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Efficient particle accelera- tion then can only occur in reverse shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (ii) We also see that the disk wind advects protons from the acceleration region be- fore they have time to cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Only electrons can cool locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='8 1 6 4 2 0 2 4 6 8 10 12 e−τγγ log10 (Eγ / eV) Star Wind photosphere Total attenuation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 8: Attenuation factors due to γγ-annihilation between high- energy nonthermal radiation and photon fields from the star and from the photosphere of the disk-driven wind in NGC 4190 ULX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The total attenuation is plotted with a black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' resulting SED is consequently dominated by synchrotron and IC radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (iii) As the acceleration region is close to the star, the local magnetic field is relatively strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Synchrotron emission reaches energies of hundreds of keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As the medium is far more dense than in stellar CWBs, free-free absorption causes this radi- ation to turnover below ∼ 24 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The total power at millimeter (mm) and submm wavelengths can be between three and five orders of magnitude higher in SCWBs than in stellar CWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (iv) IC is the dominant radiation mechanism at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The stronger thermal fields of SCWBs (wind photosphere and star) provide the seed photons, but also impose a high-energy cutoff at ∼ 1 GeV through γ − γ attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Instead, stellar CWBs can reach energies close to 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (v) The strong mag- netic fields in the acceleration region cut electromagnetic cas- Article number, page 10 of 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' : Super critical colliding wind binaries Table 3: Parameters of NGC 4190 ULX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Parameter Symbol Value Units System Inclination(1) i 0 Orbital semi-axis(2) a 15 R⊙ Distance to the source(3) d 3 Mpc Black hole Mass(1) MBH 10 M⊙ Gravitational radius(2) rg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='48 × 106 cm Accretion disk Disk semi opening angle(1) δ 30 Critical radius(2) rcrit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5 × 109 cm Eddington accretion rate ˙MEdd 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 10−7 M⊙ yr−1 Mass accretion rate(1) ˙Minput 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 10−6 M⊙ yr−1 Mass loss in winds(1) ˙Mdw 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='98 × 10−6 M⊙ yr−1 Wind velocity(2) vdw 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='95 × 109 cm s−1 Wind semi opening angle(2) θ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5 Beaming factor(2) b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='07 − B2V Star Mass(4) M∗ 8 M⊙ Radius(4) R∗ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 R⊙ Temperature(4) Teff 20600 K Mass loss in winds(4) ˙M∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 10−7 M⊙ yr−1 Wind velocity(4) v∗w 7 × 107 cm s−1 Rotation velocity(1) vrot ∗ 7 × 106 cm s−1 Magnetic field(5) B∗ 200 G Colliding winds Kinetic power of disk-driven wind(2) Ldw K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='5 × 1039 erg s−1 Kinetic power of stellar wind(2) L∗ K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='17 × 1034 erg s−1 Distance from BH to SP(2) rBH 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='68 × 1011 cm Size of acceleration region(1) ∆xac 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='68 × 1010 cm Magnetic field at SP(2) BSP 200 G Injection spectral index(1) p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 − Acceleration efficiency(2) ηac 10−2 − Molecular mean weight(1) µ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='6 − Reverse shock Velocity(2) vRS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='4 × 109 cm s−1 Temperature(2) TRS 1010 K Cold matter density(2) nRS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='9 × 1011 cm−3 Cooling length(2) RΛ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='2 × 1013 cm Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We indicate the parameters we have assumed with superscript (1) and those we have derived with (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Parameters with superscripts (3), (4), and (5) were taken from Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2013), Kobulnicky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2019), and Shultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2015), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' cades in SCWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (vi) The SED is always dominated by the X- ray component associated with the disk or its wind in SCWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Finally, (vii) stellar CWBs have wider orbits and a variable sep- aration between the components of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This produces variability related to the orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' On the contrary, the or- bits of SCWBs should be mostly circularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' In general, CWBs are weaker than SCWBs, although span a broader energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' An interesting feature of SCWBs is their potential as cosmic ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' As mentioned, the strong wind of the disk drags away the relativistic protons before they cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These protons, with maximum energies of the order of 1 PeV, are then injected into the ISM where they diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Even if a fraction of just ∼ 1 % of the wind kinetic power goes to relativistic protons, the cosmic ray output of a SCWB would be in the range 1037−39 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These protons might interact with ambient clouds at some distance from the system, producing gamma rays through pp → π0 + pp interactions and the subsequent pion decays π0 → γγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The gamma-ray emission from the illuminated clouds 30 32 34 36 38 40 42 8 6 4 2 0 2 4 6 8 10 12 Fermi CTA ALMA VLA log10 (EγLγ / erg s-1) log10 (Eγ / eV) wind photosphere beamed disk inverse Compton (star) inverse Compton (wind) synchrotron Total SED XMM Newton data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 9: Thermal and nonthermal SEDs of NGC 4190 ULX 1 in logarithmic scale (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The nonthermal SED is par- tially attenuated for energies > 1 GeV and totally attenuated for energies > 50 GeV due to annihilation of γ-rays with the photon fields of the star and the photosphere of the disk-driven wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The gray dashed lines are the nonattenuated IC contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The total SED is plotted with a solid black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Data from XMM-Newton (Epoch 3), and the sensitivity of ALMA, Fermi, VLA, and CTA are also shown (instrument sensitivities were taken from Sotomayor & Romero 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' can be even stronger than the emission from the binary itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' However, the spectrum should be softer because of propagation effects (Aharonian & Atoyan 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Recent modeling by Pit- tard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2021) of particle acceleration in colliding wind bina- ries with wind velocities of a few 103 km s−1 and mG magnetic fields in the acceleration region demonstrate that up to ∼ 30 % of the wind power can be transferred to nonthermal particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This means that, in some extreme cases, a SCWB might inject up to ∼ 1040 erg s−1 in cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Another type of CWB is the so-called gamma-ray binary (GRB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', LS 5039, PSR B1259-63, LSI +61◦ 303, PSR J2032+4127, and others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Dubus 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Chernyakova & Malyshev 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These sources are formed by a massive star (usually a Be star with a dense equatorial decretion disk and a fast wind) and a young pulsar in an eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The pul- sar ejects a relativistic pair wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The wind collision produces a broadband spectrum from electrons accelerated at the shock that cool by synchrotron and IC radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The two-peak SEDs are similar to those we estimate for SCWBs, but some differ- ences are also clearly seen: (i) GRBs are less energetic because the spin-down luminosity of the pulsar is much smaller than the power of a supercritical wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (ii) GRBs are highly variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This variability is modulated with the orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The orbital mod- ulation of the different components of the broadband spectrum is a consequence of the orbital variability of geometrical parame- ters, such as the geometry of the contact surface of the stellar and pulsar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Absorption effects are also strongly variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (iii) Hadronic interactions are likely when the pulsar crosses the equatorial disk of the star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Bykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (iv) GeV flares have been observed after the periastron passage in sources such as PSR B1259-63 (Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Chernyakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These flares are attributed to the effects of the unshocked pulsar wind interaction with photons from the stellar disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Khangulyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Article number, page 11 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' main We finally mention that some black holes accreting at super- critical rates seem to be capable of launching mildly relativis- tic jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A remarkable case in our Galaxy is the notorious mi- croquasar SS433 (Fabrika 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This object resembles a ULX source seen edge on (Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The accretion rate should be extremely high in order to explain the large jet power LK ∼ 1040 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' (2006) suggest rates of ∼ 5 × 103 ˙MEdd ∼ 5 × 10−4 M⊙ yr−1, which are consistent with estimates of equatorial mass outflows inferred from radio obser- vations (Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' These outflows, ejected toward either side of the jets, present a thermal spectrum and might well correspond to the radiation-driven wind of the hypercritical disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The contamination from the jet base makes it impossible to dis- entangle contributions from colliding winds from those coming from the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' However, the equatorial outflow might propagate well beyond the system and reveal itself if it collides with any clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The shock generated in the collision would convert the kinetic energy of the plasmoids into internal energy and relativis- tic particles, which might then cool by pp interactions with the cloud material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Such a scenario might explain the detection of a GeV source by the Fermi satellite on the side of SS433 (Bordas 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We will explore the details of this hypoth- esis elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Summary and conclusions We explored the consequences of supercritical accretion in bi- nary systems consisting of a hot star and a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We find that a fraction of the kinetic power of the radiation-driven wind released by the accretion disk is transformed into relativistic par- ticles in the region of the wind that collides with the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Elec- trons are cooled locally, mainly through synchrotron and inverse Compton radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The radiation fields of the star and wind pho- tosphere provide abundant thermal photons for the latter process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' they also absorb high-energy radiation above a few GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Free- free absorption imposes a high-frequency turnover in the ra- dio regime, suppressing centimeter radio waves, unlike the case of colliding wind binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The relativistic protons are blown away by the wind before they can cool down significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Once trapped by the outflow, these protons are transported to outer re- gions where they can interact with ambient gas away from the binary system, producing hadronic gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Our most im- portant finding is that, in addition to being strong thermal UV and X-ray sources, supercritical colliding wind binaries can be significant nonthermal sources at mm wavelengths and GeV en- ergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' The authors thank the anonymous referee for a careful and constructive review, and for his/her comments that improved this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' We thank also Daniela Pérez and Jiˇri Horák for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' This work was sup- ported by grant PIP 0554 (CONICET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' LA ackowledges the Universidad Na- cional de La Plata for the education received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' GER acknowledges the support from the Spanish Ministerio de Ciencia e Innovación (MICINN) under grant PID2019-105510GBC31 and through the Center of Excellence Mara de Maeztu 2020-2023 award to the ICCUB (CEX2019-000918-M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' References Abaroa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Sotomayor Checa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021, Boletin de la Asocia- cion Argentina de Astronomia La Plata Argentina, 62, 262 Abbott, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1978, ApJ, 225, 893 Abdo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2010, ApJ, 723, 649 Abdo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011, ApJ, 736, L11 Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Atoyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1996, A&A, 309, 917 Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Kririllov-Ugriumov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Vardanian, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1985, Ap&SS, 115, 201 Bachetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016, Astronomische Nachrichten, 337, 349 Balbo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Walter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017, A&A, 603, A111 Begelman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', King, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Pringle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2006, MNRAS, 370, 399 Benaglia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Marcote, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Moldón, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2015, A&A, 579, A99 Benaglia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2003, A&A, 399, 1121 Blundell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Mioduszewski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Muxlow, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Podsiadlowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Rupen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2001, ApJ, 562, L79 Bordas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020, Nature Astronomy, 4, 1132 Bykov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Petrov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Kalyashova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Troitsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021, ApJ, 921, L10 Chernyakova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Abdo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Neronov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2014, MNRAS, 439, 432 Chernyakova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Malyshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020, in Multifrequency Behaviour of High Energy Cosmic Sources - XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 3-8 June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Palermo, 45 De Becker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2007, A&A Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', 14, 171 De Becker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Raucq, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2013, A&A, 558, A28 del Palacio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Bosch-Ramon, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Benaglia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016, A&A, 591, A139 Drury, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1983, Reports on Progress in Physics, 46, 973 Dubus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2013, A&A Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', 21, 64 Eggleton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1983, ApJ, 268, 368 Eichler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Usov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1993, ApJ, 402, 271 Fabrika, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2004, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Space Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', 12, 1 Fabrika, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Atapin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Vinokurov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Sholukhova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021, Astrophysical Bulletin, 76, 6 Farnier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Walter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Leyder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011, A&A, 526, A57 Fukue, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2004, PASJ, 56, 569 Fukue, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2009, PASJ, 61, 1305 Ghosh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Rana, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021, MNRAS, 504, 974 Ginzburg, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Syrovatskii, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1964, The Origin of Cosmic Rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Oxford: Pergamon Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Gladstone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Copperwheat, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Heinke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2013, ApJS, 206, 14 Grimaldo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Reimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Kissmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Niederwanger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Reitberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2019, ApJ, 871, 55 Kaaret, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Feng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Roberts, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017, ARA&A, 55, 303 Kato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Fukue, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020, Fundamentals of Astrophysical Fluid Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Sin- gapore: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Khangulyan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Bogovalov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Ribó, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2012, ApJ, 752, L17 King, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2009, MNRAS, 393, L41 King, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Davies, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Ward, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Fabbiano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Elvis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2001, ApJ, 552, L109 Kobulnicky, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Chick, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Povich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2019, AJ, 158, 73 Koliopanos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Vasilopoulos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Godet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017, A&A, 608, A47 Kosec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Pinto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Fabian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Walton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2018, MNRAS, 473, 5680 Lamers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Cassinelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1999, Introduction to Stellar Winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Cambridge: Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Kang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Ryu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1996, ApJ, 464, 131 Leyder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Walter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Rauw, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2008, A&A, 477, L29 Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Torres, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020, Nature Astronomy, 4, 1177 Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Bregman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2005, ApJS, 157, 59 McCray, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Snow, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1979, ARA&A, 17, 213 McKinley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1980, American Journal of Physics, 48, 612 Mirabel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Rodríguez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1994, Nature, 371, 46 Muñoz-Darias, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Casares, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Mata Sánchez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016, Nature, 534, 75 Muijres, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Vink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', de Koter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Müller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Langer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2012, A&A, 537, A37 Müller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020, A&A, 636, A92 Myasnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Zhekov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Belov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1998, MNRAS, 298, 1021 Neilsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2009, Nature, 458, 481 O’C Drury, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Duffy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Kirk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1996, A&A, 309, 1002 Okazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Owocki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Russell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Corcoran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2008, MN- RAS, 388, L39 Pittard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Vila, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2021, MNRAS, 504, 4204 Pittard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Vila, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2020, MNRAS, 495, 2205 Protheroe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1999, in Topics in Cosmic-Ray Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' New York: Nova Science Publishing, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Duvernois, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 230 Pshirkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2016, MNRAS, 457, L99 Raymond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Cox, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Smith, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1976, ApJ, 204, 290 Reimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Pohl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Reimer, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2006, ApJ, 644, 1118 Reitberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Kissmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Reimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Reimer, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2014, ApJ, 789, 87 Reynoso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Medina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011, A&A, 531, A30 Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Del Valle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Orellana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2010, A&A, 518, A12 Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Paredes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2011, Introducción a la Astrofísica Relativista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Barcelona: Universitat de Barcelona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Vila, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2008, A&A, 485, 623 Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Vila, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2014, Introduction to Black Hole Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Heidelberg: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Rybicki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Lightman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1986, Radiative Processes in Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' New York: Wiley VCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Shakura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Sunyaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1973, A&A, 500, 33 Article number, page 12 of 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Abaroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' : Super critical colliding wind binaries Shultz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Wade, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Alecian, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & BinaMIcS Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2015, MN- RAS, 454, L1 Sotomayor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2022, A&A, 664, A178 Stevens, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Blondin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Pollock, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1992, ApJ, 386, 265 Tavani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Sabatini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Pian, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2009, ApJ, 698, L142 Tetarenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Sivakoff, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Miller-Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2017, MNRAS, 469, 3141 Tully, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Courtois, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Dolphin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2013, AJ, 146, 86 Wade, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & MiMeS Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2015, in Astronomical Society of the Pa- cific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 494, Physics and Evolution of Magnetic and Related Stars, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Balega, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Romanyuk, & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' Kudryavtsev, 30 Watarai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' & Fukue, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 1999, PASJ, 51, 725 Wolfire, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', McKee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', Hollenbach, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=', & Tielens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} +page_content=' 2003, ApJ, 587, 278 Article number, page 13 of 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FAT4oBgHgl3EQfoh2y/content/2301.08635v1.pdf'} diff --git a/7NE3T4oBgHgl3EQfRgki/content/tmp_files/2301.04421v1.pdf.txt b/7NE3T4oBgHgl3EQfRgki/content/tmp_files/2301.04421v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0638dd5514fbdf7e9bb800c85b727af9c591c64 --- /dev/null +++ b/7NE3T4oBgHgl3EQfRgki/content/tmp_files/2301.04421v1.pdf.txt @@ -0,0 +1,1476 @@ + + + +Abstract— Motion prediction is essential for safe and efficient +autonomous +driving. +However, +the +inexplicability +and +uncertainty of complex artificial intelligence models may lead to +unpredictable failures of the motion prediction module, which +may mislead the system to make unsafe decisions. Therefore, it +is necessary to develop methods to guarantee reliable +autonomous driving, where failure detection is a potential +direction. Uncertainty estimates can be used to quantify the +degree of confidence a model has in its predictions and may be +valuable for failure detection. We propose a framework of +failure detection for motion prediction from the uncertainty +perspective, considering both motion uncertainty and model +uncertainty, +and +formulate +various +uncertainty +scores +according to different prediction stages. The proposed approach +is evaluated based on different motion prediction algorithms, +uncertainty estimation methods, uncertainty scores, etc., and +the results show that uncertainty is promising for failure +detection for motion prediction but should be used with caution. +I. INTRODUCTION +Motion prediction is a hot topic in mobile robot and +autonomous vehicle communities, accurate prediction of the +future motion of surrounding traffic participants is +fundamental to robust and reliable decision-making. +Artificial intelligence (AI), especially deep learning, has been +widely used in autonomous driving tasks by its advantages in +dealing with complex problems. With the collection of +large-scale data, the improvement of computing power and +related algorithms, AI is expected to play a vital role in +autonomous driving systems in the future [1]. +However, although AI-based motion prediction has shown +statistical performance advantages, it is difficult to avoid +unpredictable failures due to the inherent inexplicability and +insufficient reliability of deep learning models, which may +cause serious autonomous driving accidents [2]. From the +uncertainty perspective, motion prediction faces the dual +challenge of uncertainty from the environment and the model. +Drivers, pedestrians, etc. in the environment have uncertainty +in their intentions and movements, which makes it difficult to +accurately predict their future in all scenarios. Additionally, +due to insufficient training data and training process, the +model may experience serious performance degradation +when faced with rare or unknown scenarios. + +*Research supported by the National Science Foundation of China Project: +U1964203 and 52072215, and the National Key R&D Program of +China:2020YFB1600303. (Corresponding authors: Hong Wang) +Wenbo Shao, Liang Peng, Jun Li and Hong Wang are with School of Veh +icle and Mobility, Tsinghua University, Beijing 100084, China. (e-mail: {sw +b19, peng-l20}@mails.tsinghua.edu.cn; {lijun1958, hong_wang}@tsinghua. +edu.cn) +Yanchao Xu is with School of Mechanical Engineering, Beijing Institute +of Technology, Beijing 100081, China. (e-mail: 3120200410@bit.edu.cn) +Failure Detector +Main Model +Maneuver +Classifier +Trajectory +Predictor +Graph +Model +UM +UT +Is there a wrong maneuver +classification or trajectory prediction? + +Fig. 1. Uncertainty-based failure detection for motion prediction. UM, UT are +the uncertainty scores extracted for maneuver classification and trajectory +prediction, respectively. +The failure detection, isolation, and recovery mechanism is +an effective way to solve the above problems [3]. Among +them, the study of failure detection for AI models has +attracted increasing interest, which is of critical significance +for the development of reliable autonomous driving systems +[4]. As shown in Fig. 1, using the information extracted from +the main model, i.e. motion prediction model, a failure +detector is built to identify maneuver classification errors and +trajectory prediction errors. Uncertainty, as a measure of the +confidence level of the model in its output, has been used by +some researchers for failure detection in tasks such as +semantic segmentation [5]. Our study exploits various +uncertainties from motion prediction and explores their +usefulness for failure detection. +In this work, we concentrate on failure detection for motion +prediction from the uncertainty perspective. The main +contributions are as follows: + A framework of failure detection using uncertainty for +motion prediction tasks, taking into account both motion +uncertainty and model uncertainty. + A series of uncertainty scores for failure detection +formulated for different motion prediction stages and +algorithms. + A detailed evaluation and comparison with multiple +motion prediction algorithms, uncertainty estimation +methods and uncertainty scores. +II. RELATED WORK +A. Motion Prediction and Motion Uncertainty Estimation +Traditional motion prediction methods predict the future +motion of the target agent (TA) based on its historical state by +explicitly modeling kinematic models, such as Kalman +Filter[5], [6], but they only apply to short-term prediction +under scenarios with few interactions. In recent years, deep +learning-based motion prediction [8]–[10] has demonstrated +promising performance by simultaneously modeling TA’s +historical state, its interactions with surrounding traffic +participants, and other environmental information in deep +Failure Detection for Motion Prediction of Autonomous Driving: +An Uncertainty Perspective* +Wenbo Shao, Yanchao Xu, Liang Peng, Jun Li, and Hong Wang + + + +neural networks. A broader review of deep learning-based +motion prediction can be found in [11]. As for the model’s +output form, some studies regard motion prediction as a +multipoint regression problem [12]–[14], so as to output the +unimodal predicted trajectory. However, due to the diversity +of intentions and the uncertainty of traffic participants’ +behaviors, the future trajectory distribution corresponding to +one model input presents multiple possibilities. Recently, +increasing researchers and prediction competitions have paid +attention to multimodal motion prediction, which is generally +divided into two stages: maneuver or target classification, and +trajectory prediction. Some studies [15], [16] define +maneuvers as specified behavior patterns, then train the +maneuver classifier through supervised learning. For example, +CS-LSTM [15] defines six maneuver modes for vehicles on +highways, where longitudinal maneuvers include normal +driving and braking, the lateral maneuvers include left lane +change, right lane change, and lane keeping. The predicted +maneuvers can serve as an important guide for future +trajectory prediction. Other studies do not explicitly define +specific behavior patterns before training, but guide the +model to learn the optimal maneuver modes through model +design and training process [17]–[19]. For example, +Trajectron++ [18] adopts the conditional variational +autoencoder (CVAE) to encode multimodality by introducing +latent variables, and relies on a bivariate Gaussian Mixture +Model (GMM) to model the final output. +B. Model Uncertainty Estimation +The above multimodal prediction algorithms model the +uncertainty in the traffic participants’ movements. In addition, +deep learning models have inherent uncertainty, generally +called model uncertainty or epistemic uncertainty [20], it is +difficult to ignore in the real world where there are +distribution shifts or out-of-distribution data. Bayesian neural +network (BNN) [21]–[23] is a representative method for +estimating model uncertainty, in which Bayesian inference +plays an important role. Methods such as Monte-Carlo +dropout [24], [25] achieve approximate inference through +sampling, and they further promote the generality and +popularity of BNN. Besides, deep ensemble [26]–[28], as a +simple and scalable method, has shown promising +performance in model uncertainty estimation and thus has +attracted many researchers and practitioners. As the +representative method requiring only a single forward pass, +evidential deep learning (EDL) [29] computes the uncertainty +of the output distribution by modeling the prior distribution +for the classification. +C. Failure Detection for Autonomous Driving +Failure detection is attracting attention as a technology to +achieve reliable autonomous driving. It uses the main model's +input, internal features, or output to diagnose whether there is +a failure. Learning-based approaches build a specialized +model to act as the failure detector, and it identifies failures of +the main model by using failure cases for supervised training +[30]–[32] or estimating reconstruction errors [33]–[35]. In +addition, uncertainty-based anomaly detection has attracted +some +interest, +such +as +detecting +misclassified +or +out-of-distribution examples through maximum softmax +probabilities directly output by classification networks [36] or +predictive entropy quantization taking into account model +uncertainty [26]. However, to the best of our knowledge, +most current research on failure detection for autonomous +driving focuses on perception tasks, such as semantic +segmentation, depth estimation, etc. [5], and failure detection +for motion prediction models from the uncertainty +perspective has been rarely discussed. +Our approach utilizes both motion uncertainty and model +uncertainty, proposes uncertainty scores for different stages +of motion prediction, and investigates the effect of motion +prediction failure detection based on different scores. +III. METHODOLOGY +A. Problem Setting +Motion prediction is a task that predicts TA’s trajectory +over a period of time in the future given input information. +Assuming the current moment +0 +t  +, the input information +may include TA’s historical state + + + + +  +1 +2 +0 +[ +, +, +, +] +h +h +t +t +s +s +s + + + + +S + + +in +the past +ht timesteps, the historical state of TA’s surrounding +traffic participants, and other contextual information such as +maps, which are uniformly represented here by C . Among +them, + ts + may contain TA’s information such as the position, +speed, and category at t . The output is the predicted position +ˆY of TA in the future +ft timesteps: + + + +ˆ +, +f + +Y +S C +(1) +with +1 +2 +ˆ ˆ +ˆ +ˆ +[ , +,..., +] +ft +d d +d + +Y + consisting of the +ft + predicted +positions ˆ +td . For multimodal motion prediction, ˆY contains +predicted trajectories under multiple maneuvers. +Failure detection for motion prediction refers to identifying +potential motion prediction failures by monitoring model’s +state, where failures may exist in the form of maneuver +misclassification or excessive error of predicted trajectories. +Uncertainty, as the measure of TA's behavior or model state, +reflects the model's confidence in its particular output and +thus has the potential to diagnose potential prediction failures. +This work proposes to detect the performance degradation of +motion prediction models, i.e. the decrease in the accuracy of +prediction results, by quantifying the uncertainty scores. +B. Motion Prediction with Motion Uncertainty Estimation +Due to the unavailability of TA's actual intentions and the +randomness of its behavior, it may have multiple possible +future trajectories. GRIP++ is an enhanced graph-based +interaction-aware trajectory prediction algorithm, it models +inter-agent interactions and temporal features but only +predicts future trajectories in a single mode. As shown in Fig. +2, we add the maneuver classification module to GRIP++, by +distinguishing different behavioral patterns to improve the +authenticity and usability of the prediction results. The new +method is called GRIP+++. +We focus on two-stage tasks in the proposed method: +maneuver classification and maneuver-based trajectory +prediction. In the maneuver classification stage, given the +TA’s historical state and scene context, feature G are +extracted through the graph convolutional model (GCN), +which includes the processing of fixed and trainable graphs. +Then TA’s maneuver + + +P +| +z +z G is inferred by multilayer +perceptron (MLP), where + + +1,2,..., +z +Z + + represents one of the + + + +defined maneuver modes. In CS-LSTM [15], the modes are +divided into three types of lateral maneuvers and two types of +longitudinal maneuvers, but they are only applicable to +vehicles driving on highway, we define a common set of +maneuver modes suitable for various scenarios. Specifically, +TA's maneuvers are divided into four categories according to +their movement direction and speed: going straight, turning +left, turning right, and stopping. In the network, we adopt the +softmax head for probabilistic maneuver classification. +Graph Convolutional Model +Maneuver +Classification Module +64 +ht +n +Trajectory Prediction Module +Predicted Trajectories +concat +Maneuver Probabilities +ht + +Fig. 2. The architecture of GRIP+++. +The maneuver-based trajectory prediction module consists +of seq2seq networks taking the concatenation of the graph +feature G and the feature vector transformed by the +maneuver z as input, and outputs the future trajectory ˆ +z +Y +under the maneuver z . +To compare the generality of uncertainty-based failure +detection in different motion prediction mechanisms, we +employ another two classes of typical prediction algorithms. +Firstly, we focus on multimodal trajectory prediction based +on the generative model, so we adopt Trajectron++ [18], it +utilizes the CVAE-based latent network framework to model +multimodal future trajectories, where the discrete categorical +latent variable z encodes high-level behavior patterns: + + + +1,2,..., +P( +, +) +P ( +, +, )P ( +, +) +ˆ +ˆ +ˆ +z +Z +z +z + + + +ψ +θ +S C +S C +Y +Y +Y +S C +∣ +∣ +∣ + (2) +where θ , ψ are deep neural network parameters. +Furthermore, we use PGP [16] as a comparison, it is a +multimodal trajectory prediction method combining graph +traversal, latent vector sampling, and clustering. It models +discrete policy for graph traversal by representing HD maps +as lane graphs, and implements diverse trajectories prediction +combined with a random sampling of latent vectors for +longitudinal variability. Furthermore, it uses K-means +clustering to obtain Z predictive trajectories. With its clever +design, PGP achieved the state-of-the-art results on almost all +metrics of the nuScenes leaderboard when proposed. +C. Model Uncertainty Estimation +As mentioned above, deep ensemble has certain +advantages in model uncertainty estimation, so we design a +prediction approach that simultaneously integrates model +uncertainty and motion uncertainty estimation based on it. +Specifically, we use random initialization of the model +parameters and random shuffling of the training data to train +K homogeneous and heterogeneous models, then estimate +uncertainty based on the K set of output ˆ k +Y , + + +1,2,..., +k +K + +. +In addition, EDL, as a method to capture multiclass +uncertainties with low computational cost, is also exploited to +estimate the model uncertainty of the maneuver classification +module. Specifically, the Dirichlet distribution is considered +the prior distribution for the classification: + +1 +1 +1 +P + for P +(P| ) +( ) +0 + otherwise +z +Z +z +Z +z +D +B +  + + + + +  + + +α +α + +(3) +where +1 +[ +,..., +] +Z + + + +α + are the distribution parameters, +1 +z +z +e + + + +is the evidence, and +Z is the Z-dimensional unit simplex. +D. Uncertainty Scores Design +In our work, different uncertainty scores are proposed for +failure detection. Considering the different problem forms of +maneuver classification and trajectory prediction tasks, we +formulate corresponding scores for both. +For maneuver classification task combined with deep +ensemble, we formulate the following uncertainty scores +referring to the definition in [37]: +Total entropy (TE) for maneuver classification is +quantified to represent the total uncertainty considering both +model uncertainty and the motion uncertainty: + + + + + + + +P +1 +1 +TE= +P +| , , +P +| , , +K +k +k +z +K +z + + + + + + +  + + + + + + + + + +θ +S C θ +S C θ +∣ + (4) +where +k +θ are the parameters of the kth model of deep +ensemble, + represents the formula for calculating entropy, + represents the training set. +Data entropy (DE) for maneuver classification is quantified +to represent the average of data uncertainty from different +models. The larger the value, the higher the motion +uncertainty estimated by deep ensemble-prediction models. + + + + + + + +P +1 +1 +DE +| , , +P +| , , +K +k +k +K +z +z + + + +  + + + + + + + +θ +S C θ +S C θ +∣ + +(5) +Mutual Information (MI) is quantified to represent the +model uncertainty. As it increases, the degree of difference +between the prediction results of multiple models increases, +which to a certain extent reflects the reduction of the +confidence of the models in their classification results. + +MI +, +, , +TE +DE +z + + + + + + + +θ S C +∣ + +(6) +The maximum predicted probability [38] is also considered +and its inverse (negative maximum softmax probability, +NMaP) is calculated as an uncertainty score. +As for the EDL-based method, the above-discussed types +of uncertainty scores are also quantified for comparison, and +their formulas are derived according to (3)-(6). Additionally, +we consider the metrics suggested in [29]: + +1 +u +Z +z +z +Z + + + + + +(7) +Trajectory prediction involves multiple trajectories output +by one or more models, where each trajectory contains +position information for multiple future moments. Referring + + + +to the usual error metrics [8, 12, 18], average displacement +error (ADE) and final displacement error (FDE), we define +two basic metrics, average predictive entropy (APE) and final +predictive entropy (FPE), to represent the uncertainty formed +by multiple trajectories: + +1 +=1 +l +ˆ +A +1 +1 +1 +ˆ +( n2 +1) +ln +2 +PE= +f +f +t +t +i +i +f +f +t +t +t +t +d + + + + + + + + + +  + + + + + + + + +(8) + + + +l +ˆ +FP +1 +ˆ +n2 +1 +ln +2 +E +f +f +t +td + + + +  + + + + + + +(9) +where for different predicted trajectories of the same input, +the predicted position ˆ +td at the same time is assumed to +follow a two-dimensional Gaussian distribution. +Based on the above two basic metrics, different types of +uncertainty scores are defined according to the source of +different predicted trajectories (such as different sub-models, +different maneuvers, or both), which may represent model +uncertainty, motion uncertainty, or both. +IV. EXPERIMENTS +A. Experimental Setup +1) Model Implementation: For the training of GRIP+++, +inspired by [15], we adopt a two-stage training approach. In +the first stage, we focus on improving the trajectory +prediction accuracy under the real maneuver, by training the +model as a regression task at each time: + +, +1 +ˆ +1 +ft +t z +t +reg +f +t +L +t + + + + Y +Y +(10) +where +,ˆ +t z +Y and +t +Y are predicted positions for true maneuver +z and ground truth at time t respectively. +In the second stage, we additionally consider the loss of +maneuver classification by adding the cross-entropy loss: + +reg +man +L +L +L + + + + +(11) +where + + + + +log P +, +| +man +L +z +  +S C +,  is the weighting factor, and +z is the true maneuver label. Besides, in the implementation +of GRIP+++, the trajectories are sampled at 2Hz, with an +observation length of 3s and a prediction horizon of 3s. +As for the implementation of Trajectron++ [18] and PGP +[16], we follow their original model design and training +scheme. For deep ensemble, we set +5 +K  +, a scheme +considered cost-controllable and sufficiently efficient. To +achieve EDL, referring to [29], we incorporate a +Kullback-Leibler (KL) divergence term into our loss function +to avoid unnecessary uncertainty reduction. +2) Dataset: The proposed motion prediction models and +failure detectors are trained and validated on real traffic +datasets. Specifically, GRIP+++ and its failure detectors are +trained on SinD and tested on SinD and INTERACTION, +respectively. Trajectron++, PGP and their failure detection +experiments are carried out on the nuScenes dataset. +The SinD [39] dataset consists of 13248 recorded +trajectories from a signalized intersection. The traffic +participant classes include car, truck, bus, tricycle, bike, +motorcycle, and pedestrian. The INTERACTION [40] dataset +contains motion data collected in four categories of scenarios, +where we adopt the TC_intersection_VA (VA) subset that +also belongs to signalized intersection. It provided 3775 +trajectories for around 60 minutes. The nuScenes [41] dataset +is a large-scale self-driving car dataset with 1000 scenes, each +scene contains 20s object annotations and HD semantic maps. +3) Evaluation methodology: We set the evaluation +methodology separately for the failure detection for the +two-stage prediction task. Maneuver classification is a +classification task, a good failure detector is considered to +assign higher uncertainty scores to misclassified cases. +Therefore, we adopt the area under the receiver operating +characteristic curve (AUROC) as the basic evaluation metric. +However, AUROC does not reflect the impact of the addition +of the uncertainty estimation module on the original +prediction algorithm. Therefore, we also plot the cut-off +curve to evaluate the average accuracy of the remaining data +after filtering out a certain percentage of data in descending +order of uncertainty. The area under the cut-off curve +(AUCOC) is regarded as an overall evaluation of the +prediction model with the failure detector, with a larger value +indicating better performance. +For trajectory prediction tasks, AUROC is not suitable, we +use the cut-off curve as the evaluation methodology. Unlike +maneuver classification, the curve here is drawn by +calculating the average prediction error of the remaining data, +so a smaller AUCOC represents better performance. +B. Failure Detection for Maneuver Classification +Regarding failure detection for maneuver classification, we +set up several experiments to answer the following questions. + +Fig. 3. Uncertainty distribution for correctly classified and misclassified +samples. Experimental results of GRIP+++ based on deep ensemble. +How different are the distributions of uncertainty +scores for correct and misclassified cases? An effective +uncertainty-based failure detector is built on the assumption +that the uncertainty score level has a strong correlation with +the correctness of the prediction. As shown in Fig. 3, the +uncertainty scores of the correctly predicted maneuvers are +generally relatively low, while the incorrectly predicted cases +generally have high uncertainty scores. Meanwhile, there is a +relatively obvious separation between the two distributions, +especially for TA, DA, and NMaP. Therefore, it is +preliminarily inferred that the uncertainty scores have the +potential for failure detection. +Differences between different uncertainty scores for +failure detection? As indicated previously, in the deep +ensemble-based maneuver classification network, we can +extract various uncertainty scores, here we set up experiments +to compare the effects of different scores as the reference for +failure detection. The second row of TABLE I shows the +results, NMaP, TE, and DE achieve better failure detection + + + +performance when used as uncertainty scores, where the total +uncertainty considering both motion and model uncertainty is +slightly better than the motion uncertainty alone. NMaP is +relatively simple to calculate and has a strong detection +ability. Furthermore, although MI, which represents the +model uncertainty, reflects the reduced confidence of the +model when faced with unknown scenarios (as in TABLE II), +its performance is relatively weak when used alone as the +reference for failure detection. In Fig. 4, the cut-off curve and +AUCOC corresponding to different uncertainty scores are +further compared. Their performance has a great advantage +over the random filtering method and is close to the optimal +situation. And the relative relationship between different +uncertainty scores is consistent with TABLE I. +TABLE I. AUROC(↑) FOR MANEUVER CLASSIFICATION STAGE OF GRIP+++ + +TE +DE +MI +NMaP +u +Ensemble +0.911 +0.903 +0.864 +0.918 +- +Model 1 +- +0.871 +- +0.867 +- +Model 2 +- +0.868 +- +0.864 +- +Model 3 +- +0.871 +- +0.867 +- +Model 4 +- +0.868 +- +0.864 +- +Model 5 +- +0.863 +- +0.858 +- +EDL +0.912 +0.909 +0.911 +0.912 +0.910 +TABLE II. AVERAGE UNCERTAINTY OBTAINED BY DEEP ENSEMBLE-BASED +GRIP+++ TRAINED ON SIND, AND TESTED ON IN-DISTRIBUTION DATA +(SIND) AND OUT-OF-DISTRIBUTION DATA (VA), RESPECTIVELY + +TE +DE +MI +NMaP +SinD +0.318 +0.250 +0.068 +-0.877 +VA +0.303 +0.198 +0.105 +-0.879 + +Fig. 4. Cut-off curves and AUCOC (↑). The optimal curve is drawn by +directly using the classification error as a filtering reference; the random +curve is drawn by filtering the data in random order. +TABLE III. AUCOC (↑) FOR MANEUVER CLASSIFICATION STAGE OF +GRIP+++. MODEL I IS THE RESULT FROM THE ITH MODEL IN DEEP ENSEMBLE + +TE +DE +MI +NMaP +u +Ensemble +0.988 +0.987 +0.984 +0.989 +- +Model 1 +- +0.981 +- +0.982 +- +Model 2 +- +0.980 +- +0.981 +- +Model 3 +- +0.981 +- +0.982 +- +Model 4 +- +0.980 +- +0.980 +- +Model 5 +- +0.979 +- +0.979 +- +EDL +0.978 +0.978 +0.978 +0.978 +0.978 +Uncertainty scores based on deep ensemble vs. +uncertainty scores based on a single model? Here, we +obtain DE and NMaP from the single model in deep ensemble, +and they are further used for failure detection for the +maneuver classification module of the corresponding model. +From the comparison of rows 2-7 of TABLE I, although the +uncertainty scores extracted from the single model has a +certain failure detection ability, they are not as good as the +failure detector based on deep ensemble. In addition, it is also +concluded from the comparison of rows 2-7 in TABLE III +that the introduction of deep ensemble is beneficial to +improve the maneuver classification performance combined +with failure detector filtering. +How well do the EDL-based uncertainty scores +perform? As a comparison, we employ EDL to extract +uncertainty scores and evaluate their performance for failure +detection. TABLE I shows that using the uncertainty scores +extracted by EDL as references for the failure detector +achieves comparable results to deep ensemble. However, +TABLE III presents that the overall accuracy after filtering +the data based on these uncertainty scores is not high. One +possible reason is that the regularization term added by EDL +during the training process causes a drop in the prediction +performance of the main model, which in turn weakens the +effect of motion prediction with failure detection. +C. Failure Detection for Trajectory Prediction +As for failure detection for trajectory prediction, we design +some experiments to answer the following questions. +How well does the failure detector based on uncertainty +scores from multiple trajectories perform? For the +prediction error, considering the K predicted trajectories +under the real maneuver z, we calculate the minimum +(minADEz, minFDEz) and mean (meanADEz, meanFDEz) of +the errors of the K trajectories, and the error of their average +trajectory (ADEz, avg, FDEz, avg). We calculate APEz and FPEz +of the above K trajectories to estimate the predictive +uncertainty. As a comparison, we calculate the uncertainty of +the average trajectories of K models in different maneuvers +(APEavg, FPEavg), which to some extent represent the motion +uncertainty. In TABLE IV, each column represents an error +metric and each row represents the corresponding uncertainty +score used for failure detection (except rows 1-3). By +comparing rows 2-5 of the 2 sub-tables, APEz and FPEz have +stronger failure detection potential than APEavg and FPEavg. +Are the uncertainty scores extracted in the maneuver +classification stage applicable to the trajectory prediction +stage? Theoretically, the uncertainty scores obtained in the +maneuver classification stage represent the confidence of the +model in the current scene, so it may be suitable for failure +detection in the trajectory prediction stage. We conduct some +experiments to explore this question, the results are recorded +in rows 6-9 of the two sub-tables of TABLE IV. Compared +with the above trajectory uncertainty scores, the uncertainty +extracted in the maneuver classification stage has limited +potential for detecting high-error trajectories. One of the +possible reasons is that the uncertainty scores calculated +directly based on the trajectories imply the consideration of +information such as the velocity and acceleration of the object, +thus having a greater correlation with the trajectory error. +How is the failure detection generalizing to scenarios +with larger distributional shifts? Here, we use the VA +dataset to test the model trained based on SinD, results are +shown in TABLE V and TABLE VI. Compared with TABLE +I, III, and IV, when faced with larger distributional shifts, +while the reduction in the prediction accuracy of the main +model leads to a worsening of AUCOC, the decrease in +failure detection ability (such as AUROC) is relatively small. + + + +TABLE IV. AUCOC (↓)/IMPROVEMENT RATIO (IR)1 (↑) FOR THE +TRAJECTORY PREDICTION STAGE OF GRIP+++ + +minADEz +meanADEz +ADEz, avg +Optimal +0.066 +0.096 +0.088 +Random +0.259 +0.345 +0.330 +APEz +0.119/0.725 +0.143/0.813 +0.139/0.790 +APEavg +0.136/0.636 +0.172/0.694 +0.166/0.677 +TE +0.170/0.459 +0.228/0.469 +0.218/0.464 +DE +0.170/0.457 +0.229/0.466 +0.218/0.462 +MI +0.169/0.462 +0.227/0.476 +0.216/0.470 +NMaP +0.170/0.461 +0.228/0.472 +0.217/0.467 + +minFDEz +meanFDEz +FDEz, avg +Optimal +0.114 +0.182 +0.164 +Random +0.522 +0.718 +0.686 +FPEz +0.249/0.670 +0.301/0.779 +0.293/0.754 +FPEavg +0.278/0.599 +0.358/0.672 +0.345/0.654 +TE +0.361/0.395 +0.493/0.420 +0.471/0.413 +DE +0.362/0.393 +0.494/0.417 +0.472/0.410 +MI +0.359/0.400 +0.489/0.428 +0.467/0.420 +NMaP +0.360/0.397 +0.491/0.423 +0.497/0.416 +TABLE V. RESULTS FOR MANEUVER CLASSIFICATION STAGE OF GRIP+++ +WITH DEEP ENSEMBLE, WHICH IS TRAINED ON SIND AND TESTES ON VA + +TE +DE +MI +NMaP +AUROC +0.914 +0.915 +0.863 +0.912 +AUCOC +0.978 +0.978 +0.971 +0.978 +TABLE VI. AUCOCOPTIMAL/AUCOCUNCERTAINTY (↓)/AUCOCRANDOM/IR(↑) FOR +TRAJECTORY PREDICTION STAGE OF GRIP+++ WITH DEEP ENSEMBLE, +WHICH IS TRAINED ON SIND AND TESTES ON VA + +minADEz +meanADEz +ADEz, avg +APEz +0.088/0.210/ +0.445/0.656 +0.125/0.238/ +0.565/0.744 +0.117/0.234/ +0.550/0.730 + +minFDEz +meanFDEz +FDEz, avg +FPEz +0.158/0.991/ +0.491/0.601 +0.243/1.262/ +0.550/0.699 +0.228/1.232/ +0.543/0.686 +TABLE VII. AUCOCOPTIMAL/AUCOCUNCERTAINTY (↓)/AUCOCRANDOM/IR(↑) FOR +TRAJECTRON++ ON NUSCENES + +Single model +Deep ensemble +(mean)minADE +0.088/0.167/0.378/0.730 +0.096/0.160/0.384/0.778 +(mean)minFDE +0.132/0.308/0.689/0.683 +0.151/0.293/0.702/0.742 +(mean)meanADE +0.322/0.386/1.045/0.912 +0.339/0.394/1.040/0.922 +(mean)meanFDE +0.608/0.754/2.096/0.902 +0.637/0.763/2.082/0.913 +minminADE +- +0.055/0.112/0.234/0.682 +meanmaxpADE +- +0.181/0.280/0.801/0.841 +TABLE VIII. AUCOCOPTIMAL/AUCOCUNCERTAINTY(↓)/AUCOCRANDOM/IR(↑) FOR +PGP ON NUSCENES, UC MEANS UNIFIED CLUSTERING + +Single model +Deep ensemble +(mean)minADE +0.498/0.837/0.945/0.242 +0.529/0.832/0.945/0.271 +(mean)minFDE +0.623/1.273/1.554/0.302 +0.747/1.249/1.548/0.373 +minminADE +- +0.367/0.628/0.708/0.234 +meanmaxpADE +- +1.538/2.497/3.115/0.392 +minADE (uc) +- +0.488/0.797/0.908/0.264 +minFDE (uc) +- +0.612/0.181/1.466/0.333 +How well does uncertainty-based failure detection +perform in generative model-based trajectory prediction? +We adopt Trajectron++ combined with deep ensemble to +extract multiple uncertainty scores as failure detection +references. The results of this investigation are provided in +TABLE VII, where minADE/minFDE/meanADE/meanFDE +for single model is calculated based on the 10 trajectories + +1 IR is calculated by (AUCOCrandom – AUCOCuncertainty)/(AUCOCrandom – +AUCOCoptimal), where AUCOCrandom, AUCOCoptimal, and AUCOCuncertainty +represent the AUCOC based on the optimal sorting, the random sorting, and +the uncertainty scores-based sorting, respectively. +predicted by the single model, and the corresponding +uncertainty scores for failure detection are APE/FPE/APE +/FPE obtained from the 10 trajectories. In contrast, +meanminADE/meanminFDE/meanmeanADE/meanmeanFD +E/minminADE/meanmaxpADE for deep ensemble are +calculated based on 50 trajectories from all 5 ensemble +models, where the first operator (mean/min) is for different +sub-models and the second operator (mean/min/maxp) is for +different maneuvers from each model’s output. The +corresponding uncertainty scores for failure detection are +meanAPE/meanFPE/meanAPE/meanFPE/APEall/APEmaxp, +where meanAPE/meanFPE are obtained by averaging APE/ +FPE from 5 sub-models, APEall is directly calculated from all +50 trajectories, APEmaxp is calculated according to the +maximum probability trajectory of each model. The results +show promising performance of the uncertainty-based failure +detector. +Can the above uncertainty-based failure detection be +simply applied to any trajectory prediction algorithms? In +addition to the typical deep neural network architecture and +modules, existing trajectory prediction algorithms may use +various tricks, which may directly affect the uncertainty +scores extracted from the output trajectories. We conduct +some exploratory experiments with PGP, a high-performance +prediction algorithm integrating special tricks including +traversal, sampling, and clustering, to analyze the +performance of applying the uncertainty scores obtained from +the output trajectories for failure detection. In addition, we +apply deep ensemble to consider model uncertainty. From the +evaluation results in TABLE VIII, we conclude that the +performance of direct uncertainty quantification based on +output results is not very outstanding. Possible reasons +include operations such as sampling latent vectors from an +unconstrained normal distribution or clustering. This result +reminds us that it is necessary to improve uncertainty +estimation methods and scores according to the prediction +algorithms’ characteristics. For example, we propose a +framework for unified clustering based on the outputs of all +sub-models of the deep ensemble, the results in the last two +rows of TABLE VIII show some improvement over the +original model in trajectory prediction performance. +V. CONCLUSION +In this work, we propose a framework to detect motion +prediction failures from the uncertainty perspective. We +divide motion prediction tasks into two stages, maneuver +classification and maneuver-based trajectory prediction, and +formulate corresponding uncertainty scores for failure +detection, where motion uncertainty and model uncertainty +are both discussed. Our experiments cover the comparison of +different prediction tasks, multiple prediction algorithms, +different uncertainty estimation methods, and various +uncertainty scores, Finally, we observe that uncertainty +quantification is promising for failure detection for motion +prediction, with the potential to generalize to environments +with larger distributional shifts. However, it is also necessary +to conduct targeted discussions and designs for different +prediction algorithms. Our future work will focus on the +integration of the proposed method with safety decision +-making for autonomous driving, and its implementation and +validation on physical vehicle platforms. + + + +REFERENCES +[1] +A. Jain, L. Del Pero, H. Grimmett, and P. Ondruska, “Autonomy 2.0: +Why is self-driving always 5 years away?” arXiv, Aug. 09, 2021. doi: +10.48550/arXiv.2107.08142. +[2] +L. Plaza, “Collision Between Vehicle Controlled by Developmental +Automated Driving System and Pedestrian,” PB2019-101402, Mar. +2018. +[3] +J. Sifakis and D. Harel, “Trustworthy Autonomous System +Development,” ACM Trans. Embed. Comput. Syst., Jun. 2022, doi: +10.1145/3545178. +[4] +L. A. Dennis and M. Fisher, “Verifiable Self-Aware Agent-Based +Autonomous Systems,” Proceedings of the IEEE, vol. 108, no. 7, pp. +1011–1026, Jul. 2020, doi: 10.1109/JPROC.2020.2991262. +[5] +B. Sun, J. Xing, H. Blum, R. Siegwart, and C. Cadena, “See Yourself in +Others: Attending Multiple Tasks for Own Failure Detection,” in 2022 +International Conference on Robotics and Automation (ICRA), 2022, +pp. 8409–8416. doi: 10.1109/ICRA46639.2022.9812310. +[6] +E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter for +nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive +Systems for Signal Processing, Communications, and Control +Symposium +(Cat. +No.00EX373), +2000, +pp. +153–158. +doi: +10.1109/ASSPCC.2000.882463. +[7] A. Cosgun et al., “Towards full automated drive in urban environments: +A demonstration in GoMentum Station, California,” in 2017 IEEE +Intelligent Vehicles Symposium (IV), Jun. 2017, pp. 1811–1818. doi: +10.1109/IVS.2017.7995969. +[8] +A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and S. +Savarese, “Social LSTM: Human Trajectory Prediction in Crowded +Spaces,” in Proceedings of the IEEE Conference on Computer Vision +and Pattern Recognition, 2016, pp. 961–971. +[9] +J. Gao et al., “VectorNet: Encoding HD Maps and Agent Dynamics +From Vectorized Representation,” in Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, 2020, pp. +11525–11533. +[10] J. Gu, C. Sun, and H. Zhao, “DenseTNT: End-to-End Trajectory +Prediction From Dense Goal Sets,” in Proceedings of the IEEE/CVF +International +Conference +on +Computer +Vision, +2021, +pp. +15303–15312. +[11] S. Mozaffari, O. Y. Al-Jarrah, M. Dianati, P. Jennings, and A. +Mouzakitis, “Deep Learning-Based Vehicle Behavior Prediction for +Autonomous Driving Applications: A Review,” IEEE Transactions on +Intelligent Transportation Systems, vol. 23, no. 1, pp. 33–47, 2022, doi: +10.1109/TITS.2020.3012034. +[12] X. Li, X. Ying, and M. C. Chuah, “GRIP++: Enhanced Graph-based +Interaction-aware Trajectory Prediction for Autonomous Driving.” +arXiv, May 19, 2020. doi: 10.48550/arXiv.1907.07792. +[13] X. Mo, Z. Huang, Y. Xing, and C. Lv, “Multi-Agent Trajectory +Prediction With Heterogeneous Edge-Enhanced Graph Attention +Network,” IEEE Transactions on Intelligent Transportation Systems, +vol. +23, +no. +7, +pp. +9554–9567, +Jul. +2022, +doi: +10.1109/TITS.2022.3146300. +[14] N. Djuric et al., “Uncertainty-aware Short-term Motion Prediction of +Traffic Actors for Autonomous Driving,” in Proceedings of the +IEEE/CVF Winter Conference on Applications of Computer Vision, +2020, pp. 2095–2104. +[15] N. Deo and M. M. Trivedi, “Convolutional Social Pooling for Vehicle +Trajectory Prediction,” in Proceedings of the IEEE Conference on +Computer Vision and Pattern Recognition Workshops, 2018, pp. +1468–1476. +[16] N. Deo, E. Wolff, and O. Beijbom, “Multimodal Trajectory Prediction +Conditioned on Lane-Graph Traversals,” in Proceedings of the 5th +Conference on Robot Learning, Jan. 2022, pp. 203–212. +[17] H. Cui et al., “Multimodal Trajectory Predictions for Autonomous +Driving using Deep Convolutional Networks,” in 2019 International +Conference on Robotics and Automation (ICRA), 2019, pp. 2090–2096. +doi: 10.1109/ICRA.2019.8793868. +[18] T. Salzmann, B. Ivanovic, P. Chakravarty, and M. Pavone, +“Trajectron++: Dynamically-Feasible Trajectory Forecasting with +Heterogeneous Data,” in Computer Vision – ECCV 2020, Cham, 2020, +pp. 683–700. doi: 10.1007/978-3-030-58523-5_40. +[19] T.-J. Hsieh, C.-S. Shih, C.-W. Lin, C.-W. Chen, and P.-K. Tsung, +“Trajectory Prediction at Unsignalized Intersections using Social +Conditional Generative Adversarial Network,” in 2021 IEEE +International Intelligent Transportation Systems Conference (ITSC), +Sep. 2021, pp. 844–851. doi: 10.1109/ITSC48978.2021.9564441. +[20] J. Gawlikowski et al., “A Survey of Uncertainty in Deep Neural +Networks.” arXiv, Jan. 18, 2022. doi: 10.48550/arXiv.2107.03342. +[21] A. Kendall and Y. Gal, “What Uncertainties Do We Need in Bayesian +Deep Learning for Computer Vision?,” in Advances in Neural +Information Processing Systems, 2017, vol. 30. +[22] C. Louizos and M. Welling, “Multiplicative Normalizing Flows for +Variational Bayesian Neural Networks,” in Proceedings of the 34th +International Conference on Machine Learning, Jul. 2017, pp. +2218–2227. +[23] C. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra, “Weight +Uncertainty in Neural Network,” in Proceedings of the 32nd +International Conference on Machine Learning, Jun. 2015, pp. +1613–1622. +[24] Y. Gal and Z. Ghahramani, “Dropout as a Bayesian Approximation: +Representing Model Uncertainty in Deep Learning,” in Proceedings of +The 33rd International Conference on Machine Learning, Jun. 2016, +pp. 1050–1059. +[25] Y. Gal, J. Hron, and A. Kendall, “Concrete Dropout,” in Advances in +Neural Information Processing Systems, 2017, vol. 30. +[26] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and +Scalable Predictive Uncertainty Estimation using Deep Ensembles,” in +Advances in Neural Information Processing Systems, 2017, vol. 30. +[27] Y. Wen, D. Tran, and J. Ba, “BatchEnsemble: An Alternative +Approach to Efficient Ensemble and Lifelong Learning.” arXiv, Feb. +19, 2020. doi: 10.48550/arXiv.2002.06715. +[28] F. Wenzel, J. Snoek, D. Tran, and R. Jenatton, “Hyperparameter +Ensembles for Robustness and Uncertainty Quantification,” in +Advances in Neural Information Processing Systems, 2020, vol. 33, pp. +6514–6527. +[29] M. Sensoy, L. Kaplan, and M. Kandemir, “Evidential deep learning to +quantify classification uncertainty,” in Advances in neural information +processing systems, 2018, vol. 31. +[30] D. Hendrycks, M. Mazeika, and T. Dietterich, “Deep Anomaly +Detection with Outlier Exposure.” arXiv, Jan. 28, 2019. doi: +10.48550/arXiv.1812.04606. +[31] C. B. Kuhn, M. Hofbauer, Z. Xu, G. Petrovic, and E. Steinbach, +“Pixel-Wise Failure Prediction For Semantic Video Segmentation,” in +2021 IEEE International Conference on Image Processing (ICIP), Sep. +2021, pp. 614–618. doi: 10.1109/ICIP42928.2021.9506552. +[32] Q. M. Rahman, N. Sünderhauf, P. Corke, and F. Dayoub, “FSNet: A +Failure Detection Framework for Semantic Segmentation,” IEEE +Robotics and Automation Letters, vol. 7, no. 2, pp. 3030–3037, Apr. +2022, doi: 10.1109/LRA.2022.3143219. +[33] K. Lis, K. Nakka, P. Fua, and M. Salzmann, “Detecting the Unexpected +via Image Resynthesis,” in Proceedings of the IEEE/CVF International +Conference on Computer Vision, 2019, pp. 2152–2161. +[34] D. Haldimann, H. Blum, R. Siegwart, and C. Cadena, “This is not what +I imagined: Error Detection for Semantic Segmentation through Visual +Dissimilarity.” arXiv, Sep. 02, 2019. doi: 10.48550/arXiv.1909.00676. +[35] L. Deecke, R. Vandermeulen, L. Ruff, S. Mandt, and M. Kloft, “Image +Anomaly Detection with Generative Adversarial Networks,” in +Machine Learning and Knowledge Discovery in Databases, Cham, +2019, pp. 3–17. doi: 10.1007/978-3-030-10925-7_1. +[36] D. Hendrycks and K. Gimpel, “A Baseline for Detecting Misclassified +and Out-of-Distribution Examples in Neural Networks,” presented at +the International Conference on Learning Representations, 2017. +[37] D. Feng, A. Harakeh, S. L. Waslander, and K. Dietmayer, “A Review +and Comparative Study on Probabilistic Object Detection in +Autonomous +Driving,” +IEEE +Transactions +on +Intelligent +Transportation Systems, vol. 23, no. 8, pp. 9961–9980, 2022, doi: +10.1109/TITS.2021.3096854. +[38] W. J. Maddox, P. Izmailov, T. Garipov, D. P. Vetrov, and A. G. Wilson, +“A Simple Baseline for Bayesian Uncertainty in Deep Learning,” in +Advances in Neural Information Processing Systems, 2019, vol. 32. +[39] Y. Xu et al., “SIND: A Drone Dataset at Signalized Intersection in +China.” arXiv, Sep. 06, 2022. doi: 10.48550/arXiv.2209.02297. +[40] W. Zhan et al., “INTERACTION Dataset: An INTERnational, +Adversarial and Cooperative moTION Dataset in Interactive Driving +Scenarios with Semantic Maps.” arXiv, Sep. 30, 2019. doi: +10.48550/arXiv.1910.03088. +[41] H. Caesar et al., “nuScenes: A Multimodal Dataset for Autonomous +Driving,” in Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2020, pp. 11621–11631. + + diff --git a/7NE3T4oBgHgl3EQfRgki/content/tmp_files/load_file.txt b/7NE3T4oBgHgl3EQfRgki/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8f4c5f7ae74132a18e8602c5568cc31d86aa7c0 --- /dev/null +++ b/7NE3T4oBgHgl3EQfRgki/content/tmp_files/load_file.txt @@ -0,0 +1,821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf,len=820 +page_content='\uf020 Abstract— Motion prediction is essential for safe and efficient autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, the inexplicability and uncertainty of complex artificial intelligence models may lead to unpredictable failures of the motion prediction module, which may mislead the system to make unsafe decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Therefore, it is necessary to develop methods to guarantee reliable autonomous driving, where failure detection is a potential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Uncertainty estimates can be used to quantify the degree of confidence a model has in its predictions and may be valuable for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We propose a framework of failure detection for motion prediction from the uncertainty perspective, considering both motion uncertainty and model uncertainty, and formulate various uncertainty scores according to different prediction stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The proposed approach is evaluated based on different motion prediction algorithms, uncertainty estimation methods, uncertainty scores, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', and the results show that uncertainty is promising for failure detection for motion prediction but should be used with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' INTRODUCTION Motion prediction is a hot topic in mobile robot and autonomous vehicle communities, accurate prediction of the future motion of surrounding traffic participants is fundamental to robust and reliable decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Artificial intelligence (AI), especially deep learning, has been widely used in autonomous driving tasks by its advantages in dealing with complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' With the collection of large-scale data, the improvement of computing power and related algorithms, AI is expected to play a vital role in autonomous driving systems in the future [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, although AI-based motion prediction has shown statistical performance advantages, it is difficult to avoid unpredictable failures due to the inherent inexplicability and insufficient reliability of deep learning models, which may cause serious autonomous driving accidents [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' From the uncertainty perspective, motion prediction faces the dual challenge of uncertainty from the environment and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Drivers, pedestrians, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' in the environment have uncertainty in their intentions and movements, which makes it difficult to accurately predict their future in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Additionally, due to insufficient training data and training process, the model may experience serious performance degradation when faced with rare or unknown scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Research supported by the National Science Foundation of China Project: U1964203 and 52072215, and the National Key R&D Program of China:2020YFB1600303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' (Corresponding authors: Hong Wang) Wenbo Shao, Liang Peng, Jun Li and Hong Wang are with School of Veh icle and Mobility, Tsinghua University, Beijing 100084, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' (e-mail: {sw b19, peng-l20}@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' {lijun1958, hong_wang}@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='cn) Yanchao Xu is with School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' (e-mail: 3120200410@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='cn) Failure Detector Main Model Maneuver Classifier Trajectory Predictor Graph Model UM UT Is there a wrong maneuver classification or trajectory prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Uncertainty-based failure detection for motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' UM, UT are the uncertainty scores extracted for maneuver classification and trajectory prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The failure detection, isolation, and recovery mechanism is an effective way to solve the above problems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Among them, the study of failure detection for AI models has attracted increasing interest, which is of critical significance for the development of reliable autonomous driving systems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1, using the information extracted from the main model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' motion prediction model, a failure detector is built to identify maneuver classification errors and trajectory prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Uncertainty, as a measure of the confidence level of the model in its output, has been used by some researchers for failure detection in tasks such as semantic segmentation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Our study exploits various uncertainties from motion prediction and explores their usefulness for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In this work, we concentrate on failure detection for motion prediction from the uncertainty perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The main contributions are as follows: \uf09f A framework of failure detection using uncertainty for motion prediction tasks, taking into account both motion uncertainty and model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' \uf09f A series of uncertainty scores for failure detection formulated for different motion prediction stages and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' \uf09f A detailed evaluation and comparison with multiple motion prediction algorithms, uncertainty estimation methods and uncertainty scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Motion Prediction and Motion Uncertainty Estimation Traditional motion prediction methods predict the future motion of the target agent (TA) based on its historical state by explicitly modeling kinematic models, such as Kalman Filter[5], [6], but they only apply to short-term prediction under scenarios with few interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In recent years, deep learning-based motion prediction [8]–[10] has demonstrated promising performance by simultaneously modeling TA’s historical state, its interactions with surrounding traffic participants, and other environmental information in deep Failure Detection for Motion Prediction of Autonomous Driving: An Uncertainty Perspective* Wenbo Shao, Yanchao Xu, Liang Peng, Jun Li, and Hong Wang neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' A broader review of deep learning-based motion prediction can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As for the model’s output form, some studies regard motion prediction as a multipoint regression problem [12]–[14], so as to output the unimodal predicted trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, due to the diversity of intentions and the uncertainty of traffic participants’ behaviors, the future trajectory distribution corresponding to one model input presents multiple possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Recently, increasing researchers and prediction competitions have paid attention to multimodal motion prediction, which is generally divided into two stages: maneuver or target classification, and trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Some studies [15], [16] define maneuvers as specified behavior patterns, then train the maneuver classifier through supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For example, CS-LSTM [15] defines six maneuver modes for vehicles on highways, where longitudinal maneuvers include normal driving and braking, the lateral maneuvers include left lane change, right lane change, and lane keeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The predicted maneuvers can serve as an important guide for future trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Other studies do not explicitly define specific behavior patterns before training, but guide the model to learn the optimal maneuver modes through model design and training process [17]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For example, Trajectron++ [18] adopts the conditional variational autoencoder (CVAE) to encode multimodality by introducing latent variables, and relies on a bivariate Gaussian Mixture Model (GMM) to model the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Model Uncertainty Estimation The above multimodal prediction algorithms model the uncertainty in the traffic participants’ movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In addition, deep learning models have inherent uncertainty, generally called model uncertainty or epistemic uncertainty [20], it is difficult to ignore in the real world where there are distribution shifts or out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Bayesian neural network (BNN) [21]–[23] is a representative method for estimating model uncertainty, in which Bayesian inference plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Methods such as Monte-Carlo dropout [24], [25] achieve approximate inference through sampling, and they further promote the generality and popularity of BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Besides, deep ensemble [26]–[28], as a simple and scalable method, has shown promising performance in model uncertainty estimation and thus has attracted many researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As the representative method requiring only a single forward pass, evidential deep learning (EDL) [29] computes the uncertainty of the output distribution by modeling the prior distribution for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Failure Detection for Autonomous Driving Failure detection is attracting attention as a technology to achieve reliable autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=" It uses the main model's input, internal features, or output to diagnose whether there is a failure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Learning-based approaches build a specialized model to act as the failure detector, and it identifies failures of the main model by using failure cases for supervised training [30]–[32] or estimating reconstruction errors [33]–[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In addition, uncertainty-based anomaly detection has attracted some interest, such as detecting misclassified or out-of-distribution examples through maximum softmax probabilities directly output by classification networks [36] or predictive entropy quantization taking into account model uncertainty [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, to the best of our knowledge, most current research on failure detection for autonomous driving focuses on perception tasks, such as semantic segmentation, depth estimation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [5], and failure detection for motion prediction models from the uncertainty perspective has been rarely discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Our approach utilizes both motion uncertainty and model uncertainty, proposes uncertainty scores for different stages of motion prediction, and investigates the effect of motion prediction failure detection based on different scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Problem Setting Motion prediction is a task that predicts TA’s trajectory over a period of time in the future given input information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Assuming the current moment 0 t \uf03d , the input information may include TA’s historical state \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 2 0 [ , , , ] h h t t s s s \uf02d \uf02b \uf02b \uf03d S in the past ht timesteps, the historical state of TA’s surrounding traffic participants, and other contextual information such as maps, which are uniformly represented here by C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Among them, \uf028 \uf029ts may contain TA’s information such as the position, speed, and category at t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The output is the predicted position ˆY of TA in the future ft timesteps: \uf028 \uf029 ˆ , f \uf03d Y S C (1) with 1 2 ˆ ˆ ˆ ˆ [ , ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', ] ft d d d \uf03d Y consisting of the ft predicted positions ˆ td .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For multimodal motion prediction, ˆY contains predicted trajectories under multiple maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Failure detection for motion prediction refers to identifying potential motion prediction failures by monitoring model’s state, where failures may exist in the form of maneuver misclassification or excessive error of predicted trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=" Uncertainty, as the measure of TA's behavior or model state, reflects the model's confidence in its particular output and thus has the potential to diagnose potential prediction failures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' This work proposes to detect the performance degradation of motion prediction models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' the decrease in the accuracy of prediction results, by quantifying the uncertainty scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=" Motion Prediction with Motion Uncertainty Estimation Due to the unavailability of TA's actual intentions and the randomness of its behavior, it may have multiple possible future trajectories." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' GRIP++ is an enhanced graph-based interaction-aware trajectory prediction algorithm, it models inter-agent interactions and temporal features but only predicts future trajectories in a single mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2, we add the maneuver classification module to GRIP++, by distinguishing different behavioral patterns to improve the authenticity and usability of the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The new method is called GRIP+++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We focus on two-stage tasks in the proposed method: maneuver classification and maneuver-based trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In the maneuver classification stage, given the TA’s historical state and scene context, feature G are extracted through the graph convolutional model (GCN), which includes the processing of fixed and trainable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Then TA’s maneuver \uf028 \uf029 P | z z G is inferred by multilayer perceptron (MLP), where \uf07b \uf07d 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', z Z \uf0ce represents one of the defined maneuver modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In CS-LSTM [15], the modes are divided into three types of lateral maneuvers and two types of longitudinal maneuvers, but they are only applicable to vehicles driving on highway, we define a common set of maneuver modes suitable for various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=" Specifically, TA's maneuvers are divided into four categories according to their movement direction and speed: going straight, turning left, turning right, and stopping." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In the network, we adopt the softmax head for probabilistic maneuver classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Graph Convolutional Model Maneuver Classification Module 64 ht n Trajectory Prediction Module Predicted Trajectories concat Maneuver Probabilities ht Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The architecture of GRIP+++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The maneuver-based trajectory prediction module consists of seq2seq networks taking the concatenation of the graph feature G and the feature vector transformed by the maneuver z as input, and outputs the future trajectory ˆ z Y under the maneuver z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' To compare the generality of uncertainty-based failure detection in different motion prediction mechanisms, we employ another two classes of typical prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Firstly, we focus on multimodal trajectory prediction based on the generative model, so we adopt Trajectron++ [18], it utilizes the CVAE-based latent network framework to model multimodal future trajectories, where the discrete categorical latent variable z encodes high-level behavior patterns: \uf07b \uf07d 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', P( , ) P ( , , )P ( , ) ˆ ˆ ˆ z Z z z \uf0ce \uf03d \uf0e5 ψ θ S C S C Y Y Y S C ∣ ∣ ∣ (2) where θ , ψ are deep neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Furthermore, we use PGP [16] as a comparison, it is a multimodal trajectory prediction method combining graph traversal, latent vector sampling, and clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' It models discrete policy for graph traversal by representing HD maps as lane graphs, and implements diverse trajectories prediction combined with a random sampling of latent vectors for longitudinal variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Furthermore, it uses K-means clustering to obtain Z predictive trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' With its clever design, PGP achieved the state-of-the-art results on almost all metrics of the nuScenes leaderboard when proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Model Uncertainty Estimation As mentioned above, deep ensemble has certain advantages in model uncertainty estimation, so we design a prediction approach that simultaneously integrates model uncertainty and motion uncertainty estimation based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Specifically, we use random initialization of the model parameters and random shuffling of the training data to train K homogeneous and heterogeneous models, then estimate uncertainty based on the K set of output ˆ k Y , \uf07b \uf07d 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', k K \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In addition, EDL, as a method to capture multiclass uncertainties with low computational cost, is also exploited to estimate the model uncertainty of the maneuver classification module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Specifically, the Dirichlet distribution is considered the prior distribution for the classification: 1 1 1 P for P (P| ) ( ) 0 otherwise z Z z Z z D B \uf061 \uf02d \uf03d \uf0ec \uf0ce \uf0ef \uf03d \uf0ed \uf0ef\uf0ee \uf0d5 α α (3) where 1 [ ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', ] Z \uf061 \uf061 \uf03d α are the distribution parameters, 1 z z e \uf061 \uf03d \uf02d is the evidence, and Z is the Z-dimensional unit simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Uncertainty Scores Design In our work, different uncertainty scores are proposed for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Considering the different problem forms of maneuver classification and trajectory prediction tasks, we formulate corresponding scores for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For maneuver classification task combined with deep ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' we formulate the following uncertainty scores referring to the definition in [37]: Total entropy (TE) for maneuver classification is quantified to represent the total uncertainty considering both model uncertainty and the motion uncertainty: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 P 1 1 TE= P | ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' P | ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' K k k z K z \uf03d \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf03d \uf0ea \uf0fa \uf0eb \uf0fb \uf0eb \uf0fb \uf0eb \uf0fb \uf0e5 θ S C θ S C θ ∣ (4) where k θ are the parameters of the kth model of deep ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' represents the formula for calculating entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' represents the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Data entropy (DE) for maneuver classification is quantified to represent the average of data uncertainty from different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The larger the value, the higher the motion uncertainty estimated by deep ensemble-prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 P 1 1 DE | , , P | , , K k k K z z \uf03d \uf03d \uf0e9 \uf0f9 \uf03d \uf0e9 \uf0f9 \uf0eb \uf0fb \uf0eb \uf0fb \uf0e5 θ S C θ S C θ ∣ (5) Mutual Information (MI) is quantified to represent the model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As it increases, the degree of difference between the prediction results of multiple models increases, which to a certain extent reflects the reduction of the confidence of the models in their classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' MI , , , TE DE z \uf03d \uf03d \uf02d \uf0e9 \uf0f9 \uf0eb \uf0fb θ S C ∣ (6) The maximum predicted probability [38] is also considered and its inverse (negative maximum softmax probability, NMaP) is calculated as an uncertainty score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As for the EDL-based method, the above-discussed types of uncertainty scores are also quantified for comparison, and their formulas are derived according to (3)-(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Additionally, we consider the metrics suggested in [29]: 1 u Z z z Z \uf061 \uf03d \uf03d \uf0e5 (7) Trajectory prediction involves multiple trajectories output by one or more models, where each trajectory contains position information for multiple future moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Referring to the usual error metrics [8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' average displacement error (ADE) and final displacement error (FDE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' we define two basic metrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' average predictive entropy (APE) and final predictive entropy (FPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' to represent the uncertainty formed by multiple trajectories: 1 =1 l ˆ A 1 1 1 ˆ ( n2 1) ln 2 PE= f f t t i i f f t t t t d \uf070 \uf03d \uf0f9 \uf0e9 \uf0f9 \uf02b \uf02b \uf0eb \uf0e9 \uf0fb \uf03d \uf0ea \uf0fa \uf0eb \uf0fb \uf053 \uf0e5 \uf0e5 (8) \uf028 \uf029 l ˆ FP 1 ˆ n2 1 ln 2 E f f t td \uf070 \uf03d \uf0e9 \uf0f9 \uf03d \uf02b \uf02b \uf053 \uf0eb \uf0fb (9) where for different predicted trajectories of the same input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' the predicted position ˆ td at the same time is assumed to follow a two-dimensional Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Based on the above two basic metrics, different types of uncertainty scores are defined according to the source of different predicted trajectories (such as different sub-models, different maneuvers, or both), which may represent model uncertainty, motion uncertainty, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Experimental Setup 1) Model Implementation: For the training of GRIP+++, inspired by [15], we adopt a two-stage training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In the first stage, we focus on improving the trajectory prediction accuracy under the real maneuver, by training the model as a regression task at each time: , 1 ˆ 1 ft t z t reg f t L t \uf03d \uf02d \uf03d \uf0e5 Y Y (10) where ,ˆ t z Y and t Y are predicted positions for true maneuver z and ground truth at time t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In the second stage, we additionally consider the loss of maneuver classification by adding the cross-entropy loss: reg man L L L \uf06c \uf03d \uf02b (11) where \uf028 \uf029 \uf028 \uf029 log P , | man L z \uf03d \uf02d S C , \uf06c is the weighting factor, and z is the true maneuver label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Besides, in the implementation of GRIP+++, the trajectories are sampled at 2Hz, with an observation length of 3s and a prediction horizon of 3s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As for the implementation of Trajectron++ [18] and PGP [16], we follow their original model design and training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For deep ensemble, we set 5 K \uf03d , a scheme considered cost-controllable and sufficiently efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' To achieve EDL, referring to [29], we incorporate a Kullback-Leibler (KL) divergence term into our loss function to avoid unnecessary uncertainty reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2) Dataset: The proposed motion prediction models and failure detectors are trained and validated on real traffic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Specifically, GRIP+++ and its failure detectors are trained on SinD and tested on SinD and INTERACTION, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Trajectron++, PGP and their failure detection experiments are carried out on the nuScenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The SinD [39] dataset consists of 13248 recorded trajectories from a signalized intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The traffic participant classes include car, truck, bus, tricycle, bike, motorcycle, and pedestrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The INTERACTION [40] dataset contains motion data collected in four categories of scenarios, where we adopt the TC_intersection_VA (VA) subset that also belongs to signalized intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' It provided 3775 trajectories for around 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The nuScenes [41] dataset is a large-scale self-driving car dataset with 1000 scenes, each scene contains 20s object annotations and HD semantic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 3) Evaluation methodology: We set the evaluation methodology separately for the failure detection for the two-stage prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Maneuver classification is a classification task, a good failure detector is considered to assign higher uncertainty scores to misclassified cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Therefore, we adopt the area under the receiver operating characteristic curve (AUROC) as the basic evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, AUROC does not reflect the impact of the addition of the uncertainty estimation module on the original prediction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Therefore, we also plot the cut-off curve to evaluate the average accuracy of the remaining data after filtering out a certain percentage of data in descending order of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The area under the cut-off curve (AUCOC) is regarded as an overall evaluation of the prediction model with the failure detector, with a larger value indicating better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For trajectory prediction tasks, AUROC is not suitable, we use the cut-off curve as the evaluation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Unlike maneuver classification, the curve here is drawn by calculating the average prediction error of the remaining data, so a smaller AUCOC represents better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Failure Detection for Maneuver Classification Regarding failure detection for maneuver classification, we set up several experiments to answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Uncertainty distribution for correctly classified and misclassified samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Experimental results of GRIP+++ based on deep ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' How different are the distributions of uncertainty scores for correct and misclassified cases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' An effective uncertainty-based failure detector is built on the assumption that the uncertainty score level has a strong correlation with the correctness of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 3, the uncertainty scores of the correctly predicted maneuvers are generally relatively low, while the incorrectly predicted cases generally have high uncertainty scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Meanwhile, there is a relatively obvious separation between the two distributions, especially for TA, DA, and NMaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Therefore, it is preliminarily inferred that the uncertainty scores have the potential for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Differences between different uncertainty scores for failure detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As indicated previously, in the deep ensemble-based maneuver classification network, we can extract various uncertainty scores, here we set up experiments to compare the effects of different scores as the reference for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The second row of TABLE I shows the results, NMaP, TE, and DE achieve better failure detection performance when used as uncertainty scores, where the total uncertainty considering both motion and model uncertainty is slightly better than the motion uncertainty alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' NMaP is relatively simple to calculate and has a strong detection ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Furthermore, although MI, which represents the model uncertainty, reflects the reduced confidence of the model when faced with unknown scenarios (as in TABLE II), its performance is relatively weak when used alone as the reference for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 4, the cut-off curve and AUCOC corresponding to different uncertainty scores are further compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Their performance has a great advantage over the random filtering method and is close to the optimal situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' And the relative relationship between different uncertainty scores is consistent with TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AUROC(↑) FOR MANEUVER CLASSIFICATION STAGE OF GRIP+++ TE DE MI NMaP u Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='918 Model 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='867 Model 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='864 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='867 Model 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='864 Model 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='858 EDL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='910 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AVERAGE UNCERTAINTY OBTAINED BY DEEP ENSEMBLE-BASED GRIP+++ TRAINED ON SIND, AND TESTED ON IN-DISTRIBUTION DATA (SIND) AND OUT-OF-DISTRIBUTION DATA (VA), RESPECTIVELY TE DE MI NMaP SinD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='877 VA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='879 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Cut-off curves and AUCOC (↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The optimal curve is drawn by directly using the classification error as a filtering reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' the random curve is drawn by filtering the data in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AUCOC (↑) FOR MANEUVER CLASSIFICATION STAGE OF GRIP+++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' MODEL I IS THE RESULT FROM THE ITH MODEL IN DEEP ENSEMBLE TE DE MI NMaP u Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='989 Model 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='982 Model 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='981 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='982 Model 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='980 Model 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='979 EDL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 Uncertainty scores based on deep ensemble vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' uncertainty scores based on a single model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Here, we obtain DE and NMaP from the single model in deep ensemble, and they are further used for failure detection for the maneuver classification module of the corresponding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' From the comparison of rows 2-7 of TABLE I, although the uncertainty scores extracted from the single model has a certain failure detection ability, they are not as good as the failure detector based on deep ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In addition, it is also concluded from the comparison of rows 2-7 in TABLE III that the introduction of deep ensemble is beneficial to improve the maneuver classification performance combined with failure detector filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' How well do the EDL-based uncertainty scores perform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As a comparison, we employ EDL to extract uncertainty scores and evaluate their performance for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' TABLE I shows that using the uncertainty scores extracted by EDL as references for the failure detector achieves comparable results to deep ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, TABLE III presents that the overall accuracy after filtering the data based on these uncertainty scores is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' One possible reason is that the regularization term added by EDL during the training process causes a drop in the prediction performance of the main model, which in turn weakens the effect of motion prediction with failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Failure Detection for Trajectory Prediction As for failure detection for trajectory prediction, we design some experiments to answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' How well does the failure detector based on uncertainty scores from multiple trajectories perform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For the prediction error, considering the K predicted trajectories under the real maneuver z, we calculate the minimum (minADEz, minFDEz) and mean (meanADEz, meanFDEz) of the errors of the K trajectories, and the error of their average trajectory (ADEz, avg, FDEz, avg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We calculate APEz and FPEz of the above K trajectories to estimate the predictive uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' As a comparison, we calculate the uncertainty of the average trajectories of K models in different maneuvers (APEavg, FPEavg), which to some extent represent the motion uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In TABLE IV, each column represents an error metric and each row represents the corresponding uncertainty score used for failure detection (except rows 1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' By comparing rows 2-5 of the 2 sub-tables, APEz and FPEz have stronger failure detection potential than APEavg and FPEavg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Are the uncertainty scores extracted in the maneuver classification stage applicable to the trajectory prediction stage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Theoretically, the uncertainty scores obtained in the maneuver classification stage represent the confidence of the model in the current scene, so it may be suitable for failure detection in the trajectory prediction stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We conduct some experiments to explore this question, the results are recorded in rows 6-9 of the two sub-tables of TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Compared with the above trajectory uncertainty scores, the uncertainty extracted in the maneuver classification stage has limited potential for detecting high-error trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' One of the possible reasons is that the uncertainty scores calculated directly based on the trajectories imply the consideration of information such as the velocity and acceleration of the object, thus having a greater correlation with the trajectory error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' How is the failure detection generalizing to scenarios with larger distributional shifts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Here, we use the VA dataset to test the model trained based on SinD, results are shown in TABLE V and TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Compared with TABLE I, III, and IV, when faced with larger distributional shifts, while the reduction in the prediction accuracy of the main model leads to a worsening of AUCOC, the decrease in failure detection ability (such as AUROC) is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AUCOC (↓)/IMPROVEMENT RATIO (IR)1 (↑) FOR THE TRAJECTORY PREDICTION STAGE OF GRIP+++ minADEz meanADEz ADEz, avg Optimal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='088 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='330 APEz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='119/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='143/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='139/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='790 APEavg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='136/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='172/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='166/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='677 TE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='170/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='228/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='218/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='464 DE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='170/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='229/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='218/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='462 MI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='169/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='227/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='216/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='470 NMaP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='170/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='228/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='217/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='467 minFDEz meanFDEz FDEz, avg Optimal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='164 Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='686 FPEz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='249/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='301/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='293/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='754 FPEavg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='278/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='358/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='345/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='654 TE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='361/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='493/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='471/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='413 DE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='362/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='494/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='472/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='410 MI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='359/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='489/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='467/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='420 NMaP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='360/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='491/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='497/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='416 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' RESULTS FOR MANEUVER CLASSIFICATION STAGE OF GRIP+++ WITH DEEP ENSEMBLE, WHICH IS TRAINED ON SIND AND TESTES ON VA TE DE MI NMaP AUROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='912 AUCOC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='978 TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AUCOCOPTIMAL/AUCOCUNCERTAINTY (↓)/AUCOCRANDOM/IR(↑) FOR TRAJECTORY PREDICTION STAGE OF GRIP+++ WITH DEEP ENSEMBLE, WHICH IS TRAINED ON SIND AND TESTES ON VA minADEz meanADEz ADEz, avg APEz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='088/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='210/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='445/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='125/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='238/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='565/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='117/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='234/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='550/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='730 minFDEz meanFDEz FDEz, avg FPEz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='158/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='991/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='491/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='243/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='262/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='550/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='228/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='232/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='543/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='686 TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AUCOCOPTIMAL/AUCOCUNCERTAINTY (↓)/AUCOCRANDOM/IR(↑) FOR TRAJECTRON++ ON NUSCENES Single model Deep ensemble (mean)minADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='088/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='167/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='378/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='096/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='160/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='384/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='778 (mean)minFDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='132/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='308/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='689/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='151/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='293/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='702/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='742 (mean)meanADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='322/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='386/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='045/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='339/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='394/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='040/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='922 (mean)meanFDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='608/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='754/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='096/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='637/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='763/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='082/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='913 minminADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='055/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='112/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='234/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='682 meanmaxpADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='181/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='280/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='801/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='841 TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' AUCOCOPTIMAL/AUCOCUNCERTAINTY(↓)/AUCOCRANDOM/IR(↑) FOR PGP ON NUSCENES, UC MEANS UNIFIED CLUSTERING Single model Deep ensemble (mean)minADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='498/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='837/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='945/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='529/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='832/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='945/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='271 (mean)minFDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='623/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='273/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='554/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='747/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='249/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='548/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='373 minminADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='367/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='628/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='708/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='234 meanmaxpADE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='538/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='497/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='115/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='392 minADE (uc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='488/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='797/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='908/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='264 minFDE (uc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='612/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='181/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='466/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='333 How well does uncertainty-based failure detection perform in generative model-based trajectory prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We adopt Trajectron++ combined with deep ensemble to extract multiple uncertainty scores as failure detection references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The results of this investigation are provided in TABLE VII, where minADE/minFDE/meanADE/meanFDE for single model is calculated based on the 10 trajectories 1 IR is calculated by (AUCOCrandom – AUCOCuncertainty)/(AUCOCrandom – AUCOCoptimal), where AUCOCrandom, AUCOCoptimal, and AUCOCuncertainty represent the AUCOC based on the optimal sorting, the random sorting, and the uncertainty scores-based sorting, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' predicted by the single model, and the corresponding uncertainty scores for failure detection are APE/FPE/APE /FPE obtained from the 10 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In contrast, meanminADE/meanminFDE/meanmeanADE/meanmeanFD E/minminADE/meanmaxpADE for deep ensemble are calculated based on 50 trajectories from all 5 ensemble models, where the first operator (mean/min) is for different sub-models and the second operator (mean/min/maxp) is for different maneuvers from each model’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The corresponding uncertainty scores for failure detection are meanAPE/meanFPE/meanAPE/meanFPE/APEall/APEmaxp, where meanAPE/meanFPE are obtained by averaging APE/ FPE from 5 sub-models, APEall is directly calculated from all 50 trajectories, APEmaxp is calculated according to the maximum probability trajectory of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' The results show promising performance of the uncertainty-based failure detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Can the above uncertainty-based failure detection be simply applied to any trajectory prediction algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In addition to the typical deep neural network architecture and modules, existing trajectory prediction algorithms may use various tricks, which may directly affect the uncertainty scores extracted from the output trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We conduct some exploratory experiments with PGP, a high-performance prediction algorithm integrating special tricks including traversal, sampling, and clustering, to analyze the performance of applying the uncertainty scores obtained from the output trajectories for failure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' In addition, we apply deep ensemble to consider model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' From the evaluation results in TABLE VIII, we conclude that the performance of direct uncertainty quantification based on output results is not very outstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Possible reasons include operations such as sampling latent vectors from an unconstrained normal distribution or clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' This result reminds us that it is necessary to improve uncertainty estimation methods and scores according to the prediction algorithms’ characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' For example, we propose a framework for unified clustering based on the outputs of all sub-models of the deep ensemble, the results in the last two rows of TABLE VIII show some improvement over the original model in trajectory prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' CONCLUSION In this work, we propose a framework to detect motion prediction failures from the uncertainty perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' We divide motion prediction tasks into two stages, maneuver classification and maneuver-based trajectory prediction, and formulate corresponding uncertainty scores for failure detection, where motion uncertainty and model uncertainty are both discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Our experiments cover the comparison of different prediction tasks, multiple prediction algorithms, different uncertainty estimation methods, and various uncertainty scores, Finally, we observe that uncertainty quantification is promising for failure detection for motion prediction, with the potential to generalize to environments with larger distributional shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' However, it is also necessary to conduct targeted discussions and designs for different prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Our future work will focus on the integration of the proposed method with safety decision making for autonomous driving, and its implementation and validation on physical vehicle platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Jain, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Del Pero, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Grimmett, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ondruska, “Autonomy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='0: Why is self-driving always 5 years away?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' arXiv, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 09, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='08142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Plaza, “Collision Between Vehicle Controlled by Developmental Automated Driving System and Pedestrian,” PB2019-101402, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Sifakis and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Harel, “Trustworthy Autonomous System Development,” ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1145/3545178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Dennis and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Fisher, “Verifiable Self-Aware Agent-Based Autonomous Systems,” Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 108, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1011–1026, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2020, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/JPROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2991262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Xing, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Blum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Siegwart, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Cadena, “See Yourself in Others: Attending Multiple Tasks for Own Failure Detection,” in 2022 International Conference on Robotics and Automation (ICRA), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 8409–8416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/ICRA46639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='9812310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Wan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Van Der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='00EX373), 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 153–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/ASSPCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='882463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Cosgun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “Towards full automated drive in urban environments: A demonstration in GoMentum Station, California,” in 2017 IEEE Intelligent Vehicles Symposium (IV), Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1811–1818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/IVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='7995969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Alahi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Goel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ramanathan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Robicquet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Fei-Fei, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Savarese, “Social LSTM: Human Trajectory Prediction in Crowded Spaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 961–971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “VectorNet: Encoding HD Maps and Agent Dynamics From Vectorized Representation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 11525–11533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Sun, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Zhao, “DenseTNT: End-to-End Trajectory Prediction From Dense Goal Sets,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 15303–15312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Mozaffari, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Al-Jarrah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Dianati, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Jennings, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Mouzakitis, “Deep Learning-Based Vehicle Behavior Prediction for Autonomous Driving Applications: A Review,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 33–47, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/TITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='3012034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ying, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Chuah, “GRIP++: Enhanced Graph-based Interaction-aware Trajectory Prediction for Autonomous Driving.” arXiv, May 19, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='07792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Mo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Xing, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Lv, “Multi-Agent Trajectory Prediction With Heterogeneous Edge-Enhanced Graph Attention Network,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 9554–9567, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/TITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='3146300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Djuric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “Uncertainty-aware Short-term Motion Prediction of Traffic Actors for Autonomous Driving,” in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2095–2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Deo and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Trivedi, “Convolutional Social Pooling for Vehicle Trajectory Prediction,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1468–1476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [16] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Deo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Wolff, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Beijbom, “Multimodal Trajectory Prediction Conditioned on Lane-Graph Traversals,” in Proceedings of the 5th Conference on Robot Learning, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 203–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “Multimodal Trajectory Predictions for Autonomous Driving using Deep Convolutional Networks,” in 2019 International Conference on Robotics and Automation (ICRA), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2090–2096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/ICRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='8793868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Salzmann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ivanovic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Chakravarty, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Pavone, “Trajectron++: Dynamically-Feasible Trajectory Forecasting with Heterogeneous Data,” in Computer Vision – ECCV 2020, Cham, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 683–700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1007/978-3-030-58523-5_40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Hsieh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Shih, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Chen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Tsung, “Trajectory Prediction at Unsignalized Intersections using Social Conditional Generative Adversarial Network,” in 2021 IEEE International Intelligent Transportation Systems Conference (ITSC), Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 844–851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/ITSC48978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='9564441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gawlikowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “A Survey of Uncertainty in Deep Neural Networks.” arXiv, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 18, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='03342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kendall and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gal, “What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=',” in Advances in Neural Information Processing Systems, 2017, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Louizos and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Welling, “Multiplicative Normalizing Flows for Variational Bayesian Neural Networks,” in Proceedings of the 34th International Conference on Machine Learning, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2218–2227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Blundell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Cornebise, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kavukcuoglu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Wierstra, “Weight Uncertainty in Neural Network,” in Proceedings of the 32nd International Conference on Machine Learning, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1613–1622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gal and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ghahramani, “Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning,” in Proceedings of The 33rd International Conference on Machine Learning, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 1050–1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Hron, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kendall, “Concrete Dropout,” in Advances in Neural Information Processing Systems, 2017, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [26] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Lakshminarayanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Pritzel, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Blundell, “Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles,” in Advances in Neural Information Processing Systems, 2017, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Wen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Tran, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ba, “BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning.” arXiv, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 19, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='06715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Wenzel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Snoek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Tran, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Jenatton, “Hyperparameter Ensembles for Robustness and Uncertainty Quantification,” in Advances in Neural Information Processing Systems, 2020, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 6514–6527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Sensoy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kaplan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kandemir, “Evidential deep learning to quantify classification uncertainty,” in Advances in neural information processing systems, 2018, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Hendrycks, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Mazeika, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Dietterich, “Deep Anomaly Detection with Outlier Exposure.” arXiv, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 28, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='04606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kuhn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Hofbauer, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Petrovic, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Steinbach, “Pixel-Wise Failure Prediction For Semantic Video Segmentation,” in 2021 IEEE International Conference on Image Processing (ICIP), Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 614–618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/ICIP42928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='9506552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [32] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Rahman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Sünderhauf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Corke, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Dayoub, “FSNet: A Failure Detection Framework for Semantic Segmentation,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 3030–3037, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='3143219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Lis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Nakka, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Fua, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Salzmann, “Detecting the Unexpected via Image Resynthesis,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 2152–2161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Haldimann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Blum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Siegwart, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Cadena, “This is not what I imagined: Error Detection for Semantic Segmentation through Visual Dissimilarity.” arXiv, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 02, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='00676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [35] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Deecke, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Vandermeulen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Ruff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Mandt, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Kloft, “Image Anomaly Detection with Generative Adversarial Networks,” in Machine Learning and Knowledge Discovery in Databases, Cham, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 3–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1007/978-3-030-10925-7_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [36] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Hendrycks and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Gimpel, “A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks,” presented at the International Conference on Learning Representations, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Feng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Harakeh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Waslander, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Dietmayer, “A Review and Comparative Study on Probabilistic Object Detection in Autonomous Driving,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 9961–9980, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1109/TITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='3096854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [38] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Maddox, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Izmailov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Garipov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Vetrov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Wilson, “A Simple Baseline for Bayesian Uncertainty in Deep Learning,” in Advances in Neural Information Processing Systems, 2019, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [39] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “SIND: A Drone Dataset at Signalized Intersection in China.” arXiv, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 06, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='02297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [40] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “INTERACTION Dataset: An INTERnational, Adversarial and Cooperative moTION Dataset in Interactive Driving Scenarios with Semantic Maps.” arXiv, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 30, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content='03088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' [41] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' Caesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=', “nuScenes: A Multimodal Dataset for Autonomous Driving,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} +page_content=' 11621–11631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE3T4oBgHgl3EQfRgki/content/2301.04421v1.pdf'} diff --git a/99AzT4oBgHgl3EQfSvs8/content/tmp_files/2301.01236v1.pdf.txt b/99AzT4oBgHgl3EQfSvs8/content/tmp_files/2301.01236v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8b8554b1db02565b1b27a55e9f0db86949a4aea --- /dev/null +++ b/99AzT4oBgHgl3EQfSvs8/content/tmp_files/2301.01236v1.pdf.txt @@ -0,0 +1,684 @@ +� For correspondence: +jens.sjolund@it.uu.se +Funding: This work was partially +supported by the Wallenberg AI, +Autonomous Systems and +Software Program (WASP) funded +by the Knut and Alice Wallenberg +Foundation. +A Tutorial on Parametric +Variational Inference +Jens Sjölund 1 � +1Department of Information Technology, Uppsala University, Sweden +Abstract +Variational inference uses optimization, rather than integration, to approximate the marginal +likelihood, and thereby the posterior, in a Bayesian model. Thanks to advances in computational +scalability made in the last decade, variational inference is now the preferred choice for many +high-dimensional models and large datasets. This tutorial introduces variational inference from +the parametric perspective that dominates these recent developments, in contrast to the +mean-field perspective commonly found in other introductory texts. +Introduction +In Bayesian machine learning and statistics, the central object of interest is the posterior distribu- +tion found by Bayesian inference—combining prior beliefs with observations according to Bayes’ +rule. In simple cases, such as in conjugate models, this can be done exactly. But, general (non- +conjugate) models require approximate inference techniques such as Monte Carlo or variational +inference. These have complementary strengths and weaknesses, hence the most appropriate +choice is application dependent. We focus on variational inference, which is on the one hand not +guaranteed to be asymptotically exact but is on the other hand computationally efficient and scal- +able to high-dimensional models and large datasets. +Notation +We use a single observation variable 풙 to denote both the observed inputs and outputs. Our pri- +mary interest is however in the latent variables 풛. Since we adhere to the Bayesian framework, +the “parameters” of a model (such as the slope and intercept in a linear regression model) that are +assigned priors are actually latent variables. We denote remaining parameters of interest by 휽. +Variational inference +So, why do we need variational inference? First, recall that to infer anything about the latent vari- +ables from our observations, we need the posterior: +푝휽(풛 ∣ 풙) = 푝휽(풙, 풛) +푝휽(풙) . +(1) +The expression in the denominator, 푝휽(풙), is called the marginal likelihood of 풙 because it can be +rewritten as a marginalization over the latent variables: +푝휽(풙) = ∫ 푝휽(풙, 풛) 푑풛. +(2) +The catch is that in practice this integral is often intractable, i.e. not computable in closed form. +Since 풙 are our observations, the marginal likelihood is a (normalizing) constant. Nevertheless, +without knowing this constant the utility of the posterior is limited. Hence the need for approximate +inference. +Sjölund +| +arXiv +| +January 4, 2023 +| +1–9 +arXiv:2301.01236v1 [stat.ML] 3 Jan 2023 + +Key idea +The key idea in variational inference is to replace the intractable marginal likelihood with a tractable +lower bound that we then maximize. Modeling mainly consists of choosing a family  of probability +distribution that are well-behaved yet sufficiently expressive. More specifically, we want there to +be a distribution 푞 ∈ , called the variational posterior, that can be used as a drop-in replacement +for the true posterior. The variational posterior should therefore be “close” to the true posterior +푝휽(풛 ∣ 풙) and at the same time (relatively) easy to find. The search procedure amounts to mathemat- +ical optimization, which is why variational inference is sometimes described as trading a difficult +integration problem for an easier optimization problem. +The evidence lower bound (ELBO) +In variational inference, the distance between the true posterior 푝(풛 ∣ 풙) and the variational poste- +rior 푞(풛) is measured using the Kullback-Leibler (KL) divergence, +KL(푞(풛) ‖ 푝(풛 ∣ 풙)) = − ∫ 푞(풛) log +(푝(풛 ∣ 풙) +푞(풛) +) +푑풛. +(3) +Other distance measures can also be used to make the variational posterior similar to the true pos- +terior, but the KL divergence has a particular benefit: through a neat trick we can simultaneously +estimate the marginal likelihood and circumvent the need to evaluate the posterior in equation 3. +To see how, we first note the two mathematical identities: +∫ 푞(풛) 푑풛 = 1, +(4) +푝(풙) = 푝(풙, 풛) +푝(풛 ∣ 풙) = 푝(풙, 풛) +푞(풛) +(푝(풛 ∣ 풙) +푞(풛) +)−1 +. +(5) +Using these we may rewrite the marginal likelihood as follows: +log 푝(풙) = +(4) log 푝(풙) ⋅ ∫ 푞(풛) 푑풛 = ∫ 푞(풛) log 푝(풙) 푑풛 += +(5) ∫ 푞(풛) log +(푝(풙, 풛) +푞(풛) +) +푑풛 − ∫ 푞(풛) log +(푝(풛 ∣ 풙) +푞(풛) +) +푑풛 += ∫ 푞(풛) log +(푝(풙, 풛) +푞(풛) +) +푑풛 + KL(푞(풛) ‖ 푝(풛 ∣ 풙)). +(6) +Because the KL divergence is always nonnegative, the first term lower bounds the log marginal +likelihood (also known as the evidence) for any 푞, and is therefore known as the evidence lower +bound (ELBO): +ELBO(푞(풛)) = ∫ 푞(풛) log +(푝(풙, 풛) +푞(풛) +) +푑풛 = 피푞(풛) +[log 푝(풙, 풛) − log 푞(풛)] . +(7) +Equation 6 can thus be written more succinctly as +log 푝(풙) = ELBO(푞(풛)) + KL(푞(풛) ‖ 푝(풛 ∣ 풙)). +(8) +For a fixed model 푝(풙, 풛), the (log) evidence is a constant. Hence—recalling that the KL divergence is +nonnegative—we conclude that maximizing the ELBO is equivalent to minimizing the KL divergence. +This is great, because to compute the KL divergence we would have to marginalize over a function +that includes the same intractable posterior that we want to estimate. In contrast, the model only +enters in the ELBO through the joint distribution 푝(풙, 풛), which means that, first, we don’t need to +compute the problematic integral in equation 2 and, second, we can factorize the joint distribution, +e.g., as encoded by a directed graphical model (Wainwright and Jordan, 2008). +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +2 of 9 + +Example 1 +Suppose we have a single observation 푥 from an Exp(휆) likelihood with a Gamma(훼, 훽) prior +on the rate parameter 휆. Assuming that 훼 and 훽 are known, the only latent variable of interest +is 푧 = {휆}. Specifically, +푝(푥 ∣ 휆) = 휆푒−휆푥, +푝(휆) = +훽훼 +Γ(훼)휆훼−1푒−훽휆, +where Γ(훼) is the Gamma function. Since the Gamma distribution is the conjugate prior for 휆, +we know that the posterior is also a Gamma distribution. Invoking Bayes’ rule and disregarding +all factors not including 휆, we find that +푝(휆 ∣ 푥) ∝ 푝(푥 ∣ 휆)푝(휆) ∝ 휆훼푒−휆(훽+푥). +Hence, we identify the posterior as 푝(휆 ∣ 푥) = Gamma(훼 + 1, 훽 + 푥). +But, let’s pretend we don’t know this and instead want to fit a Lognormal(휇, 휎2) distribution to +the posterior using variational inference, i.e. +푞(휆) = +1 +휆휎 +√ +2휋 +exp +( +−(log 휆 − 휇)2 +2휎2 +) +. +From equation 7 we have that +ELBO(푞(휆)) = 피푞(휆) +[log (푝(푥 ∣ 휆)푝(휆)) − log 푞(휆)] += 피푞(휆) +[ +log +( 훽훼 +Γ(훼) +) ++ 훼 log 휆 − 휆(훽 + 푥) + log +√ +2휋 + log 휎 + log 휆 + (log 휆 − 휇)2 +2휎2 +] += log +( +훽훼√ +2휋 +Γ(훼) +) ++ (훼 + 1)피푞(휆) +[log 휆] − (훽 + 푥)피푞(휆) [휆] + log 휎 + +1 +2휎2 피푞(휆) +[(log 휆 − 휇)2] . +The expectation 피푞(휆) [휆] = exp +( +휇 + 휎2 +2 +) +since, by definition, it is the mean of the lognormal +distribution. Furthermore, the change of integration variables 푦 = log 휆, which transforms 푞(휆) +into 푞(푦) =  (휇, 휎2), shows that +피푞(휆) +[log 휆] = 피푞(푦) [푦] = 휇, +피푞(휆) +[(log 휆 − 휇)2] = 피푞(푦) +[(푦 − 휇)2] = 휎2. +The final expression for the ELBO is thus +ELBO(푞(휆)) = log +( +훽훼√ +2휋 +Γ(훼) +) ++ (훼 + 1)휇 − (훽 + 푥)푒휇+ 휎2 +2 + log 휎 + 1 +2. +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +3 of 9 + +Modeling +How, then, do we choose the variational family ? Historically, the dominant approach has been +to assume a particular factorization of the variational posterior, and to use calculus of variations to +search for distributions that match this factorization. This is known as mean-field variational infer- +ence (Blei, Kucukelbir, and McAuliffe, 2017), and is still the approach most-often taught in classes. +However, mean-field variational inference is only applicable to a rather limited set of models. Most +of the successes of VI in the last 10–15 years have instead taken a parametric approach, where the +variational family is parameterized by a highly expressive model such as a deep neural network. +One can then use “standard” optimization techniques to search for the parameters 휽∗ that max- +imize the ELBO. In light of the above, this tutorial focuses exclusively on parametric variational +inference. +In example 1, we indeed took the parametric approach, since the variational posterior was +explicitly parameterized by a Lognormal distribution with parameters 휽 = {휇, 휎}. In example 2, we +take a closer look at the ELBO for a specific instance of this model. +To approximate the true posterior distribution accurately, we want the variational family  to +be as rich as possible so long as we maintain tractability—it is impossible to overfit! However, as +example 3 shows, there is one pitfall to be aware of: 푞(풛) needs to be zero whenever 푝(풛 ∣ 풙) is zero. +Estimating the ELBO +In the examples we’ve seen so far the expectations could be computed in closed form. But that +will rarely be the case in general (non-conjugate) models. We can, however, use a Monte Carlo +estimate to replace the expectation with a sum, +ELBO(푞(풛)) = ∫ 푞(풛) log +(푝(풙, 풛) +푞(풛) +) +푑풛 = 피푞(풛) +[log 푝(풙, 풛) − log 푞(풛)] +≈ 1 +퐿 +퐿 +∑ +푖=1 +(log 푝(풙, 풛(푖)) − log 푞(풛(푖))) . +(9) +The key requirement is that we are able to draw samples 풛(푖) from the variational posterior 푞(풛). But, +as suggested by the previous section, it is not enough to evaluate the ELBO for a given 푞 ∈ —we +want to find the best 푞! Having parameterized the variational posterior 푞휽(풛) with the parameters 휽, +we may rephrase this as finding parameter values that maximize the ELBO. For efficient optimiza- +tion, however, we need to evaluate both the objective function (the ELBO) and its gradient. +Gradient-based optimization of the ELBO +In optimization, it is standard practice to consider minimization problems. (Since a maximization +problem can be transformed into a minimization problem by negating the objective function, this +can be done without loss of generality.) We thus express our optimization problem as: +휽∗ = arg min +휽 +− 피푞휽(풛) +[log 푝(풙, 풛) − log 푞휽(풛)] . +(10) +Applying, for instance, gradient descent to this problem corresponds to the iterations +휽푘+1 = 휽푘 + 휂∇휽피푞휽(풛) +[log 푝(풙, 풛) − log 푞휽(풛)] , +푘 = 0, 1, … +(11) +where the hyperparameter 휂 > 0 is the step size. But this reveals a complication: the gradient +acts on the parameters of the distribution that we compute the expectation over. Consequently, +we cannot simply move the gradient inside the expectation, nor can we use the Monte Carlo trick +to first replace the expectation with samples and then compute the gradient on those. But there +are other, less direct, ways of applying the Monte Carlo idea that do work (incidentally, this turns +gradient descent into stochastic gradient descent). We begin by rewriting the gradient of the ELBO +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +4 of 9 + +Example 2 +To make thing more concrete, we continue with the setting from example 1 and set 훼 = 3, +훽 = 1, and 푥 = 1. The evidence 푝(푥) is the, previously neglected, proportionality constant +relating the posterior and the joint distributions, +푝(푥) = 푝(푥, 휆) +푝(휆 ∣ 푥) = Γ(훼 + 1) +Γ(훼) +훽훼 +(훽 + 푥)훼+1 = +훼훽훼 +(훽 + 푥)훼+1 . +Inserting the numerical values above gives 푝(푥 = 1) = 3∕16. +For simplicity, we fix 휎 = 0.5 in the variational posterior (this corresponds approximately to +the value found by moment matching) and study the effect of changing 휇. +Example 2—figure 1. The fit of a Lognormal(휇, 휎2 = 0.25) variational posterior to a Gamma(4, 2) posterior +for different values of 휇. +Example 2—figure 2. How well the ELBO approximates the log evidence depends on the parameter 휇. +The gap corresponds exactly to the KL divergence, hence maximizing the ELBO is equivalent to minimizing +the KL divergence. +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +5 of 9 + +0.6 +Exact posterior +Variational posterior (μ= 0.4) +0.5 +Variational posterior (μ= 0.6) +0.4 +Variational posterior (μ= 0.8) +0.3 +0.2 +0.1 +0.0 +2 +FM +0 +F51 +61.68 +KL(qμ*() II p( I X=1)) +1.70 +1.72 +1.74 +1.76 +1.78 +Inp(x= 1) +1.B0 +ELBO(μ, = 0.5) +0.40 +0.45 +0.50 +0.55 +090 +0.65 +0.70 +0.75 +0.BO +μExample 3 +Let’s return to Example 1 and see what happens if we try to use an  (휇, 휎2) distribution as the +variational posterior, i.e. +푞(휆) = +1 +√ +2휋휎2 +exp +( +−(휆 − 휇)2 +2휎2 +) +. +Deriving the ELBO as before, we have that +ELBO(푞(휆)) = 피푞(휆) +[log (푝(푥 ∣ 휆)푝(휆)) − log 푞(휆)] += 피푞(휆) +[ +log +( 훽훼 +Γ(훼) +) ++ 훼 log 휆 − 휆(훽 + 푥) + log +√ +2휋 + log 휎 + (휆 − 휇)2 +2휎2 +] += log +( +훽훼√ +2휋 +Γ(훼) +) ++ 훼 피푞(휆) +[log 휆] +⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ +undefined! +−(훽 + 푥) 피푞(휆) [휆] +⏟⏟⏟ +=휇 ++ log 휎 + +1 +2휎2 피푞(휆) +[(휆 − 휇)2] +⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ +=휎2 +The logarithm is only defined for positive values, hence 피푞(휆) +[log 휆] is undefined. This illustrates +an important caveat when choosing the variational distribution: 푞(풛) needs to be zero whenever +푝(풛 ∣ 풙) is zero. +as follows (Ranganath, Gerrish, and Blei, 2014): +∇휽피푞휽(풛) +[log 푝(풙, 풛) − log 푞휽(풛)] = ∇휽 ∫ +(log 푝(풙, 풛) − log 푞휽(풛)) 푞휽(풛) 푑풛 += ∫ +(log 푝(풙, 풛) − log 푞휽(풛)) ∇휽 푞휽(풛) 푑풛 − ∫ +(∇휽 log 푞휽(풛)) 푞휽(풛) 푑풛 +(12) +But the second term in this expression vanishes, +∫ +(∇휽 log 푞휽(풛)) 푞휽(풛) 푑풛 = ∫ +∇휃푞휽(풛) +�� +� +푞휽(풛) �� +� +푞휽(풛) 푑풛 = ∇휃 ∫ 푞휽(풛) 푑풛 +⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ +=1 += 0. +(13) +In conclusion, we have that +∇휽피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] = ∫ +(log 푝(풙, 풛) − log 푞휽(풛)) ∇휽 푞휽(풛)푑풛. +(14) +Sometimes, as in example 4, we can rewrite ∇휽푞휽(풛) (the gradient of the variational posterior) such +that we can directly use a Monte Carlo method to estimate the integral. Later, we will cover two +more general Monte Carlo-based approaches: reparameterization (Kingma and Welling, 2014) and +black-box variational inference (Ranganath, Gerrish, and Blei, 2014). +Reparameterization +The “reparameterization trick” was popularized in the work introducing the variational autoencoder +(Kingma and Welling, 2014) but the general principle has a much longer history (Devroye, 1996). +The idea is to decouple the source of randomness from the parameters by cleverly reformulating +the random variable 풛 ∼ 푞휽(풛) as a parameterized transformation 푧 = 푔휽(휖) of another random +variable 휖 ∼ 푝(휖) that is easy to sample. Effectively, this moves the randomness “outside” the model +and makes it possible to move the gradient inside the expectation, as shown in the example below. +The reparameterization trick is valid if and only if 푔(휖, 휽) is a continuous function of 휽 for all 휖 +(Schulman et al., 2015). Further, it works in the same way as in the example above also for expec- +tations 피푞휽(풛) [푓(풛)] where 푓(풛) is a general nonlinear function of 풛, +∇휽피푞휽(풛) [푓(풛)] = ∇휽피푝(휖) +[푓(푔휽(휖))] = 피푝(휖) +[∇휽푓(푔휽(휖))] . +(15) +By setting 푓(풛) = log 푝(풙, 풛) − log 푞휽(풛) we retrieve the ELBO as a special case. +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +6 of 9 + +Example 4 +Consider a univariate Normal variational posterior parameterized by the mean 휇 and standard +deviation 휎, i.e. +푞휽(푧) = +1 +√ +2휋휎2 +exp +( +−(푧 − 휇)2 +2휎2 +) +, +휽 = {휇, 휎}. +After some algebraic manipulations, the partial derivatives can be written as: +휕푞휽 +휕휇 = 푧 − 휇 +휎2 +⋅ 푞휽(푧), +휕푞휽 +휕휎 = 1 +휎 +((푧 − 휇)2 +휎2 +− 1 +) +⋅ 푞휽(푧). +Note that 푞휽(푧) appears in both of these expressions. By inserting the above in equation 14, +we thus arrive at an expectation that we can replace with a Monte Carlo estimate: +∇휽피푞휽(푧) [log 푝(풙, 푧) − log 푞휽(푧)] += ∫ +(log 푝(풙, 푧) − log 푞휽(푧)) ( +푧−휇 +휎2 , 1 +휎 +( +(푧−휇)2 +휎2 +− 1 +))⊤ +푞휽(푧)푑푧 +≈ 1 +퐿 +퐿 +∑ +푖=1 +(log 푝(풙, 푧(푖)) − log 푞휽(푧(푖))) ( +푧(푖)−휇 +휎2 , 1 +휎 +( +(푧(푖)−휇)2 +휎2 +− 1 +))⊤ +, +where 푧(푖) ∼ 푞휽(푧). +Example 5 +Suppose the variational posterior is a univariate Normal distribution parameterized by the +mean 휇 and standard deviation 휎, i.e. 푞휽(푧) =  (푧; 휽) where 휽 = {휇, 휎}. This can be reparame- +terized as 푧 = 푔휽(휖) = 휇 + 휎 ⋅ 휖 where 휖 ∼  (0, 1). +Let’s consider the effect this has on the expectation 피푞휽(푧) +[log 푧]. +(i) Original expression: +피푞휽(푧) +[log 푧] = +1 +√ +2휋휎2 ∫ log 푧 exp +( +−(푧 − 휇)2 +2휎2 +) +푑푧, +∇휽피푞휽(푧) +[log 푧] = ∫ log 푧 ∇휽푞휽(푧)푑푧 += 피푞휽(푧) +[ +log 푧 ⋅ +(( +푧−휇 +휎2 +) +, 1 +휎 +( +(푧−휇)2 +휎2 +− 1 +))⊤] +where we used the expression for ∇휽푞휽(푧) from example 4. +(ii) Reparameterized expression: +피푞휽(푧) +[log 푧] ||||푧=휇+휎휖 += +1 +√ +2휋휎2 ∫ log(휇 + 휎휖) exp +( +−(휇 + 휎휖 − 휇)2 +2휎2 +) +휎 푑휖 += +1 +√ +2휋 ∫ log(휇 + 휎휖) exp +( +−휖2 +2 +) +푑휖 = 피푝(휖) +[log(휇 + 휎휖)] , +∇휽피푝(휖) +[log(휇 + 휎휖)] = 피푝(휖) +[∇휽 log(휇 + 휎휖)] += 피푝(휖) +[( +1 +휇+휎휖 , +휖 +휇+휎휖 +)⊤] +. +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +7 of 9 + +Amortized variational inference +Many probabilistic models have local latent variables 풛푖 associated with each data point 풙푖. The +simplest case is when the joint distribution factorizes as +푝(풙, 풛) = +푁 +∏ +푖=1 +푝(풙푖 ∣ 풛푖)푝(풛푖). +(16) +Suppose we use a variational posterior that factorizes accordingly, +푞휽(풛) = +푁 +∏ +푖=1 +푞휽푖(풛푖), +(17) +then the ELBO maximization in equation 10 decomposes into a sum of local ELBOs +휽∗ = arg min +휽 +− +푁 +∑ +푖=1 +피푞휽푖 (풛푖) +[log 푝(풙푖 ∣ 풛푖) + log 푝(풛푖) − log 푞휽푖(풛푖)] . +(18) +Since the optimization variables are 휽 = {휽1, … , 휽푁}, large datasets amount to large optimization +problems, which are computationally demanding to solve. This led to the idea of amortized vari- +ational inference (Rezende, Mohamed, and Wierstra, 2014), wherein a machine learning model +(often a neural network) is trained to directly predict the solution 휽∗ of this optimization problem. +Specifically, let Λ휙 denote a neural network parameterized by 휙 that maps individual datapoints +풙푖 to corresponding parameters 휽푖 of the local variational posterior 푞휽푖(풛푖). This model is trained us- +ing the expression in equation 18 as the loss function but replacing 휽푖 = Λ휙(풙푖). Note that even +though the objective function is the same, this is a form of amortized optimization (Amos, 2022) +since we are now using 휙 as the optimization variables instead of 휽. Furthermore, the loss function +is a sum over datapoints, which means that the standard machinery for training neural networks +(stochastic gradient descent etc.) can be applied. In the context of variational autoencoders, the +model Λ휙 is referred to as the encoder, which is accompanied by a, jointly trained, decoder corre- +sponding to the probability distribution 푝(풙 ∣ 풛) (Kingma and Welling, 2019). +Black-box variational inference +The reparameterization trick lets you compute the exact gradient by automatic differentiation, +which is undoubtedly convenient. On the other hand, there are many models in which reparam- +eterization is impossible. In these cases, one can instead estimate the gradient using black-box +variational inference (BBVI) (Ranganath, Gerrish, and Blei, 2014), which is more general yet still +convenient. However, the BBVI estimator suffers from high variance. +BBVI relies on the observation that +∇휽 log 푞휽(풛) = ∇휽푞휽(풛) +푞휽(풛) , +(19) +which is sometimes referred to as the REINFORCE trick (Williams, 1992). This can be used to rewrite +equation 14 as +∇휽피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] = ∫ +(log 푝(풙, 풛) − log 푞휽(풛)) 푞휽(풛)∇휽 log 푞휽(풛) +⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ +=∇휽푞휽(풛) +푑풛 += 피푞휽(풛) [(log 푝(풙, 풛) − log 푞휽(풛)) ∇휽 log 푞휽(풛)] ≈ 1 +퐿 +퐿 +∑ +푖=1 +(log 푝(풙, 풛(푖)) − log 푞휽(풛(푖))) ∇휽 log 푞휽(풛(푖)). +(20) +Since we can often use automatic differentiation to evaluate the score function ∇휽 log 푞휽(풛), it ap- +pears that this reformulation resolves the problem of estimating the gradient of the ELBO from +samples. The catch is, however, that this estimator often has a too high variance to be useful in +practice. Arguably, the key contribution of BBVI was to adapt two variance reduction techniques— +Rao-Blackwellization and control variates—to the estimator in equation 20. Going into detail on +these variance reduction techniques would, however, take us beyond the scope of a tutorial on the +basics of variational inference. We refer the interested reader to the original work by Ranganath, +Gerrish, and Blei (2014). +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +8 of 9 + +Acknowledgments +This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Pro- +gram (WASP) funded by the Knut and Alice Wallenberg Foundation. This preprint was created using +the LaPreprint template (https://github.com/roaldarbol/lapreprint) by Mikkel Roald-Arbøl. +References +Amos, Brandon (2022). “Tutorial on amortized optimization for learning to optimize over continu- +ous domains”. In: arXiv preprint 2202.00665. +Blei, David M, Alp Kucukelbir, and Jon D McAuliffe (2017). “Variational inference: A review for statis- +ticians”. In: Journal of the American statistical Association 112.518, pp. 859–877. +Devroye, Luc (1996). “Random variate generation in one line of code”. In: Proceedings Winter Simu- +lation Conference. IEEE, pp. 265–272. +Kingma, Diederik P and Max Welling (2014). “Auto-Encoding Variational Bayes”. In: 2nd International +Conference on Learning Representations. +— (2019). “An introduction to variational autoencoders”. In: Foundations and Trends® in Machine +Learning 12.4, pp. 307–392. +Ranganath, Rajesh, Sean Gerrish, and David M Blei (2014). “Black box variational inference”. In: +Artificial intelligence and statistics. PMLR, pp. 814–822. +Rezende, Danilo Jimenez, Shakir Mohamed, and Daan Wierstra (2014). “Stochastic backpropagation +and approximate inference in deep generative models”. In: International conference on machine +learning. PMLR, pp. 1278–1286. +Schulman, John et al. (2015). “Gradient estimation using stochastic computation graphs”. In: Ad- +vances in Neural Information Processing Systems 28. +Wainwright, Martin J and Michael I Jordan (2008). “Graphical models, exponential families, and vari- +ational inference”. In: Foundations and Trends in Machine Learning 1.1–2, pp. 1–305. +Williams, Ronald J (1992). “Simple statistical gradient-following algorithms for connectionist rein- +forcement learning”. In: Machine learning 8.3, pp. 229–256. +Sjölund +| +A Tutorial on Parametric Variational Inference +arXiv +| +9 of 9 + diff --git a/99AzT4oBgHgl3EQfSvs8/content/tmp_files/load_file.txt b/99AzT4oBgHgl3EQfSvs8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fed8b0c64b5f88fed055a836da97032749e90621 --- /dev/null +++ b/99AzT4oBgHgl3EQfSvs8/content/tmp_files/load_file.txt @@ -0,0 +1,223 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf,len=222 +page_content='� For correspondence: jens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='sjolund@it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='se Funding: This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' A Tutorial on Parametric Variational Inference Jens Sjölund 1 � 1Department of Information Technology, Uppsala University, Sweden Abstract Variational inference uses optimization, rather than integration, to approximate the marginal likelihood, and thereby the posterior, in a Bayesian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Thanks to advances in computational scalability made in the last decade, variational inference is now the preferred choice for many high-dimensional models and large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This tutorial introduces variational inference from the parametric perspective that dominates these recent developments, in contrast to the mean-field perspective commonly found in other introductory texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Introduction In Bayesian machine learning and statistics, the central object of interest is the posterior distribu- tion found by Bayesian inference—combining prior beliefs with observations according to Bayes’ rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In simple cases, such as in conjugate models, this can be done exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' But, general (non- conjugate) models require approximate inference techniques such as Monte Carlo or variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' These have complementary strengths and weaknesses, hence the most appropriate choice is application dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' We focus on variational inference, which is on the one hand not guaranteed to be asymptotically exact but is on the other hand computationally efficient and scal- able to high-dimensional models and large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Notation We use a single observation variable 풙 to denote both the observed inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Our pri- mary interest is however in the latent variables 풛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Since we adhere to the Bayesian framework, the “parameters” of a model (such as the slope and intercept in a linear regression model) that are assigned priors are actually latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' We denote remaining parameters of interest by 휽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Variational inference So, why do we need variational inference?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' First, recall that to infer anything about the latent vari- ables from our observations, we need the posterior: 푝휽(풛 ∣ 풙) = 푝휽(풙, 풛) 푝휽(풙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (1) The expression in the denominator, 푝휽(풙), is called the marginal likelihood of 풙 because it can be rewritten as a marginalization over the latent variables: 푝휽(풙) = ∫ 푝휽(풙, 풛) 푑풛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (2) The catch is that in practice this integral is often intractable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' not computable in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Since 풙 are our observations, the marginal likelihood is a (normalizing) constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Nevertheless, without knowing this constant the utility of the posterior is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Hence the need for approximate inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | arXiv | January 4, 2023 | 1–9 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='01236v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='ML] 3 Jan 2023 Key idea The key idea in variational inference is to replace the intractable marginal likelihood with a tractable lower bound that we then maximize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Modeling mainly consists of choosing a family \ue23d of probability distribution that are well-behaved yet sufficiently expressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' More specifically, we want there to be a distribution 푞 ∈ \ue23d, called the variational posterior, that can be used as a drop-in replacement for the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The variational posterior should therefore be “close” to the true posterior 푝휽(풛 ∣ 풙) and at the same time (relatively) easy to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The search procedure amounts to mathemat- ical optimization, which is why variational inference is sometimes described as trading a difficult integration problem for an easier optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The evidence lower bound (ELBO) In variational inference, the distance between the true posterior 푝(풛 ∣ 풙) and the variational poste- rior 푞(풛) is measured using the Kullback-Leibler (KL) divergence, KL(푞(풛) ‖ 푝(풛 ∣ 풙)) = − ∫ 푞(풛) log (푝(풛 ∣ 풙) 푞(풛) ) 푑풛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (3) Other distance measures can also be used to make the variational posterior similar to the true pos- terior, but the KL divergence has a particular benefit: through a neat trick we can simultaneously estimate the marginal likelihood and circumvent the need to evaluate the posterior in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' To see how, we first note the two mathematical identities: ∫ 푞(풛) 푑풛 = 1, (4) 푝(풙) = 푝(풙, 풛) 푝(풛 ∣ 풙) = 푝(풙, 풛) 푞(풛) (푝(풛 ∣ 풙) 푞(풛) )−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (5) Using these we may rewrite the marginal likelihood as follows: log 푝(풙) = (4) log 푝(풙) ⋅ ∫ 푞(풛) 푑풛 = ∫ 푞(풛) log 푝(풙) 푑풛 = (5) ∫ 푞(풛) log (푝(풙, 풛) 푞(풛) ) 푑풛 − ∫ 푞(풛) log (푝(풛 ∣ 풙) 푞(풛) ) 푑풛 = ∫ 푞(풛) log (푝(풙, 풛) 푞(풛) ) 푑풛 + KL(푞(풛) ‖ 푝(풛 ∣ 풙)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (6) Because the KL divergence is always nonnegative, the first term lower bounds the log marginal likelihood (also known as the evidence) for any 푞, and is therefore known as the evidence lower bound (ELBO): ELBO(푞(풛)) = ∫ 푞(풛) log (푝(풙, 풛) 푞(풛) ) 푑풛 = 피푞(풛) [log 푝(풙, 풛) − log 푞(풛)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (7) Equation 6 can thus be written more succinctly as log 푝(풙) = ELBO(푞(풛)) + KL(푞(풛) ‖ 푝(풛 ∣ 풙)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (8) For a fixed model 푝(풙, 풛), the (log) evidence is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Hence—recalling that the KL divergence is nonnegative—we conclude that maximizing the ELBO is equivalent to minimizing the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This is great, because to compute the KL divergence we would have to marginalize over a function that includes the same intractable posterior that we want to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In contrast, the model only enters in the ELBO through the joint distribution 푝(풙, 풛), which means that, first, we don’t need to compute the problematic integral in equation 2 and, second, we can factorize the joint distribution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=', as encoded by a directed graphical model (Wainwright and Jordan, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 2 of 9 Example 1 Suppose we have a single observation 푥 from an Exp(휆) likelihood with a Gamma(훼, 훽) prior on the rate parameter 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Assuming that 훼 and 훽 are known, the only latent variable of interest is 푧 = {휆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Specifically, 푝(푥 ∣ 휆) = 휆푒−휆푥, 푝(휆) = 훽훼 Γ(훼)휆훼−1푒−훽휆, where Γ(훼) is the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Since the Gamma distribution is the conjugate prior for 휆, we know that the posterior is also a Gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Invoking Bayes’ rule and disregarding all factors not including 휆, we find that 푝(휆 ∣ 푥) ∝ 푝(푥 ∣ 휆)푝(휆) ∝ 휆훼푒−휆(훽+푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Hence, we identify the posterior as 푝(휆 ∣ 푥) = Gamma(훼 + 1, 훽 + 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' But, let’s pretend we don’t know this and instead want to fit a Lognormal(휇, 휎2) distribution to the posterior using variational inference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 푞(휆) = 1 휆휎 √ 2휋 exp ( −(log 휆 − 휇)2 2휎2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' From equation 7 we have that ELBO(푞(휆)) = 피푞(휆) [log (푝(푥 ∣ 휆)푝(휆)) − log 푞(휆)] = 피푞(휆) [ log ( 훽훼 Γ(훼) ) + 훼 log 휆 − 휆(훽 + 푥) + log √ 2휋 + log 휎 + log 휆 + (log 휆 − 휇)2 2휎2 ] = log ( 훽훼√ 2휋 Γ(훼) ) + (훼 + 1)피푞(휆) [log 휆] − (훽 + 푥)피푞(휆) [휆] + log 휎 + 1 2휎2 피푞(휆) [(log 휆 − 휇)2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The expectation 피푞(휆) [휆] = exp ( 휇 + 휎2 2 ) since, by definition, it is the mean of the lognormal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Furthermore, the change of integration variables 푦 = log 휆, which transforms 푞(휆) into 푞(푦) = \ue23a (휇, 휎2), shows that 피푞(휆) [log 휆] = 피푞(푦) [푦] = 휇, 피푞(휆) [(log 휆 − 휇)2] = 피푞(푦) [(푦 − 휇)2] = 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The final expression for the ELBO is thus ELBO(푞(휆)) = log ( 훽훼√ 2휋 Γ(훼) ) + (훼 + 1)휇 − (훽 + 푥)푒휇+ 휎2 2 + log 휎 + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 3 of 9 Modeling How, then, do we choose the variational family \ue23d?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Historically, the dominant approach has been to assume a particular factorization of the variational posterior, and to use calculus of variations to search for distributions that match this factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This is known as mean-field variational infer- ence (Blei, Kucukelbir, and McAuliffe, 2017), and is still the approach most-often taught in classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' However, mean-field variational inference is only applicable to a rather limited set of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Most of the successes of VI in the last 10–15 years have instead taken a parametric approach, where the variational family is parameterized by a highly expressive model such as a deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' One can then use “standard” optimization techniques to search for the parameters 휽∗ that max- imize the ELBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In light of the above, this tutorial focuses exclusively on parametric variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In example 1, we indeed took the parametric approach, since the variational posterior was explicitly parameterized by a Lognormal distribution with parameters 휽 = {휇, 휎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In example 2, we take a closer look at the ELBO for a specific instance of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' To approximate the true posterior distribution accurately, we want the variational family \ue23d to be as rich as possible so long as we maintain tractability—it is impossible to overfit!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' However, as example 3 shows, there is one pitfall to be aware of: 푞(풛) needs to be zero whenever 푝(풛 ∣ 풙) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Estimating the ELBO In the examples we’ve seen so far the expectations could be computed in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' But that will rarely be the case in general (non-conjugate) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' We can, however, use a Monte Carlo estimate to replace the expectation with a sum, ELBO(푞(풛)) = ∫ 푞(풛) log (푝(풙, 풛) 푞(풛) ) 푑풛 = 피푞(풛) [log 푝(풙, 풛) − log 푞(풛)] ≈ 1 퐿 퐿 ∑ 푖=1 (log 푝(풙, 풛(푖)) − log 푞(풛(푖))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (9) The key requirement is that we are able to draw samples 풛(푖) from the variational posterior 푞(풛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' But, as suggested by the previous section, it is not enough to evaluate the ELBO for a given 푞 ∈ \ue23d—we want to find the best 푞!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Having parameterized the variational posterior 푞휽(풛) with the parameters 휽, we may rephrase this as finding parameter values that maximize the ELBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' For efficient optimiza- tion, however, we need to evaluate both the objective function (the ELBO) and its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Gradient-based optimization of the ELBO In optimization, it is standard practice to consider minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (Since a maximization problem can be transformed into a minimization problem by negating the objective function, this can be done without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=') We thus express our optimization problem as: 휽∗ = arg min 휽 − 피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (10) Applying, for instance, gradient descent to this problem corresponds to the iterations 휽푘+1 = 휽푘 + 휂∇휽피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] , 푘 = 0, 1, … (11) where the hyperparameter 휂 > 0 is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' But this reveals a complication: the gradient acts on the parameters of the distribution that we compute the expectation over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Consequently, we cannot simply move the gradient inside the expectation, nor can we use the Monte Carlo trick to first replace the expectation with samples and then compute the gradient on those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' But there are other, less direct, ways of applying the Monte Carlo idea that do work (incidentally, this turns gradient descent into stochastic gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' We begin by rewriting the gradient of the ELBO Sjölund | A Tutorial on Parametric Variational Inference arXiv | 4 of 9 Example 2 To make thing more concrete, we continue with the setting from example 1 and set 훼 = 3, 훽 = 1, and 푥 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The evidence 푝(푥) is the, previously neglected, proportionality constant relating the posterior and the joint distributions, 푝(푥) = 푝(푥, 휆) 푝(휆 ∣ 푥) = Γ(훼 + 1) Γ(훼) 훽훼 (훽 + 푥)훼+1 = 훼훽훼 (훽 + 푥)훼+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Inserting the numerical values above gives 푝(푥 = 1) = 3∕16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' For simplicity, we fix 휎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='5 in the variational posterior (this corresponds approximately to the value found by moment matching) and study the effect of changing 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Example 2—figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The fit of a Lognormal(휇, 휎2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='25) variational posterior to a Gamma(4, 2) posterior for different values of 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Example 2—figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' How well the ELBO approximates the log evidence depends on the parameter 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The gap corresponds exactly to the KL divergence, hence maximizing the ELBO is equivalent to minimizing the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 5 of 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='6 Exact posterior Variational posterior (μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='5 Variational posterior (μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='4 Variational posterior (μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='0 2 FM 0 F51 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='68 KL(qμ*() II p( I X=1)) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='78 Inp(x= 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='B0 ELBO(μ, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='55 090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='BO μExample 3 Let’s return to Example 1 and see what happens if we try to use an \ue23a (휇, 휎2) distribution as the variational posterior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 푞(휆) = 1 √ 2휋휎2 exp ( −(휆 − 휇)2 2휎2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Deriving the ELBO as before, we have that ELBO(푞(휆)) = 피푞(휆) [log (푝(푥 ∣ 휆)푝(휆)) − log 푞(휆)] = 피푞(휆) [ log ( 훽훼 Γ(훼) ) + 훼 log 휆 − 휆(훽 + 푥) + log √ 2휋 + log 휎 + (휆 − 휇)2 2휎2 ] = log ( 훽훼√ 2휋 Γ(훼) ) + 훼 피푞(휆) [log 휆] ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ undefined!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' −(훽 + 푥) 피푞(휆) [휆] ⏟⏟⏟ =휇 + log 휎 + 1 2휎2 피푞(휆) [(휆 − 휇)2] ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ =휎2 The logarithm is only defined for positive values, hence 피푞(휆) [log 휆] is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This illustrates an important caveat when choosing the variational distribution: 푞(풛) needs to be zero whenever 푝(풛 ∣ 풙) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' as follows (Ranganath, Gerrish, and Blei, 2014): ∇휽피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] = ∇휽 ∫ (log 푝(풙, 풛) − log 푞휽(풛)) 푞휽(풛) 푑풛 = ∫ (log 푝(풙, 풛) − log 푞휽(풛)) ∇휽 푞휽(풛) 푑풛 − ∫ (∇휽 log 푞휽(풛)) 푞휽(풛) 푑풛 (12) But the second term in this expression vanishes, ∫ (∇휽 log 푞휽(풛)) 푞휽(풛) 푑풛 = ∫ ∇휃푞휽(풛) �� � 푞휽(풛) �� � 푞휽(풛) 푑풛 = ∇휃 ∫ 푞휽(풛) 푑풛 ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ =1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (13) In conclusion, we have that ∇휽피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] = ∫ (log 푝(풙, 풛) − log 푞휽(풛)) ∇휽 푞휽(풛)푑풛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (14) Sometimes, as in example 4, we can rewrite ∇휽푞휽(풛) (the gradient of the variational posterior) such that we can directly use a Monte Carlo method to estimate the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Later, we will cover two more general Monte Carlo-based approaches: reparameterization (Kingma and Welling, 2014) and black-box variational inference (Ranganath, Gerrish, and Blei, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Reparameterization The “reparameterization trick” was popularized in the work introducing the variational autoencoder (Kingma and Welling, 2014) but the general principle has a much longer history (Devroye, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The idea is to decouple the source of randomness from the parameters by cleverly reformulating the random variable 풛 ∼ 푞휽(풛) as a parameterized transformation 푧 = 푔휽(휖) of another random variable 휖 ∼ 푝(휖) that is easy to sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Effectively, this moves the randomness “outside” the model and makes it possible to move the gradient inside the expectation, as shown in the example below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The reparameterization trick is valid if and only if 푔(휖, 휽) is a continuous function of 휽 for all 휖 (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Further, it works in the same way as in the example above also for expec- tations 피푞휽(풛) [푓(풛)] where 푓(풛) is a general nonlinear function of 풛, ∇휽피푞휽(풛) [푓(풛)] = ∇휽피푝(휖) [푓(푔휽(휖))] = 피푝(휖) [∇휽푓(푔휽(휖))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (15) By setting 푓(풛) = log 푝(풙, 풛) − log 푞휽(풛) we retrieve the ELBO as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 6 of 9 Example 4 Consider a univariate Normal variational posterior parameterized by the mean 휇 and standard deviation 휎, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 푞휽(푧) = 1 √ 2휋휎2 exp ( −(푧 − 휇)2 2휎2 ) , 휽 = {휇, 휎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' After some algebraic manipulations, the partial derivatives can be written as: 휕푞휽 휕휇 = 푧 − 휇 휎2 ⋅ 푞휽(푧), 휕푞휽 휕휎 = 1 휎 ((푧 − 휇)2 휎2 − 1 ) ⋅ 푞휽(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Note that 푞휽(푧) appears in both of these expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' By inserting the above in equation 14, we thus arrive at an expectation that we can replace with a Monte Carlo estimate: ∇휽피푞휽(푧) [log 푝(풙, 푧) − log 푞휽(푧)] = ∫ (log 푝(풙, 푧) − log 푞휽(푧)) ( 푧−휇 휎2 , 1 휎 ( (푧−휇)2 휎2 − 1 ))⊤ 푞휽(푧)푑푧 ≈ 1 퐿 퐿 ∑ 푖=1 (log 푝(풙, 푧(푖)) − log 푞휽(푧(푖))) ( 푧(푖)−휇 휎2 , 1 휎 ( (푧(푖)−휇)2 휎2 − 1 ))⊤ , where 푧(푖) ∼ 푞휽(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Example 5 Suppose the variational posterior is a univariate Normal distribution parameterized by the mean 휇 and standard deviation 휎, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 푞휽(푧) = \ue23a (푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 휽) where 휽 = {휇, 휎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This can be reparame- terized as 푧 = 푔휽(휖) = 휇 + 휎 ⋅ 휖 where 휖 ∼ \ue23a (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Let’s consider the effect this has on the expectation 피푞휽(푧) [log 푧].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (i) Original expression: 피푞휽(푧) [log 푧] = 1 √ 2휋휎2 ∫ log 푧 exp ( −(푧 − 휇)2 2휎2 ) 푑푧, ∇휽피푞휽(푧) [log 푧] = ∫ log 푧 ∇휽푞휽(푧)푑푧 = 피푞휽(푧) [ log 푧 ⋅ (( 푧−휇 휎2 ) , 1 휎 ( (푧−휇)2 휎2 − 1 ))⊤] where we used the expression for ∇휽푞휽(푧) from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (ii) Reparameterized expression: 피푞휽(푧) [log 푧] ||||푧=휇+휎휖 = 1 √ 2휋휎2 ∫ log(휇 + 휎휖) exp ( −(휇 + 휎휖 − 휇)2 2휎2 ) 휎 푑휖 = 1 √ 2휋 ∫ log(휇 + 휎휖) exp ( −휖2 2 ) 푑휖 = 피푝(휖) [log(휇 + 휎휖)] , ∇휽피푝(휖) [log(휇 + 휎휖)] = 피푝(휖) [∇휽 log(휇 + 휎휖)] = 피푝(휖) [( 1 휇+휎휖 , 휖 휇+휎휖 )⊤] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 7 of 9 Amortized variational inference Many probabilistic models have local latent variables 풛푖 associated with each data point 풙푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The simplest case is when the joint distribution factorizes as 푝(풙, 풛) = 푁 ∏ 푖=1 푝(풙푖 ∣ 풛푖)푝(풛푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (16) Suppose we use a variational posterior that factorizes accordingly, 푞휽(풛) = 푁 ∏ 푖=1 푞휽푖(풛푖), (17) then the ELBO maximization in equation 10 decomposes into a sum of local ELBOs 휽∗ = arg min 휽 − 푁 ∑ 푖=1 피푞휽푖 (풛푖) [log 푝(풙푖 ∣ 풛푖) + log 푝(풛푖) − log 푞휽푖(풛푖)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (18) Since the optimization variables are 휽 = {휽1, … , 휽푁}, large datasets amount to large optimization problems, which are computationally demanding to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This led to the idea of amortized vari- ational inference (Rezende, Mohamed, and Wierstra, 2014), wherein a machine learning model (often a neural network) is trained to directly predict the solution 휽∗ of this optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Specifically, let Λ휙 denote a neural network parameterized by 휙 that maps individual datapoints 풙푖 to corresponding parameters 휽푖 of the local variational posterior 푞휽푖(풛푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This model is trained us- ing the expression in equation 18 as the loss function but replacing 휽푖 = Λ휙(풙푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Note that even though the objective function is the same, this is a form of amortized optimization (Amos, 2022) since we are now using 휙 as the optimization variables instead of 휽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Furthermore, the loss function is a sum over datapoints, which means that the standard machinery for training neural networks (stochastic gradient descent etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=') can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In the context of variational autoencoders, the model Λ휙 is referred to as the encoder, which is accompanied by a, jointly trained, decoder corre- sponding to the probability distribution 푝(풙 ∣ 풛) (Kingma and Welling, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Black-box variational inference The reparameterization trick lets you compute the exact gradient by automatic differentiation, which is undoubtedly convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' On the other hand, there are many models in which reparam- eterization is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In these cases, one can instead estimate the gradient using black-box variational inference (BBVI) (Ranganath, Gerrish, and Blei, 2014), which is more general yet still convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' However, the BBVI estimator suffers from high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' BBVI relies on the observation that ∇휽 log 푞휽(풛) = ∇휽푞휽(풛) 푞휽(풛) , (19) which is sometimes referred to as the REINFORCE trick (Williams, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This can be used to rewrite equation 14 as ∇휽피푞휽(풛) [log 푝(풙, 풛) − log 푞휽(풛)] = ∫ (log 푝(풙, 풛) − log 푞휽(풛)) 푞휽(풛)∇휽 log 푞휽(풛) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ =∇휽푞휽(풛) 푑풛 = 피푞휽(풛) [(log 푝(풙, 풛) − log 푞휽(풛)) ∇휽 log 푞휽(풛)] ≈ 1 퐿 퐿 ∑ 푖=1 (log 푝(풙, 풛(푖)) − log 푞휽(풛(푖))) ∇휽 log 푞휽(풛(푖)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (20) Since we can often use automatic differentiation to evaluate the score function ∇휽 log 푞휽(풛), it ap- pears that this reformulation resolves the problem of estimating the gradient of the ELBO from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' The catch is, however, that this estimator often has a too high variance to be useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Arguably, the key contribution of BBVI was to adapt two variance reduction techniques— Rao-Blackwellization and control variates—to the estimator in equation 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Going into detail on these variance reduction techniques would, however, take us beyond the scope of a tutorial on the basics of variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' We refer the interested reader to the original work by Ranganath, Gerrish, and Blei (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 8 of 9 Acknowledgments This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Pro- gram (WASP) funded by the Knut and Alice Wallenberg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' This preprint was created using the LaPreprint template (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='com/roaldarbol/lapreprint) by Mikkel Roald-Arbøl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' References Amos, Brandon (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Tutorial on amortized optimization for learning to optimize over continu- ous domains”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: arXiv preprint 2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='00665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Blei, David M, Alp Kucukelbir, and Jon D McAuliffe (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Variational inference: A review for statis- ticians”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Journal of the American statistical Association 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='518, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 859–877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Devroye, Luc (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Random variate generation in one line of code”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Proceedings Winter Simu- lation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' IEEE, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 265–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Kingma, Diederik P and Max Welling (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Auto-Encoding Variational Bayes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: 2nd International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' — (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “An introduction to variational autoencoders”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Foundations and Trends® in Machine Learning 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 307–392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Ranganath, Rajesh, Sean Gerrish, and David M Blei (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Black box variational inference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Artificial intelligence and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 814–822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Rezende, Danilo Jimenez, Shakir Mohamed, and Daan Wierstra (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Stochastic backpropagation and approximate inference in deep generative models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 1278–1286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Schulman, John et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Gradient estimation using stochastic computation graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Ad- vances in Neural Information Processing Systems 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Wainwright, Martin J and Michael I Jordan (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Graphical models, exponential families, and vari- ational inference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Foundations and Trends in Machine Learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='1–2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 1–305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Williams, Ronald J (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' “Simple statistical gradient-following algorithms for connectionist rein- forcement learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' In: Machine learning 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' 229–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} +page_content=' Sjölund | A Tutorial on Parametric Variational Inference arXiv | 9 of 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfSvs8/content/2301.01236v1.pdf'} diff --git a/9NE1T4oBgHgl3EQfCQIp/content/tmp_files/2301.02861v1.pdf.txt b/9NE1T4oBgHgl3EQfCQIp/content/tmp_files/2301.02861v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9be3f01e98b63920fdd0f002a6e424c6c041713 --- /dev/null +++ b/9NE1T4oBgHgl3EQfCQIp/content/tmp_files/2301.02861v1.pdf.txt @@ -0,0 +1,1231 @@ +arXiv:2301.02861v1 [math.NT] 7 Jan 2023 +IDENTITIES INVOLVING DEGENERATE HARMONIC AND DEGENERATE +HYPERHARMONIC NUMBERS +HYE KYUNG KIM1, DAE SAN KIM2, AND TAEKYUN KIM3,∗ +ABSTRACT. Harmonic numbers have been studied since antiquity, while hyperharmonic numbers +were intoduced by Conway and Guy in 1996. The degenerate harmonic numbers and degenerate +hyperharmonic numbers are their respective degenerate versions. The aim of this paper is to further +investigate some properties, recurrence relations and identities involving the degenerate harmonic +and degenerate hyperharmonic numbers in connection with degenerate Stirling numbers of the first +kind, degenerate Daehee numbers and degenerate derangements. +1. INTRODUCTION +In recent years, various degenerate versions of many special numbers and polynomials have +beem studied and yielded a lot of fascinating and fruitful results (see [5, 6, 7, 8, 9, 10, 11, 12] and +the references therein), which began with Carlitz’s work on the degenerate Bernoulli and degen- +erate Euler numbers (see [2]). It is worthwhile to mention that these explorations for degenerate +versions are not limited to polynomials and numbers but also extended to transcendental functions, +like gamma functions (see [9, 10]). It is also remarkable that the λ-umbral calculus and λ-q-umbral +calculus were introduced as degenerate versions of the umbral calculus and the q-umbral calculus, +respectively (see [6, 11]). As it turns out, the λ-umbral calculus and λ-q-umbral calculus are more +convenient than the umbral calculus and the q-umbral calculus when dealing with degenerate Shef- +fer polynomials and degenerate q-Sheffer polynomials. +The aim of this paper is to further investigate some properties, recurrence relations and identities +involving the degenerate harmonic numbers (see (6)) and the degenerate hyperharmonic numbers +(see (7), (8)) in connection with degenerate Stirling numbers of the first kind, degenerate Daehee +numbers and degenerate derangements. The degenerate harmonic numbers and degenerate hyper- +harmonic numbers are respectively degenerate versions of the harmonic numbers and the hyperhar- +monic numbers, of which the latter are introduced in [4]. +The outline of this paper is as follows. In Section 1, we recall the degenerate exponentials and +the degenerate logarithms. We remind the reader of the harmonic numbers, and of the hyperhar- +monic numbers together with their explicit expression due to Conway and Guy (see [4]). Then +we recall their degenerate versions, namely the degenerate harmonic numbers, and the degenerate +hyperharmonic numbers together with their explicit expression (see [7, 8]). We also mention the +recently introduced degenerate Stirling numbers of the first kind and the degenerate Daehee num- +bers of order r. Section 2 is the main result of this paper. We obtain an expression of the degenerate +hyperharmonic numbers of order r in terms of the same numbers of lower orders in Theorem 1. We +express the Daehee numbers in terms of the degenerate harmonic numbers and of the degenerate +hyperharmonic numbers, respectively in Theorem 2 and Theorem 3. In Theorem 4, the degenerate +harmonic numbers are represented in terms of the degenerate hyperharmonic numbers of order r. +2010 Mathematics Subject Classification. 05A19; 11B73; 11B83. +Key words and phrases. degenerate harmonic number; degenerate hyperharmonic number; degenerate Daehee num- +ber; degenerate logarithm; degenerate Stirling number of the first kind; degenerate derangement. +* is corresponding author. +1 + +2 +Identities involving degenerate harmonic and degenerate hyperharmonic numbers +In Theorem 5, the degenerate Daehee numbers are represented in terms of the degenerate Daehee +numbers of order r −1 and of the degenerate hyperharmonic numbers. We derive a simple relation +between the degenerate hyperharmonic numbers and the degenerate Daehee numbers in Theorem +6. We deduce an identity involving the degenerate hyperharmonic numbers and the degenerate de- +rangements in Theorem 7. The degenerate Daehee numbers are expressed in terms of the degenerate +Stirling numbers of the first kind in Theorem 8. Finally, we get an identity involving the degenerate +Stirling numbers of the first kind and the degenerate harmonic numbers in Theorem 9. +For any nonzero λ ∈ R, the degenerate exponential functions are defined by +ex +λ(t) = (1+λt) +x +λ = +∞ +∑ +n=0 +(x)n,λ +tn +n!, +eλ(t) = e1 +λ(t), +(see [2, 8]), +(1) +where +(x)0,λ = 1, (x)n,λ = x(x−λ)···(x−(n−1)λ), (n ≥ 1), +(see [8]). +Let logλ t be the compositional inverse of eλ(t) with eλ(logλ t) = logλ eλ(t) = t. It is called the +degenerate logarithm and is given by +logλ(1+t) = +∞ +∑ +k=1 +λ k−1(1)k, 1 +λ +k! +tk = 1 +λ ((1+t)λ −1), +(see [5]). +(2) +The harmonic numbers are given by +H0 = 0, Hn = 1+ 1 +2 +···+ 1 +n, +(n ∈ N), +(see [3, 4, 16]). +(3) +In 1996, Conway and Guy introduced the hyperharmonic numbers H(r) +n +of order r, (n,r ≥ 0), which +are given by +H(r) +0 += 0, (r ≥ 0), H(0) +n += 1 +n, (n ≥ 1), H(r) +n += +n +∑ +k=1 +H(r−1) +k +, (n,r ≥ 1), +(see [4]). +(4) +Thus, by (4), we get +H(r) +n += +�n+r −1 +n +� +(Hn+r−1 −Hr−1), +(r ≥ 1), +(see [4]). +(5) +Recently, the degenerate harmonic numbers are defined by +H0,λ = 0, Hn,λ = +n +∑ +k=1 +1 +λ +�λ +k +� +(−1)k−1, +(n ≥ 1), +(see [8]). +(6) +Note that limλ→0 Hn,λ = Hn. The degenerate hyperharmonic numbers H(r) +n,λ of order r, (n,r ≥ 0), +are defined by +H(r) +0,λ = 0, (r ≥ 0), H(0) +n,λ = 1 +λ +�λ +n +� +(−1)n−1, (n ≥ 1), H(r) +n,λ = +n +∑ +k=1 +H(r−1) +k,λ +, (n,r ≥ 1), +(see [7]). +(7) +We see from (6) and (7) that H(1) +n,λ = Hn,λ. From (7), we note that +H(r) +n,λ = (−1)r−1 +�λ−1 +r−1 +� +�n+r −1 +n +� +(Hn+r−1,λ −Hr−1,λ), +(see [7]), +(8) +where n, r are positive numbers. Here we observe from (5) and (8) that limλ→0 H(r) +n,λ = H(r) +n . + +H. K. Kim, D. S. Kim, and T. Kim +3 +In [5], the degenerate Stirling numbers of the first kind are defined by +(x)n = +n +∑ +k=0 +S1,λ(n,k)(x)k,λ , +(n ≥ 0), +(see [5, 8]), +(9) +where (x)0 = 1, (x)n = x(x−1)···(x−n+1), (n ≥ 1). +For r ∈ N, the degenerate Daehee numbers of order r are defined by +�logλ(1+t) +t +�r += +∞ +∑ +n=0 +D(r) +n,λ +tn +n!, +(see [11]). +(10) +In particular, for r = 1, Dn,λ = D(1) +n,λ are called the degenerate Daehee numbers +2. IDENTITIES INVOLVING DEGENERATE HARMONIC AND DEGENERATE HYPERHARMONIC +NUMBERS +From (6) and (7), we note that +−logλ(1−t) +(1−t) += +∞ +∑ +n=1 +Hn,λtn, +(see [7]), +(11) +and +−logλ(1−t) +(1−t)r += +∞ +∑ +n=1 +H(r) +n,λtn, +(see [7]), +(12) +where r is a nonnegative integer. +By (12), we get +∞ +∑ +n=1 +H(r−1) +n,λ +tn = −logλ(1−t) +(1−t)r +(1−t) = +∞ +∑ +n=1 +H(r) +n,λtn(1−t) += +∞ +∑ +n=1 +H(r) +n,λtn − +∞ +∑ +n=1 +H(r) +n,λtn+1 = +∞ +∑ +n=1 +(H(r) +n,λ −H(r) +n−1,λ)tn. +(13) +By comparing the coefficients on both sides of (13), we get +(14) +H(r) +n,λ = H(r) +n−1,λ +H(r−1) +n,λ +. +For 1 ≤ s ≤ r, by (12), we get +∞ +∑ +n=1 +H(r) +n,λtn = −logλ(1−t) +(1−t)r += −logλ(1−t) +(1−t)r−s +1 +(1−t)s += +∞ +∑ +l=1 +H(r−s) +l,λ +tl +∞ +∑ +k=0 +�k +s−1 +k +� +tk += +∞ +∑ +n=1 +n +∑ +l=1 +H(r−s) +l,λ +�n−l +s−1 +s−1 +� +tn. +(15) +By comparing the coefficients on both sides of (15), we get +H(r) +n,λ = +n +∑ +l=1 +H(r−s) +l,λ +�n−l +s−1 +s−1 +� +, +(16) +where r, s ∈ Z with 1 ≤ s ≤ r. In particular, for r = s, we have +H(r) +n,λ = +n +∑ +l=1 +H(0) +l,λ +�n−l +r −1 +r −1 +� += +n +∑ +l=1 +1 +λ +�λ +l +� +(−1)l−1 +�n−l +r −1 +r −1 +� +. +(17) +Therefore, by (16) and (17), we obtain the following theorem. + +4 +Identities involving degenerate harmonic and degenerate hyperharmonic numbers +Theorem 1. For r, s ∈ Z with 1 ≤ s ≤ r, we have +H(r) +n,λ = +n +∑ +l=1 +H(r−s) +l,λ +�n−l +s−1 +s−1 +� +, +and +H(r) +n,λ = +n +∑ +l=1 +1 +λ +�λ +l +� +(−1)l−1 +�n−l +r −1 +r −1 +� +. +From (11) and (14), we note that +∞ +∑ +n=0 +Dn,λ +tn +n! = logλ(1+t) +t += logλ(1+t) +1+t +1+t +t += +� ∞ +∑ +k=1 +(−1)k+1Hk,λtk +�� +1+ 1 +t +� += +∞ +∑ +n=1 +(−1)n+1Hn,λtn + +∞ +∑ +n=0 +(−1)nHn+1,λtn += 1+ +∞ +∑ +n=1 +(−1)n(Hn+1,λ −Hn,λ)tn. +(18) +Therefore, by comparing the coefficients on both sides of (18), we have the following theorem. +Theorem 2. For n ≥ 0, we have +D0,λ = 1, Dn,λ = (−1)nn!(Hn+1,λ −Hn,λ), (n ≥ 1). +From (12), we note that +∞ +∑ +n=0 +Dn,λ +tn +n! = logλ(1+t) +t += logλ(1+t) +t(1+t)r (1+t)r += +∞ +∑ +k=0 +H(r) +k+1,λ(−1)ktk +∞ +∑ +l=0 +�r +l +� +tl += +∞ +∑ +n=0 +� n +∑ +k=0 +H(r) +k+1,λ +� r +n−k +� +(−1)k +� +tn. +(19) +Therefore, by (19), we obtain the following theorem +Theorem 3. For n ≥ 0, we have +Dn,λ = n! +n +∑ +k=0 +H(r) +k+1,λ +� r +n−k +� +(−1)k. +Now, we observe from (2) that +(20) +∞ +∑ +n=0 +Dn,λ +tn +n! = logλ(1+t) +t += +∞ +∑ +n=1 +�λ +n +� 1 +λ tn−1 = +∞ +∑ +n=0 +� λ +n+1 +� 1 +λ tn. +Thus, by (20), we get +Dn,λ = n! 1 +λ +� λ +n+1 +� += (λ −1)n +n+1 , +(n ≥ 0). +(21) + +H. K. Kim, D. S. Kim, and T. Kim +5 +From (11), we have +∞ +∑ +n=1 +Hn,λtn = −logλ(1−t) +1−t += −logλ(1−t) +t +t +1−t += +∞ +∑ +l=0 +Dl,λ(−1)l tl +l! +∞ +∑ +m=1 +tm += +∞ +∑ +n=1 +� n−1 +∑ +l=0 +Dl,λ +(−1)l +l! +� +tn. +(22) +Thus, by Theorem 3 and (22), we get +Hn,λ = +n−1 +∑ +l=0 +Dl,λ +(−1)l +l! += +n−1 +∑ +l=0 +(−1)l +l! +l! +l +∑ +k=0 +H(r) +k+1,λ +� r +l −k +� +(−1)k += +n−1 +∑ +l=0 +l +∑ +k=0 +(−1)k+lH(r) +k+1,λ +� r +l −k +� +, +(n ≥ 1). +(23) +Therefore, by (23), we obtain the following theorem. +Theorem 4. For n ≥ 1, we have +Hn,λ = +n−1 +∑ +l=0 +l +∑ +k=0 +(−1)k+l +� r +l −k +� +H(r) +k+1,λ. +By (10), we get +∞ +∑ +n=0 +D(r) +n,λ +tn +n! = +�logλ(1+t) +t +�r += logλ(1+t) +t(1+t)k +�logλ(1+t) +t +�r−1 +(1+t)k += +∞ +∑ +i=1 +(−1)i+1H(k) +i,λ ti−1 +∞ +∑ +j=0 +D(r−1) +j,λ +t j +j! +∞ +∑ +l=0 +�k +l +� +tl += +∞ +∑ +i=0 +(−1)iH(k) +i+1,λti +∞ +∑ +m=0 +� m +∑ +j=0 +�m +j +� +D(r−1) +j,λ +(k)m− j +� tm +m! += +∞ +∑ +n=0 +� n +∑ +i=0 +n−i +∑ +j=0 +(−1)i +�n−i +j +�(k)n−i− j +(n−i)! D(r−1) +j,λ +H(k) +i+1,λ +� +tn. +(24) +Therefore, by comparing the coefficients on both sides of (24), we obtain the following theorem. +Theorem 5. For n,k ≥ 0 and r ≥ 1, we have +D(r) +n,λ = n! +n +∑ +i=0 +n−i +∑ +j=0 +(−1)i +�n−i +j +�(k)n−i− j +(n−i)! D(r−1) +j,λ +H(k) +i+1,λ. +By (11), we get +∞ +∑ +n=1 +Hn,λtn = −logλ(1−t) +1−t += logλ(1−t) +−t +t +1−t += +∞ +∑ +l=0 +(−1)lDl,λ +tl +l! +∞ +∑ +j=1 +t j += +∞ +∑ +n=1 +� n−1 +∑ +l=0 +(−1)l Dl,λ +l! +� +tn. +(25) + +6 +Identities involving degenerate harmonic and degenerate hyperharmonic numbers +Thus, by comparing the coefficients on both sides of (25), we get +Hn,λ = +n−1 +∑ +l=0 +(−1)l Dl,λ +l! , +(n ≥ 1). +(26) +From (12), we can derive the following. +∞ +∑ +n=1 +H(r) +n,λtn = −logλ(1−t) +t +t +(1−t)r += +∞ +∑ +l=0 +Dl,λ(−1)l tl +l! +∞ +∑ +m=1 +�r +m−2 +m−1 +� +tm += +∞ +∑ +n=1 +� +n +∑ +m=1 +�r +m−2 +r −1 +� Dn−m,λ +(n−m)!(−1)n−m +� +tn. +(27) +Therefore, by (26) and (27), we obtain the following theorem. +Theorem 6. For n ∈ N, we have +Hn,λ = +n−1 +∑ +l=0 +(−1)l Dl,λ +l! , +(n ≥ 1), +and +H(r) +n,λ = +n +∑ +m=1 +�r +m−2 +r −1 +� Dn−m,λ +(n−m)!(−1)n−m. +The degenerate derangements are defined by +1 +1−t eλ(−t) = +∞ +∑ +n=0 +dn,λ +tn +n!. +(28) +Thus, we note that +dn,λ = n! +n +∑ +k=0 +(1)k,λ +(−1)k +k! +, +(n ≥ 0). +Now, we observe that +−logλ(1−t) +(1−t)r +eλ(−t) = +∞ +∑ +l=1 +H(r) +l,λtl +∞ +∑ +k=0 +(1)k,λ +k! +(−1)ktk += +∞ +∑ +n=1 +� n +∑ +l=1 +H(r) +l,λ +(1)n−l,λ +(n−l)! (−1)n−l +� +tn. +(29) +On the other hand, by (28), we get +−logλ(1−t) +(1−t)r +eλ(−t) = −logλ(1−t) +(1−t)r−1 +1 +1−t eλ(−t) += +∞ +∑ +l=1 +H(r−1) +l,λ +tl +∞ +∑ +k=0 +dk,λ +tk +k! = +∞ +∑ +n=1 +� n +∑ +l=1 +H(r−1) +l,λ +dn−l,λ +(n−l)! +� +tn. +(30) +Therefore, by (29) and (30), we obtain the following theorem. +Theorem 7. For n ∈ N, we have +n +∑ +l=1 +H(r) +l,λ +(1)n−l,λ +(n−l)! (−1)n−l = +n +∑ +l=1 +H(r−1) +l,λ +dn−l,λ +(n−l)!. + +H. K. Kim, D. S. Kim, and T. Kim +7 +We let Y = logλ(1+t). Then, for N ≥ 1, we have +� d +dt +�N +Y = (λ −1)(λ −2)···(λ −N +1)(1+t)λ−N += N! +λ +�λ +N +� +eλ−N +λ +(logλ(1+t)) += N! +λ +�λ +N +� ∞ +∑ +k=0 +(λ −N)k,λ +1 +k!(logλ(1+t))k += N! +λ +�λ +N +� ∞ +∑ +k=0 +(λ −N)k,λ +∞ +∑ +n=k +S1,λ(n,k)tn +n! += +∞ +∑ +n=0 +�N! +λ +�λ +N +� n +∑ +k=0 +S1,λ(n,k)(λ −N)k,λ +�tn +n!, +(31) +where N is a positive integer. +On the other hand, by (10), we get +Y = logλ(1+t) = logλ(1+t) +t +t = +∞ +∑ +n=1 +nDn−1,λ +tn +n!. +(32) +Thus, by (32), we get +� d +dt +�N +Y = +∞ +∑ +n=N +nDn−1,λn(n−1)···(n−N +1)tn−N +n! += +∞ +∑ +n=0 +(n+N)Dn+N−1,λ +tn +n!. +(33) +Therefore, by (31) and (33), we obtain the following theorem. +Theorem 8. For N ∈ N and n ≥ N −1, we have +Dn,λ = +N! +n+1 +1 +λ +�λ +N +� n−N+1 +∑ +k=0 +S1,λ(n−N +1,k)(λ −N)k,λ. +Next, we let F = −logλ(1−t). Then, for N ≥ 1, we have +� d +dt +�N +F = (−1)N+1(λ −1)(λ −2)···(λ −N +1)(1−t)λ−N += (−1)N+1 N! +λ +�λ +N +� +eλ−N +λ +(logλ(1−t)) += (−1)N+1N! 1 +λ +�λ +N +� ∞ +∑ +k=0 +(λ −N)k,λ +1 +k!(logλ(1−t))k += (−1)N+1N! 1 +λ +�λ +N +� ∞ +∑ +k=0 +(λ −N)k,λ +∞ +∑ +n=k +S1,λ(n,k)(−1)n tn +n! += +∞ +∑ +n=0 +� +N! 1 +λ +�λ +N +� n +∑ +k=0 +(−1)n−N−1(λ −N)k,λS1,λ(n,k) +�tn +n!. +(34) +On the other hand, by (11), we get +(35) +F = −logλ(1−t) = −logλ(1−t) +1−t +(1−t) = +∞ +∑ +n=1 +(Hn,λ −Hn−1,λ)tn. + +8 +Identities involving degenerate harmonic and degenerate hyperharmonic numbers +Thus, by (35) and for N ≥ 1, we have +� d +dt +�N +F = +∞ +∑ +n=N +n(n−1)···(n−N +1)(Hn,λ −Hn−1,λ)tn−N += +∞ +∑ +n=0 +(n+N)(n+N −1)···(n+1)(Hn+N,λ −Hn+N−1,λ)tn += +∞ +∑ +n=0 +N! +�n+N +N +� +(Hn+N,λ −Hn+N−1,λ)tn. +(36) +Therefore, by (34) and (36), we obtain the following theorem. +Theorem 9. For N ∈ N and n ≥ 0, we have +1 +n! +1 +λ +�λ +N +� n +∑ +k=0 +(−1)n−N−1(λ −N)k,λS1,λ(n,k) = +�n+N +N +� +(Hn+N,λ −Hn+N−1,λ). +By Theorem 9 and (6), we get +1 +n! +n +∑ +k=0 +(−1)n−N−1(λ −N)k,λS1,λ(n,k) = +�n+N +N +� +1 +λ +�λ +N +� (Hn+N,λ −Hn+N−1,λ) += +�n+N +N +� +1 +λ +�λ +N +� 1 +λ +� +λ +n+N +� +(−1)n+N−1 = (−1)n+N−1 +� λ +N+n +� +�λ +N +� +�n+N +N +� +. +(37) +Therefore, by (37), we obtain the following corollary. +Corollary 10. For n ≥ 0 and N ∈ N, we have +1 +n! +n +∑ +k=0 +(λ −N)k,λS1,λ(n,k) = +� λ +n+N +� +�λ +N +� +�n+N +N +� +. +Remark 11. From Corollary 10 and letting λ → 0, we obtain +(−1)n +N +n+N +�n+N +N +� += 1 +n! +n +∑ +k=0 +(−1)kNkS1(n,k). +Remark 12. Recently, on the Daehee numbers and their related topics various studies have been +conducted by several researchers. Interested readers may refer to [1, 12, 13, 14, 15, 17, 18]. +3. CONCLUSION +Many different tools have been used in the explorations for degenerate versions of some special +numbers and polynomials, which include generating functions, combinatorial methods, umbral cal- +culus, p-adic analysis, differential equations, probability theory, operator theory, special functions +and analytic number theory (see [5, 6, 7, 8, 9, 10, 11, 12] and the references therein). In this paper, +we used the elementary methods of generating functions in order to study the degenerate harmonic +and degenerate hyperharmonic numbers. Some properties, recurrence relations and identities relat- +ing to those numbers were derived in connection with the degenerate Stirling numbers of the first +kind, the degenerate Daehee numbers and the degenerate derangement. +We would like to continue to investigate various degenerate versions of certain special numbers +and polynomials, especially their applications to physics, science and engineering. + +H. K. Kim, D. S. Kim, and T. Kim +9 +Acknowledgments +The authors thank Jangjeon Institute for Mathematical Sciences for the support of this research. +Availability of data and material +Not applicable. +Funding +This work was supported by the Basic Science Research Program, the National Research Founda- +tion of Korea, (NRF-2021R1F1A1050151). +Ethics approval and consent to participate +All authors declare that there is no ethical problem in the production of this paper. +Competing interests +All authors declare no conflict of interest. +Consent for publication +All authors want to publish this paper in this journal. +Author’ Contributions +All authors read and approved the final manuscript. +REFERENCES +[1] S. Araci, U. Duran and M. Acikgoz, On weighted q-Daehee polynomials with their applications. Indag. Math. (N.S.) +30 (2019), no. 2, 365-374. +[2] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math. 15 (1979), 51-88. +[3] L. Comtet, Advanced combinatorics. The art of finite and infinite expansions. Revised and enlarged edition. D. Reidel +Publishing Co., Dordrecht, 1974. xi+343 pp. ISBN: 90-277-0441-4. +[4] J. H. Conway and R. K. Guy, The book of numbers. Copernicus, New York, 1996. x+310 pp. ISBN: 0-387-97993-X. +[5] D. S. Kim and T. Kim, A note on a new type of degenerate Bernoulli numbers. Russ. J. Math. Phys. 27 (2020), no. 2, +227-235. +[6] D. S. Kim and T. Kim, Degenerate Sheffer sequence and λ-Sheffer sequence. J. Math. Anal. Appl. 493 (2021), no. +1, 124521. +[7] T. Kim and D. S. Kim, Some identities on degenerate hyperharmonic numbers. Georgian Math. J., 2022 (2022). +https://doi.org/10.1515/gmj-2022-2203 +[8] T. Kim and D. S. Kim, On some degenerate differential and degenerate difference operators. Russ. J. Math. Phys. 29 +(2022), no. 1, 37-46. +[9] T. Kim and D. S. Kim, Degenerate Laplace transform and degenerate gamma function. Russ. J. Math. Phys. 24 +(2017), no. 2, 241–248 . +[10] T. Kim and D. S. Kim, Note on the degenerate gamma function Russ. J. Math. Phys. 27 (2020), no. 3, 352-358. +[11] T. Kim, D. S. Kim and H. K. Kim, λ-q-Sheffer sequence and its applications. Demonstr. Math. 55 (2022), 843–865. +[12] T. Kim, D. S. Kim, H. Lee and J. Kwon, Representations by degenerate Daehee polynomials. Open Math. 20 (2022), +no. 1, 179-194. +[13] J. Kwon, W. J. Kim and S.-H. Rim, On the some identities of the type 2 Daehee and Changhee polynomials arising +from p-adic integrals on Zp. Proc. Jangjeon Math. Soc. 22 (2019), no. 3, 487-497. +[14] J. G. Lee, J. Kwon, G.-W. Jang and L.-C. Jang, Some identities of λ-Daehee polynomials. J. Nonlinear Sci. Appl. +10 (2017), no. 8, 4137-4142. +[15] J.-W. Park, B. M. Kim and J. Kwon, On a modified degenerate Daehee polynomials and numbers. J. Nonlinear Sci. +Appl. 10 (2017), no. 3, 1108-1115. +[16] S. Roman, The umbral calculus. Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jo- +vanovich, Publishers], New York, 1984. x+193 pp. ISBN: 0-12-594380-6 +[17] S. K. Sharma, W. A. Khan, S. Araci and S. S. Ahmed, New type of degenerate Daehee polynomials of the second +kind. Adv. Difference Equ. 2020 (2020), Paper No. 428, 14 pp. +[18] S. J. Yun and J.-W. Park, On fully degenerate Daehee numbers and polynomials of the second kind. J. Math. 2020 +(2020), Art. ID 7893498, 9 pp. + +10 +Identities involving degenerate harmonic and degenerate hyperharmonic numbers +DEPARTMENT OF MATHEMATICS EDUCATION, DAEGU CATHOLIC UNIVERSITY, GYEONGSAN 38430, REPUB- +LIC OF KOREA +Email address: hkkim@cu.ac.kr +DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY, SEOUL 121-742, REPUBLIC OF KOREA +Email address: dskim@sogang.ac.kr +DEPARTMENT OF MATHEMATICS, KWANGWOON UNIVERSITY, SEOUL 139-701, REPUBLIC OF KOREA +Email address: tkkim@kw.ac.kr + diff --git a/9NE1T4oBgHgl3EQfCQIp/content/tmp_files/load_file.txt b/9NE1T4oBgHgl3EQfCQIp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b30c1a6d2ea41523d24226425ff07c9635dff0a4 --- /dev/null +++ b/9NE1T4oBgHgl3EQfCQIp/content/tmp_files/load_file.txt @@ -0,0 +1,381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf,len=380 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='02861v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='NT] 7 Jan 2023 IDENTITIES INVOLVING DEGENERATE HARMONIC AND DEGENERATE HYPERHARMONIC NUMBERS HYE KYUNG KIM1, DAE SAN KIM2, AND TAEKYUN KIM3,∗ ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Harmonic numbers have been studied since antiquity, while hyperharmonic numbers were intoduced by Conway and Guy in 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The degenerate harmonic numbers and degenerate hyperharmonic numbers are their respective degenerate versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The aim of this paper is to further investigate some properties, recurrence relations and identities involving the degenerate harmonic and degenerate hyperharmonic numbers in connection with degenerate Stirling numbers of the first kind, degenerate Daehee numbers and degenerate derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' INTRODUCTION In recent years, various degenerate versions of many special numbers and polynomials have beem studied and yielded a lot of fascinating and fruitful results (see [5, 6, 7, 8, 9, 10, 11, 12] and the references therein), which began with Carlitz’s work on the degenerate Bernoulli and degen- erate Euler numbers (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' It is worthwhile to mention that these explorations for degenerate versions are not limited to polynomials and numbers but also extended to transcendental functions, like gamma functions (see [9, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' It is also remarkable that the λ-umbral calculus and λ-q-umbral calculus were introduced as degenerate versions of the umbral calculus and the q-umbral calculus, respectively (see [6, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' As it turns out, the λ-umbral calculus and λ-q-umbral calculus are more convenient than the umbral calculus and the q-umbral calculus when dealing with degenerate Shef- fer polynomials and degenerate q-Sheffer polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The aim of this paper is to further investigate some properties, recurrence relations and identities involving the degenerate harmonic numbers (see (6)) and the degenerate hyperharmonic numbers (see (7), (8)) in connection with degenerate Stirling numbers of the first kind, degenerate Daehee numbers and degenerate derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The degenerate harmonic numbers and degenerate hyper- harmonic numbers are respectively degenerate versions of the harmonic numbers and the hyperhar- monic numbers, of which the latter are introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' In Section 1, we recall the degenerate exponentials and the degenerate logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We remind the reader of the harmonic numbers, and of the hyperhar- monic numbers together with their explicit expression due to Conway and Guy (see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Then we recall their degenerate versions, namely the degenerate harmonic numbers, and the degenerate hyperharmonic numbers together with their explicit expression (see [7, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We also mention the recently introduced degenerate Stirling numbers of the first kind and the degenerate Daehee num- bers of order r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Section 2 is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We obtain an expression of the degenerate hyperharmonic numbers of order r in terms of the same numbers of lower orders in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We express the Daehee numbers in terms of the degenerate harmonic numbers and of the degenerate hyperharmonic numbers, respectively in Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' In Theorem 4, the degenerate harmonic numbers are represented in terms of the degenerate hyperharmonic numbers of order r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 05A19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 11B73;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 11B83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' degenerate harmonic number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' degenerate hyperharmonic number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' degenerate Daehee num- ber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' degenerate logarithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' degenerate Stirling number of the first kind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' degenerate derangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' is corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1 2 Identities involving degenerate harmonic and degenerate hyperharmonic numbers In Theorem 5, the degenerate Daehee numbers are represented in terms of the degenerate Daehee numbers of order r −1 and of the degenerate hyperharmonic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We derive a simple relation between the degenerate hyperharmonic numbers and the degenerate Daehee numbers in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We deduce an identity involving the degenerate hyperharmonic numbers and the degenerate de- rangements in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The degenerate Daehee numbers are expressed in terms of the degenerate Stirling numbers of the first kind in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Finally, we get an identity involving the degenerate Stirling numbers of the first kind and the degenerate harmonic numbers in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For any nonzero λ ∈ R, the degenerate exponential functions are defined by ex λ(t) = (1+λt) x λ = ∞ ∑ n=0 (x)n,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=', eλ(t) = e1 λ(t), (see [2, 8]), (1) where (x)0,λ = 1, (x)n,λ = x(x−λ)···(x−(n−1)λ), (n ≥ 1), (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Let logλ t be the compositional inverse of eλ(t) with eλ(logλ t) = logλ eλ(t) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' It is called the degenerate logarithm and is given by logλ(1+t) = ∞ ∑ k=1 λ k−1(1)k, 1 λ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' tk = 1 λ ((1+t)λ −1), (see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (2) The harmonic numbers are given by H0 = 0, Hn = 1+ 1 2 +···+ 1 n, (n ∈ N), (see [3, 4, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (3) In 1996, Conway and Guy introduced the hyperharmonic numbers H(r) n of order r, (n,r ≥ 0), which are given by H(r) 0 = 0, (r ≥ 0), H(0) n = 1 n, (n ≥ 1), H(r) n = n ∑ k=1 H(r−1) k , (n,r ≥ 1), (see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (4) Thus, by (4), we get H(r) n = �n+r −1 n � (Hn+r−1 −Hr−1), (r ≥ 1), (see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (5) Recently, the degenerate harmonic numbers are defined by H0,λ = 0, Hn,λ = n ∑ k=1 1 λ �λ k � (−1)k−1, (n ≥ 1), (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (6) Note that limλ→0 Hn,λ = Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The degenerate hyperharmonic numbers H(r) n,λ of order r, (n,r ≥ 0), are defined by H(r) 0,λ = 0, (r ≥ 0), H(0) n,λ = 1 λ �λ n � (−1)n−1, (n ≥ 1), H(r) n,λ = n ∑ k=1 H(r−1) k,λ , (n,r ≥ 1), (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (7) We see from (6) and (7) that H(1) n,λ = Hn,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' From (7), we note that H(r) n,λ = (−1)r−1 �λ−1 r−1 � �n+r −1 n � (Hn+r−1,λ −Hr−1,λ), (see [7]), (8) where n, r are positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Here we observe from (5) and (8) that limλ→0 H(r) n,λ = H(r) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim 3 In [5], the degenerate Stirling numbers of the first kind are defined by (x)n = n ∑ k=0 S1,λ(n,k)(x)k,λ , (n ≥ 0), (see [5, 8]), (9) where (x)0 = 1, (x)n = x(x−1)···(x−n+1), (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For r ∈ N, the degenerate Daehee numbers of order r are defined by �logλ(1+t) t �r = ∞ ∑ n=0 D(r) n,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=', (see [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (10) In particular, for r = 1, Dn,λ = D(1) n,λ are called the degenerate Daehee numbers 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' IDENTITIES INVOLVING DEGENERATE HARMONIC AND DEGENERATE HYPERHARMONIC NUMBERS From (6) and (7), we note that −logλ(1−t) (1−t) = ∞ ∑ n=1 Hn,λtn, (see [7]), (11) and −logλ(1−t) (1−t)r = ∞ ∑ n=1 H(r) n,λtn, (see [7]), (12) where r is a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' By (12), we get ∞ ∑ n=1 H(r−1) n,λ tn = −logλ(1−t) (1−t)r (1−t) = ∞ ∑ n=1 H(r) n,λtn(1−t) = ∞ ∑ n=1 H(r) n,λtn − ∞ ∑ n=1 H(r) n,λtn+1 = ∞ ∑ n=1 (H(r) n,λ −H(r) n−1,λ)tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (13) By comparing the coefficients on both sides of (13), we get (14) H(r) n,λ = H(r) n−1,λ +H(r−1) n,λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For 1 ≤ s ≤ r, by (12), we get ∞ ∑ n=1 H(r) n,λtn = −logλ(1−t) (1−t)r = −logλ(1−t) (1−t)r−s 1 (1−t)s = ∞ ∑ l=1 H(r−s) l,λ tl ∞ ∑ k=0 �k +s−1 k � tk = ∞ ∑ n=1 n ∑ l=1 H(r−s) l,λ �n−l +s−1 s−1 � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (15) By comparing the coefficients on both sides of (15), we get H(r) n,λ = n ∑ l=1 H(r−s) l,λ �n−l +s−1 s−1 � , (16) where r, s ∈ Z with 1 ≤ s ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' In particular, for r = s, we have H(r) n,λ = n ∑ l=1 H(0) l,λ �n−l +r −1 r −1 � = n ∑ l=1 1 λ �λ l � (−1)l−1 �n−l +r −1 r −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (17) Therefore, by (16) and (17), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 4 Identities involving degenerate harmonic and degenerate hyperharmonic numbers Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For r, s ∈ Z with 1 ≤ s ≤ r, we have H(r) n,λ = n ∑ l=1 H(r−s) l,λ �n−l +s−1 s−1 � , and H(r) n,λ = n ∑ l=1 1 λ �λ l � (−1)l−1 �n−l +r −1 r −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' From (11) and (14), we note that ∞ ∑ n=0 Dn,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = logλ(1+t) t = logλ(1+t) 1+t 1+t t = � ∞ ∑ k=1 (−1)k+1Hk,λtk �� 1+ 1 t � = ∞ ∑ n=1 (−1)n+1Hn,λtn + ∞ ∑ n=0 (−1)nHn+1,λtn = 1+ ∞ ∑ n=1 (−1)n(Hn+1,λ −Hn,λ)tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (18) Therefore, by comparing the coefficients on both sides of (18), we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n ≥ 0, we have D0,λ = 1, Dn,λ = (−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (Hn+1,λ −Hn,λ), (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' From (12), we note that ∞ ∑ n=0 Dn,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = logλ(1+t) t = logλ(1+t) t(1+t)r (1+t)r = ∞ ∑ k=0 H(r) k+1,λ(−1)ktk ∞ ∑ l=0 �r l � tl = ∞ ∑ n=0 � n ∑ k=0 H(r) k+1,λ � r n−k � (−1)k � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (19) Therefore, by (19), we obtain the following theorem Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n ≥ 0, we have Dn,λ = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n ∑ k=0 H(r) k+1,λ � r n−k � (−1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Now, we observe from (2) that (20) ∞ ∑ n=0 Dn,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = logλ(1+t) t = ∞ ∑ n=1 �λ n � 1 λ tn−1 = ∞ ∑ n=0 � λ n+1 � 1 λ tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Thus, by (20), we get Dn,λ = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1 λ � λ n+1 � = (λ −1)n n+1 , (n ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (21) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim 5 From (11), we have ∞ ∑ n=1 Hn,λtn = −logλ(1−t) 1−t = −logλ(1−t) t t 1−t = ∞ ∑ l=0 Dl,λ(−1)l tl l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ∞ ∑ m=1 tm = ∞ ∑ n=1 � n−1 ∑ l=0 Dl,λ (−1)l l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (22) Thus, by Theorem 3 and (22), we get Hn,λ = n−1 ∑ l=0 Dl,λ (−1)l l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = n−1 ∑ l=0 (−1)l l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' l ∑ k=0 H(r) k+1,λ � r l −k � (−1)k = n−1 ∑ l=0 l ∑ k=0 (−1)k+lH(r) k+1,λ � r l −k � , (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (23) Therefore, by (23), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n ≥ 1, we have Hn,λ = n−1 ∑ l=0 l ∑ k=0 (−1)k+l � r l −k � H(r) k+1,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' By (10), we get ∞ ∑ n=0 D(r) n,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = �logλ(1+t) t �r = logλ(1+t) t(1+t)k �logλ(1+t) t �r−1 (1+t)k = ∞ ∑ i=1 (−1)i+1H(k) i,λ ti−1 ∞ ∑ j=0 D(r−1) j,λ t j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ∞ ∑ l=0 �k l � tl = ∞ ∑ i=0 (−1)iH(k) i+1,λti ∞ ∑ m=0 � m ∑ j=0 �m j � D(r−1) j,λ (k)m− j � tm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = ∞ ∑ n=0 � n ∑ i=0 n−i ∑ j=0 (−1)i �n−i j �(k)n−i− j (n−i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' D(r−1) j,λ H(k) i+1,λ � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (24) Therefore, by comparing the coefficients on both sides of (24), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n,k ≥ 0 and r ≥ 1, we have D(r) n,λ = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n ∑ i=0 n−i ∑ j=0 (−1)i �n−i j �(k)n−i− j (n−i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' D(r−1) j,λ H(k) i+1,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' By (11), we get ∞ ∑ n=1 Hn,λtn = −logλ(1−t) 1−t = logλ(1−t) −t t 1−t = ∞ ∑ l=0 (−1)lDl,λ tl l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ∞ ∑ j=1 t j = ∞ ∑ n=1 � n−1 ∑ l=0 (−1)l Dl,λ l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (25) 6 Identities involving degenerate harmonic and degenerate hyperharmonic numbers Thus, by comparing the coefficients on both sides of (25), we get Hn,λ = n−1 ∑ l=0 (−1)l Dl,λ l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' , (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (26) From (12), we can derive the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ∞ ∑ n=1 H(r) n,λtn = −logλ(1−t) t t (1−t)r = ∞ ∑ l=0 Dl,λ(−1)l tl l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ∞ ∑ m=1 �r +m−2 m−1 � tm = ∞ ∑ n=1 � n ∑ m=1 �r +m−2 r −1 � Dn−m,λ (n−m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (−1)n−m � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (27) Therefore, by (26) and (27), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n ∈ N, we have Hn,λ = n−1 ∑ l=0 (−1)l Dl,λ l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' , (n ≥ 1), and H(r) n,λ = n ∑ m=1 �r +m−2 r −1 � Dn−m,λ (n−m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (−1)n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The degenerate derangements are defined by 1 1−t eλ(−t) = ∞ ∑ n=0 dn,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='. (28) Thus, we note that dn,λ = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n ∑ k=0 (1)k,λ (−1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' , (n ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Now, we observe that −logλ(1−t) (1−t)r eλ(−t) = ∞ ∑ l=1 H(r) l,λtl ∞ ∑ k=0 (1)k,λ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (−1)ktk = ∞ ∑ n=1 � n ∑ l=1 H(r) l,λ (1)n−l,λ (n−l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (−1)n−l � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (29) On the other hand, by (28), we get −logλ(1−t) (1−t)r eλ(−t) = −logλ(1−t) (1−t)r−1 1 1−t eλ(−t) = ∞ ∑ l=1 H(r−1) l,λ tl ∞ ∑ k=0 dk,λ tk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = ∞ ∑ n=1 � n ∑ l=1 H(r−1) l,λ dn−l,λ (n−l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' � tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (30) Therefore, by (29) and (30), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n ∈ N, we have n ∑ l=1 H(r) l,λ (1)n−l,λ (n−l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (−1)n−l = n ∑ l=1 H(r−1) l,λ dn−l,λ (n−l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='. H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim 7 We let Y = logλ(1+t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Then, for N ≥ 1, we have � d dt �N Y = (λ −1)(λ −2)···(λ −N +1)(1+t)λ−N = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' λ �λ N � eλ−N λ (logλ(1+t)) = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' λ �λ N � ∞ ∑ k=0 (λ −N)k,λ 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (logλ(1+t))k = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' λ �λ N � ∞ ∑ k=0 (λ −N)k,λ ∞ ∑ n=k S1,λ(n,k)tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = ∞ ∑ n=0 �N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' λ �λ N � n ∑ k=0 S1,λ(n,k)(λ −N)k,λ �tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=', (31) where N is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' On the other hand, by (10), we get Y = logλ(1+t) = logλ(1+t) t t = ∞ ∑ n=1 nDn−1,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='. (32) Thus, by (32), we get � d dt �N Y = ∞ ∑ n=N nDn−1,λn(n−1)···(n−N +1)tn−N n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = ∞ ∑ n=0 (n+N)Dn+N−1,λ tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='. (33) Therefore, by (31) and (33), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For N ∈ N and n ≥ N −1, we have Dn,λ = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n+1 1 λ �λ N � n−N+1 ∑ k=0 S1,λ(n−N +1,k)(λ −N)k,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Next, we let F = −logλ(1−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Then, for N ≥ 1, we have � d dt �N F = (−1)N+1(λ −1)(λ −2)···(λ −N +1)(1−t)λ−N = (−1)N+1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' λ �λ N � eλ−N λ (logλ(1−t)) = (−1)N+1N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1 λ �λ N � ∞ ∑ k=0 (λ −N)k,λ 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (logλ(1−t))k = (−1)N+1N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1 λ �λ N � ∞ ∑ k=0 (λ −N)k,λ ∞ ∑ n=k S1,λ(n,k)(−1)n tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' = ∞ ∑ n=0 � N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1 λ �λ N � n ∑ k=0 (−1)n−N−1(λ −N)k,λS1,λ(n,k) �tn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='. (34) On the other hand, by (11), we get (35) F = −logλ(1−t) = −logλ(1−t) 1−t (1−t) = ∞ ∑ n=1 (Hn,λ −Hn−1,λ)tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 8 Identities involving degenerate harmonic and degenerate hyperharmonic numbers Thus, by (35) and for N ≥ 1, we have � d dt �N F = ∞ ∑ n=N n(n−1)···(n−N +1)(Hn,λ −Hn−1,λ)tn−N = ∞ ∑ n=0 (n+N)(n+N −1)···(n+1)(Hn+N,λ −Hn+N−1,λ)tn = ∞ ∑ n=0 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' �n+N N � (Hn+N,λ −Hn+N−1,λ)tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (36) Therefore, by (34) and (36), we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For N ∈ N and n ≥ 0, we have 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1 λ �λ N � n ∑ k=0 (−1)n−N−1(λ −N)k,λS1,λ(n,k) = �n+N N � (Hn+N,λ −Hn+N−1,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' By Theorem 9 and (6), we get 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n ∑ k=0 (−1)n−N−1(λ −N)k,λS1,λ(n,k) = �n+N N � 1 λ �λ N � (Hn+N,λ −Hn+N−1,λ) = �n+N N � 1 λ �λ N � 1 λ � λ n+N � (−1)n+N−1 = (−1)n+N−1 � λ N+n � �λ N � �n+N N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (37) Therefore, by (37), we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' For n ≥ 0 and N ∈ N, we have 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n ∑ k=0 (λ −N)k,λS1,λ(n,k) = � λ n+N � �λ N � �n+N N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' From Corollary 10 and letting λ → 0, we obtain (−1)n N n+N �n+N N � = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' n ∑ k=0 (−1)kNkS1(n,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Recently, on the Daehee numbers and their related topics various studies have been conducted by several researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Interested readers may refer to [1, 12, 13, 14, 15, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' CONCLUSION Many different tools have been used in the explorations for degenerate versions of some special numbers and polynomials, which include generating functions, combinatorial methods, umbral cal- culus, p-adic analysis, differential equations, probability theory, operator theory, special functions and analytic number theory (see [5, 6, 7, 8, 9, 10, 11, 12] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' In this paper, we used the elementary methods of generating functions in order to study the degenerate harmonic and degenerate hyperharmonic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Some properties, recurrence relations and identities relat- ing to those numbers were derived in connection with the degenerate Stirling numbers of the first kind, the degenerate Daehee numbers and the degenerate derangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' We would like to continue to investigate various degenerate versions of certain special numbers and polynomials, especially their applications to physics, science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim 9 Acknowledgments The authors thank Jangjeon Institute for Mathematical Sciences for the support of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Availability of data and material Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Funding This work was supported by the Basic Science Research Program, the National Research Founda- tion of Korea, (NRF-2021R1F1A1050151).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Ethics approval and consent to participate All authors declare that there is no ethical problem in the production of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Competing interests All authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Consent for publication All authors want to publish this paper in this journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Author’ Contributions All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Araci, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Duran and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Acikgoz, On weighted q-Daehee polynomials with their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Indag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=') 30 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 2, 365-374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Utilitas Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 15 (1979), 51-88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Comtet, Advanced combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' The art of finite and infinite expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Revised and enlarged edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Reidel Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=', Dordrecht, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' xi+343 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ISBN: 90-277-0441-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Conway and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Guy, The book of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Copernicus, New York, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' x+310 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ISBN: 0-387-97993-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, A note on a new type of degenerate Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 27 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 2, 227-235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, Degenerate Sheffer sequence and λ-Sheffer sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 493 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1, 124521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, Some identities on degenerate hyperharmonic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Georgian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=', 2022 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='1515/gmj-2022-2203 [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, On some degenerate differential and degenerate difference operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 29 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1, 37-46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, Degenerate Laplace transform and degenerate gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 24 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 2, 241–248 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, Note on the degenerate gamma function Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 27 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 3, 352-358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, λ-q-Sheffer sequence and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Demonstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 55 (2022), 843–865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Lee and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kwon, Representations by degenerate Daehee polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Open Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 20 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 1, 179-194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kwon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Rim, On the some identities of the type 2 Daehee and Changhee polynomials arising from p-adic integrals on Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Jangjeon Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 22 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 3, 487-497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kwon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Jang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Jang, Some identities of λ-Daehee polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 10 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 8, 4137-4142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Park, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kim and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Kwon, On a modified degenerate Daehee polynomials and numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 10 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 3, 1108-1115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Roman, The umbral calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Pure and Applied Mathematics, 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Academic Press, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [Harcourt Brace Jo- vanovich, Publishers], New York, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' x+193 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ISBN: 0-12-594380-6 [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Sharma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Khan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Araci and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Ahmed, New type of degenerate Daehee polynomials of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Difference Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 2020 (2020), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 428, 14 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Yun and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Park, On fully degenerate Daehee numbers and polynomials of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 2020 (2020), Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' ID 7893498, 9 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content=' 10 Identities involving degenerate harmonic and degenerate hyperharmonic numbers DEPARTMENT OF MATHEMATICS EDUCATION, DAEGU CATHOLIC UNIVERSITY, GYEONGSAN 38430, REPUB- LIC OF KOREA Email address: hkkim@cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='kr DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY, SEOUL 121-742, REPUBLIC OF KOREA Email address: dskim@sogang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='kr DEPARTMENT OF MATHEMATICS, KWANGWOON UNIVERSITY, SEOUL 139-701, REPUBLIC OF KOREA Email address: tkkim@kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} +page_content='kr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfCQIp/content/2301.02861v1.pdf'} diff --git a/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf b/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d126cb3cb6df551c50ef4020ed0e883708267cef --- /dev/null +++ b/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:01262dbda26eb60a40ab2d73bbb0d61831a1ec2b3f16a31595e401692e63df5a +size 193782 diff --git a/A9FAT4oBgHgl3EQfrx7P/vector_store/index.faiss b/A9FAT4oBgHgl3EQfrx7P/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ce9c59ef2e5b7a57dd09dc68d528eae96b6e0d04 --- /dev/null +++ b/A9FAT4oBgHgl3EQfrx7P/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:af582328fb9199499516cc030f6e3225ceb2b4864a6902c013d2b6702e173fc2 +size 1441837 diff --git a/A9FAT4oBgHgl3EQfrx7P/vector_store/index.pkl b/A9FAT4oBgHgl3EQfrx7P/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..42a2f016f1fef6e9c6ba64f00ab24cc5574462af --- /dev/null +++ b/A9FAT4oBgHgl3EQfrx7P/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:02e56db9b963589338936f99f9a6a55410754150363c75f83c66ab84877d3975 +size 64777 diff --git a/BNE1T4oBgHgl3EQfVgSF/content/tmp_files/2301.03103v1.pdf.txt b/BNE1T4oBgHgl3EQfVgSF/content/tmp_files/2301.03103v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..44b539d09e39ee72cdb896836bba3f97b913c854 --- /dev/null +++ b/BNE1T4oBgHgl3EQfVgSF/content/tmp_files/2301.03103v1.pdf.txt @@ -0,0 +1,2084 @@ +A Multi-Site Accelerator-Rich Processing Fabric for +Scalable Brain-Computer Interfacing +Karthik Sriram, Raghavendra Pradyumna Pothukuchi, Michał Gerasimiuk, Oliver Ye, Muhammed Ugur +Rajit Manohar, Anurag Khandelwal, Abhishek Bhattacharjee +Yale University, New Haven, USA +Abstract—Hull1 is an accelerator-rich distributed implantable +brain-computer interface (BCI) that reads biological neurons at +data rates that are 2-3 orders of magnitude higher than the +prior art, while supporting many neuroscientific applications. +Prior approaches have restricted brain interfacing to tens of +megabits per second in order to meet two constraints necessary +for effective operation and safe long-term implantation—power +dissipation under tens of milliwatts and response latencies in +the tens of milliseconds. Hull also adheres to these constraints, +but is able to interface with the brain at much higher data +rates, thereby enabling, for the first time, BCI-driven research on +and clinical treatment of brain-wide behaviors and diseases that +require reading and stimulating many brain locations. Central to +Hull’s power efficiency is its realization as a distributed system of +BCI nodes with accelerator-rich compute. Hull balances modular +system layering with aggressive cross-layer hardware-software +co-design to integrate compute, networking, and storage. The +result is a lesson in designing networked distributed systems with +hardware accelerators from the ground up. +I. I N T RO D U C T I O N +Brain-computer interfaces (BCIs) sense the electrical activity +of biological neurons and electrically stimulate them to “rewire” +neuronal circuits. By directly connecting brains to computers, +BCIs help advance our understanding of the brain and the +mind [1, 2], offer treatment of neurological disorders [2–6], +enable industrial robotics [7], permit novel modes of personal +entertainment [8], and more. +BCIs can be realized as surface electrodes (i.e., electrical +sensors) placed on the scalp above the skull to measure brain +activity [2, 3]. While such wearable BCIs do not require surgical +deployment, the signals they collect are muffled by the skull, +making them noisy, low-resolution, and less ideal for forward- +looking BCI applications [5, 9–12]. +Instead, this work focuses on implantable BCIs that are +surgically embedded directly on, around, and in the brain +tissue [13, 14]. Implantable BCIs directly record from and +stimulate neurons with high fidelity, spatial resolution, and in +real time [2, 5]. Hundreds of individuals use clinically approved +implantable BCIs to treat epilepsy, movement disorders, as well +as impaired vision [15–17]. Implantable BCIs are also being +studied in clinical trials to assess their effectiveness in treating +brain stroke, memory disorders, paralysis, anxiety/depression, +addiction, and more [14, 18, 19]. +Conflicting constraints make it challenging to design hard- +ware for implantable BCIs. BCIs cannot overheat brain regions +1A hull is the protective outer covering of grain. We call our design Hull +since it similarly protects the brain. +by >1 ◦C to avoid cellular damage [20, 21] and must therefore +be ultra-low-power. But, BCI designers are also seeking to +leverage improvements in sensor technology that are reading +exponentially increasing neuronal data [22]. It is challenging to +constrain power/energy while processing such large data, espe- +cially to respond to neuronal activity in real-time (i.e., in ms). +Hardware over-specialization is not a viable way to reduce BCI +power; to enable many research and clinical studies, BCIs must +be adequately programmable to personalize algorithms, support +several computational methods to treat multiple disorders, and +enable deployment of maturing/emerging algorithms [23, 24]. +Complicating BCI design further is the emergence of +applications that read and process neural activity from many +brain sites over time [1, 13, 25, 26]. This is because the brain’s +functions (and disorders) are ultimately based on physical and +functional connectivity between brain regions that evolve over +time [1, 13, 25, 26]. +Existing BCIs [16, 17, 23, 27, 28] are designed for single- +site implantation and lack the ability to store adequately long +historical neural data. Most BCIs [29, 30] have additional +limitations in that they have historically eschewed a subset of +programmability, data rates, and flexibility to meet safe power +constraints. These BCIs are specialized to a specific task and/or +limit personalization of the algorithm [29, 30]. Some support +more programmability by sacrificing high data rates [16, 17, +27, 28]. Consequently, none support the distributed and rapidly +evolving neural data processing that emerging BCI applications +require. Recent work on HALO balances flexibility, data rates, +and power, but is limited to one brain site. At best, distributed +BCI applications have been studied in prior work that consists +of multiple sensor implants that offload processing to external +devices with higher power budgets [31, 32]. But, this is not +a panacea because of the long network latency, privacy, and +mobility limitations [33]. +Our work is the first to offer a path toward scalable whole- +brain interfacing across multiple brain sites. Our solution, Hull, +is a distributed BCI consisting of multiple BCI nodes that +communicate with one another wirelessly and interface with +the brain with an aggregate data rate 2-3 orders of magnitude +higher than the prior state-of-art [23]. Each BCI node consists +of flexible compute made up of reconfigurable power-efficient +domain-specific hardware accelerators customized for important +neural processing applications. These accelerators are tightly +co-designed with storage and networking to ensure system-wide +adherence to power and response latency constraints. +1 +arXiv:2301.03103v1 [cs.DC] 8 Jan 2023 + +Hull uses a scheduler based on an integer linear program +(ILP) to optimally map tasks and algorithms to the hardware +accelerators across all of Hull’s nodes, and to create the network +and storage access schedules to feed the accelerators. Hull +supports three types of BCI applications [34, 35]: +The first category consists of applications that continuously +monitor brain activity and respond to aberrant behavior without +engaging with agents external to Hull [34–36]. This includes, +for example, detection of seizures [26], prediction of their +spread, and finally, to mitigate symptoms, electrical stimulation +of regions where seizures are expected to migrate. +The second category also monitor the brain continuously but +rely on agents external to Hull to respond. This includes, for +example, the detection of an individual’s intended movement, +and relaying of this data to prostheses or assistive devices [3, 33– +35]. These applications enable paralyzed individuals to actuate +machines and restore (partial) sensory function. +The third category interactively queries Hull to analyze data +from multiple brain sites. These queries may be used “in the +loop” or “out of the loop”. With the former, clinicians may need +to modify treatment options based on data that the BCI reads; +e.g., confirming that Hull has correctly detected seizures and +responded appropriately [37, 38]. The latter refers to interactive +queries used by technicians to debug system operations, by +clinicians to glean the individual’s medical history, and by +researchers to better understand brain function. +In supporting these types of applications, Hull offers research +insights on building end-to-end computer systems centered on +hardware accelerators. Specifically, our choice of hardware +accelerators follows several important design principles: +First, to maximize power efficiency, as well as simplicity of +hardware design, we identify algorithmic kernels within our +applications that accelerate not only their computation but also +their networking and storage latency needs. This is a non-trivial +exercise that requires domain-specific knowledge to convert BCI +applications into equivalent forms that are amenable to exposing +these kernels. We reformulate known computational pipelines +for seizure propagation prediction/treatment and movement +intent – two BCI applications that Hull focuses on – to use hash- +based similarity measures that identify the neural signals from +different brain sites likeliest to be correlated, before applying +heavy-weight correlation measures. The same hashes can drive +the design of domain-specific layouts of data in our storage +stack. Co-designing our hashes with our storage layout permit +several power/latency-efficient data retrievals from our storage +layer. The hashes also enable reducing the data communicated +in the intra-BCI network between the Hull nodes. +Second, we design our accelerators to be predictable in +latency and power for our target data rates. Predictable perfor- +mance and power characteristics facilitate the optimal design of +compute and network schedules—the enabling feature for our +ILP scheduler. Designing accelerators with predictable perfor- +mance and power requires care. For some accelerators, whose +data generation rate is input-dependent (e.g., data compression), +we use theoretically-derived worst-case latency and throughput +estimates. Furthermore, we design our accelerators in their own +clock domains to enable a range of operating frequencies with +well-defined power and performance characteristics. +Third, we design our accelerators to be reconfigurable. This +permits the repurposing of our hash accelerators, for example, +to act as hash indices for our storage layer, as filters that reduce +our networking traffic, and to be tuned differently depending +on the target application being supported. +Overall, Hull scales brain-computer interface bandwidth +beyond what was previously achievable. Hull is flexible, recon- +figurable, and supports real-time distributed neural processing. +We use a detailed physical synthesis flow in a 28 nm CMOS +process (including tapeouts of partial hardware at 12 nm) +coupled with network and storage models to evaluate Hull’s +power and performance. We show that Hull can process an +aggregate of 460 Mbps wireless data from multiple regions in +only 10 ms and dissipates no more than 15 mW at any node, +confirming its suitability for autonomous use. In interactive +mode, it can support 10 queries per second over 6 MB of +data over 10 implants. Existing designs support two orders of +magnitude lower data and need intrusive wiring. In summary, +our specific contributions are: +1) Hull, the first distributed and wireless BCI system that scales +to multi-region neural processing in real time. +2) The cross-layer co-design of BCI applications, processing, +and storage for scalable and distributed neural processing. +Hull is the first to support long-term storage of data, hashing, +and database indices to enable distributed signal processing +on a single BCI platform. +3) An evaluation of Hull’s flexibility and performance on +epileptic seizure propagation and detection of movement +intent, in various deployment scenarios, as well as in +the support of more arbitrary queries that may be used +interactively by clinicians/technicians. +Hull furthers the elevation of hardware accelerators to first- +class compute citizens that, like CPUs, can directly engage +with networking and storage. This trend will be crucial to +future IoT, swarm, and intermittent computing environments +that sustain adaptive and complex functionality while meeting +strict safe-use (i.e., power, latency, throughput) constraints. +II. BAC K G RO U N D +A. Brain-Computer Interface Design +BCI applications typically perform signal measurement, +feature extraction, classification/decision-making, and when +applicable, neural feedback/stimulation [2, 5]. The hardware +organization of BCIs reflects these four aspects. Signal mea- +surement is performed by electrodes that read the electrical +activity of clusters of biological neurons, and analog to digital +converters (ADCs) then digitize this data. State-of-the-art +sensors consist of 96-256 electrode arrays per implant. ADCs +typically encode measured signal samples with 8-16 bits at a +rate of 20-50 K samples per second per electrode. Digitized data +is then relayed to compute logic for feature extraction, based +on which classification/decision-making proceeds. If needed, +the electrode arrays are repurposed, after a digital-to-analog +(DAC) conversion step, to electrically stimulate the brain. +2 + +Modern BCIs also use radios to communicate with external +agents (e.g., servers, prostheses), presenting an evolution from +the surgical cables (which were susceptible to infections +and restricted free movement) used in early BCIs [39, 40]. +Finally, BCIs are powered with rechargeable batteries and/or +inductive power transfer. These components are packaged in +hermetically-fused silica or titanium capsules. While the power +limit considered safe for permanent implantation varies on +the implant’s target location and depth, we use 15 mW as a +conservative limit [23, 41] for each of Hull’s constituent BCI +nodes. +B. Neural Processing Applications & Algorithmic Kernels +Future BCI applications will collect data across multiple +brain sites, and compare histories of stored neural signals +across them. Many applications exhibit these needs, including +algorithms for neuromuscular rehabilitation and neuropsychi- +atric disorders [2, 4, 13, 42–44], but we focus on epileptic +seizure propagation and detection of movement intent as they +form the bulk of emerging BCI use [2, 5, 45–47]. In addition, +we also consider spike sorting, a crucial kernel widely used +in many applications [48, 49]. Spike sorting differs from +seizure propagation and movement intent in that it is not a full +application in itself. Nevertheless, we study it because it is a +prime candidate for wide use in a distributed manner. +1) Epileptic seizure propagation application: Seizures often +migrate across brain regions [26]. Predicting seizure spread +can help explain seizure dynamics and offer treatment options. +When a seizure is detected at a brain site, seizure propagation +algorithms compare neural signals from the originating site +against current and past signals collected from other brain sites +of interest. Correlation measures are used to detect whether +there is a seizure match across brain sites; i.e., whether a seizure +is likely to propagate to another brain region. +Figure 1a shows the steps used (but unsupported in their +entirety in any existing BCI) in standard seizure propagation +pipelines [25, 26]. First, seizure signals are detected in the +signals from each electrode in all the brain regions that the +electrodes probe. This step typically uses band-pass filters or a +fast Fourier transform (FFT) on continuous signal windows to +generate features, followed by a classifier like a support vector +machine (SVM) [50]. Alternatively, clinicians may manually +annotate the onset of a seizure. +Once a seizure is detected in a region at a specific point in +time, the signal window from that region is compared with all +the concurrent and previous windows from all other regions, +up to a chosen time in the past. +2) Detection of movement intent application: BCIs can infer +coarse-grained movement from reading single sites of the motor +cortex region [51, 52], but more fine-grained movement intent +(e.g., the movement of individual fingers grasping an object) +requires reading neural activity from multiple brain regions [45, +53, 54]. Figure 1b shows a typical computational pipeline +that infers fine-grained movement intent [47, 55–57]. Neural +signals from all electrodes in all target brain sites are first +filtered or converted into the frequency domain using FFT +(a) Seizure propagation analysis. +(b) Decoding movement intent and stimulating response to it. +(c) Spike sorting to separate the combined electrode activity. +Fig. 1: Main BCI application steps. BCIs do not yet support +on-device seizure propagation or multi-site movement intent. +for feature extraction. Then, the features are all pushed into +a classifier to deduce intended movement. Linear SVMs are +commonly used for classification because they are effective, +and because their parameters are intuitive for neuroscientists to +reason about [3, 55, 58, 59]. Intended movement is then relayed +to an external agent like a prosthetic arm. The prosthetic arm’s +movement then has to be conveyed to the brain regions (e.g., +the sensorimotor cortex) responsible for sensing the individual’s +environment using neural stimulation patterns [60, 61]. +3) Spike sorting algorithmic kernel: Spike sorting is an +exemplar of key signal transformations that comprise important +applications, and that benefit from engagement with multiple +brain sites. Most sensor arrays used in existing BCIs have +electrodes that measure the combined electrical activity of +a cluster of neurons, rather than that of individual neurons. +Spike sorting detects all the peaks in the combined electrode +activity and separates them into a series of constituent signal +spikes from distinct neurons. Figure 1c shows this algorithm. +It measures the distance of each signal peak from several +spike templates, and the nearest template is chosen as the +peak’s spike. In some variants [62], the templates are obtained +dynamically from clustering the peaks. Spike distances are +measured with dynamic time warping (DTW) or earth movers +distance (EMD) [63, 64], which are computationally expensive. +Modern spike sorting methods are too slow to be deployed +online; distributed spike sorting has even higher overheads. +No existing BCIs support the signal processing needed for +historical analysis of seizure and movement intent activity +emanating from multiple brain sites, and for distributed spike +sorting. Most designs use a single implanted device that +senses and processes information from the brain region probed +by the implant [16, 17, 23, 27, 28]. Some designs use +distributed sensors that do not directly connect to computational +support [31, 32], and offload data to an external device. But, +the lack of on-device distributed processing precludes BCI +support for applications that require ms-scale decisions, such +as preempting propagation of seizures, or control of prosthetics. +C. Locality-Sensitive Hashing for Signal Comparison +All the applications described previously use signal com- +parison that is expensive. We use locality-sensitive hashing +for fast time series matching [65] to meet Hull’s ms-scale +latency constraints. We face two challenges in using locality- +3 + +Propagation +Electrode +Seizure +Signal similarity in +Data +Detection +all other regionsStimulation +Electrode +Feature +SVM +Data +ExtractionSorted Spikes +Electrode +Spike +Template match +Data +Detection +for every spike(a) Hull overview. +(b) The processor fabric in each of Hull’s nodes. +Fig. 2: The Hull BCI is made up of nodes that are implanted in distinct brain sites. The nodes communicate wirelessly with +each other and external agents. Each Hull node has sensors, radios, analog/digital conversion, processing fabric, and storage; +the processing fabric contains hardware accelerators and configurable switches that can be used to create different pipelines. +sensitive hashing. The first is the presence of variable-latency +computations involving randomization, and the other is the +need to support multiple comparison measures—the choice of +measure varies across BCI uses [63, 64, 66]. We leverage prior +work on two locality-sensitive hashing schemes developed for +DTW [67] and EMD [68]. Subsequent sections describe how +we modify them to suit the needs of Hull’s target applications. +The DTW hash generation process [67] first creates sketches +of the signal by using the dot product of a random vector with +sliding windows in the signal. If the dot product is positive, +the sketch value for the window is 1; otherwise, it is 0. Next, +it counts the occurrences of n-grams formed by n consecutive +sketch values. The n-grams and their counts are used by a +randomized weighted min-hash to produce the final hash. +The original EMD hash [68] is obtained by first calculating +the dot product of the entire signal with a random vector, and +computing a linear function of the dot product’s square root. +D. Flexibility as a Goal in Brain-Computer Interface Design +A key takeaway from Sections II-B and II-C is the need for +flexible support of compute on emerging BCIs. Indeed, this +is a topic explored in recent work on the HALO architecture +for BCIs [23, 69, 70]. Prior to HALO, power efficiency was +achieved by specializing BCIs to offer a specific type of +computation for a specific brain region. However, flexibility is +an important requirement for future BCIs for several reasons: +First, there is no single best signal processing pipeline for +a task; instead, there exist several distinct signal processing +pipelines with different tradeoffs [24, 35, 71]. For Hull, this +means that the specific hardware accelerators needed to support +target computational pipelines (e.g., DTW vs cross-correlation), +and the configuration of key parameters in these accelerators +(e.g., window sizes, thresholds) must be customizable to users. +Second, BCIs may be used in different ways [34, 35]. One +use is autonomous operation, monitoring neural activity and +stimulating neurons when a harmful event occurs. An example +is epileptic seizure monitoring and deep brain stimulation to +preempt the seizure before its onset [71]. Alternatively, BCIs +may translate neural activity into commands for an external +device [33] (e.g., the commands to move a prosthetic) or the +letters to be displayed on a screen [5]. It is common for the +BCI to also translate the external activity into neural feedback +(e.g., to recreate the sense of touch and movement) [72]. +Third, beyond clinical uses, the same BCI platform should +support algorithmic deployment and data collection for research +and exploration of the brain sciences [5, 35, 73]. In these +cases, many applications and usage modes may be necessary +depending on the desired experiment.Some of these uses may +require interactive monitoring, where the BCI and a clinician +are part of the decision-making loop [37]. In this case, the +BCI operates autonomously until it detects abnormal activity, +such as the onset of a seizure. When this happens, it alerts +a clinician, who can use additional data from the individual +to determine the course of action [37]. A useful BCI system +must be customizable to support these different scenarios. +Beyond these scenarios, there are many practical reasons +that BCIs should be flexible, such as changes in the individ- +ual’s neurological conditions (which may require modifying +treatment protocols), changes in electrode behavior from the +immune response of the brain to the BCI etc. [5, 35, 71]. +Supporting high performance with flexibility under extreme +power and latency constraints is challenging. Like HALO, +Hull relies on modular hardware accelerators (henceforth +referred to as processing elements or PEs) to form various +signal processing pipelines. Unlike HALO and any existing +BCI, however, Hull supports the distributed signal processing +applications in Section II-B for the first time. +III. T H E D E S I G N O F T H E H U L L S Y S T E M +Figure 2 shows the Hull BCI and its constituent Hull +nodes implanted in different regions of the brain. Hull nodes +communicate with one another wirelessly. An ILP scheduler +maps applications and interactive queries onto Hull’s nodes. +Each Hull node contains 16-bit ADCs/DACs, a reconfigurable +processor with several PEs, an integrated physical storage layer +made of non-volatile memory (NVM), separate radios for Hull’s +4 + +Closed-loop +Prosthesis +ILP +NVM +Autonomous +Operation +Queries +External +Write +Radio +DAC +Configuration +Read +ADC +Intra-BCI +Processor +Radio +Brain +Interactive +Tissue +Power Supply +Monitoring +Data↑ +CSEL +MC +SC +NGRAM +DTW +HCONV +EMDH +GATE +CCHECK +NEO +THR +HFREQ +DCOMP +FFT +BBF +SVM +HCOMP +UNPACK +XCOR +NPACKnodes to communicate with one another (i.e., intra-BCI radios) +and externally (i.e., external radio), and a power supply. +A. Rewriting Applications for On-Device Processing +We make three changes to existing BCI applications to +run them on Hull (to meet real-time constraints), rather than +relying on external processing. First, we rewrite the signal +processing pipelines to use fast hash-based signal comparison +in the common case, falling back to more time-consuming +approaches (e.g., cross-correlation or DTW) only when more +accurate computation is really necessary. Second, we allow our +applications to use memory. Third, we observe that classifiers +commonly used in neuroscience are linear (e.g., SVMs), and +therefore compute classifier outputs hierarchically across Hull’s +nodes in a manner that reduces network communication. +Figure 3a shows our newly created seizure propagation +application. While functionally equivalent to the standard +version, our application is made up of three phases—seizure +detection, hash comparison, and exact signal matching. On +every sample at all electrodes, we generate new hashes for +each sliding signal window (e.g., one hash for a 120-sample +window), and store them on the on-device non-volatile memory +in each Hull node (Section III-B). When a Hull node detects +a seizure locally (i.e., in the brain region that it probes), it +broadcasts the hashes of the signal windows that were classified +as a seizure. All other Hull nodes check if these hashes match +with any of their recently stored local hashes, and respond when +a match is found. A match indicates that a seizure experienced +in one brain region likely has a correlated seizure in another +region. To ascertain this, the Hull node that initially detected +the seizure broadcasts the entire signal window for the signals +that resulted in a hash collision. Seizure propagation is then +confirmed by running an exact comparison with these signals at +the nodes that had the hash collision. Since the full signal data +and exact similarity matches are performed only when necessary, +computation per Hull node and communication among Hull +nodes is reduced by two orders of magnitude compared to the +baseline application pipelines in Section II-B. +Figure 3c shows that we use a similar approach to enable, +for the first time, an online version of spike sorting even in +distributed scenarios. Like seizure propagation, spike sorting +benefits from hash-based signal processing and memory. The +templates are stored in NVM, and distance computation is +replaced with hash collision checks. Because spike sorting is a +precursor to many neural processing algorithms [35, 48], this +online realization of it for the first time unlocks the ability to +support many spike sorting-centered applications. +Finally, movement intent also benefits from computing our +linear classifier hierarchically. Figure 3b shows the pipeline +that Hull supports. Each Hull node computes a partial classifier +output from the signals it receives and transmits the output. +One node, the leader, computes the final SVM classification and +communicates it to an external prosthetic device. The prosthetic +device’s movements are broadcast back to Hull; each node then +electrically stimulates the sensorimotor cortex of the brain to +simulate the “feeling” of having moved a natural limb. +(a) Seizure propagation. +(b) Decoding movement intent and stimulating response to it. +(c) Spike sorting. +Fig. 3: High-level overview of the BCI applications supported +for online distributed processing in Hull. +B. Flexible & Energy-Efficient Accelerator Design +Figure 2b shows the processing fabric that we design for +each of Hull’s nodes. Several accelerators or PEs are connected +via programmable switches to realize many signal processing +pipelines. A low-power microcontroller (MC) support mis- +cellaneous workloads for which there are no PEs. The PEs +are designed for flexibility to support various computational +functions, power/energy- and area-efficient acceleration, and +deterministic latency and energy consumption to enable our ILP +scheduler to optimally map application tasks onto our acceler- +ators. We use the recently-published HALO architecture [23] +as a starting point to realize a set of PEs that are useful for +single-implant scenarios, and then go beyond to realize PEs +that accelerate our distributed neural applications. +Hull includes PEs for single-site spike detection (NEO– +non-linear energy operator; DWT–discrete wavelet transform), +compression (LZ4; LZMA), feature extraction (FFT–discrete +fast Fourier transform; XCOR–cross-correlation measure; BBF– +Butterworth bandpass filtering), thresholding (THR), conditional +(GATE), classification (SVM–linear support vector machine), +and the radio for communication with systems outside of Hull. +Hull then integrates several new PEs to support distributed +computation, fine-grained wireless communication, and access +to per-node NVM. Each PE has appropriately sized SRAM +buffers to support its processing. The PEs include support for: +1) Hash generation: +Hull supports +Euclidean, cross- +correlation, DTW, and EMD; we support configurability of +hash settings for all four measures. +First, we identify that important parameters of the DTW hash +(e.g., size and step of the sliding window), and n-gram length +(Section II-C) can be modified to also support Euclidean, and +cross-correlation measures. There is no need for new hardware +to support additional means of configurability beyond what is +already needed for the DTW-hash parameters. +5 + +Signal +Source +Device +Hash +Hash +Generation +Seizure +Broadcast +Transmit +Sensor +Detection +Hash +Signal +Remote +Collision +Signal +Devices +Check +SimilarityIntent +Source +Feature +Local +Global +Al +Sensor +Device +Extraction +SVM +SVM +Devices +Remote +Feature +Local +Sensor +Devices +Extraction +SVMSorted +Spikes +Spike +Hash +Nearest +Sensor +Generation +Template Lookup +Detection +TemplateSecond, we identify that the DTW and EMD hashes share +dot product computation of the signal with a random vector +(Section II-C), enabling the reuse of hardware. +Finally, we select a different weighted min-hash algorithm +for the last step of the DTW hash than the one originally +proposed in prior work [67]. Our approach [74] preserves hash +properties while achieving deterministic latency and power. +Our hash generation uses three PEs: HCONV, to obtain the +dot product of a configurable signal window with a random +vector; NGRAM, to compute the n-gram counts in a signal and +generate the DTW-based hash; and EMDH, to square root the +dot product, and other operations to generate the EMD hash. +2) Hash collision check: To determine signal similarity +across multiple brain sites, the hashes received over the network +by the Hull nodes must be compared with the locally generated +hashes in the recent past (e.g., 100 ms). Each Hull node uses +a CCHECK PE that receives decompressed hashes from the +network, stores them in SRAM registers, and sorts them in +place. The PE requests the storage controller (SC) to read +the hashes to be compared from the NVM. These hashes are +compared with those in the registers using binary search. +3) Signal similarity: CSEL identifies signals for exact +signal comparison using DTW, EMD, and Euclidean distance. +For DTW, we build a pipelined implementation that uses +the standard DTW algorithm [75] with a Sakoe-Chiba band +parameter for faster computation [76]. This PE can also support +Euclidean distance computation by using the Sakoe-Chiba band +parameter to be 1. We use the microcontroller to run EMD [77] +for now, although we will build custom PEs in the future. +4) Intra-BCI network compression and packing: The intra- +Hull network transmits hashes and signals. We compress the +hashes but transmit uncompressed raw signals. Compression +makes data more vulnerable to bit errors. Because the hashes +are used only for approximate matching, bit errors are not +as critical to the quality of signal correlation. But, the raw +signals are used for accurate matching. Measures like DTW +are naturally resilient to single-bit errors in the signal, but their +quality worsens rapidly with erroneous compressed signals. +Compression PEs (i.e., LZ/LZMA) built for HALO do not +meet Hull’s power and latency constraints for hashes. Instead, +we build PEs customized to our particular data/communication +needs. The HFREQ PE collects the hash values (and sorts them +by frequency of occurrence) that a Hull node must transmit. The +HCOMP PE encodes the hashes first with dictionary coding, +then uses run-length encoding of the dictionary indexes [78], +and finally uses Elias-γ coding [79] on the run-length counts. +HCOMP’s compression ratio is only 10% lower than that of +LZ4/LZMA, but consumes ≈7× less power. +Compressed data is sent to the NPACK PE, which adds +checksums before transmission. The UNPACK and DCOMP +PEs decode and decompress packets on the receiving side. +5) Storage control: An SC PE manages NVM access. SC +uses SRAM to buffer data before NVM writes in 4 KB pages. +The SRAM also permits data buffering during NVM erase +operations when writes cannot be accepted. Finally, SC (and +the SRAM) permits data reorganization to accelerate future +reads from the NVM (Section III-C). SC uses registers to store +metadata about data written by the ADC and hash PEs (e.g., the +last written page and the size of written data). This accelerates, +for example, the search for recent common signal data. +6) Microcontroller: +The MC runs at low frequency +(20 MHz), and integrates 8 KB memory. It configures individual +PEs into target pipelines (Section IV) and receives commands to +stimulate neurons either for stopping a seizure or for conveying +neural feedback from a prosthetic. The MC can be used for +general-purpose computation not supported by any PEs such +as new algorithms, or infrequently run system operations such +as clock synchronization (Section III-F). +7) Well-defined throughput: Each PE operates in its own +clock domain, like prior work [23], but also supports multiple +frequencies. This enables each PE to lower operating frequency +(and reduce power) to the minimum necessary to sustain the +PE’s target data rate, for varying input electrode counts. This +feature also ensures fixed latency even when PEs process a +variable number of input electrode signals. We design each PE +to support a maximum frequency f P E +max which is high enough +to support the maximum data processing rate required. We use +a configurable register that can be used to set the frequency to +f P E +max/k, where k is user-programmable. The clock frequency +is varied using a simple state machine that uses a counter to +only pass through every k clock pulses. The power consumed +by this counter is in the µW range [80], much lower than the +per-PE power. Overall, the dynamic power of the PEs scales +linearly with the frequency. This also enables deterministic +power and latency and helps optimal scheduling (Section III-E). +C. On-device non-volatile memory +Each Hull node integrates 128 GB on-device NVM to store +raw neural signals, hashes of these signals, and pre-loaded data +needed by applications (e.g., templates for spike sorting). We +divide the NVM into four partitions, one for each of these +classes of data, and another for use by the MC. The sizes +of the partitions are configurable through registers. When a +partition is full, the oldest data in the partition is overwritten. +We optimize the layout of signal and hash data in the NVM +for performance and power. Hull’s ADCs (and hash generation +PEs) process electrode samples sequentially. If the data is stored +in this manner, extracting a contiguous segment of one signal +would require long-latency reads from multiple discontinuous +NVM locations. Instead, we store contiguous chunks (where +a chunk size is user-specified) of each signal. Retrieving the +signal (or hashes) at a particular electrode and time-step need +only offset calculations. SC enables this reorganization as it +buffers data in 4 KB SRAM pages before NVM writes. +D. Networking +We use separate radios for intra-Hull and external device +communication as the required distances and communication +needs are different. For intra-Hull communication, we use a +custom network protocol with a fixed schedule across the nodes. +The schedule is decided by an ILP based on application goals +6 + +Fig. 4: Seizure detection and propagation on Hull. The colors of the PEs are matched with the high-level tasks from Figure 3a. +(Section III-E). To coordinate intra-Hull communication, we +use TDMA for its simplicity and deterministic behavior. +Each network packet has an 84-bit header, and a variable data +size up to a maximum packet size of 256 bytes. The header and +the data have 32-bit CRC32 [81] checksums. On a checksum +mismatch, the receiver simply discards the packet and does not +participate in the pipeline for processing the current sample. +However, we find that while it is best to discard erroneous +packets with hashes, erroneous packets carrying raw signal +data can still be used without adversely affecting the overall +application because of the resiliency of measures like DTW. +E. Task Scheduling on Accelerators, Storage, & Networking +As input to our ILP scheduler, users provide a description of +the desired computation as a dataflow pipeline using functions +of the PEs, or as an interactive query from which the dataflow +can be extracted (Section IV). They also provide the priorities +of the tasks in the application (e.g., seizure detection versus +signal comparison), and constraints like the overall response +latency. A higher priority task ensures that the system processes +more neural signals in this task relative to the others when all +signals cannot be processed for all tasks due to power or latency +constraints. The ILP maps each function to the corresponding +PE in one or more of Hull’s nodes. +The ILP considers each possible mapping of application +tasks (e.g., seizure detection, hash comparison) to PEs as a flow, +and maximizes the weighted sum of the number of channels +processed in each flow. It uses three major constraints: +Latency: End-to-end latencies through the PEs and communi- +cation links must be below a specified limit. +Power: The power consumed by all the PEs and links at all +times must be below a specified limit. +Communication: Only one flow is allowed to use the radio at +any time because of TDMA. +Our ILP setup is simple because of the behavior of the PEs. +With variable throughput processing, the latency of processing +any number of input signals is the same. The dynamic power +consumed by a PE scales predictably linearly with the input +size (since frequency scales linearly). Finally, the system allows +two flows to share the same PE. When this occurs and electrode +signals to be processed are allocated to both flows, the signals +from each flow are interleaved so that they are all run at the +same frequency—completing within the same time as if they +were run independently. The hardware tags the signals from +each flow so that they are routed to the correct destinations. +F. Clock Synchronization +Hull’s distributed processing requires the clocks in each BCI +node to be synchronized up to a few µs of precision. Hull’s +clocks are based on pausable clock generators and clock control +units [82, 83] that suffer only picoseconds of clock uncertainty, +a scale much smaller than our µs target. Hull operates at the +temperature of the human body and does not experience clock +drift due to temperature variance. Nevertheless, Hull runs clock +synchronization once a day using SNTP [84]. +One of the Hull nodes is set up to act as the SNTP server, to +which all other nodes send messages to synchronize their time. +The clients send their previously synchronized clock times, +and current times, while the server sends its corresponding +times. The difference between these values is used to adjust +the clocks. This process repeats a few times until all the +clocks are synchronized within the desired precision. During +clock synchronization, the intra-Hull network is unavailable for +application use. However, operations that do not require the +network (e.g., seizure detection) or NVM access can continue. +IV. D E P L OY I N G H U L L F O R BCI A P P L I C AT I O N S +Hull supports autonomous epileptic seizure propagation +in autonomous, movement intent detection for closed-loop +prosthesis, online spike sorting, and interactive querying. +Autonomous seizure propagation and detection: Figure 4 +shows Hull’s implementation of autonomous seizure detection +and propagation. The choice of the PE functions is based +on prior work [50]. This implementation uses XCOR, BBF, +and FFT to extract features from the ADC measurements and +uses an SVM to detect a seizure. When a seizure is detected, +the nodes exchange hashes for comparison. To confirm that a +seizure is indeed likely being propagated, Hull uses the DTW +distance of the signals across nodes, and electrically stimulates +the brain in response to predicted propagation within 10 ms. +The dataflow in Figure 4 is fed to the ILP to schedule this +application on Hull. The ILP generates an optimal mapping of +the functions and generates a configuration code. This code is +run by each Hull node’s microcontroller to configure the PEs. +Online spike sorting: Figure 5 shows the mapping of online +spike sorting to Hull. The template-matching version pre-loads +the NVM in the nodes with templates and their hashes. +Fig. 5: Spike sorting on Hull. +7 + +NEO +sC +ADC +HCONV +EMDH +GATE +CCHECK +MCSource +Remote +Device +FFT +Devices +SVM +ADC +BBF +THR +SC +XCOR +UNPACK +DCOMP +CCHECK +DTW +THR +SC +NGRAM +NPACK +HCONV +GATE +HFREQ +HCOMP +CSELMovement intent detection and feedback: Figure 6 shows +how Hull implements detection of movement intent and +feedback, augmented from prior work[55]. Each node extracts +features from its local signals and computes a partial SVM +output. Then, one node receives the partial SVM outputs and +computes the commands for the prosthetic. The movements of +the prosthetic are transmitted wirelessly, and each node runs a +stimulation algorithm for its region to provide neural feedback. +Fig. 6: Query pipeline for movement intent application +Interactive querying: +Interactive queries are used to read +multi-site data or modify system configuration. The general +format for an interactive query follows a select-project structure, +akin to SQL queries [85]: +from [set of devices] select +data[electrodes][time range] where condition +The query specifies select criteria, i.e., the range of time +from which data is requested, along with the nodes from which +the data should be returned, and the project criteria, i.e., the +conditions that the selected data must satisfy. Similar to select- +project-based SQL queries, Hull’s interactive query interface +can support a wide range of complex queries. The project +conditions are evaluated on the PEs when possible, and on the +microcontroller otherwise. The following illustrates an example +query to fetch ±100 ms of data from all devices from the time +they detected a seizure in the last 5 s. This example requires +seizure detection using 120 ms windows of the raw signal data. +from * select data[:][t-100:t+100] where +seizure_detect(data[t-120:t]) and t >= -5000 +and t <= 0 +Complex examples can supply template signals and request +data from nodes that recorded signals similar to the templates. +Queries are separately compiled and the extracted dataflow is +sent to the ILP, which finalizes query execution schedules. +Users can also set up the pipelines of specific tasks; e.g., +a clinician may modify seizure_detect to use only FFT +for feature extraction instead of FFT, BBF and XCOR as in +Figure 4. Such a configuration does not need the ILP. +Interactive queries use a power-hungry radio, precluding +simultaneous execution of queries and autonomous tasks +in some cases. Some of these are either slowed down or +temporarily paused; e.g., when a clinician responds to a seizure +alert and requests recent signal data, seizure propagation has +to be paused to send the data to the clinician. +V. M E T H O D O L O G Y +Processing fabric: Hull’s PEs are designed with a commercial +28 nm fully-depleted silicon-on-insulator (FD-SOI) CMOS +process and synthesized using the Cadence® suite of tools. +We use standard cell libraries from STMicroelectronic and +foundry-supplied memory macros that are interpolated to 40 °C, +which is close to human body temperature. We design each +PE for its highest frequency, and scale the power when using +them at lower frequency. We run multi-corner, physically-aware +synthesis, and use latency and power measurements from the +worst variation corner. Table I shows these values. We taped +out early designs of the PEs at 12 nm to confirm these values. +TABLE I: Latency and Power of the PEs. +Processing +Max Freq +Power (µW ) +Latency +Elements +(MHz) +Leakage +Dyn/Elec +(mS) +FFT +15.7 +141.97 +9.02 +4.00 +XCOR +85 +377.00 +44.11 +4.00 +BBF +6 +66.00 +0.35 +4.00 +SVM +3 +99.00 +0.53 +1.67 +THR +16 +2.00 +0.11 +0.06 +NEO +3 +12.00 +0.03 +4.00 +HCONV +3 +89.89 +0.80 +1.50 +NGRAM +0.2 +15.69 +0.08 +1.50 +EMDH +0.03 +10.47 +0.00 +0.04 +GATE +5 +67.00 +0.63 +0.00 +HFREQ +2.88 +61.98 +0.52 +4.00 +HCOMP +2.88 +77.00 +0.65 +4.00 +NPACK +3 +3.53 +5.49 +0.008 +UNPACK +3 +3.53 +5.49 +0.008 +DCOMP +16.393 +7.20 +0.14 +0.50 +CCHECK +16.393 +7.20 +0.14 +0.50 +CSEL +0.1 +4.00 +6.00 +0.04 +SC +3.2 +95.30 +1.64 +0.03-4 +DTW +50 +167.93 +26.94 +0.003 +We assume that each node uses a standard 96-electrode +array [86] to sense neural activity, and a configurable 16-bit +ADC [87] generating 30 K samples per second per electrode. +The ADC dissipates 2.88 mW per sample from all 96 electrodes. +Each node has a DAC to support electrical stimulation of brain +tissue [88], a process that consumes ≈0.6 mW of power. +Radio parameters. We use a radio that can transmit/receive +up to 10 m to external devices, at 46 Mbps, 250 MHz frequency, +and which consumes 9.2 mW. For intra-Hull communication, we +consider a state-of-the-art radio designed for safe implantation +in the brain [89]. While the radio was originally designed for +asymmetric transmission/reception, we modify it for symmetric +communication. Our intra-Hull radio supports a transmission +distance of 20 cm (i.e., > 90th percentile head breadth [90]). To +estimate the power and data rates, we use path-loss models [91], +with a path-loss parameter of 3.5 for transmission through the +brain, skull, and skin, consistent with prior studies [92, 93]. +We calculate that our radio can transmit/receive 7 Mbps at +4.12 GHz and consumes 1.721 mW of power. +Non-volatile memory. We use NVMs with 4 KB page sizes +and 1 MB block sizes. The NVMs can read 8 bytes, write a +page, or erase a block in one operation. We use SLC NAND +parameters like erase time (1.5 ms), program time (350 us), +and voltage (2.7 V) from industrial technical reports [94] with +NVSim [95]. We choose a low operating power transistor type +in NVSim, and use a temperature of 40 °C. NVSim assesses +a leakage power of 0.252 mW, dynamic energies of 164.4 nJ +and 261.143 nJ per page for reads and writes, respectively. We +also use these parameters to size our SC buffers to 24 KB. +Electrophysiological data. We use publicly available electro- +physiological data for our evaluation [96, 97]. For seizure +detection and propagation, we use a data from the Mayo +8 + +Source +ADC +FFT +SVM +UNPACK +SVM +THR +MC +MC +Device +Remote +MC +ADC +FFT +SVM +NPACK +DevicesSeizure +Detection +Spike +Sorting +Signal +Similarity +Movement +Intent +100 +101 +102 +103 +104 +Max Aggregate Throughput (Mbps) +Central No-Hash +Central +Hull No-Hash +Hull +(a) Maximum aggregate throughput of Hull versus +alternative BCI architectures. +1 +2 +4 +8 +16 +32 +64 +Number of devices +1 +10 +100 +1000 +10000 +Max Aggregate Throughput (Mbps) +DTW Comparison +Hash Comparison +DTW One-All +Movement Intent +(b) Maximum +aggregate +throughput +of +communication-dependent tasks in Hull. +0 +50 +100 +150 +Sensor Data Rate (Mbps) +0 +25 +50 +75 +100 +125 +Throughput (Mbps) +Seizure Detection +Spike Sorting +(c) Maximum throughput of tasks without inter- +node communication, using re-designed PEs. +Fig. 7: Experimental quantification of Hull’s benefits. +Clinic [97] of a patient (label “I001 P013”) with 76 electrodes +implanted in the parietal/occipital lobes. This data-set was +recorded for 4 days at 5 KHz, and is annotated with hundreds +of seizure instances. We upscaled the sampling frequency to +30 KHz, and split the dataset to emulate multiple BCI devices. +We use consecutive and overlapping 4 ms windows (120 +samples) from the electrodes to detect seizures [98]. For +propagation, we check similarity with a seizure-positive signal +in the last 100 ms from electrode data in all nodes [98]. For +hash pipelines, we use one 8-bit hash for 120 sample data. +For spike sorting, we use the Spikeforest dataset [96, 99]. +This dataset contains recordings collected from the CA1 region +of a rat hippocampus using tetrode electrodes recorded at +30 KHz sampling frequency. The dataset contains spikes from +10 neurons, with 65, 000 spikes that were manually sorted. +Alternative system architectures. Table II shows the systems +that we compare Hull against. Hull No-Hash uses the same +Hull architecture but does not use hashes. The power saved by +removing the hash processing PEs is allocated to the remaining +tasks optimally. Hull No-Hash does not require re-writing +the applications for hash-based processing. Central uses one +processing node with the same processor as Hull, and multiple +sensors that are connected using wires. Finally, Central No- +Hash is a centralized design without hash processing, like +most existing BCIs [27, 31, 100]. We do not consider wireless +centralized designs as they need a radio and have lesser compute +available than the wired ones. We also do not consider designs +without memory as they do not support seizure propagation. We +map our applications onto all systems using the ILP, ensuring +that each node consumes < 15 mW. +TABLE II: Alternative BCI designs. +Design +Architecture +Comparison +Communication +Hull (Proposed) +Distributed +Hash, Signal +Wireless +Hull-No hash +Distributed +Signal +Wireless +Central +Centralized +Hash, Signal +Wired +Central-No hash +Centralized +Signal +Wired +VI. E VA L UAT I O N +A. Comparing BCI Architectures +Figure 7a shows the maximum aggregate throughput of the +systems in Table II. A task’s maximum aggregate throughput is +achieved when it is the only task running in the system, summed +over all nodes. Central No-Hash has the worst throughput for +all tasks. This design suffers from having just one processor +and from using expensive signal processing. Central increases +throughput by an order of magnitude for tasks that benefit +from hashing (spike sorting and signal similarity). However, +the single processor remains the bottleneck for all tasks. +Hull No-Hash has distributed processors and enjoys higher +aggregate seizure detection and movement intent. However, it +performs poorly for tasks that need signal comparison (signal +similarity, spike sorting). For these tasks, Hull No-Hash has +lower throughput than Central because it does not use hashes. +Hull uses distributed hash-based processing and has the highest +aggregate throughput for all tasks. Compared to Central-No +hash, which is closest to state-of-the-art BCIs, Hull’s data +rates are an order of magnitude higher for seizure detection, +and movement intent detection, and are nearly three orders of +magnitude higher for signal similarity and spike sorting. +B. Throughput for Communication-Dependent Tasks +Figure 7b shows the maximum aggregate throughput of +the communication-dependent task (hash comparison, DTW +comparison, and movement intent), with various node counts. +DTW Comparison uses all-to-all comparison of raw signals. +It has a lower throughput than the other tasks because only +16 out of 96 electrode signals can be transmitted for all-to- +all comparison. The reason is that new electrode samples are +obtained at 47 Mbps from the ADC, but the intra-Hull radio can +only transmit about 7 Mbps. Increasing the number of nodes +decreases the throughput further because of the communication +delays. Because Hull uses a TDMA network, where slots for +network access are serialized, DTW Comparison has the worst +throughput and scales poorly with node count. +An alternative DTW One-All, which only uses one-to-all +DTW comparison, scales better since its communication latency +9 + +doesn’t increase with the number of nodes. However, a one-to- +all comparison is insufficient for general BCI applications. +Hash Comparison uses all-to-all hash communication to +check for collisions. Its throughput increases to 470 Mbps until +10 devices, after which it begins to decrease. When the number +of nodes is small, few TDMA slots are required to exchange all +hashes, enabling a linear increase in throughput as a function of +node count. But, as node counts keep increasing, it takes longer +to communicate all hashes and overall throughput reduces. +Finally, Movement Intent uses all-to-one communication of +the partial SVM products. However, as the product is small, its +throughput scales linearly with the number of nodes (note that +the Y-axis in Figure 7b is logarithmic). It also has the highest +aggregate throughput because it needs the least communication. +Figure 7b shows that hashing, and distributing the SVM +computation in Hull enables it to scale to many regions and +with higher data rates than what has been possible. +C. Throughput for Non-Communicating Tasks +We design our PEs for a maximum sensor rate of 47 Mbps +per node (Section V). However, we study potential PE re-design +to support higher processing rates for tasks that do not need +communication. Figure 7c shows the throughput of Seizure +Detection and Spike Sorting for varying per-node signal sensor +rates. Task throughput increases linearly up to 105 Mbps for +spike sorting, and 70 Mbps for seizure detection. Beyond this +sensing rate, the higher frequency of the PEs and ADCs results +in exceeding the device power limit. Nonetheless, these values +are nearly twice as supported by existing single-implant BCIs +and show the robustness of our methodology. +D. Application Level Throughput +The throughput achieved at the application level depends +on the number of implanted nodes. Additionally, when there +are multiple tasks, it depends on the priorities assigned to the +application tasks. Recall that the ILP schedules applications +to optimize a priority-weighted sum of the signals processed +in each task. For seizure detection propagation, Figure 8a +shows the weighted aggregate throughput as a function of the +number of devices, for various weight choices (in the format +seizure detection:hash comparison:DTW comparison). For an +equal priority to seizure detection, DTW processing, and hash +comparison, we find that the maximum throughput is achieved +for 11 nodes. Other weight choices have different optimal node +counts. Note that there is no comparable system for on-device +seizure propagation—Hull is the first design with this feature. +Movement intent has only one task, and its throughput (in +number of intents detected per second), is shown in Figure 8b. +This metric accounts for only movement intent detection, and +not for the variable response latency of the prosthetic device. +Hull spike sorts up to 12, 250 spikes per second per node +with 82% accuracy, comparing well to the state of the art [96]. +E. Interactive Queries +We consider three types of common queries applied on data +ranging from the past 100 ms (≈7 MB over all nodes) to the +0 +20 +40 +60 +80 +Number of Devices +0 +10 +20 +30 +40 +50 +Weighted Throughput (Mbps) +1 +1:1:1 +3:1:1 +1:3:1 +0 +20 +40 +60 +80 +Number of Devices +0 +50 +100 +150 +Movement Intents / Second +2 +(a) Weighted throughput of seizure +propagation tasks. +0 +20 +40 +60 +80 +Number of Devices +0 +10 +20 +30 +40 +50 +Weighted Throughput (Mbps) +1 +1:1:1 +3:1:1 +1:3:1 +0 +20 +40 +60 +80 +Number of Devices +0 +50 +100 +150 +Movement Intents / Second +2 +(b) Movement intents +per second +(without device movement time). +Fig. 8: Application level metrics on Hull. +past 1 s (≈60 MB). They are: Q1, which returns all signals that +were detected as a seizure; Q2, which returns all signals that +matched with a template using a hash; and Q3, which returns +all data in the timeframe. For Q1 and Q2, we vary the fraction +of data that tests positive for their condition. +Figure 9 shows Hull’s throughput with 11 nodes for our +queries. Hull supports up to 10 queries per second (QPS) for +Q1 and Q2 over the last 100 ms data (the common case). If Q2 +is run with DTW instead of hash-based search, we see a QPS +of 8, which is only slightly lower, but the power consumption +increases from 3.57 mW for the hash vs the entire 15 mW for +DTW based matching. Thus, DTW-based matching is unsuitable +when interactively querying in response to a seizure. +Q3 on this data takes 1.21 s, yielding a throughput of ≈0.8. +In interactive querying, the external radio, which consumes +high power, is the bottleneck. +As the data to be searched increases, the query latency +increases linearly due to the radio latency. However, Hull +can still process 1 QPS for Q1 and Q2 for the past 1 s data +(≈60 MB), making it suitable for real-time use. +7 (110 ms) +24 (400 ms) +42 (700 ms) +60 (1000 ms) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +5% +5% +50% +50% +100% +100% +100% +5% +5% +50% +50% +100% +100% +100% +5% +5% +50% +50% +100% +100% +100% +5% +5% +50% +50% +100% +100% +100% +Query Data Size (MB) (Time Range) +0.1 +1 +10 +Queries per second +Q1 +Q2 +Q3 +Fig. 9: Interactive query throughput on Hull with 11 nodes. +F. Hashing +Accuracy: We vary the parameters of all our hash functions +and show the performance of the best configuration for seizure +propagation and spike sorting. Figure 11 shows the accuracy +(TP: True positive, TN: True negative, FP: False positive, FN: +False negative) for the four hash functions. XCOR and EMD +hashes have ≈ 85% accuracy while Euclidean and DTW have +over 90% accuracy. The high true positive rate of our DTW +10 + +0 +25 +50 +75 +100 +125 +Window Size +0 +1 +2 +3 +4 +5 +6 +Ngram Size +XCOR +DTW +Euclidean +XCOR Euclidean DTW +EMD +0 +50 +100 +Percentage (%) +TP +FN +TN +FP +Fig. 10: Hash accuracy. +0 +25 +50 +75 +100 +125 +Window Size +0 +1 +2 +3 +4 +5 +6 +Ngram Size +XCOR +DTW +Euclidean +XCOR Euclidean DTW +EMD +0 +25 +50 +75 +100 +Percentage (%) +True PositiveTrue Negative +Fig. 11: Hash flexibility. +Standard Bit Error Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Percentage of Packets with Error (%) +30 +50 +70 +90 +100 +Hash Packets +Signal Packets +DTW Failure +10 +4 +10 +5 +10 +6 +0 +5 +10 +15 +Fig. 12: Bit error rates. +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 13 14 15 +Number of devices +0.01 +0.1 +1 +10 +100 +1000 +10000 +100000 +Time (seconds) +Full ILP +Reduced +Fig. 13: Time to solve the ILP. +hash is particularly beneficial for the seizure propagation (note +that false positives are removed using exact DTW). +Parameter selection: Figure 11 shows the best parameters +of our hash implementation (window size and n-gram size— +Section II-C) to approximate each of Euclidean, cross correla- +tion, and DTW similarity. We also show parameters (with lighter +colors in the figure) that are within 90% of the true positive +rate achieved by the corresponding best configuration. This +flexibility enables reusing a single fast hardware accelerator +for different measures. +G. Impact of Network Bit Error Rate +The intra-Hull network protocol drops packets carrying +hashes when there is a checksum error but allows signal packets +to flow into PEs since signal similarity measures are naturally +resilient to a few errors. We simulate various bit-error ratios +(BERs) using uniformly random bit flips in the packet header +and data. Figure 12 shows the fraction of hash or signal packets +with an error at different BERs, and the fraction of erroneous +signal packets that flipped the similarity measure (DTW). For +reference, the BER is <10−4 for the radio we use [89]. +Figure 12 shows that signals and hashes suffer errors as +BER increases, but signals are more susceptible since they are +longer. But, even though several signal packets suffer errors, +they have no impact on the final signal similarity outcome. +H. ILP Performance +The complexity of the ILP increases with the number of +pipeline stages in the application and the number of Hull nodes. +When all nodes are the same and have the same power/energy +constraints, the schedule of one node can be replicated (with +a constant offset) on all other nodes and remain optimal. We +call this method Reduced ILP. However, we cannot apply +this method when the nodes are different or have different +constraints. Figure 13 shows the time taken to solve the ILP +and the reduced version for varying numbers of devices for the +seizure propagation application. We measure this time when +using GLPK, an open-source ILP solver, with default settings +on an Intel-Xeon E5-2620 v3 machine with 93 GB RAM. +As expected, the solver time for the standard ILP increases +exponentially with the number of devices, taking ≈2 hours +with 11 devices. For >12 devices, the ILP did not finish within +24 hours and was terminated. The reduced ILP however, can +be solved in less than 10ms for any number of devices. +VII. R E L AT E D W O R K +Commercial and research BCIs have focused largely on single +brain location monitoring and stimulation [16, 17, 23, 27, 28], +and have no support for distributed systems, making them +inhospitable for the applications that we target. +Most implantable BCIs offer little to no storage capacity +and stream data out continuously instead. NeuroChip [100] is +an exception, but is wired to an external case storing a 128 +GB SD card that must be physically extracted for offline data +analysis. Hull is the first to use storage for pre-processing and +reduce computation by using the hash. +A growing interest in distributed analyses of the brain [1, +13, 25, 26] has motivated the design of rudimentary multi-site +BCIs [31, 32, 101]. Prior studies [31, 32] propose microchips +that stream sensor data wirelessly to a central hub outside the +skull using back-scattering radio techniques. Unfortunately, +these approaches are restricted in their interfacing bandwidth +as they rely on centralized processing and communication. +Although recent work has studied unary neural networks on +single-site BCIs [102], we will study distributed neural network +models for seizure detection, propagation, spike sorting, and +movement intent for multi-side BCIs going forward. Hull can +support any algorithm with linear computational complexity +without significant changes to the ILP formulation. However, +neural network inference, which is super-linear, may require +non-linear formulations for scheduling. Using MILP and +approximations for such PEs may be a suitable extension. +VIII. C O N C L U S I O N & F U T U R E W O R K +Hull enables distributed BCI interfacing that can scale to +multiple regions, and provides for the first time, on-device +computation for important BCI applications. Hull offers two +orders of magnitude higher task throughput, and real-time +support for interactive querying with up to 10 QPS over 7 MB +data or 1 QPS over 60 MB data. +Hull will influence the wider field of IoT devices, ranging +from low-power temperature and voltage sensors [103], AR/VR +devices, to devices in smart home, factory, and vehicle settings. +These devices must collect and process large volumes of data +on the edge, as communicating this data to centralized locations +is likely to be near impossible for today’s cloud infrastructure. +Similar to Hull, a network of power-constrained devices will +need to process large volumes of data, often with flexible +processing requirements to support rapidly evolving use cases. +11 + +Hull’s design principles– i.e., its modular PE architecture, +fast-but-approximate hash-based approach to signal similarity, +support for low-power and efficiently-indexed non-volatile +storage, and a centralized planner that produces near-optimal +mapping of task schedules to devices – can be instrumental to +success in other IoT environments as well. +R E F E R E N C E S +[1] R. A. Andersen, T. Aflalo, L. Bashford, D. Bj˚anes, and +S. Kellis, “Exploring Cognition with Brain–Machine +Interfaces,” Annual Review of Psychology, vol. 73, pp. +131–158, 2022. +[2] M. A. Lebedev and M. A. Nicolelis, “Brain-machine +interfaces: From basic science to neuroprostheses and +neurorehabilitation,” Physiological reviews, vol. 97, no. 2, +pp. 767–837, 2017. +[3] S. Chandrasekaran, M. Fifer, S. Bickel, L. Osborn, +J. Herrero, B. Christie, J. Xu, R. K. Murphy, S. Singh, +M. F. Glasser et al., “Historical perspectives, challenges, +and future directions of implantable brain-computer +interfaces for sensorimotor applications,” Bioelectronic +medicine, vol. 7, no. 1, pp. 1–11, 2021. +[4] A. S. Widge, D. D. Dougherty, and C. T. Moritz, “Af- +fective brain-computer interfaces as enabling technology +for responsive psychiatric stimulation,” Brain-Computer +Interfaces, vol. 1, no. 2, pp. 126–136, 2014. +[5] J. J. Shih, D. J. Krusienski, and J. R. Wolpaw, “Brain- +computer interfaces in medicine,” in Mayo clinic pro- +ceedings, vol. 87, no. 3. +Elsevier, 2012, pp. 268–279. +[6] D. J. McFarland, J. Daly, C. Boulay, and M. A. Parvaz, +“Therapeutic applications of BCI technologies,” Brain- +Computer Interfaces, vol. 4, no. 1-2, pp. 37–52, 2017. +[7] L. Huang and G. van Luijtelaar, “Brain computer inter- +face for epilepsy treatment,” Brain-Computer Interface +Systems-Recent Progress and Future Prospects, 2013. +[8] C. M¨uhl, B. Allison, A. Nijholt, and G. Chanel, “A survey +of affective brain computer interfaces: principles, state- +of-the-art, and challenges,” Brain-Computer Interfaces, +vol. 1, no. 2, pp. 66–84, 2014. +[9] G. Buzs´aki, C. A. Anastassiou, and C. Koch, “The Origin +of Extracellular Fields and Currents — EEG, ECoG, LFP +and Spikes,” Nature Reviews Neuroscience, vol. 13, pp. +407–420, May 2012. +[10] B. Pesaran, M. Vinck, G. T. Einevoll, A. Sirota, P. Fries, +M. Siegel, W. Truccolo, C. E. Schroeder, and R. Srini- +vasan, “Investigating large-scale brain dynamics using +field potential recordings: analysis and interpretation,” +Nature neuroscience, vol. 21, no. 7, pp. 903–919, 2018. +[11] J. del R. Milan and J. M. Carmena, “Invasive or +Noninvasive: Understanding Brain-Machine Interface +Technology [Conversations in BME],” IEEE Engineering +in Medicine and Biology Magazine, vol. 29, pp. 16–22, +Jan 2010. +[12] E. Musk et al., “An integrated brain-machine interface +platform with thousands of channels,” Journal of medical +Internet research, vol. 21, no. 10, p. e16194, 2019. +[13] K. M. Szostak, P. Feng, F. Mazza, and T. G. Con- +standinou, “Distributed Neural Interfaces: Challenges and +Trends in Scaling Implantable Technology,” Handbook +of Neuroengineering, pp. 1–37, 2021. +[14] A. B. Rapeaux and T. G. Constandinou, “Implantable +brain machine interfaces: first-in-human studies, tech- +nology challenges and trends,” Current opinion in +biotechnology, vol. 72, pp. 102–111, 2021. +[15] U.S. Food and Drug Administration, “FDA authorizes +marketing of device to facilitate muscle rehabilitation in +stroke patients,” https://www.fda.gov/news-events/press- +announcements/fda-authorizes-marketing-device- +facilitate-muscle-rehabilitation-stroke-patients, +April +2021. +[16] F. T. Sun and M. J. Morrell, “The RNS System: +responsive cortical stimulation for the treatment of +refractory partial epilepsy,” Expert review of medical +devices, vol. 11, no. 6, pp. 563–572, 2014. +[17] Medtronic, “Deep Brain Stimulation Systems - Activa +PC,” +https://www.medtronic.com/us-en/healthcare- +professionals/products/neurological/deep-brain- +stimulation-systems/activa-pc.html, November 2018, +Retrieved August 10, 2019. +[18] U.S. +Food and Drug +Administration, “Implanted +Brain-Computer Interface (BCI) Devices for Patients +with Paralysis or Amputation - Non-clinical Testing and +Clinical Considerations,” https://www.fda.gov/regulatory- +information/search-fda-guidance-documents/implanted- +brain-computer-interface-bci-devices-patients- +paralysis-or-amputation-non-clinical-testing, February +2019, Retrieved August 10, 2019. +[19] J. J. Han, “Synchron receives FDA approval to be- +gin early feasibility study of their endovascular, brain- +computer interface device,” Artificial Organs, vol. 45, +no. 10, pp. 1134–1135, 2021. +[20] C. Serrano-Amenos, F. Hu, P. T. Wang, S. Kellis, +R. A. Andersen, C. Y. Liu, P. Heydari, A. H. Do, and +Z. Nenadic, “Thermal analysis of a skull implant in brain- +computer interfaces,” in 2020 42nd Annual International +Conference of the IEEE Engineering in Medicine & +Biology Society (EMBC). +IEEE, 2020, pp. 3066–3069. +[21] P. D. Wolf, “Thermal Considerations for the Design of +an Implanted Cortical Brain-Machine Interface (BMI),” +Indwelling Neural Implants: Strategies for Contending +with the In Vivo Environment, 2008. +[22] I. Stevenson and K. Kording, “How Advances in Neural +Recording Affect Data Analysis,” Nature neuroscience, +vol. 14, pp. 139–42, 02 2011. +[23] I. Karageorgos, K. Sriram, J. Vesel`y, M. Wu, M. Powell, +D. Borton, R. Manohar, and A. Bhattacharjee, “Hardware- +software co-design for brain-computer interfaces,” in +2020 ACM/IEEE 47th Annual International Symposium +on Computer Architecture (ISCA). +IEEE, 2020, pp. +391–404. +[24] R. Zelmann, A. C. Paulk, I. Basu, A. Sarma, A. Yousefi, +B. Crocker, E. Eskandar, Z. Williams, G. R. Cosgrove, +12 + +D. S. Weisholtz et al., “CLoSES: A platform for closed- +loop intracranial stimulation in humans,” NeuroImage, +vol. 223, p. 117314, 2020. +[25] V. K. Jirsa, T. Proix, D. Perdikis, M. M. Woodman, +H. Wang, J. Gonzalez-Martinez, C. Bernard, C. B´enar, +M. Guye, P. Chauvel et al., “The Virtual Epileptic Patient: +Individualized whole-brain models of epilepsy spread,” +Neuroimage, vol. 145, pp. 377–388, 2017. +[26] F. Bartolomei, S. Lagarde, F. Wendling, A. McGonigal, +V. Jirsa, M. Guye, and C. B´enar, “Defining epileptogenic +networks: Contribution of SEEG and signal analysis,” +Epilepsia, vol. 58, no. 7, pp. 1131–1147, 2017. +[27] Hossein Kassiri, Sana Tonekaboni, M. Tariqus Salam, +Nima Soltani, karim Abdelhalim, Jose Luis Perez Ve- +lasquez, Roman Genov, “Closed-Loop Neurostimulators: +A Survey and A Seizure-Predicting Design Example +for Intractable Epilepsy Treatment,” IEEE Transactions +on Biomedical Circuits and Systems, vol. 11, no. 5, pp. +1026–1040, 2017. +[28] G. O’Leary and D. M. Groppe and T. A. Valiante +and N. Verma and R. Genov, “Nurip: Neural interface +processor for brain-state classification and programmable- +waveform neurostimulation,” IEEE Journal of Solid State +Circuits, vol. 53, 2018. +[29] J. N. Y. Aziz, K. Abdelhalim, R. Shulyzki, R. Genov, B. L. +Bardakjian, M. Derchansky, D. Serletis, and P. L. Carlen, +“256-Channel Neural Recording and Delta Compression +Microsystem With 3D Electrodes,” IEEE Journal of +Solid-State Circuits, vol. 44, no. 3, pp. 995–1005, March +2009. +[30] T. Chen, C. Jeng, S. Chang, H. Chiueh, S. Liang, Y. Hsu, +and T. Chien, “A Hardware Implementation of Real- +Time Epileptic Seizure Detector on FPGA,” 2011 IEEE +Biomedical Circuits and Systems Conference (BioCAS), +pp. 25–28, Nov 2011. +[31] N. Ahmadi, M. L. Cavuto, P. Feng, L. B. Leene, +M. Maslik, F. Mazza, O. Savolainen, K. M. Szostak, C.-S. +Bouganis, J. Ekanayake et al., “Towards a Distributed, +Chronically-Implantable Neural Interface,” in 2019 9th +International IEEE/EMBS Conference on Neural Engi- +neering (NER). +IEEE, 2019, pp. 719–724. +[32] J. Lee, V. Leung, A.-H. Lee, J. Huang, P. Asbeck, P. P. +Mercier, S. Shellhammer, L. Larson, F. Laiwalla, and +A. Nurmikko, “Neural recording and stimulation using +wireless networks of microimplants,” Nature Electronics, +vol. 4, no. 8, pp. 604–614, 2021. +[33] B. Zhu, U. Shin, and M. Shoaran, “Closed-Loop Neural +Prostheses With On-Chip Intelligence: A Review and a +Low-Latency Machine Learning Model for Brain State +Detection,” IEEE Transactions on Biomedical Circuits +and Systems, 2021. +[34] C. Zrenner, P. Belardinelli, F. M¨uller-Dahlhaus, and +U. Ziemann, “Closed-Loop Neuroscience and Non- +Invasive Brain Stimulation: A Tale of Two Loop,” +Frontiers in cellular neuroscience, vol. 10, p. 92, 2016. +[35] M. D. Murphy, D. J. Guggenmos, D. T. Bundy, and +R. J. Nudo, “Current Challenges Facing the Translation +of Brain Computer Interfaces from Preclinical Trials to +Use in Human Patients,” Front Cell Neurosci, vol. 9, p. +497, 2015. +[36] A. O. Hebb, J. J. Zhang, M. H. Mahoor, C. Tsiokos, +C. Matlack, H. J. Chizeck, and N. Pouratian, “Creating +the feedback loop: Closed loop neurostimulation,” Neu- +rosurgery Clinics of North America, vol. 25, no. 1, pp. +187–204, 2014. +[37] V. Sladky, P. Nejedly, F. Mivalt, B. H. Brinkmann, I. Kim, +E. K. St. Louis, N. M. Gregg, B. N. Lundstrom, C. M. +Crowe, T. P. Attia et al., “Distributed brain co-processor +for tracking spikes, seizures and behaviour during elec- +trical brain stimulation,” Brain Communications, vol. 4, +no. 3, p. fcac115, 2022. +[38] M. A. Kural, J. Jing, F. F¨urbass, H. Perko, E. Qerama, +B. Johnsen, S. Fuchs, M. B. Westover, and S. Beniczky, +“Accurate identification of EEG recordings with interictal +epileptiform discharges using a hybrid approach: Artifi- +cial intelligence supervised by human experts,” Epilepsia, +vol. 63, no. 5, pp. 1064–1073, 2022. +[39] J. D. Simeral, T. Hosman, J. Saab, S. N. Flesher, +M. Vilela, B. Franco, J. N. Kelemen, D. M. Brandman, +J. G. Ciancibello, P. G. Rezaii et al., “Home Use of +a Percutaneous Wireless Intracortical Brain-Computer +Interface by Individuals With Tetraplegia,” IEEE Trans- +actions on Biomedical Engineering, vol. 68, no. 7, pp. +2313–2325, 2021. +[40] M. Yin, D. A. Borton, J. Komar, N. Agha, Y. Lu, +H. Li, J. Laurens, Y. Lang, Q. Li, C. Bull et al., “Wire- +less Neurosensor for Full-Spectrum Electrophysiology +Recordings during Free Behavior,” Neuron, vol. 84, no. 6, +pp. 1170–1182, 2014. +[41] S. M. Kim, P. Tathireddy, R. Normann, and F. Solzbacher, +“Thermal Impact of an Active 3-D Microelectrode Array +Implanted in the Brain,” IEEE Transactions on Neural +Systems and Rehabilitation Engineering, vol. 15, pp. +493–501, Dec 2007. +[42] O. Sporns, “Graph theory methods: applications in brain +networks,” Dialogues in clinical neuroscience, 2022. +[43] M. T. Bilge, A. K. Gosai, and A. S. Widge, “Deep Brain +Stimulation in Psychiatry,” Psychiatric Clinics, vol. 41, +no. 3, pp. 373–383, 2018. +[44] G. Deco and M. L. Kringelbach, “Great Expectations: +Using Whole-Brain Computational Connectomics for +Understanding Neuropsychiatric Disorders,” Neuron, +vol. 84, no. 5, pp. 892–905, 2014. +[45] J. A. Gallego, T. R. Makin, and S. D. McDougle, “Going +beyond primary motor cortex to improve brain–computer +interfaces,” Trends in Neurosciences, vol. 45, no. 3, pp. +176–183, 2022. +[46] D. Camargo-Vargas, M. Callejas-Cuervo, and S. Maz- +zoleni, “Brain-Computer Interfaces Systems for Upper +and Lower Limb Rehabilitation: A Systematic Review,” +Sensors, vol. 21, no. 13, p. 4312, 2021. +[47] S. J. Bensmaia and L. E. Miller, “Restoring sensorimotor +13 + +function through intracortical interfaces: progress and +looming challenges,” Nature Reviews Neuroscience, +vol. 15, no. 5, pp. 313–325, 2014. +[48] S. Todorova, P. Sadtler, A. Batista, S. Chase, and V. Ven- +tura, “To sort or not to sort: the impact of spike-sorting +on neural decoding performance,” Journal of neural +engineering, vol. 11, no. 5, p. 056005, 2014. +[49] H. G. Rey, C. Pedreira, and R. Q. Quiroga, “Past, present +and future of spike sorting techniques,” Brain research +bulletin, vol. 119, pp. 106–117, 2015. +[50] H. Shiao, V. Cherkassky, J. Lee, B. Veber, E. E. Patterson, +B. H. Brinkmann, and G. A. Worrell, “SVM-Based +System for Prediction of Epileptic Seizures From iEEG +Signal,” IEEE Transactions on Biomedical Engineering, +vol. 64, no. 5, pp. 1011–1022, May 2017. +[51] I. K. Niazi, N. Jiang, O. Tiberghien, J. F. Nielsen, +K. Dremstrup, and D. Farina, “Detection of movement +intention from single-trial movement-related cortical +potentials,” Journal of Neural Engineering, vol. 8, no. 6, +p. 066009, 2011. +[52] M. Jochumsen, I. K. Niazi, H. Rovsing, C. Rovsing, G. A. +Nielsen, T. K. Andersen, N. Dong, M. E. Sørensen, +N. Mrachacz-Kersting, N. Jiang et al., “Detection of +movement intentions through a single channel of elec- +troencephalography,” in Replace, Repair, Restore, Relieve– +Bridging Clinical and Engineering Solutions in Neurore- +habilitation. +Springer, 2014, pp. 465–472. +[53] C. L. Ojakangas, A. Shaikhouni, G. M. Friehs, A. H. +Caplan, M. D. Serruya, M. Saleh, D. S. Morris, and J. P. +Donoghue, “Decoding Movement Intent From Human +Premotor Cortex Neurons for Neural Prosthetic Appli- +cations,” Journal of Clinical Neurophysiology: Official +Publication of the American Electroencephalographic +Society, vol. 23, no. 6, p. 577, 2006. +[54] J. F. Kalaska, “From Intention to Action: Motor Cortex +and the Control of Reaching Movements,” Progress in +Motor Control, pp. 139–178, 2009, series Title: Advances +in Experimental Medicine and Biology. +[55] A. Shakeel, M. S. Navid, M. N. Anwar, S. Mazhar, +M. Jochumsen, and I. K. Niazi, “A Review of Techniques +for Detection of Movement Intention Using Movement- +Related Cortical Potentials,” Computational and Mathe- +matical Methods in Medicine, vol. 2015, 2015. +[56] D. M. Taylor, S. I. H. Tillery, and A. B. Schwartz, +“Direct Cortical Control of 3D Neuroprosthetic Devices,” +Science, vol. 296, no. 5574, pp. 1829–1832, 2002. +[57] E. E. Fetz, “Restoring motor function with bidirectional +neural interfaces,” Progress in Brain Research, vol. 218, +pp. 241–252, 2015. +[58] M. Capogrosso, T. Milekovic, D. Borton, F. Wagner, E. M. +Moraud, J.-B. Mignardot, N. Buse, J. Gandar, Q. Barraud, +D. Xing et al., “A Brain–Spine Interface Alleviating Gait +Deficits after Spinal Cord Injury in Primates,” Nature, +vol. 539, no. 7628, pp. 284–288, 2016. +[59] K.-R. Muller, C. Anderson, and G. Birch, “Linear and +nonlinear methods for brain-computer interfaces,” IEEE +Transactions on Neural Systems and Rehabilitation +Engineering, vol. 11, no. 2, pp. 165–169, 2003. +[60] R. Darie, M. Powell, and D. Borton, “Delivering the +Sense of Touch to the Human Brain,” Neuron, vol. 93, +no. 4, pp. 728–730, 2017. +[61] S. Tafazoli, C. J. MacDowell, Z. Che, K. C. Letai, C. R. +Steinhardt, and T. J. Buschman, “Learning to control the +brain through adaptive closed-loop patterned stimulation,” +Journal of Neural Engineering, vol. 17, no. 5, p. 056007, +2020. +[62] U. Rutishauser, E. M. Schuman, and A. N. Mamelak, +“Online detection and sorting of extracellularly recorded +action potentials in human medial temporal lobe record- +ings, in vivo,” Journal of Neuroscience Methods, vol. +154, no. 1-2, pp. 204–224, 2006. +[63] B. Sotomayor-G´omez, F. P. Battaglia, and M. Vinck, +“SpikeShip: A method for fast, unsupervised discovery +of high-dimensional neural spiking patterns,” bioRxiv, +pp. 2020–06, 2021. +[64] L. Grossberger, F. P. Battaglia, and M. Vinck, “Unsuper- +vised clustering of temporal patterns in high-dimensional +neuronal ensembles using a novel dissimilarity measure,” +PLoS Comput Biol, vol. 14, no. 7, p. e1006283, 07 2018. +[65] Y. B. Kim, “Physiological Time Series Retrieval and +Prediction with Locality-Sensitive Hashing,” Ph.D. dis- +sertation, Massachusetts Institute of Technology, 2017. +[66] Y. Cao, N. Rakhilin, P. H. Gordon, X. Shen, and E. C. +Kan, “A real-time spike classification method based on +dynamic time warping for extracellular enteric neural +recording with large waveform variability,” Journal of +Neuroscience Methods, vol. 261, pp. 97–109, 2016. +[67] C. Luo and A. Shrivastava, “SSH (Sketch, Shingle, & +Hash) for Indexing Massive-Scale Time Series,” in NIPS +2016 Time Series Workshop. +PMLR, 2017, pp. 38–58. +[68] D. Gorisse, M. Cord, and F. Precioso, “Locality-Sensitive +Hashing for Chi2 Distance,” IEEE Transactions on +Pattern Analysis and Machine Intelligence, vol. 34, no. 2, +pp. 402–409, 2011. +[69] A. Bhattacharjee and R. Manohar, “HALO: A Flexible +and Low Power Processing Fabric for Brain-Computer +Interfaces,” in 2022 IEEE Hot Chips 34 Symposium +(HCS). +IEEE Computer Society, 2022, pp. 1–37. +[70] I. Karageorgos, K. Sriram, J. Vesel´y, N. Lindsay, X. Wen, +M. Wu, M. Powell, D. Borton, R. Manohar, and A. Bhat- +tacharjee, “Balancing specialized versus flexible compu- +tation in brain–computer interfaces,” IEEE Micro, vol. 41, +no. 3, pp. 87–94, 2021. +[71] F. T. Sun and M. J. Morrell, “Closed-loop Neurostim- +ulation: The Clinical Experience,” Neurotherapeutics, +vol. 11, no. 3, pp. 553–563, 2014. +[72] G. W. V. Vidal, M. L. Rynes, Z. Kelliher, and S. J. +Goodwin, “Review of Brain-Machine Interfaces Used in +Neural Prosthetics with New Perspective on Somatosen- +sory Feedback through Method of Signal Breakdown,” +Scientifica, vol. 2016, 2016. +[73] C. Herff, D. J. Krusienski, and P. Kubben, “The Poten- +14 + +tial of Stereotactic-EEG for Brain-Computer Interfaces: +Current Progress and Future Directions,” Frontiers in +Neuroscience, vol. 14, p. 123, 2020. +[74] D. Karger, E. Lehman, T. Leighton, R. Panigrahy, +M. Levine, and D. Lewin, “Consistent Hashing and Ran- +dom Trees: Distributed Caching Protocols for Relieving +Hot Spots on the World Wide Web,” in ACM Symposium +on Theory of Computing (STOC), 1997. +[75] J. Kruskall and M. Liberman, “The symmetric time +warping algorithm: From continuous to discrete,” Time +Warps, String Edits and Macromolecules: The Theory +and Practice of Sequence Comparison, 1983. +[76] H. Sakoe and S. Chiba, “Dynamic programming al- +gorithm optimization for spoken word recognition,” +IEEE Transactions on Acoustics, Speech, and Signal +Processing, vol. 26, pp. 159–165, 1978. +[77] O. Pele and M. Werman, “Fast and robust Earth Mover’s +Distances,” in 2009 IEEE 12th international conference +on computer vision. +IEEE, 2009, pp. 460–467. +[78] I. M. Pu, Fundamental data compression. +Butterworth- +Heinemann, 2005. +[79] P. Elias, “Universal codeword sets and representations of +the integers,” IEEE Transactions on Information Theory, +vol. 21, no. 2, pp. 194–203, 1975. +[80] N. Bingham and R. Manohar, “Qdi constant-time coun- +ters,” IEEE Transactions on Very Large Scale Integration +(VLSI) Systems, vol. 27, no. 1, pp. 83–91, 2018. +[81] W. W. Peterson and D. T. Brown, “Cyclic codes for +error detection,” Proceedings of the IRE, vol. 49, no. 1, +pp. 228–235, 1961. +[82] K. Yun and R. Donohue, “Pausible clocking: a first +step toward heterogeneous systems,” in Proceedings +International Conference on Computer Design. VLSI +in Computers and Processors, 1996, pp. 118–123. +[83] A. Moreno and J. Cortadella, “Synthesis of all-digital +delay lines,” in 2017 23rd IEEE International Symposium +on Asynchronous Circuits and Systems (ASYNC), 2017, +pp. 75–82. +[84] D. Mills, “Simple Network Time Protocol (SNTP),” RFC +Editor, RFC 1769, 1995. +[85] D. D. Chamberlin and R. F. Boyce, “SEQUEL: A +structured English query language,” in Proceedings of +the 1974 ACM SIGFIDET (now SIGMOD) workshop on +Data description, access and control. +Association for +Computing Machinery, 1974, pp. 249–264. +[86] Blackrock +Microsystems, +“The +Benchmark +for +Multichannel, High-density Neural Recording,” https: +//www.blackrockmicro.com/electrode-types/utah-array/, +Retrieved August 10, 2019. +[87] J. Shen, A. Shikata, L. D. Fernando, N. Guthrie, B. Chen, +M. Maddox, N. Mascarenhas, R. Kapusta, and M. C. W. +Coln, “A 16-bit 16-ms/s sar adc with on-chip calibration +in 55-nm cmos,” IEEE Journal of Solid-State Circuits, +vol. 53, no. 4, pp. 1149–1160, 2018. +[88] Medtronic, +“Medtronic +Activa +PC +Multi- +program +neurostimulator +implant +manual,” +http://www.neuromodulation.ch/sites/default/files/ +pictures/activa PC DBS implant manuel.pdf, +2008, +Retrieved August 10, 2019. +[89] H. Rahmani and A. Babakhani, “A wirelessly powered +reconfigurable fdd radio with on-chip antennas for multi- +site neural interfaces,” IEEE Journal of Solid-State +Circuits, vol. 56, no. 10, pp. 3177–3190, 2021. +[90] J. W. Young, “Head and Face Anthropometry of Adult +U.S. Citizens,” Federal Aviation Administration, Tech. +Rep., 1993. +[91] A. F. Molisch, K. Balakrishnan, C.-C. Chong, S. Emami, +A. Fort, J. Karedal, J. Kunisch, H. Schantz, U. Schuster, +and K. Siwiak, “IEEE 802.15. 4a channel model-final +report,” IEEE P802, vol. 15, no. 04, p. 0662, 2004. +[92] M. S¨arest¨oniemi, C. Pomalaza-Raez, K. Sayrafian, +T. Myllyl¨a, and J. Iinatti, “A Preliminary Study of +RF Propagation for High Data Rate Brain Telemetry,” +in Body Area Networks. Smart IoT and Big Data +for Intelligent Health Management. +Cham: Springer +International Publishing, 2022, pp. 126–138. +[93] A. Taparugssanagorn, A. Rabbachin, M. H¨am¨al¨ainen, +J. Saloranta, J. Iinatti et al., “A Review of Channel +Modelling for Wireless Body Area Network in Wire- +less Medical Communications,” The 11th International +Symposium on Wireless Personal Multimedia Communi- +cations (WPMC, 2008. +[94] “MT29F128G08AKCABH2-10,” +https://www. +micron.com/products/nand-flash/slc-nand/part- +catalog/mt29f128g08akcabh2-10, Retrieved November +17, 2022. +[95] X. Dong, C. Xu, Y. Xie, and N. P. Jouppi, “Nvsim: +A circuit-level performance, energy, and area model +for emerging nonvolatile memory,” IEEE Transactions +on Computer-Aided Design of Integrated Circuits and +Systems, vol. 31, no. 7, pp. 994–1007, 2012. +[96] J. Magland, J. J. Jun, E. Lovero, A. J. Morley, C. L. +Hurwitz, A. P. Buccino, S. Garcia, and A. H. Barnett, +“SpikeForest, reproducible web-facing ground-truth vali- +dation of automated neural spike sorters,” Elife, vol. 9, +p. e55167, 2020. +[97] “IEEG.ORG,” https://www.ieeg.org, Retrieved November +17, 2022. +[98] P. Shah, A. Ashourvan, F. Mikhail, A. Pines, L. Kini, +K. Oechsel, S. R. Das, J. M. Stein, R. T. Shinohara, D. S. +Bassett et al., “Characterizing the role of the structural +connectome in seizure dynamics,” Brain, vol. 142, no. 7, +pp. 1955–1972, 2019. +[99] J. E. Chung, J. F. Magland, A. H. Barnett, V. M. Tolosa, +A. C. Tooker, K. Y. Lee, K. G. Shah, S. H. Felix, +L. M. Frank, and L. F. Greengard, “A Fully Automated +Approach to Spike Sorting,” Neuron, vol. 95, no. 6, pp. +1381–1394, 2017. +[100] L. E. Shupe, F. P. Miles, G. Jones, R. Yun, J. Mish- +ler, I. Rembado, R. L. Murphy, S. I. Perlmutter, and +E. E. Fetz, “Neurochip3: An Autonomous Multichannel +Bidirectional Brain-Computer Interface for Closed-Loop +15 + +Activity-Dependent Stimulation,” Frontiers in Neuro- +science, vol. 15, 2021. +[101] J. E. Chung, H. R. Joo, J. L. Fan, D. F. Liu, A. H. +Barnett, S. Chen, C. Geaghan-Breiner, M. P. Karlsson, +M. Karlsson, K. Y. Lee et al., “High-Density, Long- +Lasting, and Multi-region Electrophysiological Record- +ings Using Polymer Electrode Arrays,” Neuron, vol. 101, +no. 1, pp. 21–31, 2019. +[102] D. Wu, J. Li, Z. Pan, Y. Kim, and J. S. Miguel, “uBrain: +A Unary Brain Computer Interface,” in Proceedings of +the 49th Annual International Symposium on Computer +Architecture. +Association for Computing Machinery, +2022, pp. 468–481. +[103] E. M. Stewart, A. Liao, and C. Roberts, “Open µpmu: +A real world reference distribution micro-phasor mea- +surement unit data set for research and application +development,” IEEE Power Engineering Letters, 2016. +16 + diff --git a/BNE1T4oBgHgl3EQfVgSF/content/tmp_files/load_file.txt b/BNE1T4oBgHgl3EQfVgSF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0597edb9172cf5bc145050d822d9dbda96c9b56f --- /dev/null +++ b/BNE1T4oBgHgl3EQfVgSF/content/tmp_files/load_file.txt @@ -0,0 +1,1678 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf,len=1677 +page_content='A Multi-Site Accelerator-Rich Processing Fabric for Scalable Brain-Computer Interfacing Karthik Sriram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Raghavendra Pradyumna Pothukuchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Michał Gerasimiuk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Oliver Ye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Muhammed Ugur Rajit Manohar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Anurag Khandelwal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Abhishek Bhattacharjee Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' USA Abstract—Hull1 is an accelerator-rich distributed implantable brain-computer interface (BCI) that reads biological neurons at data rates that are 2-3 orders of magnitude higher than the prior art,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' while supporting many neuroscientific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Prior approaches have restricted brain interfacing to tens of megabits per second in order to meet two constraints necessary for effective operation and safe long-term implantation—power dissipation under tens of milliwatts and response latencies in the tens of milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull also adheres to these constraints, but is able to interface with the brain at much higher data rates, thereby enabling, for the first time, BCI-driven research on and clinical treatment of brain-wide behaviors and diseases that require reading and stimulating many brain locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Central to Hull’s power efficiency is its realization as a distributed system of BCI nodes with accelerator-rich compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull balances modular system layering with aggressive cross-layer hardware-software co-design to integrate compute, networking, and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The result is a lesson in designing networked distributed systems with hardware accelerators from the ground up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' I N T RO D U C T I O N Brain-computer interfaces (BCIs) sense the electrical activity of biological neurons and electrically stimulate them to “rewire” neuronal circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' By directly connecting brains to computers, BCIs help advance our understanding of the brain and the mind [1, 2], offer treatment of neurological disorders [2–6], enable industrial robotics [7], permit novel modes of personal entertainment [8], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' BCIs can be realized as surface electrodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', electrical sensors) placed on the scalp above the skull to measure brain activity [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' While such wearable BCIs do not require surgical deployment, the signals they collect are muffled by the skull, making them noisy, low-resolution, and less ideal for forward- looking BCI applications [5, 9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Instead, this work focuses on implantable BCIs that are surgically embedded directly on, around, and in the brain tissue [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Implantable BCIs directly record from and stimulate neurons with high fidelity, spatial resolution, and in real time [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hundreds of individuals use clinically approved implantable BCIs to treat epilepsy, movement disorders, as well as impaired vision [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Implantable BCIs are also being studied in clinical trials to assess their effectiveness in treating brain stroke, memory disorders, paralysis, anxiety/depression, addiction, and more [14, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Conflicting constraints make it challenging to design hard- ware for implantable BCIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' BCIs cannot overheat brain regions 1A hull is the protective outer covering of grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We call our design Hull since it similarly protects the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' by >1 ◦C to avoid cellular damage [20, 21] and must therefore be ultra-low-power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' But, BCI designers are also seeking to leverage improvements in sensor technology that are reading exponentially increasing neuronal data [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It is challenging to constrain power/energy while processing such large data, espe- cially to respond to neuronal activity in real-time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', in ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hardware over-specialization is not a viable way to reduce BCI power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' to enable many research and clinical studies, BCIs must be adequately programmable to personalize algorithms, support several computational methods to treat multiple disorders, and enable deployment of maturing/emerging algorithms [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Complicating BCI design further is the emergence of applications that read and process neural activity from many brain sites over time [1, 13, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This is because the brain’s functions (and disorders) are ultimately based on physical and functional connectivity between brain regions that evolve over time [1, 13, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Existing BCIs [16, 17, 23, 27, 28] are designed for single- site implantation and lack the ability to store adequately long historical neural data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Most BCIs [29, 30] have additional limitations in that they have historically eschewed a subset of programmability, data rates, and flexibility to meet safe power constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These BCIs are specialized to a specific task and/or limit personalization of the algorithm [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Some support more programmability by sacrificing high data rates [16, 17, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Consequently, none support the distributed and rapidly evolving neural data processing that emerging BCI applications require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Recent work on HALO balances flexibility, data rates, and power, but is limited to one brain site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' At best, distributed BCI applications have been studied in prior work that consists of multiple sensor implants that offload processing to external devices with higher power budgets [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' But, this is not a panacea because of the long network latency, privacy, and mobility limitations [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Our work is the first to offer a path toward scalable whole- brain interfacing across multiple brain sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Our solution, Hull, is a distributed BCI consisting of multiple BCI nodes that communicate with one another wirelessly and interface with the brain with an aggregate data rate 2-3 orders of magnitude higher than the prior state-of-art [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each BCI node consists of flexible compute made up of reconfigurable power-efficient domain-specific hardware accelerators customized for important neural processing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These accelerators are tightly co-designed with storage and networking to ensure system-wide adherence to power and response latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='03103v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='DC] 8 Jan 2023 Hull uses a scheduler based on an integer linear program (ILP) to optimally map tasks and algorithms to the hardware accelerators across all of Hull’s nodes, and to create the network and storage access schedules to feed the accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull supports three types of BCI applications [34, 35]: The first category consists of applications that continuously monitor brain activity and respond to aberrant behavior without engaging with agents external to Hull [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This includes, for example, detection of seizures [26], prediction of their spread, and finally, to mitigate symptoms, electrical stimulation of regions where seizures are expected to migrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The second category also monitor the brain continuously but rely on agents external to Hull to respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This includes, for example, the detection of an individual’s intended movement, and relaying of this data to prostheses or assistive devices [3, 33– 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These applications enable paralyzed individuals to actuate machines and restore (partial) sensory function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The third category interactively queries Hull to analyze data from multiple brain sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These queries may be used “in the loop” or “out of the loop”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' With the former, clinicians may need to modify treatment options based on data that the BCI reads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', confirming that Hull has correctly detected seizures and responded appropriately [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The latter refers to interactive queries used by technicians to debug system operations, by clinicians to glean the individual’s medical history, and by researchers to better understand brain function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In supporting these types of applications, Hull offers research insights on building end-to-end computer systems centered on hardware accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Specifically, our choice of hardware accelerators follows several important design principles: First, to maximize power efficiency, as well as simplicity of hardware design, we identify algorithmic kernels within our applications that accelerate not only their computation but also their networking and storage latency needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This is a non-trivial exercise that requires domain-specific knowledge to convert BCI applications into equivalent forms that are amenable to exposing these kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We reformulate known computational pipelines for seizure propagation prediction/treatment and movement intent – two BCI applications that Hull focuses on – to use hash- based similarity measures that identify the neural signals from different brain sites likeliest to be correlated, before applying heavy-weight correlation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The same hashes can drive the design of domain-specific layouts of data in our storage stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Co-designing our hashes with our storage layout permit several power/latency-efficient data retrievals from our storage layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The hashes also enable reducing the data communicated in the intra-BCI network between the Hull nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Second, we design our accelerators to be predictable in latency and power for our target data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Predictable perfor- mance and power characteristics facilitate the optimal design of compute and network schedules—the enabling feature for our ILP scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Designing accelerators with predictable perfor- mance and power requires care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For some accelerators, whose data generation rate is input-dependent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', data compression), we use theoretically-derived worst-case latency and throughput estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Furthermore, we design our accelerators in their own clock domains to enable a range of operating frequencies with well-defined power and performance characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Third, we design our accelerators to be reconfigurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This permits the repurposing of our hash accelerators, for example, to act as hash indices for our storage layer, as filters that reduce our networking traffic, and to be tuned differently depending on the target application being supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Overall, Hull scales brain-computer interface bandwidth beyond what was previously achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull is flexible, recon- figurable, and supports real-time distributed neural processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use a detailed physical synthesis flow in a 28 nm CMOS process (including tapeouts of partial hardware at 12 nm) coupled with network and storage models to evaluate Hull’s power and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We show that Hull can process an aggregate of 460 Mbps wireless data from multiple regions in only 10 ms and dissipates no more than 15 mW at any node, confirming its suitability for autonomous use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In interactive mode, it can support 10 queries per second over 6 MB of data over 10 implants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Existing designs support two orders of magnitude lower data and need intrusive wiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In summary, our specific contributions are: 1) Hull, the first distributed and wireless BCI system that scales to multi-region neural processing in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2) The cross-layer co-design of BCI applications, processing, and storage for scalable and distributed neural processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull is the first to support long-term storage of data, hashing, and database indices to enable distributed signal processing on a single BCI platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3) An evaluation of Hull’s flexibility and performance on epileptic seizure propagation and detection of movement intent, in various deployment scenarios, as well as in the support of more arbitrary queries that may be used interactively by clinicians/technicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull furthers the elevation of hardware accelerators to first- class compute citizens that, like CPUs, can directly engage with networking and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This trend will be crucial to future IoT, swarm, and intermittent computing environments that sustain adaptive and complex functionality while meeting strict safe-use (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', power, latency, throughput) constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' BAC K G RO U N D A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Brain-Computer Interface Design BCI applications typically perform signal measurement, feature extraction, classification/decision-making, and when applicable, neural feedback/stimulation [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The hardware organization of BCIs reflects these four aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Signal mea- surement is performed by electrodes that read the electrical activity of clusters of biological neurons, and analog to digital converters (ADCs) then digitize this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' State-of-the-art sensors consist of 96-256 electrode arrays per implant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' ADCs typically encode measured signal samples with 8-16 bits at a rate of 20-50 K samples per second per electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Digitized data is then relayed to compute logic for feature extraction, based on which classification/decision-making proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' If needed, the electrode arrays are repurposed, after a digital-to-analog (DAC) conversion step, to electrically stimulate the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2 Modern BCIs also use radios to communicate with external agents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', servers, prostheses), presenting an evolution from the surgical cables (which were susceptible to infections and restricted free movement) used in early BCIs [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, BCIs are powered with rechargeable batteries and/or inductive power transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These components are packaged in hermetically-fused silica or titanium capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' While the power limit considered safe for permanent implantation varies on the implant’s target location and depth, we use 15 mW as a conservative limit [23, 41] for each of Hull’s constituent BCI nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Neural Processing Applications & Algorithmic Kernels Future BCI applications will collect data across multiple brain sites, and compare histories of stored neural signals across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Many applications exhibit these needs, including algorithms for neuromuscular rehabilitation and neuropsychi- atric disorders [2, 4, 13, 42–44], but we focus on epileptic seizure propagation and detection of movement intent as they form the bulk of emerging BCI use [2, 5, 45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In addition, we also consider spike sorting, a crucial kernel widely used in many applications [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Spike sorting differs from seizure propagation and movement intent in that it is not a full application in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nevertheless, we study it because it is a prime candidate for wide use in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1) Epileptic seizure propagation application: Seizures often migrate across brain regions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Predicting seizure spread can help explain seizure dynamics and offer treatment options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When a seizure is detected at a brain site, seizure propagation algorithms compare neural signals from the originating site against current and past signals collected from other brain sites of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Correlation measures are used to detect whether there is a seizure match across brain sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', whether a seizure is likely to propagate to another brain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 1a shows the steps used (but unsupported in their entirety in any existing BCI) in standard seizure propagation pipelines [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' First, seizure signals are detected in the signals from each electrode in all the brain regions that the electrodes probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This step typically uses band-pass filters or a fast Fourier transform (FFT) on continuous signal windows to generate features, followed by a classifier like a support vector machine (SVM) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Alternatively, clinicians may manually annotate the onset of a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Once a seizure is detected in a region at a specific point in time, the signal window from that region is compared with all the concurrent and previous windows from all other regions, up to a chosen time in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2) Detection of movement intent application: BCIs can infer coarse-grained movement from reading single sites of the motor cortex region [51, 52], but more fine-grained movement intent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', the movement of individual fingers grasping an object) requires reading neural activity from multiple brain regions [45, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 1b shows a typical computational pipeline that infers fine-grained movement intent [47, 55–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Neural signals from all electrodes in all target brain sites are first filtered or converted into the frequency domain using FFT (a) Seizure propagation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (b) Decoding movement intent and stimulating response to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (c) Spike sorting to separate the combined electrode activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1: Main BCI application steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' BCIs do not yet support on-device seizure propagation or multi-site movement intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Then, the features are all pushed into a classifier to deduce intended movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Linear SVMs are commonly used for classification because they are effective, and because their parameters are intuitive for neuroscientists to reason about [3, 55, 58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Intended movement is then relayed to an external agent like a prosthetic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The prosthetic arm’s movement then has to be conveyed to the brain regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', the sensorimotor cortex) responsible for sensing the individual’s environment using neural stimulation patterns [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3) Spike sorting algorithmic kernel: Spike sorting is an exemplar of key signal transformations that comprise important applications, and that benefit from engagement with multiple brain sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Most sensor arrays used in existing BCIs have electrodes that measure the combined electrical activity of a cluster of neurons, rather than that of individual neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Spike sorting detects all the peaks in the combined electrode activity and separates them into a series of constituent signal spikes from distinct neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 1c shows this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It measures the distance of each signal peak from several spike templates, and the nearest template is chosen as the peak’s spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In some variants [62], the templates are obtained dynamically from clustering the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Spike distances are measured with dynamic time warping (DTW) or earth movers distance (EMD) [63, 64], which are computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Modern spike sorting methods are too slow to be deployed online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' distributed spike sorting has even higher overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' No existing BCIs support the signal processing needed for historical analysis of seizure and movement intent activity emanating from multiple brain sites, and for distributed spike sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Most designs use a single implanted device that senses and processes information from the brain region probed by the implant [16, 17, 23, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Some designs use distributed sensors that do not directly connect to computational support [31, 32], and offload data to an external device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' But, the lack of on-device distributed processing precludes BCI support for applications that require ms-scale decisions, such as preempting propagation of seizures, or control of prosthetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Locality-Sensitive Hashing for Signal Comparison All the applications described previously use signal com- parison that is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use locality-sensitive hashing for fast time series matching [65] to meet Hull’s ms-scale latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We face two challenges in using locality- 3 Propagation Electrode Seizure Signal similarity in Data Detection all other regionsStimulation Electrode Feature SVM Data ExtractionSorted Spikes Electrode Spike Template match Data Detection for every spike(a) Hull overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (b) The processor fabric in each of Hull’s nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2: The Hull BCI is made up of nodes that are implanted in distinct brain sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The nodes communicate wirelessly with each other and external agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each Hull node has sensors, radios, analog/digital conversion, processing fabric, and storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' the processing fabric contains hardware accelerators and configurable switches that can be used to create different pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' sensitive hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The first is the presence of variable-latency computations involving randomization, and the other is the need to support multiple comparison measures—the choice of measure varies across BCI uses [63, 64, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We leverage prior work on two locality-sensitive hashing schemes developed for DTW [67] and EMD [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Subsequent sections describe how we modify them to suit the needs of Hull’s target applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The DTW hash generation process [67] first creates sketches of the signal by using the dot product of a random vector with sliding windows in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' If the dot product is positive, the sketch value for the window is 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' otherwise, it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Next, it counts the occurrences of n-grams formed by n consecutive sketch values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The n-grams and their counts are used by a randomized weighted min-hash to produce the final hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The original EMD hash [68] is obtained by first calculating the dot product of the entire signal with a random vector, and computing a linear function of the dot product’s square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Flexibility as a Goal in Brain-Computer Interface Design A key takeaway from Sections II-B and II-C is the need for flexible support of compute on emerging BCIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Indeed, this is a topic explored in recent work on the HALO architecture for BCIs [23, 69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Prior to HALO, power efficiency was achieved by specializing BCIs to offer a specific type of computation for a specific brain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, flexibility is an important requirement for future BCIs for several reasons: First, there is no single best signal processing pipeline for a task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' instead, there exist several distinct signal processing pipelines with different tradeoffs [24, 35, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For Hull, this means that the specific hardware accelerators needed to support target computational pipelines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', DTW vs cross-correlation), and the configuration of key parameters in these accelerators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', window sizes, thresholds) must be customizable to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Second, BCIs may be used in different ways [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' One use is autonomous operation, monitoring neural activity and stimulating neurons when a harmful event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' An example is epileptic seizure monitoring and deep brain stimulation to preempt the seizure before its onset [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Alternatively, BCIs may translate neural activity into commands for an external device [33] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', the commands to move a prosthetic) or the letters to be displayed on a screen [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It is common for the BCI to also translate the external activity into neural feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', to recreate the sense of touch and movement) [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Third, beyond clinical uses, the same BCI platform should support algorithmic deployment and data collection for research and exploration of the brain sciences [5, 35, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In these cases, many applications and usage modes may be necessary depending on the desired experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='Some of these uses may require interactive monitoring, where the BCI and a clinician are part of the decision-making loop [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In this case, the BCI operates autonomously until it detects abnormal activity, such as the onset of a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When this happens, it alerts a clinician, who can use additional data from the individual to determine the course of action [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A useful BCI system must be customizable to support these different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Beyond these scenarios, there are many practical reasons that BCIs should be flexible, such as changes in the individ- ual’s neurological conditions (which may require modifying treatment protocols), changes in electrode behavior from the immune response of the brain to the BCI etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [5, 35, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Supporting high performance with flexibility under extreme power and latency constraints is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Like HALO, Hull relies on modular hardware accelerators (henceforth referred to as processing elements or PEs) to form various signal processing pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Unlike HALO and any existing BCI, however, Hull supports the distributed signal processing applications in Section II-B for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T H E D E S I G N O F T H E H U L L S Y S T E M Figure 2 shows the Hull BCI and its constituent Hull nodes implanted in different regions of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull nodes communicate with one another wirelessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' An ILP scheduler maps applications and interactive queries onto Hull’s nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each Hull node contains 16-bit ADCs/DACs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' a reconfigurable processor with several PEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' an integrated physical storage layer made of non-volatile memory (NVM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' separate radios for Hull’s 4 Closed-loop Prosthesis ILP NVM Autonomous Operation Queries External Write Radio DAC Configuration Read ADC Intra-BCI Processor Radio Brain Interactive Tissue Power Supply Monitoring Data↑ CSEL MC SC NGRAM DTW HCONV EMDH GATE CCHECK NEO THR HFREQ DCOMP FFT BBF SVM HCOMP UNPACK XCOR NPACKnodes to communicate with one another (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', intra-BCI radios) and externally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', external radio), and a power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rewriting Applications for On-Device Processing We make three changes to existing BCI applications to run them on Hull (to meet real-time constraints), rather than relying on external processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' First, we rewrite the signal processing pipelines to use fast hash-based signal comparison in the common case, falling back to more time-consuming approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', cross-correlation or DTW) only when more accurate computation is really necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Second, we allow our applications to use memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Third, we observe that classifiers commonly used in neuroscience are linear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', SVMs), and therefore compute classifier outputs hierarchically across Hull’s nodes in a manner that reduces network communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 3a shows our newly created seizure propagation application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' While functionally equivalent to the standard version, our application is made up of three phases—seizure detection, hash comparison, and exact signal matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' On every sample at all electrodes, we generate new hashes for each sliding signal window (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', one hash for a 120-sample window), and store them on the on-device non-volatile memory in each Hull node (Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When a Hull node detects a seizure locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', in the brain region that it probes), it broadcasts the hashes of the signal windows that were classified as a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' All other Hull nodes check if these hashes match with any of their recently stored local hashes, and respond when a match is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A match indicates that a seizure experienced in one brain region likely has a correlated seizure in another region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' To ascertain this, the Hull node that initially detected the seizure broadcasts the entire signal window for the signals that resulted in a hash collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Seizure propagation is then confirmed by running an exact comparison with these signals at the nodes that had the hash collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Since the full signal data and exact similarity matches are performed only when necessary, computation per Hull node and communication among Hull nodes is reduced by two orders of magnitude compared to the baseline application pipelines in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 3c shows that we use a similar approach to enable, for the first time, an online version of spike sorting even in distributed scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Like seizure propagation, spike sorting benefits from hash-based signal processing and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The templates are stored in NVM, and distance computation is replaced with hash collision checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Because spike sorting is a precursor to many neural processing algorithms [35, 48], this online realization of it for the first time unlocks the ability to support many spike sorting-centered applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, movement intent also benefits from computing our linear classifier hierarchically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 3b shows the pipeline that Hull supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each Hull node computes a partial classifier output from the signals it receives and transmits the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' One node, the leader, computes the final SVM classification and communicates it to an external prosthetic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The prosthetic device’s movements are broadcast back to Hull;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' each node then electrically stimulates the sensorimotor cortex of the brain to simulate the “feeling” of having moved a natural limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (a) Seizure propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (b) Decoding movement intent and stimulating response to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (c) Spike sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3: High-level overview of the BCI applications supported for online distributed processing in Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Flexible & Energy-Efficient Accelerator Design Figure 2b shows the processing fabric that we design for each of Hull’s nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Several accelerators or PEs are connected via programmable switches to realize many signal processing pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A low-power microcontroller (MC) support mis- cellaneous workloads for which there are no PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The PEs are designed for flexibility to support various computational functions, power/energy- and area-efficient acceleration, and deterministic latency and energy consumption to enable our ILP scheduler to optimally map application tasks onto our acceler- ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use the recently-published HALO architecture [23] as a starting point to realize a set of PEs that are useful for single-implant scenarios, and then go beyond to realize PEs that accelerate our distributed neural applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull includes PEs for single-site spike detection (NEO– non-linear energy operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' DWT–discrete wavelet transform), compression (LZ4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' LZMA), feature extraction (FFT–discrete fast Fourier transform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' XCOR–cross-correlation measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' BBF– Butterworth bandpass filtering), thresholding (THR), conditional (GATE), classification (SVM–linear support vector machine), and the radio for communication with systems outside of Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull then integrates several new PEs to support distributed computation, fine-grained wireless communication, and access to per-node NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each PE has appropriately sized SRAM buffers to support its processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The PEs include support for: 1) Hash generation: Hull supports Euclidean, cross- correlation, DTW, and EMD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' we support configurability of hash settings for all four measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' First, we identify that important parameters of the DTW hash (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', size and step of the sliding window), and n-gram length (Section II-C) can be modified to also support Euclidean, and cross-correlation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' There is no need for new hardware to support additional means of configurability beyond what is already needed for the DTW-hash parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5 Signal Source Device Hash Hash Generation Seizure Broadcast Transmit Sensor Detection Hash Signal Remote Collision Signal Devices Check SimilarityIntent Source Feature Local Global Al Sensor Device Extraction SVM SVM Devices Remote Feature Local Sensor Devices Extraction SVMSorted Spikes Spike Hash Nearest Sensor Generation Template Lookup Detection TemplateSecond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' we identify that the DTW and EMD hashes share dot product computation of the signal with a random vector (Section II-C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' enabling the reuse of hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, we select a different weighted min-hash algorithm for the last step of the DTW hash than the one originally proposed in prior work [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Our approach [74] preserves hash properties while achieving deterministic latency and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Our hash generation uses three PEs: HCONV, to obtain the dot product of a configurable signal window with a random vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' NGRAM, to compute the n-gram counts in a signal and generate the DTW-based hash;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' and EMDH, to square root the dot product, and other operations to generate the EMD hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2) Hash collision check: To determine signal similarity across multiple brain sites, the hashes received over the network by the Hull nodes must be compared with the locally generated hashes in the recent past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', 100 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each Hull node uses a CCHECK PE that receives decompressed hashes from the network, stores them in SRAM registers, and sorts them in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The PE requests the storage controller (SC) to read the hashes to be compared from the NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These hashes are compared with those in the registers using binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3) Signal similarity: CSEL identifies signals for exact signal comparison using DTW, EMD, and Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For DTW, we build a pipelined implementation that uses the standard DTW algorithm [75] with a Sakoe-Chiba band parameter for faster computation [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This PE can also support Euclidean distance computation by using the Sakoe-Chiba band parameter to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use the microcontroller to run EMD [77] for now, although we will build custom PEs in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4) Intra-BCI network compression and packing: The intra- Hull network transmits hashes and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We compress the hashes but transmit uncompressed raw signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Compression makes data more vulnerable to bit errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Because the hashes are used only for approximate matching, bit errors are not as critical to the quality of signal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' But, the raw signals are used for accurate matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Measures like DTW are naturally resilient to single-bit errors in the signal, but their quality worsens rapidly with erroneous compressed signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Compression PEs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', LZ/LZMA) built for HALO do not meet Hull’s power and latency constraints for hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Instead, we build PEs customized to our particular data/communication needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The HFREQ PE collects the hash values (and sorts them by frequency of occurrence) that a Hull node must transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The HCOMP PE encodes the hashes first with dictionary coding, then uses run-length encoding of the dictionary indexes [78], and finally uses Elias-γ coding [79] on the run-length counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' HCOMP’s compression ratio is only 10% lower than that of LZ4/LZMA, but consumes ≈7× less power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Compressed data is sent to the NPACK PE, which adds checksums before transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The UNPACK and DCOMP PEs decode and decompress packets on the receiving side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5) Storage control: An SC PE manages NVM access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' SC uses SRAM to buffer data before NVM writes in 4 KB pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The SRAM also permits data buffering during NVM erase operations when writes cannot be accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, SC (and the SRAM) permits data reorganization to accelerate future reads from the NVM (Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' SC uses registers to store metadata about data written by the ADC and hash PEs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', the last written page and the size of written data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This accelerates, for example, the search for recent common signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6) Microcontroller: The MC runs at low frequency (20 MHz), and integrates 8 KB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It configures individual PEs into target pipelines (Section IV) and receives commands to stimulate neurons either for stopping a seizure or for conveying neural feedback from a prosthetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The MC can be used for general-purpose computation not supported by any PEs such as new algorithms, or infrequently run system operations such as clock synchronization (Section III-F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7) Well-defined throughput: Each PE operates in its own clock domain, like prior work [23], but also supports multiple frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This enables each PE to lower operating frequency (and reduce power) to the minimum necessary to sustain the PE’s target data rate, for varying input electrode counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This feature also ensures fixed latency even when PEs process a variable number of input electrode signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We design each PE to support a maximum frequency f P E max which is high enough to support the maximum data processing rate required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use a configurable register that can be used to set the frequency to f P E max/k, where k is user-programmable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The clock frequency is varied using a simple state machine that uses a counter to only pass through every k clock pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The power consumed by this counter is in the µW range [80], much lower than the per-PE power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Overall, the dynamic power of the PEs scales linearly with the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This also enables deterministic power and latency and helps optimal scheduling (Section III-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' On-device non-volatile memory Each Hull node integrates 128 GB on-device NVM to store raw neural signals, hashes of these signals, and pre-loaded data needed by applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', templates for spike sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We divide the NVM into four partitions, one for each of these classes of data, and another for use by the MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The sizes of the partitions are configurable through registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When a partition is full, the oldest data in the partition is overwritten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We optimize the layout of signal and hash data in the NVM for performance and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull’s ADCs (and hash generation PEs) process electrode samples sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' If the data is stored in this manner, extracting a contiguous segment of one signal would require long-latency reads from multiple discontinuous NVM locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Instead, we store contiguous chunks (where a chunk size is user-specified) of each signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Retrieving the signal (or hashes) at a particular electrode and time-step need only offset calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' SC enables this reorganization as it buffers data in 4 KB SRAM pages before NVM writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Networking We use separate radios for intra-Hull and external device communication as the required distances and communication needs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For intra-Hull communication, we use a custom network protocol with a fixed schedule across the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The schedule is decided by an ILP based on application goals 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4: Seizure detection and propagation on Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The colors of the PEs are matched with the high-level tasks from Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' (Section III-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' To coordinate intra-Hull communication, we use TDMA for its simplicity and deterministic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each network packet has an 84-bit header, and a variable data size up to a maximum packet size of 256 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The header and the data have 32-bit CRC32 [81] checksums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' On a checksum mismatch, the receiver simply discards the packet and does not participate in the pipeline for processing the current sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, we find that while it is best to discard erroneous packets with hashes, erroneous packets carrying raw signal data can still be used without adversely affecting the overall application because of the resiliency of measures like DTW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Task Scheduling on Accelerators, Storage, & Networking As input to our ILP scheduler, users provide a description of the desired computation as a dataflow pipeline using functions of the PEs, or as an interactive query from which the dataflow can be extracted (Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' They also provide the priorities of the tasks in the application (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', seizure detection versus signal comparison), and constraints like the overall response latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A higher priority task ensures that the system processes more neural signals in this task relative to the others when all signals cannot be processed for all tasks due to power or latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The ILP maps each function to the corresponding PE in one or more of Hull’s nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The ILP considers each possible mapping of application tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', seizure detection, hash comparison) to PEs as a flow, and maximizes the weighted sum of the number of channels processed in each flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It uses three major constraints: Latency: End-to-end latencies through the PEs and communi- cation links must be below a specified limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Power: The power consumed by all the PEs and links at all times must be below a specified limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Communication: Only one flow is allowed to use the radio at any time because of TDMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Our ILP setup is simple because of the behavior of the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' With variable throughput processing, the latency of processing any number of input signals is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The dynamic power consumed by a PE scales predictably linearly with the input size (since frequency scales linearly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, the system allows two flows to share the same PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When this occurs and electrode signals to be processed are allocated to both flows, the signals from each flow are interleaved so that they are all run at the same frequency—completing within the same time as if they were run independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The hardware tags the signals from each flow so that they are routed to the correct destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Clock Synchronization Hull’s distributed processing requires the clocks in each BCI node to be synchronized up to a few µs of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull’s clocks are based on pausable clock generators and clock control units [82, 83] that suffer only picoseconds of clock uncertainty, a scale much smaller than our µs target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull operates at the temperature of the human body and does not experience clock drift due to temperature variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nevertheless, Hull runs clock synchronization once a day using SNTP [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' One of the Hull nodes is set up to act as the SNTP server, to which all other nodes send messages to synchronize their time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The clients send their previously synchronized clock times, and current times, while the server sends its corresponding times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The difference between these values is used to adjust the clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This process repeats a few times until all the clocks are synchronized within the desired precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' During clock synchronization, the intra-Hull network is unavailable for application use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, operations that do not require the network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', seizure detection) or NVM access can continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D E P L OY I N G H U L L F O R BCI A P P L I C AT I O N S Hull supports autonomous epileptic seizure propagation in autonomous, movement intent detection for closed-loop prosthesis, online spike sorting, and interactive querying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Autonomous seizure propagation and detection: Figure 4 shows Hull’s implementation of autonomous seizure detection and propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The choice of the PE functions is based on prior work [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This implementation uses XCOR, BBF, and FFT to extract features from the ADC measurements and uses an SVM to detect a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When a seizure is detected, the nodes exchange hashes for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' To confirm that a seizure is indeed likely being propagated, Hull uses the DTW distance of the signals across nodes, and electrically stimulates the brain in response to predicted propagation within 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The dataflow in Figure 4 is fed to the ILP to schedule this application on Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The ILP generates an optimal mapping of the functions and generates a configuration code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This code is run by each Hull node’s microcontroller to configure the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Online spike sorting: Figure 5 shows the mapping of online spike sorting to Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The template-matching version pre-loads the NVM in the nodes with templates and their hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5: Spike sorting on Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7 NEO sC ADC HCONV EMDH GATE CCHECK MCSource Remote Device FFT Devices SVM ADC BBF THR SC XCOR UNPACK DCOMP CCHECK DTW THR SC NGRAM NPACK HCONV GATE HFREQ HCOMP CSELMovement intent detection and feedback: Figure 6 shows how Hull implements detection of movement intent and feedback, augmented from prior work[55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each node extracts features from its local signals and computes a partial SVM output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Then, one node receives the partial SVM outputs and computes the commands for the prosthetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The movements of the prosthetic are transmitted wirelessly, and each node runs a stimulation algorithm for its region to provide neural feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6: Query pipeline for movement intent application Interactive querying: Interactive queries are used to read multi-site data or modify system configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The general format for an interactive query follows a select-project structure, akin to SQL queries [85]: from [set of devices] select data[electrodes][time range] where condition The query specifies select criteria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', the range of time from which data is requested, along with the nodes from which the data should be returned, and the project criteria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', the conditions that the selected data must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Similar to select- project-based SQL queries, Hull’s interactive query interface can support a wide range of complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The project conditions are evaluated on the PEs when possible, and on the microcontroller otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The following illustrates an example query to fetch ±100 ms of data from all devices from the time they detected a seizure in the last 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This example requires seizure detection using 120 ms windows of the raw signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' from * select data[:][t-100:t+100] where seizure_detect(data[t-120:t]) and t >= -5000 and t <= 0 Complex examples can supply template signals and request data from nodes that recorded signals similar to the templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Queries are separately compiled and the extracted dataflow is sent to the ILP, which finalizes query execution schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Users can also set up the pipelines of specific tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', a clinician may modify seizure_detect to use only FFT for feature extraction instead of FFT, BBF and XCOR as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Such a configuration does not need the ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Interactive queries use a power-hungry radio, precluding simultaneous execution of queries and autonomous tasks in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Some of these are either slowed down or temporarily paused;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', when a clinician responds to a seizure alert and requests recent signal data, seizure propagation has to be paused to send the data to the clinician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M E T H O D O L O G Y Processing fabric: Hull’s PEs are designed with a commercial 28 nm fully-depleted silicon-on-insulator (FD-SOI) CMOS process and synthesized using the Cadence® suite of tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use standard cell libraries from STMicroelectronic and foundry-supplied memory macros that are interpolated to 40 °C, which is close to human body temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We design each PE for its highest frequency, and scale the power when using them at lower frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We run multi-corner, physically-aware synthesis, and use latency and power measurements from the worst variation corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Table I shows these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We taped out early designs of the PEs at 12 nm to confirm these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' TABLE I: Latency and Power of the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Processing Max Freq Power (µW ) Latency Elements (MHz) Leakage Dyn/Elec (mS) FFT 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='7 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='97 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 XCOR 85 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 BBF 6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 SVM 3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='67 THR 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='06 NEO 3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 HCONV 3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='50 NGRAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='50 EMDH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='03 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='04 GATE 5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 HFREQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='88 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 HCOMP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='88 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 NPACK 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='008 UNPACK 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='008 DCOMP 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='393 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='50 CCHECK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='393 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='50 CSEL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='04 SC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='03-4 DTW 50 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='93 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='003 We assume that each node uses a standard 96-electrode array [86] to sense neural activity, and a configurable 16-bit ADC [87] generating 30 K samples per second per electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The ADC dissipates 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='88 mW per sample from all 96 electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Each node has a DAC to support electrical stimulation of brain tissue [88], a process that consumes ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='6 mW of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Radio parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use a radio that can transmit/receive up to 10 m to external devices, at 46 Mbps, 250 MHz frequency, and which consumes 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='2 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For intra-Hull communication, we consider a state-of-the-art radio designed for safe implantation in the brain [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' While the radio was originally designed for asymmetric transmission/reception, we modify it for symmetric communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Our intra-Hull radio supports a transmission distance of 20 cm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', > 90th percentile head breadth [90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' To estimate the power and data rates, we use path-loss models [91], with a path-loss parameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='5 for transmission through the brain, skull, and skin, consistent with prior studies [92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We calculate that our radio can transmit/receive 7 Mbps at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='12 GHz and consumes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='721 mW of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Non-volatile memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use NVMs with 4 KB page sizes and 1 MB block sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The NVMs can read 8 bytes, write a page, or erase a block in one operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use SLC NAND parameters like erase time (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='5 ms), program time (350 us), and voltage (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='7 V) from industrial technical reports [94] with NVSim [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We choose a low operating power transistor type in NVSim, and use a temperature of 40 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' NVSim assesses a leakage power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='252 mW, dynamic energies of 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='4 nJ and 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='143 nJ per page for reads and writes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We also use these parameters to size our SC buffers to 24 KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Electrophysiological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use publicly available electro- physiological data for our evaluation [96, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For seizure detection and propagation, we use a data from the Mayo 8 Source ADC FFT SVM UNPACK SVM THR MC MC Device Remote MC ADC FFT SVM NPACK DevicesSeizure Detection Spike Sorting Signal Similarity Movement Intent 100 101 102 103 104 Max Aggregate Throughput (Mbps) Central No-Hash Central Hull No-Hash Hull (a) Maximum aggregate throughput of Hull versus alternative BCI architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1 2 4 8 16 32 64 Number of devices 1 10 100 1000 10000 Max Aggregate Throughput (Mbps) DTW Comparison Hash Comparison DTW One-All Movement Intent (b) Maximum aggregate throughput of communication-dependent tasks in Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 0 50 100 150 Sensor Data Rate (Mbps) 0 25 50 75 100 125 Throughput (Mbps) Seizure Detection Spike Sorting (c) Maximum throughput of tasks without inter- node communication, using re-designed PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7: Experimental quantification of Hull’s benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Clinic [97] of a patient (label “I001 P013”) with 76 electrodes implanted in the parietal/occipital lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This data-set was recorded for 4 days at 5 KHz, and is annotated with hundreds of seizure instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We upscaled the sampling frequency to 30 KHz, and split the dataset to emulate multiple BCI devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We use consecutive and overlapping 4 ms windows (120 samples) from the electrodes to detect seizures [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For propagation, we check similarity with a seizure-positive signal in the last 100 ms from electrode data in all nodes [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For hash pipelines, we use one 8-bit hash for 120 sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For spike sorting, we use the Spikeforest dataset [96, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This dataset contains recordings collected from the CA1 region of a rat hippocampus using tetrode electrodes recorded at 30 KHz sampling frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The dataset contains spikes from 10 neurons, with 65, 000 spikes that were manually sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Alternative system architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Table II shows the systems that we compare Hull against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull No-Hash uses the same Hull architecture but does not use hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The power saved by removing the hash processing PEs is allocated to the remaining tasks optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull No-Hash does not require re-writing the applications for hash-based processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Central uses one processing node with the same processor as Hull, and multiple sensors that are connected using wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, Central No- Hash is a centralized design without hash processing, like most existing BCIs [27, 31, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We do not consider wireless centralized designs as they need a radio and have lesser compute available than the wired ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We also do not consider designs without memory as they do not support seizure propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We map our applications onto all systems using the ILP, ensuring that each node consumes < 15 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' TABLE II: Alternative BCI designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Design Architecture Comparison Communication Hull (Proposed) Distributed Hash, Signal Wireless Hull-No hash Distributed Signal Wireless Central Centralized Hash, Signal Wired Central-No hash Centralized Signal Wired VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E VA L UAT I O N A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Comparing BCI Architectures Figure 7a shows the maximum aggregate throughput of the systems in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A task’s maximum aggregate throughput is achieved when it is the only task running in the system, summed over all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Central No-Hash has the worst throughput for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This design suffers from having just one processor and from using expensive signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Central increases throughput by an order of magnitude for tasks that benefit from hashing (spike sorting and signal similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, the single processor remains the bottleneck for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull No-Hash has distributed processors and enjoys higher aggregate seizure detection and movement intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, it performs poorly for tasks that need signal comparison (signal similarity, spike sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For these tasks, Hull No-Hash has lower throughput than Central because it does not use hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull uses distributed hash-based processing and has the highest aggregate throughput for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Compared to Central-No hash, which is closest to state-of-the-art BCIs, Hull’s data rates are an order of magnitude higher for seizure detection, and movement intent detection, and are nearly three orders of magnitude higher for signal similarity and spike sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Throughput for Communication-Dependent Tasks Figure 7b shows the maximum aggregate throughput of the communication-dependent task (hash comparison, DTW comparison, and movement intent), with various node counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' DTW Comparison uses all-to-all comparison of raw signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It has a lower throughput than the other tasks because only 16 out of 96 electrode signals can be transmitted for all-to- all comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The reason is that new electrode samples are obtained at 47 Mbps from the ADC, but the intra-Hull radio can only transmit about 7 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Increasing the number of nodes decreases the throughput further because of the communication delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Because Hull uses a TDMA network, where slots for network access are serialized, DTW Comparison has the worst throughput and scales poorly with node count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' An alternative DTW One-All, which only uses one-to-all DTW comparison, scales better since its communication latency 9 doesn’t increase with the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, a one-to- all comparison is insufficient for general BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hash Comparison uses all-to-all hash communication to check for collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Its throughput increases to 470 Mbps until 10 devices, after which it begins to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When the number of nodes is small, few TDMA slots are required to exchange all hashes, enabling a linear increase in throughput as a function of node count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' But, as node counts keep increasing, it takes longer to communicate all hashes and overall throughput reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Finally, Movement Intent uses all-to-one communication of the partial SVM products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, as the product is small, its throughput scales linearly with the number of nodes (note that the Y-axis in Figure 7b is logarithmic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' It also has the highest aggregate throughput because it needs the least communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 7b shows that hashing, and distributing the SVM computation in Hull enables it to scale to many regions and with higher data rates than what has been possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Throughput for Non-Communicating Tasks We design our PEs for a maximum sensor rate of 47 Mbps per node (Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, we study potential PE re-design to support higher processing rates for tasks that do not need communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 7c shows the throughput of Seizure Detection and Spike Sorting for varying per-node signal sensor rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Task throughput increases linearly up to 105 Mbps for spike sorting, and 70 Mbps for seizure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Beyond this sensing rate, the higher frequency of the PEs and ADCs results in exceeding the device power limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nonetheless, these values are nearly twice as supported by existing single-implant BCIs and show the robustness of our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Application Level Throughput The throughput achieved at the application level depends on the number of implanted nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Additionally, when there are multiple tasks, it depends on the priorities assigned to the application tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Recall that the ILP schedules applications to optimize a priority-weighted sum of the signals processed in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For seizure detection propagation, Figure 8a shows the weighted aggregate throughput as a function of the number of devices, for various weight choices (in the format seizure detection:hash comparison:DTW comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For an equal priority to seizure detection, DTW processing, and hash comparison, we find that the maximum throughput is achieved for 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Other weight choices have different optimal node counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Note that there is no comparable system for on-device seizure propagation—Hull is the first design with this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Movement intent has only one task, and its throughput (in number of intents detected per second), is shown in Figure 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This metric accounts for only movement intent detection, and not for the variable response latency of the prosthetic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull spike sorts up to 12, 250 spikes per second per node with 82% accuracy, comparing well to the state of the art [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Interactive Queries We consider three types of common queries applied on data ranging from the past 100 ms (≈7 MB over all nodes) to the 0 20 40 60 80 Number of Devices 0 10 20 30 40 50 Weighted Throughput (Mbps) 1 1:1:1 3:1:1 1:3:1 0 20 40 60 80 Number of Devices 0 50 100 150 Movement Intents / Second 2 (a) Weighted throughput of seizure propagation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 0 20 40 60 80 Number of Devices 0 10 20 30 40 50 Weighted Throughput (Mbps) 1 1:1:1 3:1:1 1:3:1 0 20 40 60 80 Number of Devices 0 50 100 150 Movement Intents / Second 2 (b) Movement intents per second (without device movement time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 8: Application level metrics on Hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' past 1 s (≈60 MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' They are: Q1, which returns all signals that were detected as a seizure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Q2, which returns all signals that matched with a template using a hash;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' and Q3, which returns all data in the timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For Q1 and Q2, we vary the fraction of data that tests positive for their condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 9 shows Hull’s throughput with 11 nodes for our queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull supports up to 10 queries per second (QPS) for Q1 and Q2 over the last 100 ms data (the common case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' If Q2 is run with DTW instead of hash-based search, we see a QPS of 8, which is only slightly lower, but the power consumption increases from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='57 mW for the hash vs the entire 15 mW for DTW based matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Thus, DTW-based matching is unsuitable when interactively querying in response to a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Q3 on this data takes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='21 s, yielding a throughput of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' In interactive querying, the external radio, which consumes high power, is the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' As the data to be searched increases, the query latency increases linearly due to the radio latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, Hull can still process 1 QPS for Q1 and Q2 for the past 1 s data (≈60 MB), making it suitable for real-time use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7 (110 ms) 24 (400 ms) 42 (700 ms) 60 (1000 ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='0 5% 5% 50% 50% 100% 100% 100% 5% 5% 50% 50% 100% 100% 100% 5% 5% 50% 50% 100% 100% 100% 5% 5% 50% 50% 100% 100% 100% Query Data Size (MB) (Time Range) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='1 1 10 Queries per second Q1 Q2 Q3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 9: Interactive query throughput on Hull with 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hashing Accuracy: We vary the parameters of all our hash functions and show the performance of the best configuration for seizure propagation and spike sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 11 shows the accuracy (TP: True positive, TN: True negative, FP: False positive, FN: False negative) for the four hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' XCOR and EMD hashes have ≈ 85% accuracy while Euclidean and DTW have over 90% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The high true positive rate of our DTW 10 0 25 50 75 100 125 Window Size 0 1 2 3 4 5 6 Ngram Size XCOR DTW Euclidean XCOR Euclidean DTW EMD 0 50 100 Percentage (%) TP FN TN FP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 10: Hash accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 0 25 50 75 100 125 Window Size 0 1 2 3 4 5 6 Ngram Size XCOR DTW Euclidean XCOR Euclidean DTW EMD 0 25 50 75 100 Percentage (%) True PositiveTrue Negative Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11: Hash flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Standard Bit Error Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='0 Percentage of Packets with Error (%) 30 50 70 90 100 Hash Packets Signal Packets DTW Failure 10 4 10 5 10 6 0 5 10 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 12: Bit error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of devices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='1 1 10 100 1000 10000 100000 Time (seconds) Full ILP Reduced Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 13: Time to solve the ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' hash is particularly beneficial for the seizure propagation (note that false positives are removed using exact DTW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Parameter selection: Figure 11 shows the best parameters of our hash implementation (window size and n-gram size— Section II-C) to approximate each of Euclidean, cross correla- tion, and DTW similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We also show parameters (with lighter colors in the figure) that are within 90% of the true positive rate achieved by the corresponding best configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' This flexibility enables reusing a single fast hardware accelerator for different measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Impact of Network Bit Error Rate The intra-Hull network protocol drops packets carrying hashes when there is a checksum error but allows signal packets to flow into PEs since signal similarity measures are naturally resilient to a few errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We simulate various bit-error ratios (BERs) using uniformly random bit flips in the packet header and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 12 shows the fraction of hash or signal packets with an error at different BERs, and the fraction of erroneous signal packets that flipped the similarity measure (DTW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For reference, the BER is <10−4 for the radio we use [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 12 shows that signals and hashes suffer errors as BER increases, but signals are more susceptible since they are longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' But, even though several signal packets suffer errors, they have no impact on the final signal similarity outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' ILP Performance The complexity of the ILP increases with the number of pipeline stages in the application and the number of Hull nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' When all nodes are the same and have the same power/energy constraints, the schedule of one node can be replicated (with a constant offset) on all other nodes and remain optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We call this method Reduced ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, we cannot apply this method when the nodes are different or have different constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Figure 13 shows the time taken to solve the ILP and the reduced version for varying numbers of devices for the seizure propagation application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' We measure this time when using GLPK, an open-source ILP solver, with default settings on an Intel-Xeon E5-2620 v3 machine with 93 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' As expected, the solver time for the standard ILP increases exponentially with the number of devices, taking ≈2 hours with 11 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' For >12 devices, the ILP did not finish within 24 hours and was terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' The reduced ILP however, can be solved in less than 10ms for any number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R E L AT E D W O R K Commercial and research BCIs have focused largely on single brain location monitoring and stimulation [16, 17, 23, 27, 28], and have no support for distributed systems, making them inhospitable for the applications that we target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Most implantable BCIs offer little to no storage capacity and stream data out continuously instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' NeuroChip [100] is an exception, but is wired to an external case storing a 128 GB SD card that must be physically extracted for offline data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull is the first to use storage for pre-processing and reduce computation by using the hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A growing interest in distributed analyses of the brain [1, 13, 25, 26] has motivated the design of rudimentary multi-site BCIs [31, 32, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Prior studies [31, 32] propose microchips that stream sensor data wirelessly to a central hub outside the skull using back-scattering radio techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Unfortunately, these approaches are restricted in their interfacing bandwidth as they rely on centralized processing and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Although recent work has studied unary neural networks on single-site BCIs [102], we will study distributed neural network models for seizure detection, propagation, spike sorting, and movement intent for multi-side BCIs going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull can support any algorithm with linear computational complexity without significant changes to the ILP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' However, neural network inference, which is super-linear, may require non-linear formulations for scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Using MILP and approximations for such PEs may be a suitable extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C O N C L U S I O N & F U T U R E W O R K Hull enables distributed BCI interfacing that can scale to multiple regions, and provides for the first time, on-device computation for important BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull offers two orders of magnitude higher task throughput, and real-time support for interactive querying with up to 10 QPS over 7 MB data or 1 QPS over 60 MB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hull will influence the wider field of IoT devices, ranging from low-power temperature and voltage sensors [103], AR/VR devices, to devices in smart home, factory, and vehicle settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' These devices must collect and process large volumes of data on the edge, as communicating this data to centralized locations is likely to be near impossible for today’s cloud infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Similar to Hull, a network of power-constrained devices will need to process large volumes of data, often with flexible processing requirements to support rapidly evolving use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11 Hull’s design principles– i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', its modular PE architecture, fast-but-approximate hash-based approach to signal similarity, support for low-power and efficiently-indexed non-volatile storage, and a centralized planner that produces near-optimal mapping of task schedules to devices – can be instrumental to success in other IoT environments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R E F E R E N C E S [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Andersen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Aflalo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bashford, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bj˚anes, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kellis, “Exploring Cognition with Brain–Machine Interfaces,” Annual Review of Psychology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 73, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 131–158, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lebedev and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nicolelis, “Brain-machine interfaces: From basic science to neuroprostheses and neurorehabilitation,” Physiological reviews, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 97, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 767–837, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chandrasekaran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fifer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bickel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Osborn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Herrero, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Christie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Murphy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Singh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Historical perspectives, challenges, and future directions of implantable brain-computer interfaces for sensorimotor applications,” Bioelectronic medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1–11, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Widge, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Dougherty, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Moritz, “Af- fective brain-computer interfaces as enabling technology for responsive psychiatric stimulation,” Brain-Computer Interfaces, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 126–136, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shih, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Krusienski, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wolpaw, “Brain- computer interfaces in medicine,” in Mayo clinic pro- ceedings, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 87, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Elsevier, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 268–279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' McFarland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Daly, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Boulay, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Parvaz, “Therapeutic applications of BCI technologies,” Brain- Computer Interfaces, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 37–52, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Huang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' van Luijtelaar, “Brain computer inter- face for epilepsy treatment,” Brain-Computer Interface Systems-Recent Progress and Future Prospects, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M¨uhl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Allison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nijholt, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chanel, “A survey of affective brain computer interfaces: principles, state- of-the-art, and challenges,” Brain-Computer Interfaces, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 66–84, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Buzs´aki, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Anastassiou, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Koch, “The Origin of Extracellular Fields and Currents — EEG, ECoG, LFP and Spikes,” Nature Reviews Neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 13, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 407–420, May 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pesaran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vinck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Einevoll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sirota, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fries, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Siegel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Truccolo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Schroeder, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Srini- vasan, “Investigating large-scale brain dynamics using field potential recordings: analysis and interpretation,” Nature neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 903–919, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' del R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Milan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Carmena, “Invasive or Noninvasive: Understanding Brain-Machine Interface Technology [Conversations in BME],” IEEE Engineering in Medicine and Biology Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 16–22, Jan 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Musk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “An integrated brain-machine interface platform with thousands of channels,” Journal of medical Internet research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' e16194, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Szostak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Feng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mazza, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Con- standinou, “Distributed Neural Interfaces: Challenges and Trends in Scaling Implantable Technology,” Handbook of Neuroengineering, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1–37, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rapeaux and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Constandinou, “Implantable brain machine interfaces: first-in-human studies, tech- nology challenges and trends,” Current opinion in biotechnology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 72, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 102–111, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [15] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Food and Drug Administration, “FDA authorizes marketing of device to facilitate muscle rehabilitation in stroke patients,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='gov/news-events/press- announcements/fda-authorizes-marketing-device- facilitate-muscle-rehabilitation-stroke-patients, April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [16] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sun and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Morrell, “The RNS System: responsive cortical stimulation for the treatment of refractory partial epilepsy,” Expert review of medical devices, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 563–572, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [17] Medtronic, “Deep Brain Stimulation Systems - Activa PC,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='medtronic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='com/us-en/healthcare- professionals/products/neurological/deep-brain- stimulation-systems/activa-pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='html, November 2018, Retrieved August 10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [18] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Food and Drug Administration, “Implanted Brain-Computer Interface (BCI) Devices for Patients with Paralysis or Amputation - Non-clinical Testing and Clinical Considerations,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='gov/regulatory- information/search-fda-guidance-documents/implanted- brain-computer-interface-bci-devices-patients- paralysis-or-amputation-non-clinical-testing, February 2019, Retrieved August 10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Han, “Synchron receives FDA approval to be- gin early feasibility study of their endovascular, brain- computer interface device,” Artificial Organs, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 45, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1134–1135, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Serrano-Amenos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Andersen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Heydari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Do, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nenadic, “Thermal analysis of a skull implant in brain- computer interfaces,” in 2020 42nd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3066–3069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wolf, “Thermal Considerations for the Design of an Implanted Cortical Brain-Machine Interface (BMI),” Indwelling Neural Implants: Strategies for Contending with the In Vivo Environment, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Stevenson and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kording, “How Advances in Neural Recording Affect Data Analysis,” Nature neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 139–42, 02 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [23] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Karageorgos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sriram, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vesel`y, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Powell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Borton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Manohar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bhattacharjee, “Hardware- software co-design for brain-computer interfaces,” in 2020 ACM/IEEE 47th Annual International Symposium on Computer Architecture (ISCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 391–404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Zelmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Paulk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sarma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Yousefi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Crocker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Eskandar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Williams, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cosgrove, 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Weisholtz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “CLoSES: A platform for closed- loop intracranial stimulation in humans,” NeuroImage, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 223, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 117314, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jirsa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Proix, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Perdikis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Woodman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gonzalez-Martinez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bernard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B´enar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Guye, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chauvel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “The Virtual Epileptic Patient: Individualized whole-brain models of epilepsy spread,” Neuroimage, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 145, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 377–388, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [26] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bartolomei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lagarde, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wendling, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' McGonigal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jirsa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Guye, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B´enar, “Defining epileptogenic networks: Contribution of SEEG and signal analysis,” Epilepsia, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1131–1147, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [27] Hossein Kassiri, Sana Tonekaboni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tariqus Salam, Nima Soltani, karim Abdelhalim, Jose Luis Perez Ve- lasquez, Roman Genov, “Closed-Loop Neurostimulators: A Survey and A Seizure-Predicting Design Example for Intractable Epilepsy Treatment,” IEEE Transactions on Biomedical Circuits and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1026–1040, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [28] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' O’Leary and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Groppe and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Valiante and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Verma and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Genov, “Nurip: Neural interface processor for brain-state classification and programmable- waveform neurostimulation,” IEEE Journal of Solid State Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 53, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Aziz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Abdelhalim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shulyzki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Genov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bardakjian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Derchansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Serletis, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Carlen, “256-Channel Neural Recording and Delta Compression Microsystem With 3D Electrodes,” IEEE Journal of Solid-State Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 995–1005, March 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jeng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chiueh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hsu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chien, “A Hardware Implementation of Real- Time Epileptic Seizure Detector on FPGA,” 2011 IEEE Biomedical Circuits and Systems Conference (BioCAS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 25–28, Nov 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [31] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ahmadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cavuto, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Feng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Leene, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Maslik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mazza, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Savolainen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Szostak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bouganis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ekanayake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Towards a Distributed, Chronically-Implantable Neural Interface,” in 2019 9th International IEEE/EMBS Conference on Neural Engi- neering (NER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 719–724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lee, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Leung, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Huang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Asbeck, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mercier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shellhammer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Larson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Laiwalla, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nurmikko, “Neural recording and stimulation using wireless networks of microimplants,” Nature Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 604–614, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [33] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Zhu, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shoaran, “Closed-Loop Neural Prostheses With On-Chip Intelligence: A Review and a Low-Latency Machine Learning Model for Brain State Detection,” IEEE Transactions on Biomedical Circuits and Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Zrenner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Belardinelli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M¨uller-Dahlhaus, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ziemann, “Closed-Loop Neuroscience and Non- Invasive Brain Stimulation: A Tale of Two Loop,” Frontiers in cellular neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 92, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Murphy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Guggenmos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bundy, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nudo, “Current Challenges Facing the Translation of Brain Computer Interfaces from Preclinical Trials to Use in Human Patients,” Front Cell Neurosci, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 497, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hebb, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mahoor, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tsiokos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Matlack, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chizeck, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pouratian, “Creating the feedback loop: Closed loop neurostimulation,” Neu- rosurgery Clinics of North America, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 187–204, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [37] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sladky, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nejedly, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mivalt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Brinkmann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kim, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Louis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gregg, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lundstrom, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Crowe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Attia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Distributed brain co-processor for tracking spikes, seizures and behaviour during elec- trical brain stimulation,” Brain Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' fcac115, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kural, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jing, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F¨urbass, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Perko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Qerama, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Johnsen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fuchs, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Westover, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Beniczky, “Accurate identification of EEG recordings with interictal epileptiform discharges using a hybrid approach: Artifi- cial intelligence supervised by human experts,” Epilepsia, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1064–1073, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Simeral, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hosman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Saab, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Flesher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vilela, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Franco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kelemen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Brandman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ciancibello, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rezaii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Home Use of a Percutaneous Wireless Intracortical Brain-Computer Interface by Individuals With Tetraplegia,” IEEE Trans- actions on Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2313–2325, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Yin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Borton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Komar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Agha, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Laurens, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Wire- less Neurosensor for Full-Spectrum Electrophysiology Recordings during Free Behavior,” Neuron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1170–1182, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tathireddy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Normann, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Solzbacher, “Thermal Impact of an Active 3-D Microelectrode Array Implanted in the Brain,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 493–501, Dec 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [42] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sporns, “Graph theory methods: applications in brain networks,” Dialogues in clinical neuroscience, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bilge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gosai, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Widge, “Deep Brain Stimulation in Psychiatry,” Psychiatric Clinics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 373–383, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Deco and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kringelbach, “Great Expectations: Using Whole-Brain Computational Connectomics for Understanding Neuropsychiatric Disorders,” Neuron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 892–905, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gallego, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Makin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' McDougle, “Going beyond primary motor cortex to improve brain–computer interfaces,” Trends in Neurosciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 45, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 176–183, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [46] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Camargo-Vargas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Callejas-Cuervo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Maz- zoleni, “Brain-Computer Interfaces Systems for Upper and Lower Limb Rehabilitation: A Systematic Review,” Sensors, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4312, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bensmaia and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Miller, “Restoring sensorimotor 13 function through intracortical interfaces: progress and looming challenges,” Nature Reviews Neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 313–325, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Todorova, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sadtler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Batista, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chase, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ven- tura, “To sort or not to sort: the impact of spike-sorting on neural decoding performance,” Journal of neural engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 056005, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [49] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pedreira, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Quiroga, “Past, present and future of spike sorting techniques,” Brain research bulletin, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 119, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 106–117, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shiao, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cherkassky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Veber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Patterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Brinkmann, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Worrell, “SVM-Based System for Prediction of Epileptic Seizures From iEEG Signal,” IEEE Transactions on Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1011–1022, May 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [51] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Niazi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jiang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tiberghien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nielsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Dremstrup, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Farina, “Detection of movement intention from single-trial movement-related cortical potentials,” Journal of Neural Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 066009, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jochumsen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Niazi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rovsing, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rovsing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Nielsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Andersen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Dong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sørensen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mrachacz-Kersting, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Detection of movement intentions through a single channel of elec- troencephalography,” in Replace, Repair, Restore, Relieve– Bridging Clinical and Engineering Solutions in Neurore- habilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Springer, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 465–472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [53] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ojakangas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shaikhouni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Friehs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Caplan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Serruya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Saleh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Morris, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Donoghue, “Decoding Movement Intent From Human Premotor Cortex Neurons for Neural Prosthetic Appli- cations,” Journal of Clinical Neurophysiology: Official Publication of the American Electroencephalographic Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 577, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kalaska, “From Intention to Action: Motor Cortex and the Control of Reaching Movements,” Progress in Motor Control, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 139–178, 2009, series Title: Advances in Experimental Medicine and Biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [55] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shakeel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Navid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Anwar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mazhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jochumsen, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Niazi, “A Review of Techniques for Detection of Movement Intention Using Movement- Related Cortical Potentials,” Computational and Mathe- matical Methods in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2015, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [56] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Taylor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tillery, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Schwartz, “Direct Cortical Control of 3D Neuroprosthetic Devices,” Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 296, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5574, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1829–1832, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [57] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fetz, “Restoring motor function with bidirectional neural interfaces,” Progress in Brain Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 218, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 241–252, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [58] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Capogrosso, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Milekovic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Borton, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wagner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Moraud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mignardot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Buse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gandar, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Barraud, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “A Brain–Spine Interface Alleviating Gait Deficits after Spinal Cord Injury in Primates,” Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 539, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7628, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 284–288, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [59] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Muller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Anderson, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Birch, “Linear and nonlinear methods for brain-computer interfaces,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 165–169, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [60] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Darie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Powell, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Borton, “Delivering the Sense of Touch to the Human Brain,” Neuron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 93, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 728–730, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [61] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tafazoli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' MacDowell, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Che, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Letai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Steinhardt, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Buschman, “Learning to control the brain through adaptive closed-loop patterned stimulation,” Journal of Neural Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 056007, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [62] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rutishauser, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Schuman, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mamelak, “Online detection and sorting of extracellularly recorded action potentials in human medial temporal lobe record- ings, in vivo,” Journal of Neuroscience Methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 154, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 204–224, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [63] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sotomayor-G´omez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Battaglia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vinck, “SpikeShip: A method for fast, unsupervised discovery of high-dimensional neural spiking patterns,” bioRxiv, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2020–06, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [64] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Grossberger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Battaglia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vinck, “Unsuper- vised clustering of temporal patterns in high-dimensional neuronal ensembles using a novel dissimilarity measure,” PLoS Comput Biol, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' e1006283, 07 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [65] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kim, “Physiological Time Series Retrieval and Prediction with Locality-Sensitive Hashing,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' dis- sertation, Massachusetts Institute of Technology, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [66] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rakhilin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gordon, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shen, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kan, “A real-time spike classification method based on dynamic time warping for extracellular enteric neural recording with large waveform variability,” Journal of Neuroscience Methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 261, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 97–109, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [67] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Luo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shrivastava, “SSH (Sketch, Shingle, & Hash) for Indexing Massive-Scale Time Series,” in NIPS 2016 Time Series Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' PMLR, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 38–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [68] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Gorisse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cord, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Precioso, “Locality-Sensitive Hashing for Chi2 Distance,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 402–409, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [69] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bhattacharjee and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Manohar, “HALO: A Flexible and Low Power Processing Fabric for Brain-Computer Interfaces,” in 2022 IEEE Hot Chips 34 Symposium (HCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' IEEE Computer Society, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [70] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Karageorgos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sriram, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vesel´y, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lindsay, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Powell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Borton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Manohar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bhat- tacharjee, “Balancing specialized versus flexible compu- tation in brain–computer interfaces,” IEEE Micro, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 87–94, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [71] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sun and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Morrell, “Closed-loop Neurostim- ulation: The Clinical Experience,” Neurotherapeutics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 553–563, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [72] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Vidal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rynes, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kelliher, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Goodwin, “Review of Brain-Machine Interfaces Used in Neural Prosthetics with New Perspective on Somatosen- sory Feedback through Method of Signal Breakdown,” Scientifica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2016, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [73] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Herff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Krusienski, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kubben, “The Poten- 14 tial of Stereotactic-EEG for Brain-Computer Interfaces: Current Progress and Future Directions,” Frontiers in Neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 123, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [74] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Karger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lehman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Leighton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Panigrahy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Levine, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lewin, “Consistent Hashing and Ran- dom Trees: Distributed Caching Protocols for Relieving Hot Spots on the World Wide Web,” in ACM Symposium on Theory of Computing (STOC), 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [75] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kruskall and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Liberman, “The symmetric time warping algorithm: From continuous to discrete,” Time Warps, String Edits and Macromolecules: The Theory and Practice of Sequence Comparison, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [76] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sakoe and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chiba, “Dynamic programming al- gorithm optimization for spoken word recognition,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 26, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 159–165, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [77] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pele and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Werman, “Fast and robust Earth Mover’s Distances,” in 2009 IEEE 12th international conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' IEEE, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 460–467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [78] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pu, Fundamental data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Butterworth- Heinemann, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [79] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Elias, “Universal codeword sets and representations of the integers,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 194–203, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [80] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bingham and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Manohar, “Qdi constant-time coun- ters,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 83–91, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [81] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Peterson and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Brown, “Cyclic codes for error detection,” Proceedings of the IRE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 228–235, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [82] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Yun and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Donohue, “Pausible clocking: a first step toward heterogeneous systems,” in Proceedings International Conference on Computer Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' VLSI in Computers and Processors, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 118–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Moreno and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cortadella, “Synthesis of all-digital delay lines,” in 2017 23rd IEEE International Symposium on Asynchronous Circuits and Systems (ASYNC), 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 75–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [84] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mills, “Simple Network Time Protocol (SNTP),” RFC Editor, RFC 1769, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [85] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chamberlin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Boyce, “SEQUEL: A structured English query language,” in Proceedings of the 1974 ACM SIGFIDET (now SIGMOD) workshop on Data description, access and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Association for Computing Machinery, 1974, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 249–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [86] Blackrock Microsystems, “The Benchmark for Multichannel, High-density Neural Recording,” https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='blackrockmicro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='com/electrode-types/utah-array/, Retrieved August 10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [87] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shikata, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fernando, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Guthrie, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Maddox, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mascarenhas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kapusta, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Coln, “A 16-bit 16-ms/s sar adc with on-chip calibration in 55-nm cmos,” IEEE Journal of Solid-State Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1149–1160, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [88] Medtronic, “Medtronic Activa PC Multi- program neurostimulator implant manual,” http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='neuromodulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='ch/sites/default/files/ pictures/activa PC DBS implant manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='pdf, 2008, Retrieved August 10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [89] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rahmani and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Babakhani, “A wirelessly powered reconfigurable fdd radio with on-chip antennas for multi- site neural interfaces,” IEEE Journal of Solid-State Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 56, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 3177–3190, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [90] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Young, “Head and Face Anthropometry of Adult U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Citizens,” Federal Aviation Administration, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [91] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Molisch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Balakrishnan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Emami, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fort, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Karedal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kunisch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Schantz, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Schuster, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Siwiak, “IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 4a channel model-final report,” IEEE P802, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 04, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 0662, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [92] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S¨arest¨oniemi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pomalaza-Raez, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Sayrafian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Myllyl¨a, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Iinatti, “A Preliminary Study of RF Propagation for High Data Rate Brain Telemetry,” in Body Area Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Smart IoT and Big Data for Intelligent Health Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Cham: Springer International Publishing, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 126–138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [93] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Taparugssanagorn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rabbachin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H¨am¨al¨ainen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Saloranta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Iinatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “A Review of Channel Modelling for Wireless Body Area Network in Wire- less Medical Communications,” The 11th International Symposium on Wireless Personal Multimedia Communi- cations (WPMC, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [94] “MT29F128G08AKCABH2-10,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' micron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='com/products/nand-flash/slc-nand/part- catalog/mt29f128g08akcabh2-10, Retrieved November 17, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [95] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Xie, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jouppi, “Nvsim: A circuit-level performance, energy, and area model for emerging nonvolatile memory,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 994–1007, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [96] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Magland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jun, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lovero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Morley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Hurwitz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Buccino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Garcia, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Barnett, “SpikeForest, reproducible web-facing ground-truth vali- dation of automated neural spike sorters,” Elife, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' e55167, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [97] “IEEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='ORG,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='ieeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content='org, Retrieved November 17, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [98] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Ashourvan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mikhail, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pines, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kini, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Oechsel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Das, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Stein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shinohara, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Bassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “Characterizing the role of the structural connectome in seizure dynamics,” Brain, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 142, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1955–1972, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [99] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Magland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Barnett, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tolosa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Tooker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shah, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Felix, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Frank, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Greengard, “A Fully Automated Approach to Spike Sorting,” Neuron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 95, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1381–1394, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [100] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Shupe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Miles, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Yun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Mish- ler, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Rembado, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Murphy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Perlmutter, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fetz, “Neurochip3: An Autonomous Multichannel Bidirectional Brain-Computer Interface for Closed-Loop 15 Activity-Dependent Stimulation,” Frontiers in Neuro- science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [101] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Joo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Fan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Barnett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Geaghan-Breiner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Karlsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Karlsson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=', “High-Density, Long- Lasting, and Multi-region Electrophysiological Record- ings Using Polymer Electrode Arrays,” Neuron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 101, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 21–31, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [102] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Kim, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Miguel, “uBrain: A Unary Brain Computer Interface,” in Proceedings of the 49th Annual International Symposium on Computer Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Association for Computing Machinery, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 468–481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' [103] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Stewart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Liao, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' Roberts, “Open µpmu: A real world reference distribution micro-phasor mea- surement unit data set for research and application development,” IEEE Power Engineering Letters, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfVgSF/content/2301.03103v1.pdf'} diff --git a/BtE5T4oBgHgl3EQfTQ-D/vector_store/index.faiss b/BtE5T4oBgHgl3EQfTQ-D/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..be10b770c2d6d6509a0991712a60b5ac34df1da9 --- /dev/null +++ b/BtE5T4oBgHgl3EQfTQ-D/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2f88bf288ead57d18ca1c677782893709bc3780043189d23d152370239a7b580 +size 2293805 diff --git a/CNE1T4oBgHgl3EQfDwNk/vector_store/index.faiss b/CNE1T4oBgHgl3EQfDwNk/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b60b2cadb510134c77bb837b547d68eb8baaadfa --- /dev/null +++ b/CNE1T4oBgHgl3EQfDwNk/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2494581eeb464decb1038b755bf34e72c25be7b61c343273019c7b90256f6d87 +size 3538989 diff --git a/CNFJT4oBgHgl3EQfASxa/content/2301.11419v1.pdf b/CNFJT4oBgHgl3EQfASxa/content/2301.11419v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..67595c75edba22df0abe02324b3768e1575d0b16 --- /dev/null +++ b/CNFJT4oBgHgl3EQfASxa/content/2301.11419v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:efe35316418bc3a45a60f2aa18e9e5dffdc1f5d7a1a5c61a2ab4d43f2e055586 +size 1696742 diff --git a/CNFJT4oBgHgl3EQfASxa/vector_store/index.faiss b/CNFJT4oBgHgl3EQfASxa/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3ca9b17c44da2bfcf42dfff1bcf472b02b1bb493 --- /dev/null +++ b/CNFJT4oBgHgl3EQfASxa/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f3ef0febf4e2b5dc5322e0ead6534829dfb063854f43b8f841b09582d00d47bd +size 2293805 diff --git a/CNFJT4oBgHgl3EQfASxa/vector_store/index.pkl b/CNFJT4oBgHgl3EQfASxa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..936eecf87768b236c98be27572f62f40e2531610 --- /dev/null +++ b/CNFJT4oBgHgl3EQfASxa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f47c1916c7fb2e83b61925986e80072f341e2de20ad01df43e69747e9fff0595 +size 103997 diff --git a/DNE0T4oBgHgl3EQfggEA/content/tmp_files/2301.02417v1.pdf.txt b/DNE0T4oBgHgl3EQfggEA/content/tmp_files/2301.02417v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ec668354c842b8e10d0e3dc7d68a2d1edc7a423 --- /dev/null +++ b/DNE0T4oBgHgl3EQfggEA/content/tmp_files/2301.02417v1.pdf.txt @@ -0,0 +1,2744 @@ +arXiv:2301.02417v1 [cs.IT] 6 Jan 2023 +1 +Uplink Precoding Design for Cell-Free Massive +MIMO with Iteratively Weighted MMSE +Zhe Wang, Jiayi Zhang, Senior Member, IEEE, Hien Quoc Ngo, Senior Member, IEEE, +Bo Ai, Fellow, IEEE and M´erouane Debbah, Fellow, IEEE +Abstract +In this paper, we investigate a cell-free massive multiple-input multiple-output system with both access points +and user equipments equipped with multiple antennas over the Weichselberger Rayleigh fading channel. We study +the uplink spectral efficiency (SE) for the fully centralized processing scheme and large-scale fading decoding +(LSFD) scheme. To further improve the SE performance, we design the uplink precoding schemes based on +the weighted sum SE maximization. Since the weighted sum SE maximization problem is not jointly over all +optimization variables, two efficient uplink precoding schemes based on Iteratively Weighted sum-Minimum Mean +Square Error (I-WMMSE) algorithms, which rely on the iterative minimization of weighted MSE, are proposed for +two processing schemes investigated. Furthermore, with maximum ratio combining applied in the LSFD scheme, +we derive novel closed-form achievable SE expressions and optimal precoding schemes. Numerical results validate +the proposed results and show that the I-WMMSE precoding schemes can achieve excellent sum SE performance +with a large number of UE antennas. +Index Terms +Cell-free massive MIMO, uplink precoding, weighted sum-rate maximization, spectral efficiency. +I. INTRODUCTION +Cell-free massive multiple-input multiple-output (CF mMIMO) has attracted a lot of research interest +and is regarded as a promising technology for future wireless communications, for its ability to achieve +This article was presented in part at IEEE International Conference on Communications 2022 [1]. +Z. Wang and J. Zhang are with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China, +and also with the Frontiers Science Center for Smart High-speed Railway System, Beijing Jiaotong University, Beijing 100044, China (e-mail: +{zhewang 77, jiayizhang}@bjtu.edu.cn). +H. Q. Ngo is with the Institute of Electronics, Communications, and Information Technology, Queen’s University Belfast, BT3 9DT Belfast, +U.K. (email: hien.ngo@qub.ac.uk). +B. Ai is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China, also with +the Frontiers Science Center for Smart High-Speed Railway System and the Henan Joint International Research Laboratory of Intelligent +Networking and Data Analysis, Zhengzhou University, Zhengzhou 450001, China, and also with the Research Center of Networks and +Communications, Peng Cheng Laboratory, Shenzhen 518066, China (e-mail: boai@bjtu.edu.cn). +M. Debbah is with the Technology Innovation Institute, Abu Dhabi, United Arab Emirates, and also with CentraleSup´elec, University +Paris-Saclay, 91192 Gif-sur-Yvette, France (e-mail: merouane.debbah@tii.ae). + +2 +uniformly high spectral efficiency (SE) [2]–[7]. Basically, a large number of access points (APs), arbitrarily +distributed in a wide coverage area and connected to one or several central processing units (CPUs), jointly +serve all user equipments (UEs) on the same time-frequency resource. Compared with the traditional +cellular mMIMO system, the CF mMIMO system operates with no cell boundaries and many more APs +than UEs [8]–[10]. Relying upon the prominent network topology of CF mMIMO, four uplink (UL) +signal processing schemes, distinguished from levels of the mutual cooperation between all APs and the +assistance from the CPU, can be implemented as [5]. Among these schemes, the “Level 4” and “Level 3” are +viewed as efficient processing techniques. The so-called Level 4 is a fully-centralized processing scheme +where all the pilot and data signals received at APs are transmitted to the CPU via the fronthaul links +and the CPU performs channel estimation and data detection. The similar scheme was also investigated +in [11]–[13]. The so-called Level 3 stands for a two layer decoding scheme: in the first layer, each AP +estimates channels and decodes the UE data locally by applying an arbitrary combining scheme based +on the local channel state information (CSI); in the second layer, all the local estimates of the UE data +are gathered at the CPU in which they are linearly weighted by the optimal large-scale fading decoding +(LSFD) coefficient to obtain the final decoding data. The LSFD scheme has been widely investigated in +[14]–[17] since it can make full use of the prominent network topology for CF mMIMO and achieve +excellent performance. +To promote the practical implementation of the CF mMIMO network, a new framework of scalable CF +mMIMO system and its respective processing algorithms were proposed in [9] by exploiting the dynamic +cooperation cluster (DCC) concept. Besides, the scalability aspects in a realistic scenario with multiple +CPUs were considered in [18], where the data processing, network topology and power control strategies +with multiple CPUs were discussed. Moreover, the authors of [19] considered the uplink of a radio-strip- +based CF mMIMO network architecture with sequential fronthaul links between APs and proposed MMSE- +based sequential processing schemes, which significantly reduced the fronthaul requirement. However, +when the CF mMIMO network is operated in practice, a more practical capacity-constrained fronthaul +network would have a great effect on the system performance. The authors of [20] and [21] discussed +the uplink performance of a CF mMIMO system with limited capacity fronthaul links. Furthermore, it +is worth noting that the CF mMIMO architecture has been co-designed with another promising future +wireless technology: Reconfigurable Intelligent Surface (RIS) [22], [23], which would undoubtedly provide +vital tutorials for the future wireless network design. + +3 +The vast majority of scientific papers on CF mMIMO focus on the scenario with single-antenna UEs. +However, in practice, contemporary UEs with moderate physical sizes have already been equipped with +multiple antennas to achieve higher multiplexing gain and boost the system reliability. The authors of [24] +investigated the UL performance of a CF mMIMO system with multi-antenna UEs over maximum ratio +(MR) combining and zero-forcing (ZF) combining. The authors of [25] considered a user-centric (UC) +approach for CF mMIMO with multi-antenna UEs and proposed power allocation strategies for either sum- +rate maximization or minimum-rate maximization. Besides, the authors of [26] analyzed the downlink SE +performance for a CF mMIMO system with multi-antenna UEs and computed SE expressions in closed- +form. Then, the SE performance for a CF mMIMO system with multi-antenna UEs and low-resolution +DACs was investigated in [27]. Nevertheless, these works only investigated a simple distributed processing +scheme and are based on the overly idealistic assumption of independent and identically distributed (i.i.d.) +Rayleigh fading channels, neglecting the spatial correlation that has a significant impact on practical CF +mMIMO systems [15], [16]. The authors of [28] considered a CF mMIMO system with multi-antenna +UEs over the jointly-correlated Weichselberger model [29] and analyzed four UL processing schemes. +As observed in [26], [28], increasing the number of antennas per UE may not always benefit the SE +performance. The SE would reach the maximum value with particular number of antennas per UE, then +decrease with the increase of number of antennas per UE. One main reason for this phenomenon is +that the UEs cannot make full use of the benefit of equipping with multiple antennas to achieve higher +SE performance without UL precoding schemes. So it is undoubtedly vital to design the UL precoding +scheme to further improve the performance of systems. However, it is worth noting that the design of +UL precoding for CF mMIMO has not been investigated. For the traditional mMIMO or MIMO systems, +one popular optimization objective for the uplink/downlink precoding design is to maximize the weighted +sum rate (WSR) [30]–[33]. The authors of [30] and [32] discussed the equivalence between the WSR +maximization problem and the Weighted sum-Minimum Mean Square Error (WMMSE) problem in MIMO +systems and proposed an iteratively downlink transceiver design algorithm for the WSR maximization. +Note that the algorithm relies on the iterative minimization of weighted MSE since the WMMSE problem +are not jointly convex over all optimization variables. Moreover, the authors of [31] investigated the UL +precoding scheme optimization based on [30] under sum-power-constraint or individual-power-constraint. +Motivated by the above observations, we investigate a CF mMIMO system with both multi-antenna +APs and UEs over the Weichselberger Rayleigh fading channel. Two pragmatic processing schemes: 1) + +4 +the fully centralized processing scheme; 2) the large-scale fading decoding scheme are implemented. The +main contributions are given as follows. +• We design an efficient UL precoding scheme to maximize the WSR for the fully centralized processing +scheme based on an iteratively WMMSE (I-WMMSE) algorithm. Note that the design of I-WMMSE +precoding scheme for the fully centralized processing scheme is implemented at the CPU and based +on the instantaneous CSI. +• For the LSFD processing scheme, we derive a UL precoding scheme for the WSR maximization +based on an iteratively WMMSE algorithm. The design of I-WMMSE precoding scheme for the +LSFD scheme is implemented at the CPU but based only on channel statistics. More importantly, +we compute achievable SE expressions and optimal precoding schemes in novel closed-form for the +LSFD scheme with MR combining. +• We analyze the practical implementation and computation complexity for the proposed I-WMMSE +precoding schemes. It is found that the proposed I-WMMSE precoding schemes can be guaranteed +to converge. More importantly, the proposed UL precoding schemes are efficient to achieve excellent +sum SE/rate performance and the average rate benefits from the multiple antennas at the UE-side, +which undoubtedly provides vital insights for the practical implementation of multi-antenna UEs. +Note that this paper differs from the conference version [1] in the following aspects: i) we investigate +the fully centralized processing/LSFD schemes and design their respective UL precoding schemes, while +only the LSFD scheme was considered in [1]; ii) we provide details for the derivation of the I-WMMSE +precoding schemes, which are omitted in [1] due to the lack of space; iii) we analyze the practical im- +plementation and convergence behavior of the proposed precoding schemes. More importantly, numerical +results show vital insights for the CF mMIMO system with the proposed UL precoding schemes. +The rest of this paper is organized as follows. In Section II, we consider a CF mMIMO system with the +Weichselberger Rayleigh fading channel, and describe the channel estimation and data detection. Then, +Section III introduces the fully centralized processing and LSFD processing schemes, and provides their +respective achievable SE expressions. Novel closed-form SE expressions for the LSFD scheme with MR +combining are derived. More importantly, based on the achievable SE expressions, we propose UL I- +WMMSE precoding schemes for two processing schemes. Then, Section IV provides some insights for +the practical implementation and computation complexity of proposed I-WMMSE precoding schemes. In +Section V, numerical results and performance analysis for the I-WMMSE precoding schemes are provided. + +5 +CPU +Fronthaul +Fig. 1. A cell-free massive MIMO system. +Finally, the major conclusions and future directions are drawn in Section VI. +Notation: Lowercase letters x and boldface uppercase letters X denote the column vectors and matrices, +respectively. E {·}, tr {·} and ≜ are the expectation operator, the trace operator, and the definitions, +respectively. |·|, ∥·∥ and ∥·∥F are the determinant of a matrix or the absolute value of a number, the +Euclidean norm and the Frobenious norm, respectively. vec (A) denotes a column vector formed by the +stack of the columns of A. The n×n identity matrix is represented by In×n. The Kronecker products and +the element-wise products are denoted by ⊗ and ⊙, respectively. Finally, x ∼ NC (0, R) is a circularly +symmetric complex Gaussian distribution vector with correlation matrix R. +II. SYSTEM MODEL +In this paper, we investigate a CF mMIMO system consisting of M APs and K UEs, where all APs +are connected to one or several CPUs via fronthaul links as shown in Fig. 1. For simplicity, there is only +one CPU and all APs serve all UEs1. The numbers of antennas per AP and UE are L and N, respectively. +A standard block fading model is investigated, in which the channel response is constant and frequency +flat in a coherence block of τc-length (channel uses). Let τp and τc − τp denote channel uses dedicated +for the channel estimation and data transmission, respectively. We denote by Hmk ∈ CL×N the channel +response between AP m and UE k. We assume that Hmk for different AP-UE pairs are independent. +A. Channel Model +Based on the jointly-correlated (also known as the Weichselberger model [29]) Rayleigh fading channel2, +Hmk is modeled as +Hmk = Umk,r +� +˜Ωmk ⊙ Hmk,iid +� +UH +mk,t +(1) +1As shown in Fig. 1, a more practical network topology is with multiple CPUs and dynamic cooperation clusters, where each UE is only +served by a cluster of APs and the APs are grouped into cell-centric clusters. Each cell-centric cluster is connected to a particular CPU. +2Note that the Rayleigh fading channel is a special case of the Rician fading channel. And the performance gap between the Rician +channel and the Rayleigh channel is small [34]. However, the focus of this paper is not on the channel model but on the UL precoding +scheme design. So for the simplicity of analysis, we investigate an essential Rayleigh fading channel by assuming there is no line-of-sight +(LoS) link between each UE and AP. + +0000000000006 +where Umk,r = [umk,r,1, · · · , umk,r,L] ∈ CL×L and Umk,t = [umk,t,1, · · · , umk,t,N] ∈ CN×N are the eigen- +vector matrices of the one-sided correlation matrices Rmk,r ≜ E +� +HmkHH +mk +� +and Rmk,t ≜ E +� +HT +mkH∗ +mk +� +, +and Hmk,iid ∈ CL×N is composed of i.i.d. NC (0, 1) random entries, respectively. Besides, we denote +by Ωmk ≜ ˜Ωmk ⊙ ˜Ωmk ∈ RL×N the “eigenmode coupling matrix” with the (l, n)-th element [Ωmk]ln +specifying the average amount of power coupling from umk,r,l to umk,t,n. Hmk can also be formed as +Hmk = [hmk,1, · · · , hmk,N] with hmk,n ∈ CL being the channel between AP m and n-th antenna of UE +k. By stacking the columns of Hmk on each other, we define hmk ≜ vec (Hmk) = [hT +mk,1, · · · , hT +mk,N]T ∼ +NC (0, Rmk), where Rmk ≜ E{hmkhH +mk} is the full correlation matrix +Rmk = (U∗ +mk,t ⊗ Umk,r)diag (vec (Ωmk)) (U∗ +mk,t ⊗ Umk,r)H. +(2) +Moreover, note that Rmk can be structured into the block form as [28] with the (n, i)-th submatrix +being Rni +mk = E{hmk,nhH +mk,i}. Besides, the large-scale fading coefficient βmk can be extracted from Rmk +as βmk = +1 +LN tr (Rmk) = +1 +LN ∥Ωmk∥1. It is worth mentioning that the motivations for adopting the +Weichselberger channel model are: 1) The Weichselberger model investigated in (1) not only captures +the correlation features at both the AP-side and UE-side but models the joint correlation dependence +between each AP-UE pair through the coupling matrix; 2) The coupling matrix Ωmk reflects the practical +spatial arrangement of scattering objects between AP m and UE k. More significantly, the Weichselberger +model can reduce to most channel models of great interest by adjusting the coupling Ωmk to particular +formulation, such as the Kronecker model and i.i.d. Rayleigh fading model [28], [29]; 3) Compared with +other stochastic channel models, the Weichselberger model displays significantly less modeling error, +which is validated based on the practical measurement in [29]. +B. Channel Estimation +For the channel estimation, mutually orthogonal pilot matrices are constructed and each pilot matrix +is composed of N mutually orthogonal pilot sequences. We denote by Φk the pilot matrix assigned to +UE k with ΦH +k Φl = τpIN, if +l = k and 0 otherwise. And Pk is the index subset of UEs using the +same pilot matrix as UE k including itself. When all UEs transmit their pilot matrices, the received +signal at AP m Yp +mk ∈ CL×τp is Yp +m = �K +k=1 HmkFk,pΦT +k + Np +m, where Fk,p ∈ CN×N is the precoding +matrix for UE k under the phase of pilot transmission, Np +m ∈ CL×τp is the additive noise at AP m with +independent NC(0, σ2) entries and σ2 being the noise power, respectively. The pilot transmission should +be implemented under the power constraint as tr(Fk,pFH +k,p) ⩽ pk, where pk is the maximum transmit +power for UE k. To derive sufficient statistics for hmk, AP m projects Yp +mk onto Φ∗ +k as Yp +mk = Yp +mΦ∗ +k = + +7 +�K +l=1 HmlFl,p +� +ΦT +l Φ∗ +k +� ++Np +mΦ∗ +k = � +l∈Pk τpHmlFl,p + Qp +mk, where Qp +mk ≜ Np +mΦ∗ +k. Then, following the +standard MMSE estimation steps in [35] and [36], AP m can compute the MMSE estimation of hmk as +ˆhmk = vec( ˆHmk) = Rmk˜FH +k,pΨ−1 +mkyp +mk, +(3) +where ˆHmk is the MMSE estimation of Hmk, ˜Fk,p = FT +k,p⊗IL, yp +mk ≜ vec (Yp +mk) = � +l∈Pk τp˜Fl,phml + qp +m, +qp +m = vec (Qp +mk) and Ψmk = � +l∈Pk τp˜Fl,pRml˜FH +l,p + σ2ILN, respectively. Note that the estimate ˆhmk and +estimation error ˜hmk = hmk − ˆhmk are independent random vectors distributed as ˆhmk ∼ NC(0, ˆRmk) +and ˜hmk ∼ NC(0, Cmk), where ˆRmk ≜ τpRmk˜FH +k,pΨ−1 +mk˜Fk,pRmk and Cmk ≜ Rmk − ˆRmk. We can also +form ˆRmk and Cmk in the block structure with the (n, i)-th submatrix being ˆRni +mk = E{ˆhmk,nˆhH +mk,i} and +Cni +mk = E{˜hmk,n˜hH +mk,i}, respectively. +C. Data Transmission +For the data transmission, all antennas of all UEs simultaneously transmit their data symbols to all +APs. The received signal ym ∈ CL at AP m is +ym = +K +� +k=1 +Hmksk + nm, +(4) +where nm ∼ NC(0, σ2IL) is the independent receiver noise. The transmitted signal from UE k sk ∈ CN +can be constructed as sk = Fk,uxk, where xk ∼ NC(0, IN) is the data symbol for UE k and Fk,u ∈ CN×N +is the precoding matrix for the data transmission which should satisfy the power constraint of UE k as +tr(Fk,uFH +k,u) ⩽ pk. +III. SPECTRAL EFFICIENCY ANALYSIS AND I-WMMSE PRECODING DESIGN +In this section, we investigate two promising signal processing schemes, called “fully centralized pro- +cessing” and “LSFD processing”, and analyze their corresponding SE performance and design respective +iteratively WMMSE precoding schemes3. +A. Fully Centralized Processing +1) Spectral Efficiency Analysis: For the fully centralized processing scheme, all M APs send all the +received pilot signals and data signals to the CPU. Indeed, both the channel estimation and data detection +are implemented at the CPU. The collective channel hk ∈ CMLN for UE k can be constructed as hk = +[vec(H1k)T, · · · , vec(HMk)T]T ∼ NC(0, Rk) with Rk = diag (R1k, · · · , RMk) ∈ CMLN×MLN being the +3We only optimize the precoding matrices for the phase of data transmission Fk,u. The optimization of Fk,p is left for future research. +Although we do not design Fk,p in this paper, we try to keep the derived equations more generalized. So a scenario with arbitrary Fk,p +instead of limiting Fk,p to a particular form is investigated. It is worth noting that all equations in this paper hold for any Fk,p so undoubtedly +provide some important guidelines for the investigation of optimization design for Fk,p in the future work. + +8 +whole block-diagonal correlation matrix for UE k. Similar to (3), the CPU can derive the channel estimate +for UE k as4 ˆhk ≜ +� +ˆhT +1k, . . . , ˆhT +Mk +�T +∼ NC +� +0, τpRk¯FH +k,pΨ−1 +k ¯Fk,pRk +� +where ¯Fk,p = diag(˜Fk,p, . . . , ˜Fk,p +� +�� +� +M +) +and Ψ−1 +k += diag(Ψ−1 +1k , . . . , Ψ−1 +Mk). The channel estimation error is ˜hk ∼ NC (0, Ck) where Ck ≜ Rk − +τpRk¯FH +k,pΨ−1 +k ¯Fk,pRk. Moreover, the received data signal at the CPU can be denoted as +[yT +1 , · · · , yT +M]T +� +�� +� += y += +K +� +k=1 +[HT +1k, . . . , HT +Mk]T +� +�� +� += Hk +Fk,uxk + [nT +1 , . . . , nT +M]T +� +�� +� += n +, +(5) +or a compact form as y = �K +k=1 HkFk,uxk + n. +Under the setting of “fully centralized processing”, we assume that UL precoding matrices (Fk,u and +Fk,p) are available at the CPU. Based on the collective channel estimates, the CPU designs an arbitrary +receive combining matrix Vk ∈ CLM×N for UE k to detect xk as +ˇxk = VH +k y = VH +k ˆHkFk,uxk + VH +k ˜HkFk,uxk + +K +� +l̸=k +VH +k HlFl,uxl + VH +k n, +(6) +and the conditional MSE matrix for UE k is +Ek,(1) = E{(xk − ˇxk)(xk − ˇxk)H|{ ˆHk}, {Fk,u}} += IN − VH +k ˆHkFk,u − FH +k,u ˆHH +k Vk + VH +k +� K +� +l=1 +� +ˆHl¯Fl,u ˆHH +l + C′ +l +� ++ σ2IML +� +Vk +(7) +where ¯Fl,u ≜ Fl,uFH +l,u, C′ +l ≜ diag (C′ +1l, · · · , C′ +Ml) ∈ CML×ML and C′ +ml = E{ ˜Hml¯Fl,u ˜HH +ml} ∈ CL×L with +the (j, q)-th element of C′ +ml being [C′ +ml]jq = �N +p1=1 +�N +p2=1 +�¯Fl +� +p2p1 [Cp2p1 +ml ]jq. +By implementing the per-user-basis minimum mean-squared error-based successive interference cancel- +lation (MMSE-SIC) detector while treating co-user interference as uncorrelated Gaussian noise, we derive +the achievable SE for UE k as follows. +Corollary 1. An achievable for UE k under the setting of “fully centralized processing” with the MMSE +estimator is +SEk,(1) = +� +1 − τp +τc +� +E +� +log2 +���IN + DH +k,(1)Σ−1 +k,(1)Dk,(1) +��� +� +, +(8) +where Dk,(1) ≜ VH +k ˆHkFk,u and Σk,(1) ≜ VH +k +��K +l=1 ˆHl¯Fl,u ˆHH +l − ˆHk¯Fk,u ˆHH +k + �K +l=1 C′ +l + σ2IML +� +Vk. +4Note that the pilot signals received at the APs are first transmitted to the CPU and then the CPU estimates the channels, where τpML +complex scalars are sent from the APs to the CPU at each coherence block. Alternatively, all APs can first estimate the channels as (3), and +then send their channel estimates to the CPU, where MKLN complex scalars are sent from the APs to the CPU at each coherence block. +Since the pilot contamination is investigated (τp < KN) in this paper, we consider the first transmission protocol due to its lower fronthaul +overhead. + +9 +The expectations are with respect to all sources of randomness. +Proof. The proof follows from the similar approach as [28, Corollary 1] and is therefore omitted. +We notice that Corollary 1 holds for any combining schemes. One promising combining scheme is the +MMSE combining as +VMMSE +k += +� K +� +l=1 +� +ˆHl¯Fl,u ˆHH +l + C′ +l +� ++ σ2IML +�−1 +ˆHkFk,u, +(9) +which can minimize the mean-squared error MSEk,(1) = tr(Ek,(1)). With the MMSE combining scheme, +the conditional MSE matrix in (7) is +Eopt +k,(1) = IN − FH +k,u ˆHH +k +� K +� +l=1 +� +ˆHl¯Fl,u ˆHH +l + C′ +l +� ++ σ2IML +�−1 +ˆHkFk,u +(10) +More importantly, the MMSE combining in (9) can also maximize the achievable SE in (8) as follows. +Corollary 2. The achievable SE for UE k in (8) can be maximized by the MMSE combining scheme in +(9) with the maximum value +SEopt +k,(1) = +� +1 − τp +τc +� +E + + +log2 +������ +IN + FH +k,u ˆHH +k +� K +� +l=1 +� +ˆHl¯Fl,u ˆHH +l + C′ +l +� +− ˆHk¯Fk,u ˆHH +k + σ2IML +�−1 +ˆHkFk,u +������ + + + . +(11) +Proof. The proof can be found in [28, Appendix B] and is therefore omitted. +2) Iteratively WMMSE Precoding Design: In this part, we design the uplink precoding scheme for the +“fully centralized processing”. One popular weighted sum-rate maximization problem is investigated as5 +max +{F} +K +� +k=1 +µk,(1)SEk,(1) +s.t. +��Fk,u,(1) +��2 ⩽ pk ∀k = 1, . . . , K +(12) +where µk,(1) represents the priority weight of UE k and SEk,(1) is given by (8). +5The notation F is short for {Fk,u}k=1,...,K, denoting all variables Fk,u with k = 1, . . . , K. Similar definitions are applied for V, A, W, +S in the following. In this section, we denote by Fk,u,(1) and Fk,u,(2) the UL precoding matrix of UE k for the fully centralized processing +and LSFD scheme, respectively. + +10 +As [30] and [32], the matrix-weighted sum-MSE minimization problem as +min +{F,V,W} +K +� +k=1 +µk,(1) +� +tr +� +Wk,(1)Ek,(1) +� +− log2 +��Wk,(1) +��� +s.t. +��Fk,u,(1) +��2 ⩽ pk ∀k = 1, . . . , K +(13) +is equivalent to the weighted sum-rate maximization problem (12), where Wk,(1) ∈ CN×N is the weight +matrix for UE k. We notice that (13) is convex over each optimization variable F, V, W but is not jointly +convex over all optimization variables. Following the method in [30], we can solve (13) by sequentially +fixing two of the three optimization variables F, V, W and updating the third. +Fixing the other variables, the update of Vk is given by the MMSE solution as (9). Under the MMSE +combining, the MSE matrix is given by (10). Then, note that optimal Wk,(1) for (13) is +Wopt +k,(1) = E−1 +k,(1), +(14) +which can be easily derived through the first order optimality condition for Wk,(1) by fixing F and V. +Remark 1. When the MMSE combining VMMSE +k +and Wopt +k,(1) for all UEs are implemented in (13), we have +tr(Wk,(1)Ek,(1))−log2 +��Wk,(1) +�� = tr (IN)−log2 |(Eopt +k,(1))−1|. So the matrix-weighted sum-MSE minimization +problem in (13) would reduce to the equivalent optimization problem of (12) as6: +max +{F} +K +� +k=1 +µk,(1) log2 +���� +� +Eopt +k,(1) +�−1���� +s.t. +��Fk,u,(1) +��2 ⩽ pk ∀k = 1, . . . , K +(15) +which is a well-known relationship between Eopt +k,(1) and SEopt +k,(1). +Finally, fixing V and W, the update of Fk,u,(1) for (13) results in the optimization problem as7 +6Note that “SE” is equivalent to “rate” except from having one scaling factor (τc − τp)/τc. Since τc and τp are constants in this paper, +so we ignore the difference between SE and rate in the optimization problem. +7It is worth mentioning that the updates of optimization variables are based on the preliminary of fixing the other optimization variables. +For instance, when updating Fk,u,(1), we should fix the other optimization variables (Vk and Wk,(1)) but not only limited to their respective +optimal solutions VMMSE +k +and Wopt +k,(1). So we update Fk,u,(1) based on (16) with generalized Vk and Wk,(1) instead of (15) with optimal +VMMSE +k +and Wopt +k,(1). + +11 +min +{F} +K +� +k=1 +µk,(1)tr +� +Wk,(1) +� +IN − VH +k ˆHkFk,u,(1) +� � +IN − VH +k ˆHkFk,u,(1) +�H� ++ +K +� +k=1 +µk,(1)tr +� +Wk,(1)VH +k +� K +� +l̸=k +ˆHlFl,u,(1)FH +l,u,(1) ˆHH +l +� +Vk +� +− +K +� +k=1 +µk,(1) log2 +��Wk,(1) +�� ++ +K +� +k=1 +µk,(1)tr +� +Wk,(1)VH +k +� K +� +l=1 +E +� +˜HlFl,u,(1)FH +l,u,(1) ˜HH +l +��� F +� ++ σ2IML +� +Vk +� +s.t. +��Fk,u,(1) +��2 ⩽ pk ∀k = 1, . . . , K +(16) +which is a convex quadratic optimization problem. So the classic Lagrange multipliers methods and +Karush-Kuhn-Tucker (KKT) conditions can be applied to derive an optimal solution. The Lagrange +function of (16) is +f +� +F1,u,(1), . . . , FK,u,(1) +� += +K +� +k=1 +µk,(1)tr +� +Wk,(1) +� +IN − VH +k ˆHkFk,u,(1) +� � +IN − VH +k ˆHkFk,u,(1) +�H� ++ +K +� +k=1 +µk,(1)tr +� +Wk,(1)VH +k +� K +� +l̸=k +ˆHlFl,u,(1)FH +l,u,(1) ˆHH +l +� +Vk +� ++ +K +� +k=1 +µk,(1)tr +� +Wk,(1)VH +k +� K +� +l=1 +E +� +˜HlFl,u,(1)FH +l,u,(1) ˜HH +l +��� F +� ++ σ2IML +� +Vk +� ++ +K +� +k=1 +λk,(1) +� +tr +� +Fk,u,(1)FH +k,u,(1) +� +− pk +� +(17) +Finally, we derive the optimal precoding scheme as the following theorem. +Theorem 1. By fixing other optimization variables and applying the first-order optimality condition of +(17) with respect to each Fk,u,(1), the optimal precoding scheme is given by +Fopt +k,u,(1) = µk,(1) +� K +� +l=1 +µl,(1) +� +ˆHH +k VlWl,(1)VH +l ˆHk + E +� +˜HH +k VlWl,(1)VH +l ˜Hk +��� V, W +�� ++ λk,(1)IN +�−1 +ˆHH +k VkWk,(1) += µk,(1) +� K +� +l=1 +µl,(1) +� +ˆHH +k VlWl,(1)VH +l ˆHk + ¯Ckl +� ++ λk,(1)IN +�−1 +ˆHH +k VkWk,(1), +(18) +where λk,(1) ⩾ 0 is the Lagrangian multiplier and the (i, n)-th element of ¯Ckl ≜ E{ ˜HH +k VlWl,(1)VH +l ˜Hk|V, W} +∈ CN×N is +�¯Ckl +� +in = tr( ¯VlE{˜hk,n˜hH +k,i}) = tr +�¯VlCk,in +� +with ¯Vl ≜ VlWl,(1)VH +l and Ck,ni ≜ E{˜hk,n˜hH +k,i} = + +12 +diag (Cni +1k, . . . , Cni +Mk) ∈ CML×ML. According to the KKT condition, λk,(1) and Fk,u,(1) should also satisfy +��Fk,u,(1) +��2 ⩽ pk, +λk,(1) +���Fk,u,(1) +��2 − pk +� += 0, +λk,(1) ⩾ 0. +(19) +Proof: The proof is given in Appendix D. +We denote by Fk,u,(1)(λk,(1)) the right-hand side of (18), when �K +l=1 µl,(1)( ˆHH +k VlWl,(1)VH +l ˆHk + ¯Ckl) +is invertible and tr[Fk,u,(1)(0)Fk,u,(1)(0)H] ⩽ pk, then Fopt +k,u,(1) = Fk,u,(1) (0), otherwise we have +tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] = pk +to satisfy (19). +Corollary 3. tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] is a monotonically decreasing function of λk,(1). +Proof: Let DΛDH denote the eigendecomposition of �K +l=1 µl,(1)( ˆHH +k VlWl,(1)VH +l ˆHk + ¯Ckl). Fol- +lowing the method in [30], we define Φ = µ2 +k,(1)DH ˆHH +k VkW2 +k,(1) ˆHkVH +k D and we have +tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] = tr +�� +DΛDH + λk,(1)IN +�−1 DΦDH � +DΛDH + λk,(1)IN +�−1� += tr +�� +DΛDH + λk,(1)IN +�−2 DΦDH� += tr +�� +Λ + λk,(1)IN +�−2� += +N +� +n=1 +[Φ]nn +� +[Λ]nn + λk,(1) +�2, +(20) +so tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] is a monotonically decreasing function of λk,(1). +Based on Corollary 3, optimum λk,(1) (denoted by λopt +k,(1)) can be easily obtained by a one-dimensional +(1-D) bisection algorithm so we derive the solution for Fk,u,(1)(λopt +k,(1)). Furthermore, an iterative opti- +mization algorithm for Fk,u,(1), called “iteratively WMMSE (I-WMMSE) algorithm”, is summarized in +Algorithm 18. The convergence of Algorithm 1 is proven in [30, Theorem 3]. +Remark 2. Note that the design of Fk,p is a valuable future direction to further improve the system +performance. One valuable optimization problem is to minimize the total MSE of the channel estimators +of all UEs as +min +{Fk,p} +K +� +k=1 +tr (Ck) +s.t. ∥Fk,p∥2 ⩽ pk ∀k = 1, . . . , K +(21) +where the optimization goal is only based on the statistical knowledge so Fk,p is also based on the +statistical knowledge. +8To balance the efficiency and the computational complexity of the proposed algorithm, we also include the stopping criterion “R(i) +(1) < +R(i−1) +(1) +. Moreover, the I-WMMSE precoding scheme is derived at iteration (i − 1), which may achieve higher sum SE than the one at +iteration i. + +13 +Algorithm 1: I-WMMSE Algorithm for the Design of Fk,u,(1) +Input: Collective channel estimates ˆHk for all UEs; Estimation error covariance matrices Cml for +all possible pairs; UE weights µk,(1) for all UEs; +Output: Optimal precoding matrices Fk,u,(1) for all UEs (F(i) +k,u,(1) for the first or third stopping +criterion and F(i−1) +k,u,(1) for the second stopping criterion); +1 Initiation: i = 0, F(0) +k,u,(1) and R(0) +(1) = �K +k=1 µk,(1)SE(0) +k,(1) for all UEs; maximum iteration number +I(1),max and threshold ε(1); +2 repeat +3 +i = i + 1 +4 +Update the MMSE combining scheme V(i) +k +with F(i−1) +k,u,(1) based on (9); +5 +Update optimal MSE matrix E(i) +k,(1) with F(i−1) +l,u,(1) based on (10), and update W(i) +k,(1) based on +(14); +6 +Update optimal precoding matrix F(i) +l,u,(1) with V(i) +k +and W(i) +k,(1) based on (18), where λ(i) +k,(1) is +found by a bisection algorithm; +7 +Update sum weighted rate R(i) +(1) = �K +k=1 µk,(1)SE(i) +k,(1); +8 until +���R(i) +(1) − R(i−1) +(1) +��� /R(i−1) +(1) +⩽ ε(1) or R(i) +(1) < R(i−1) +(1) +or i ⩾ I(1),max; +B. Large-Scale Fading Decoding +1) Spectral Efficiency Analysis: Another promising processing scheme is “large-scale fading decoding”, +which is a two-layer decoding scheme to decode the data symbol. Note that UL precoding matrices (Fk,u +and Fk,p) are assumed to be available at all APs and the CPU. In the first layer, AP m applies an arbitrary +combining matrix Vmk ∈ CL×N to derive local detection of xk as +˜xmk = VH +mkym = VH +mkHmkFk,uxk + +K +� +l=1,l̸=k +VH +mkHmlFl,uxl + VH +mknm. +(22) +We notice that Vmk is designed based on local channel estimates at AP m and one handy choice is +MR combining Vmk = ˆHmk. Moreover, local MMSE (L-MMSE) combining +Vmk = +� K +� +l=1 +� +ˆHml¯Fl,u ˆHH +ml + C′ +ml +� ++ σ2IL +�−1 +ˆHmkFk,u, +(23) +is also regarded as a promising scheme, since (23) can minimize E{∥ xk − VH +mkym ∥2 |{ ˆHmk}, {Fk,u}}. +In the second layer, the “LSFD” method is implemented at the CPU [5]. The CPU weights all the local +estimates ˜xmk from all APs by the LSFD coefficient matrix as +ˆxk = +M +� +m=1 +AH +mk˜xmk = +M +� +m=1 +AH +mkVH +mkHmkFk,uxk + +M +� +m=1 +K +� +l=1,l̸=k +AH +mkVH +mkHmlFl,uxl+n′ +k, +(24) +where Amk ∈ CN×N is the complex LSFD coefficient matrix for AP m-UE k and n′ +k = �M +m=1 AH +mkVH +mknm. + +14 +Moreover, we can rewrite ˆxk in a more compact form as +ˆxk = AH +k GkkFk,uxk + +K +� +l=1,l̸=k +AH +k GklFl,uxl + n′ +k = AH +k +� +GkkFk,uxk + +K +� +l=1,l̸=k +GklFl,uxl + ˜n′ +k +� +� +�� +� +˜xk +(25) +where Ak ≜ [AT +1k, . . . , AT +Mk]T ∈ CMN×N, Gkl ≜ [VH +1kH1l; . . . ; VH +MkHMl] ∈ CMN×N and +˜n′ +k = +� +VH +1kn1; . . . ; VH +MknM +� +∈ CMN×N. +Note that the CPU does not have the knowledge of channel estimates and is only aware of channel +statistics [5]. The conditional MSE matrix for UE k Ek,(2) ≜ E +� +(xk − ˆxk) (xk − ˆxk)H |Θ +� +is +Ek,(2) = IN − FH +k,uE{GH +kk}Ak − AH +k E{Gkk}Fk,u + AH +k +� K +� +l=1 +E{Gkl¯Fl,uGH +kl} + σ2Sk +� +Ak, +(26) +where Θ denotes all the channel statistics and Sk = diag(E{VH +1kV1k}, · · · , E{VH +MkVMk}) ∈ CMN×MN. +Then, we apply classical use-and-then-forget bound to obtain the following ergodic achievable SE. +Corollary 4. For the “LSFD” scheme, an achievable SE for UE k can be written as +SEk,(2) = +� +1 − τp +τc +� +log2 +���IN + DH +k,(2)Σ−1 +k,(2)Dk,(2) +��� , +(27) +where Σk,(2) = �K +l=1 AH +k E{Gkl¯Fl,uGH +kl}Ak − Dk,(2)DH +k,(2) + σ2AH +k SkAk and Dk,(2) = AH +k E{Gkk}Fk,u. +Proof. The proof follows similar steps as the proof of [28, Corollary 2] and is therefore omitted. +Note that Ak can be optimized by the CPU based on channel statistics to maximize the achievable +SE in (27). Based on the theory of optimal receivers as in [37], we derive the optimal LSFD coefficient +matrix, which not only maximizes the achievable SE but minimizes the conditional MSE, as follows. +Corollary 5. The achievable SE in (27) is maximized by +Aopt +k += +� K +� +l=1 +E{Gkl¯Fl,uGH +kl} + σ2Sk +�−1 +E{Gkk}Fk,u, +(28) +leading to the maximum value as +SEopt +k,(2) += +� +1 − τp +τc +� +log2 +������ +IN + FH +k,uE {Gkk} +� K +� +l=1 +E +� +Gkl¯Fl,uGH +kl +� +− E {Gkk} ¯Fk,uE +� +GH +kk +� ++ σ2Sk +�−1 +E {Gkk} Fk,u +������ +. +(29) + +15 +Note that the optimal LSFD coefficient matrix in (28) can also minimize the conditional MSE for UE k +MSEk,(2) = tr(Ek,(2)). +Proof. The proof is given in Appendix B. +If the optimal LSFD coefficient matrix is applied, the MSE matrix for UE k can be written as +Eopt +k,(2) = IN − FH +k,uE +� +GH +kk +� +� K +� +l=1 +E{Gkl¯Fl,uGH +kl} + σ2Sk +�−1 +E{Gkk}Fk,u. +(31) +Furthermore, if MR combining Vmk = ˆHmk is applied, we derive closed-form SE expressions as follows. +Theorem 2. For MR combining Vmk = ˆHmk, (27) can be computed in closed-form as +SEk,(2),c = +� +1 − τp +τc +� +log2 +���IN + DH +k,(2),cΣ−1 +k,(2),cDk,(2),c +��� , +(32) +where Σk,(2),c = AH +k (�K +l=1 Tkl,(1) + � +l∈Pk Tkl,(2))Ak − Dk,(2),cDH +k,(2),c + σ2AH +k Sk,cAk and Dk,(2),c = +AH +k ZkFk,u, with E{Gkk} = Zk = [ZT +1k, . . . , ZT +Mk]T and Sk,c = diag(Z1k, · · · , ZMk) with the (n, n′)- +th element of Zmk ∈ CN×N being [Zmk]nn′ = tr(ˆRn′n +mk). Moreover, Tkl,(1) ≜ diag(Γ(1) +kl,1, · · · , Γ(1) +kl,M) ∈ +CMN×MN and Tmm′ +kl,(2) = Γ(2) +kl,m − Γ(1) +kl,m if m = m′ and Λmkl¯Fl,uΛm′lk otherwise, where Tmm′ +kl,(2) de- +notes (m, m′)-submatrix of Tkl,(2) ∈ CMN×MN, the (n, n′)-th element of N × N-dimension complex +matrices Λmkl, Λm′lk, Γ(1) +kl,m and Γ(2) +kl,m are [Λmkl]nn′ = tr(Ξn′n +mkl), [Λm′lk]nn′ = tr(Ξn′n +m′lk), [Γ(1) +mkl]nn′ = +�N +i=1 +�N +i′=1 [¯Fl,u]i′itr(Ri′i +ml ˆRn′n +mk) and [Γ(2) +kl,m]nn′ given by +� +Γ(2) +kl,m +� +nn′ = +N +� +i=1 +N +� +i′=1 +�¯Fl +� +i′i +� +tr +� +Ri′i +mlPn′n +mkl,(1) +� ++τ 2 +p +N +� +q1=1 +N +� +q2=1 +� +tr +� +˜Pq1n +mkl,(2) ˜Ri′q2 +ml ˜Rq2i +ml ˜Pn′q1 +mkl,(2) +� ++ tr +� +˜Pq1n +mkl,(2) ˜Ri′q2 +ml +� +tr +� +˜Pn′q2 +mkl,(2) ˜Rq2i +ml +��� +(33) +with Ξmkl = τpRml˜FH +l,pΨ−1 +mk˜Fk,pRmk, Ξm′lk = τpRm′k˜FH +k,pΨ−1 +m′k˜Fl,pRm′l, Pmkl,(1) = τpSmk(Ψmk − +τp˜Fl,pRml˜FH +l,p)SH +mk, Smk = Rmk˜FH +k,pΨ−1 +mk, Pmkl,(2) = Smk˜Fl,pRml˜FH +l,pSH +mk, ˜Rni +ml and ˜Pni +mkl,(2) being (n, i)- +submatrix of R +1 +2 +ml and P +1 +2 +mkl,(2), respectively. Furthermore, the optimal LSFD coefficient matrix in (28) +and MSE matrix in (31) can also be computed in closed-form as + + + + + +Aopt +k,c = +��K +l=1 Tkl,(1) + � +l∈Pk Tkl,(2) + σ2Sk,c +�−1 +ZkFk,u, +Eopt +k,(2),c = IN − FH +k,uZH +k +��K +l=1 Tkl,(1) + � +l∈Pk Tkl,(2) + σ2Sk,c +�−1 +ZkFk,u. +(34) +Proof: The proof is given in Appendix C. + +16 +2) Iteratively WMMSE Precoding Design: For the LSFD scheme, we also investigate a weighted sum- +rate maximization problem as +max +{F} +K +� +k=1 +µk,(2)SEk,(2) +s.t. +��Fk,u,(2) +��2 ⩽ pk ∀k = 1, . . . , K +(35) +where µk,(2) represents the priority weight of UE k for the “LSFD” scheme and SEk,(2) is given in (27) +with arbitrary combining structure in the first decoding layer. +Similarly, the matrix-weighted sum-MSE minimization problem as9 +min +{F,A,W,G,S} +K +� +k=1 +µk,(2) +� +tr +� +Wk,(2)Ek,(2) +� +− log2 +��Wk,(2) +��� +s.t. +��Fk,u,(2) +��2 ⩽ pk ∀k = 1, . . . , K +(36) +is equivalent to the weighted sum-rate maximization problem (35), where Wk,(2) is the weight matrix for +UE k. Note that (36) is convex over each optimization variable F, A, W, G, S but is not jointly convex +over all optimization variables. So we can solve (36) by sequentially fixing four of the five optimization +variables F, A, W, G, S and updating the fifth.10 +The update of Ak and Ek,(2) are given by the optimal LSFD scheme (28) and MSE matrix with optimal +LSFD scheme (31). Note that optimal Wk,(2) for (36) is Wopt +k,(2) = E−1 +k,(2). +Remark 3. When Aopt +k +and Wopt +k,(2) for all UEs are applied in (36), we notice that (36) becomes to the +equivalent optimization problem of (35) as +max +{F,G,S} +K +� +k=1 +µk,(2) log2 +���� +� +Eopt +k,(2) +�−1���� +s.t. +��Fk,u,(2) +��2 ⩽ pk ∀k = 1, . . . , K +(37) +which is a well-known relationship between Eopt +k,(2) and SEopt +k,(2) and proven in Appendix E. +Last but not least, fixing other variables, the update of Fk,u,(2) for (36) results in the optimization +9The notation G denotes all G-relevant variables, like E{Gkl¯Fl,u,(2)GH +kl} and E{Gkk}, etc. +10As for G and S, if L-MMSE combining scheme applied, E {Gkk} and Sk are relevant to Fk,u,(2) so we should also update them. On +the contrary, E{Gkk} and Sk with MR combining structure are irrelevant to F so we only need to update E{Gkl¯Fl,u,(2)GH +kl}. + +17 +problem as +min +{F} +K +� +k=1 +µk,(2) +� +tr +� +Wk +� +IN − FH +k,u,(2)E +� +GH +kk +� +Ak +� � +IN − FH +k,u,(2)E +� +GH +kk +� +Ak +�H�� ++ +K +� +k=1 +µk,(2) +� +tr +� +Wk,(2)AH +k +� K +� +l̸=k +E +� +Gkl¯Fl,u,(2)GH +kl +� ++ σ2Sk +� +Ak +�� +s.t. +��Fk,u,(2) +��2 ⩽ pk ∀k = 1, . . . , K +(38) +which is a convex quadratic optimization problem. Thus, we can also derive the optimal precoding scheme +by applying classic Lagrange multipliers methods and KKT conditions. The Lagrange function of (38) is +f +� +F1,u,(2), . . . , FK,u,(2) +� += +K +� +k=1 +µk,(2) +� +tr +� +Wk,(2) +� +IN − FH +k,u,(2)E +� +GH +kk +� +Ak +� � +IN − FH +k,u,(2)E +� +GH +kk +� +Ak +�H�� ++ +K +� +k=1 +µk,(2) +� +tr +� +Wk,(2)AH +k +� K +� +l̸=k +E +� +Gkl¯Fl,u,(2)GH +kl +� ++ σ2Sk +� +Ak +�� ++ +K +� +k=1 +λk,(2) +� +tr +� +Fk,u,(2)FH +k,u,(2) +� +− pk +� +. +(39) +Theorem 3. By applying the first-order optimality condition of (39) with respect to each Fk,u,(2) and +fixing other optimization variables, we obtain the optimal precoding scheme as +Fopt +k,u,(2) = µk,(2) +� K +� +l=1 +µl,(2)E +� +GH +lkAlE−1 +l,(2)AH +l Glk +� ++ λk,(2)IN +�−1 +E +� +GH +kk +� +AkE−1 +k,(2), +(40) +where λk,(2) ⩾ 0 is the Lagrangian multiplier during the phase of “LSFD” scheme. According to the KKT +condition, λk,(2) and Fk,u,(2) should also satisfy +��Fk,u,(2) +��2 ⩽ pk, +λk,(2) +���Fk,u,(2) +��2 − pk +� += 0, +λk,(2) ⩾ 0. +(41) +Note that when �K +l=1 µl,(2)E{GH +lkAlE−1 +l,(2)AH +l Glk} is invertible and tr +� +Fk,u,(2)(0)Fk,u,(2)(0 +�H] ⩽ pk, +then Fopt +k,u,(2) = Fk,u,(2) (0), otherwise we must have tr[Fk,u,(2)(λk,(2))Fk,u,(2)(λk,(2))H] = pk. Following the +similar method in Corollary 3, we notice that λk,(2) can be easily found by a 1-D bisection algorithm +since tr[Fk,u,(2)(λk,(2))Fk,u,(2)(λk,(2))H] is a monotonically decreasing function of λk,(2). +Moreover, if MR combining Vmk = ˆHmk is applied in the first layer, we can compute expectations in +(40) in closed-form as following theorem. +Theorem 4. With MR combining Vmk = ˆHmk and the optimal LSFD scheme applied, we can compute + +18 +E{GH +kk}, Aopt +k , and Eopt +k,(2) in closed-form as Theorem 2. Moreover, we have ¯Tlk = E{GH +lkAlE−1 +l,(2)AH +l Glk} ∈ +CN×N where the (i, n)-th element of ¯Tlk is tr(¯Al ¯Glk,ni) with ¯Al ≜ AlE−1 +l,(2)AH +l +and the [(m − 1) N + +p, (m′ − 1) N + p′]-th (or [o, j]-th briefly) entry of ¯Glk,ni ≜ E{glk,ngH +lk,i} ∈ CMN×MN being +E{glk,ngH +lk,i}oj = + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +0, +l /∈ Pk, m ̸= m′ +tr(Rni +mk ˆRp′p +ml), +l /∈ Pk, m = m′ +tr(Ξnp +mlk)tr(Ξp′i +m′kl), +l ∈ Pk, m ̸= m′ +tr +� +Rni +mkPp′p +mlk,(1) +� ++ τ 2 +p +�N +q1=1 +�N +q2=1 tr +� +˜Pq1p +mlk,(2) ˜Rnq2 +mk ˜Rq2i +mk ˜Pp′q1 +mlk,(2) +� ++τ 2 +p +�N +q1=1 +�N +q2=1 tr +� +˜Pq1n +mlk,(2) ˜Rnq1 +mk +� +tr +� +˜Pp′q2 +mlk,(2) ˜Rq2i +mk +� +, +l ∈ Pk, m = m′ +(42) +where Ξmlk = τpRmk˜FH +k,pΨ−1 +mk˜Fl,pRml, Ξm′kl = τpRm′l˜FH +l,pΨ−1 +m′l˜Fk,pRm′k, Sml = Rml˜FH +l,pΨ−1 +ml, Pmlk,(1) = +τpSml(Ψml − τp˜Fk,pRmk˜FH +k,p)SH +ml and Pmlk,(2) = Sml˜Fk,pRmk˜FH +k,pSH +ml. Plugging the derived results into +(40), we can compute Fopt +k,u,(2) in closed-form as +Fopt +k,u,(2),c = µk,(2) +� K +� +l=1 +µl,(2) ¯Tlk + λk,(2)IN +�−1 +ZH +k Aopt +k,c Eopt,−1 +k,(2),c . +(43) +Proof: The proof is given in Appendix F. +Furthermore, an iterative optimization algorithm for Fk,u,(2) is summarized in Algorithm 2. The con- +vergence of Algorithm 2 is proven in [30, Theorem 3]. +Remark 4. Relying on the iterative minimization of weighted MSE, two efficient uplink I-WMMSE +precoding schemes to maximize the weighted sum SE are proposed. The I-WMMSE precoding schemes +for the “FCP” and “LSFD” schemes are investigated in Algorithm 1 and Algorithm 2, respectively. Note +that the design of I-WMMSE precoding scheme for the FCP/LSFD is based on instantaneous CSI/channel +statistics, respectively. More importantly, we can compute I-WMMSE precoding schemes in novel closed- +form only for the LSFD scheme with MR combining based on Theorm 4. +IV. PRECODING IMPLEMENTATION AND COMPLEXITY ANALYSIS +In this section, we discuss the practical implementation and analyze computational complexity for the +UL precoding schemes investigated in Section III. +A. Precoding Implementation +1) Precoding Characteristics: As described above, we investigate a standard block fading model, where +the channel response is constant and frequency flat in a coherence block, which contains τc channel + +19 +Algorithm 2: I-WMMSE Algorithm for the Design of Fk,u,(2) +Input: Channel statistics Θ for all possible pairs; UE weights µk,(2) for all UEs; +Output: Optimal precoding matrices Fk,u,(2) for all UEs (F(i) +k,u,(2) for the first or third stopping +criterion and F(i−1) +k,u,(2) for the second stopping criterion); +1 Initiation: i = 0, F(0) +k,u,(2) and R(0) +(2) = �K +k=1 µk,(2)SE(0) +k,(2) for all UEs; maximum iteration number +I(2),max and threshold ε(2); +2 repeat +3 +i = i + 1 +4 +Update channel statistics Θ(i), such as E{G(i) +kk}, E{G(i) +kl ¯F(i−1) +l,u +(G(i) +kl )H} and S(i) +k ; +5 +Update optimal LSFD matrix A(i) +k +with F(i−1) +l,u,(2) and Θ(i) based on (28); +6 +Update optimal MSE matrix E(i) +k,(2) with F(i−1) +l,u,(2), A(i) +k +and E{G(i) +kk} based on (31) and update +W(i) +k,(2); +7 +Update optimal precoding matrix F(i) +k,u,(2) with A(i) +k , W(i) +k,(2) and Θ(i) based on (40), where +λ(i), +k,(2) is found by a bisection algorithm; +8 +Update sum weighted rate R(i) +(2) = �K +k=1 µk,(2)SE(i) +k,(2); +9 until |R(i) +(2) − R(i−1) +(2) +|/R(i−1) +(2) +⩽ ε(2) or R(i) +(2) < R(i−1) +(2) +or i ⩾ I(2),max; +uses. For the “fully centralized processing” scheme, we notice that the I-WMMSE precoding design is +implemented at the CPU based on the instantaneous CSI as (18). Moreover, to guarantee the convergence +of Algorithm 1, only MMSE combining as (9) is advocated to detect the UL data since the equivalent +relationship between Eopt +k,(1) and SEopt +k,(1), which only satisfies with MMSE combining, should be guaranteed. +As for the LSFD scheme, the optimal design of Fopt +k,u,(2) as (40) can only be implemented at the CPU, but +relies only on channel statistics. Besides, L-MMSE or MR combining can be applied at each AP. When +MR combining is applied, all terms in Algorithm 2 can be computed in closed-form as Theorem 4. +2) Fronthaul Requirements: For the FCP scheme with the I-WMMSE precoding, in each coherence +block, all APs should relay their received signals to the CPU and the CPU requires precoding matrices +Fk,u,(1) feedback to all UEs. All APs need to send τcML complex scalars (τpML complex scalars for the +pilot signals and (τc − τp)ML complex scalars for the received data signals). Besides, the full correlation +matrices {Rmk} are available at the CPU, which contains MKL2N2/2 complex scalars for each realization +of the AP/UE locations/statistics11. Moreover, the CPU transmits optimal precoding matrices to all UEs, +which are described by KN2 complex scalars per coherence block. In summary, for the FCP scheme +with the I-WMMSE precoding implemented, total τcMLNr + MKL2N2/2 + KN2Nr complex scalars +are transmitted via fronthaul links for each realization of the AP/UE locations. For comparison, when +11Note that the channel statistics remain constant for each realization of the AP/UE locations and each realization of the AP/UE locations +contains Nr channel realizations (coherence blocks). + +20 +the FCP scheme without the I-WMMSE precoding is implemented, all APs should also transmit τcML +complex scalars for the received signals to the CPU in each coherence block and MKL2N2/2 complex +scalars for {Rmk} to the CPU for each realization of the AP/UE locations. So for the CFP scheme without +the I-WMMSE precoding, total τcMLNr + MKL2N2/2 complex scalars are transmitted via fronthaul +links for each realization of the AP/UE locations. +As for the LSFD scheme with the I-WMMSE precoding, all APs transmit their local data estimates +˜xmk, described by (τc − τp)MKN complex scalars, to the CPU per coherence block. Besides, E{Gkk} ∈ +CMN×N, described by MKN2 complex scalars for each realization of the AP/UE locations, are also +required at the CPU. As for E{Gkl¯Fl,u,(2)GH +kl} ∈ CMN×MN, following the formulation method investigated +in Appendix F, the optimization of ¯Fl,u,(2) requires the knowledge of +� +E +� +vH +ml,phmk,nhH +m′k,ivm′l,p′�� +, +described by M2K2N4/2 complex scalars for each realization of the AP/UE locations, where vml,p denotes +the p-th column of Vml. Moreover, the CPU requires optimal precoding matrices Fk,u,(2) feedback to all +APs and UEs only for each realization of the AP/UE locations, which are KN2 complex scalars. As for the +LSFD scheme without the I-WMMSE precoding, local data estimates ˜xmk, described by (τc − τp)MKN +complex scalars per coherence block, E{Gkk}, described by MKN2 complex scalars for each realization +of the AP/UE locations, and E{GklGH +kl} ∈ CMN×MN, described by M2K2N2/2 complex scalars for each +realization of the AP/UE locations, are required. That is total (τc −τp)MKNNr + MKN2 + M2K2N2/2 +complex scalars transmitted via fronthaul links for each realization of the AP/UE locations. +3) Practical Implementation: Note that the basic motivation of the investigated I-WMMSE precoding +schemes is to achieve as good the sum uplink SE performance as possible so we ignore some practical +issues, which are vital for the realistic implementation of the investigated precoding schemes. When the +precoding schemes are implemented in practice, these realistic issues should be considered. +• Capacity-constrained fronthaul network +As discussed above, the I-WMMSE precoding require more fronthaul requirements than the case without +the I-WMMSE precoding. It is quite vital to consider a more practical capacity-constrained fronthaul +network [38]. Moreover, the wireless fronthaul [39], which is more flexible than the conventional wire +fronthaul, would also be regarded as a promising solution to boost the practical implementation of the +I-WMMSE precoding. +• Scalability aspects with dynamic cooperation clusters +When the precoding schemes are implemented in practice, a more realistic network architecture with + +21 +multiple CPUs and dynamic cooperation clusters should be advocated, where each UE is only served by +a cluster of APs (that a is user-centric cluster) and the APs are grouped into cell-centric clusters as shown +in Fig. 1. Note a user-centric cluster might consist of APs connecting with different CPUs. Based on the +signal processing schemes in [9], [18], the analytical framework in this paper can be implemented in a +scalable paradigm where the fronthaul requirements and computational complexity can be relieved with +an anticipated modest performance loss compared with canonical architecture. The I-WMMSE precoding +design with these two practical aspects is left in future work. To bring valuable technical insights for +the study of I-WMMSE precoding schemes with the DCC strategy and the capacity-constrained fronthaul +link, we provide two tutorials for the FCP and LSFD in Fig. 2 based on [9], [10], [38]. +Tutorials to investigate the I-WMMSE precoding scheme with the DCC strategy and the capacity-constrained fronthaul +1: Joint initial access, pilot assignment, and cluster formation for the DCC topology +based on a classical algorithm in [9, Sec V. A] or a more efficient algorithm as [10, +Algorithm 1]. +2: Each AP transmits the quantized versions of the local detection signals in (22) to the +CPU based on Case 2 in [38] called "Quantized Weighted Signal Available at the CPU" +as [38, eq. (20)]. +3: Based on Section Ⅲ. B. (1), generate the DCC based processing scheme for the +LSFD (the local combining design, P-LSFD, and achievable SE computation) motivated +by [10, Sec Ⅱ. B]. +4: Based on Section Ⅲ. B. (2), formulate the I-WMMSE precoding design optimization +problem with a capacity-constrained fronthaul motivated by [38, eq. (24)] and [38, eq. +(26)]. +5: Obtain the optimal precoding scheme based on potential methods. +Tutorial 2. The I-WMMSE precoding for the LSFD with the DCC strategy and +the capacity-constrained fronthaul +1: Joint initial access, pilot assignment, and cluster formation for the DCC +topology based on a classical algorithm in [9, Sec V. A] or a more efficient +algorithm as [10, Algorithm 1]. +2: Each AP transmits the quantized versions of its received pilot signals and data +signals to the CPU based on Case 1 in [38] called "Quantized Estimate of the +Channel and Quantized Signal Available at the CPU" as [38, eq. (11) ]. +3: Based on Section Ⅲ. A. (1), generate the DCC based processing scheme for +the FCP (the receive combining and achievable SE computation) motivated by +[9, Sec V. B]. +4: Based on Section Ⅲ. A. (2), formulate the I-WMMSE precoding design +optimization problem with a capacity-constrained fronthaul motivated by [38, +eq. (24)] and [38, eq. (26)] . +5: Obtain the optimal precoding scheme based on potential methods. +Tutorial 1. The I-WMMSE precoding for the FCP with the DCC strategy +and the capacity-constrained fronthaul +Fig. 2. Two tutorials to investigate the I-WMMSE precoding schemes with the DCC strategy and the capacity-constrained fronthaul. +B. Complexity Analysis +In this subsection, we analyze the computational complexity of two precoding schemes investigated. +Since the bisection step for λk,{(1),(2)} generally takes few iterations compared with other steps, we +ignore bisection steps for λk,{(1),(2)} in the complexity analysis. For the fully centralized processing +scheme and each realization of the AP/UE locations, the per-iteration complexity of iterative optimiza- +tion is O (M3K2N5Nr). For the LSFD scheme and each realization of the AP/UE locations, the per- +iteration complexity of iterative optimization based on L-MMSE combining with the Monte-Carlo method, +MR combining with the Monte-Carlo method and MR combining with the closed-form expressions are +O (M2K2N3Nr), O (M2K2N3Nr + M3KN3) and O (M3K2N5), respectively. To further reduce the +computation complexity, it’s quite necessary to apply the asymptotic analysis method [40], [41] to compute +the terms, which cannot be computed in closed-form, in approximation results. + +22 +TABLE I +COMPARISON OF TWO PRECODING SCHEMES IN THIS PAPER. THE NUMBER OF COMPLEX SCALARS IS COMPUTED FOR EACH +REALIZATION OF THE AP/UE LOCATIONS. THE SUM SE IMPROVEMENT IS COMPUTED WITH M = 20, K = 10, L = 1 AND N = 4. +FCP +LSFD +CSI +Instantaneous CSI +Statistical CSI +Detection scheme +MMSE combining +L-MMSE/MR combining + Optimal LSFD scheme +Number of complex scalars +sent from APs to the CPU +with I-WMMSE precoding +τcMLNr + MKL2N 2/2 +(τc − τp)MKNNr + MKN 2 + M 2K2N 4/2 +Number of complex scalars +sent from APs to the CPU +without I-WMMSE precoding +τcMLNr + MKL2N 2/2 +(τc − τp)MKNNr + MKN 2 + M 2K2N 2/2 +Number of complex scalars +feedback sent from the CPU +KN 2Nr +KN 2 +Per-iteration computational +complexity +O +� +M 3K2N 5Nr +� +L-MMSE: O +� +M 2K2N 3Nr +� +MR (Monte-Carlo): O +� +M 2K2N 3Nr + M 3KN 3� +MR (Analytical): O +� +M 3K2N 5� +Sum SE improvement +28.93% +L-MMSE: 46.74% +MR: 15.13% +V. NUMERICAL RESULTS +In this paper, a CF mMIMO system is investigated, where all APs and UEs are uniformly distributed in +a 1×1 km2 area with a wrap-around scheme [42]. The pathloss and shadow fading are modeled similarly as +[28]. In practice, Umk,r, Umk,t and Ωmk are estimated through measurements [29]. However, we generate +them randomly in this paper, where the coupling matrix Ωmk consists of one strong transmit eigendirection +capturing dominant power [43]12. Besides, we have Fk,p = F(0) +k,u,{(1),(2)} = +� pk +N IN. As for Algorithm 1 +and Algorithm 2, balancing the convergence and accuracy, we assume that I(1),max = I(2),max = 20, +ε(1) = ε(2) = 5 × 10−4, and weights for all UEs are equal (µk,(1) = µk,(2) = 1) without losing generality, +respectively. Moreover, we consider communication with 20 MHz bandwidth and σ2 = −94 dBm noise +power. All UEs transmit with 200 mW power constraint. Each coherence block contains τc = 200 channel +uses and τp = KN/2. Besides, a pilot assignment approach similar as that in [28] is investigated. +Figure 3 shows the cumulative distribution function (CDF) of the achievable sum SE over different +realizations of the AP/UE locations for two processing schemes investigated (we shortly call “fully +centralized processing” as “FCP” in the following) over “I-WMMSE precoding” or “w/o precoding”13. +We notice that the FCP scheme undoubtedly achieves higher SE than that of the LSFD scheme since the +12In this paper, we choose one eigendirection capturing dominant channel power (randomly accounting for 80% ∼ 95% of the total +channel power) and other eigendirections contain the remaining power. +13The “w/o precoding” scenario denotes that identity precoding matrices Fk,u,{(1),(2)} = +� pk +N IN are implemented without optimization. + +23 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +0 +0.2 +0.4 +0.6 +0.8 +1 +Fig. 3. CDF of the sum SE over different processing schemes and +precoding schemes with M = 20, K = 10, L = 2, and N = 4. +1 +2 +3 +4 +5 +6 +0 +20 +40 +60 +80 +100 +Fig. 4. Sum SE against the number antennas per AP L over different +processing schemes and precoding schemes with M = 20, K = 10, +and N = 4. +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +7 +Fig. 5. Average rate against the number of antennas per UE N over +different processing schemes and precoding schemes with M = 20, +K = 10, and L = 2. +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +7 +Fig. 6. Average SE with I-WMMSE precoding schemes against the +number of antennas per UE N over different τc with M = 20, +K = 10, and L = 2. +FCP with MMSE combining is a competitive scheme in CF mMIMO [5]. More importantly, the proposed +I-WMMSE schemes are efficient to improve the respective achievable sum SE performance, e.g., 12.78%, +19.54% and 28.13% sum SE improvement for the FCP, the LSFD with MR combining and the LSFD +with L-MMSE combining, respectively. Besides, for the LSFD scheme with MR combining, markers “◦” +generated by analytical results overlap with the curves generated by simulations, respectively, validating +our derived closed-form expressions. +Figure 4 shows the achievable sum SE as a function of the number of antennas per AP with two +processing schemes investigated and different precoding schemes14. We notice that, for the FCP or LSFD +with (L-)MMSE combining, the performance gap between the “I-WMMSE” and “w/o precoding” becomes +smaller with the increase of L, which implies that (L-)MMSE combining can use all antennas on each +14Note that the achievable sum SE investigated is the average sum SE value taken over many AP/UE locations. + +24 +10 +20 +30 +40 +50 +60 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Fig. 7. +Average SE against the number of APs M for the LSFD +scheme with K = 10, L = 4, and N = 4. +2 +3 +4 +5 +6 +8 +10 +12 +2 +3 +4 +5 +6 +2.5 +3 +3.5 +4 +3.38 +3.4 +Fig. 8. Average SE against the number of antennas per UE N for +different channel models with M = 40, K = 8, and L = 2. +2 +4 +6 +8 +10 +12 +14 +16 +18 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +(a) FCP +2 +4 +6 +8 +10 +12 +14 +16 +16 +18 +20 +22 +24 +26 +28 +(b) LSFD +Fig. 9. Convergence examples of the I-WMMSE algorithm for the FCP and LSFD with M = 20, K = 10, L = 2, and N = 4. +AP to suppress interference and achieve excellent SE performance even without any precoding scheme. +For instance, the performance gap between the “I-WMMSE” and “w/o precoding” for the LSFD with +L-MMSE combining is 46.74% and 6.17% over L = 1 and L = 6, respectively. Meanwhile, for the LSFD +with MR combining, the performance gap between the “I-WMMSE” and “w/o precoding” becomes large +with the increase of L, e.g. 15.13% and 25.48% for L = 1 and L = 6, respectively. Besides, for the +LSFD scheme with MR combining, markers “✷” generated by analytical results overlap with the curves +generated by simulations, respectively, validating our derived closed-form expressions. +To further show the advantage of the proposed I-WMMSE precoding schemes, Fig. 5 shows the average +rate15 as a function of the number of antennas per UE. We find that the average rates for all schemes with +I-WMMSE precoding schemes grow with N and the average rates for the case without UL precoding +may also suffer the degradation with the increase of N. The implementation of the I-WMMSE precoding +15Note that one main reason for the phenomenon that additional UE antennas may give rise to the SE degradation is that increasing N will +increase the channel estimation overhead and reduce the pre-log factor “(τc − τp) /τc” in all SE expressions [26], [28]. So we investigate +“the average rate” in Fig. 5, ignoring the effect of “(τc − τp) /τc”. + +25 +50 +100 +150 +200 +250 +300 +3.4 +3.6 +3.8 +4 +104 +50 +100 +150 +200 +250 +300 +0 +2 +4 +6 +105 +Fig. 10. Total number of complex scalars sent via the fronthaul per channel use for each realization of the AP/UE locations with M = 20, +K = 10, L = 2, and N = 4. +schemes undoubtedly makes UEs benefit from multiple antennas and achieve excellent rate performance. +Moreover, we observe that the I-WMMSE precoding schemes perform more efficiently with a larger +number of UE antennas. For instance, the average rate improvements achieved by the I-WMMSE precoding +for the LSFD with L-MMSE combining are 31.91% and 9.43% for N = 6 and N = 2, respectively. +However, the average SE (with scaling factor (τc − τp)/τc) with I-WMMSE precoding implemented may +also degrade with the increase of N as the Fig. 2 in [1] since, with the increase of N, the prerequisite +of “mutually orthogonal pilot matrices” still requires huge channel uses for the pilot transmission and +the inter-user interference also increases. So the design of non-orthogonal pilot matrices and per-antenna +power control scheme are quite necessary, which are regarded as promising ways to reduce the cost of +pilot transmission and further improve the SE performance [44]. +Figure 6 discusses the average SE with I-WMMSE precoding schemes against N over different τc. +Note that Fig. 5 can be viewed as a special case in Fig. 6 with the coherence block with infinite length +τc = ∞. We observe that the average SE with I-WMMSE precoding schemes increases with N over +τc = 500 or ∞, which means the SE performance can benefit from having additional UE antennas when +the coherence block resource is abundant. +Figure 7 investigates the average SE as a function of M for the LSFD scheme over different precoding +schemes16. For MR combining, markers “✷” generated by analytical results overlap with the curves +generated by simulations, respectively, validating our derived closed-form expressions again. Besides, the +I-WMMSE algorithm is more efficient to improve the SE performance for MR combining than that of +L-MMSE combining for the scenario over large L and M, e.g., 4.03% and 24.21% SE improvement for L- +MMSE combining and MR combining with M = 60, respectively, implying that the L-MMSE combining +16The “WMMSE precoding” denotes the precoding schemes generated by the I-WMMSE algorithm with only single iteration. + +26 +based on large L and M can achieve excellent SE performance even without any precoding scheme and +the proposed I-WMMSE precoding scheme is handy to mitigate the weakness of MR combining17. +Figure 8 considers the average SE as a function of N over the i.i.d. and the Weichselberger Rayleigh +fading channel. As observed, the proposed I-WMMSE precoding schemes are more efficient over the +Weichselberger Rayleigh fading channel. For instance, 24.89% and 9.77% average SE improvement can +be achieved when N = 6 over the “Weichselberger” scenario for the LSFD scheme with MR combining +and the FCP scheme, respectively, but only 0.29% and 6.63% average SE improvement can be achieved +for “I.I.D. Rayleigh channel”. Moreover, compared with Fig. 5, we notice that the I-WMMSE precoding +scheme for the FCP scheme is more efficient in the highly loaded system (the scenario in Fig. 5) where +the number of total AP-antennas is comparable with the number of total UE-antennas. +Figure 9 illustrates the convergence behavior of the I-WMMSE algorithms for the FCP scheme and +the LSFD scheme with L-MMSE/MR combining. Note the convergence example in Fig. 9 (a) for the +FCP is given by a particular channel realization and the convergence example for the LSFD in Fig. 9 (b) +is given by a particular realization of the AP/UE locations. Note that the algorithms investigated can be +guaranteed to converge and are efficient to achieve excellent sum SE performance. Besides, Fig. 9 (b) for +the LSFD scheme with MR combining validates our derived closed-form expressions in Algorithm 2. +Figure 10 investigates the total number of complex scalars sent via the fronthaul per channel use against +τc for each realization of the AP/UE locations. As observed, total number of complex scalars per channel +use for the FCP/LSFD scheme becomes smaller/larger, which can also be easily found from Table I. +Besides, the LSFD scheme requires more fronthaul signaling than the FCP scheme since APs under the +LSFD scheme need to transmit all received data signals to the CPU, which requires a huge fronthaul load. +More importantly, with the increase of τc, the gap between “I-WMMSE precoding” and “W/O precoding” +becomes smaller for either the FCP scheme or the LSFD scheme. Considering the SE performance +improvement of the I-WMMSE precoding, additional fronthaul loads can be acceptable, especially when +the coherence resource is abundant. Although the computational complexity of Algorithm 1 for the FCP +scheme is much higher than that of Algorithm 2 for the LSFD scheme, the FCP scheme needs much +less fronthaul signaling than that of the LSFD scheme and can achieve better SE performance. So two +processing schemes and their respective precoding schemes can be chosen based on different requirements. +17MR combining is a simple combining scheme but cannot efficiently suppress the interference. + +27 +VI. CONCLUSION +We consider a CF mMIMO system with both APs and UEs equipped with multiple antennas over +the Weichselberger Rayleigh fading channel. The FCP scheme and LSFD scheme are implemented. To +further improve the sum SE performance, efficient UL precoding schemes based on iteratively WMMSE +algorithms are investigated to maximize weighted sum SE for the two processing schemes. Note that we +compute achievable SE expressions and optimal precoding schemes in novel closed-form for the LSFD +scheme with MR combining. Numerical results show that the investigated I-WMMSE precoding schemes +are efficient to achieve excellent sum SE performance. More importantly, it can be seen that the proposed +I-WMMSE precoding schemes are more efficient with a larger number of UE antennas, which means +the I-WMMSE precoding schemes can achieve excellent performance even with a large number of UE +antennas. The derived results undoubtedly provides vital insights for the practical implementation of multi- +antenna UEs in CF mMIMO systems. In future work, we will investigate the design of UL precoding +scheme for the phase of pilot transmission and consider the practical implementation of the investigated +I-WMMSE precoding schemes with capacity-constrained fronthaul network and dynamic cooperation +clusters. Moreover, the non-orthogonal pilot matrix design will also be considered to further improve the +performance for the CF mMIMO system with multi-antenna UEs. Last but not least, the UL precoding +performance over a more practical Rician fading channel with phase-shifts will also be analyzed. +APPENDIX A +SOME USEFUL LEMMAS +Lemma 1. Let X ∈ CM×N be a random matrix and Y is a deterministic M × M matrix. The (n, i)-th +element of E +� +XHYX +� +is tr +� +Y · E +� +xixH +n +�� +where xi and xn are the i-th and n-th column of X. +Lemma 2. For matrices A ∈ CN1×N1, B ∈ CN1×N2, C ∈ CN2×N2, and D ∈ CN2×N1, we have +(A + BCD)−1 = A−1−A−1B +� +DA−1B + C−1�−1 DA−1, which is a well-known matrix inversion lemma +[36, Lemma B.3]. +APPENDIX B +PROOF COROLLARY 5 +Since the CPU is only aware of channel statistics, we need to treat E{Gkk}Fk,u as the true deterministic +channel and rewrite ˜xk in (25) as ˜xk = E {Gkk} Fk,uxk+(GkkFk,u − E {Gkk} Fk,u) xk + +K +� +l=1,l̸=k +GklFl,uxl + n′ +k +� +�� +� +v +where v is a complex circular symmetric noise with an invertible covariance matrix Ξk = E{vvH|Θ} = + +28 +�K +l=1 E{GklFk,uFH +k,uGH +kl} − E{Gkk}Fk,uFH +k,uE{GH +kk} + σ2Sk. Firstly, we whiten the noise as Ξ +− 1 +2 +k ˆxk = +Ξ +− 1 +2 +k E {Gkk} Fk,uxk + ˜v, where ˜v ≜ Ξ +− 1 +2 +k v becomes white. Next, we project Ξ +− 1 +2 +k ˆxk in the direction of +Ξ +− 1 +2 +k E {Gkk} Fk,u to obtain an effective scalar channel as +� +Ξ +− 1 +2 +k E {Gkk} Fk,u +�H +Ξ +− 1 +2 +k ˆxk = (E {Gkk} Fk,u)H Ξ−1 +k E {Gkk} Fk,uxk + (E {Gkk} Fk,u)H Ξ−1 +k v. +(44) +Based on theories of optimal receivers [37], we derive optimal LSFD matrix Ak=Ξ−1 +k E {Gkk}Fk,u as +Ak = +� K +� +l=1 +E +� +GklFk,uFH +k,uGH +kl +� +− E {Gkk} Fk,uFH +k,uE +� +GH +kk +� ++ σ2Sk +�−1 +E {Gkk} Fk,u. +(45) +Moreover, based on the the standard results of matrix derivation in [45], we can easily obtain the LSFD +matrix minimizing the conditional MSE for UE k MSE(2) +k += tr(E(2) +k ) as +Ak = +� K +� +l=1 +E{Gkl¯Fl,uGH +kl} + σ2Sk +�−1 +E{Gkk}Fk,u. +(46) +We notice that the LSFD matrix in (45) is equivalent to the LSFD matrix in (46), except from having +another scaling matrix IN − +� +CHB−1C + IN +�−1 CHB−1C on the right side, which would not affect the +value of (27), where B = �K +l=1 E{GklFk,uFH +k,uGH +kl} + σ2Sk and C = E{Gkk}Fk,u. So the LSFD matrix +in (46) cannot maximize the achievable SE but minimize the MSE for UE k. +APPENDIX C +PROOF OT THEOREM 2 +In this part, we compute terms of (27) in closed-form for the LSFD scheme with MR combining Vmk = +ˆHmk. For the first term Dk,(2) = AH +k E{Gkk}Fk,u, we have E{Gkk} = [E{VH +1kH1k}; . . . ; E{VH +MkHMk}] = +[ZT +1k, . . . , ZT +Mk]T ≜ Zk, where Zmk = E{VH +mkHmk} = E{ ˆHH +mk ˆHmk} ∈ CN×N and the (n, n′)-th el- +ement of Zmk can be denoted as [Zmk]nn′ = E{ˆhH +mk,nˆhmk,n′} = tr(ˆRn′n +mk). So we derive the closed- +form for Dk,(2) as Dk,(2),c = AH +k ZkFk,u. As for the second term Sk ∈ CMN×MN, we have Sk = +diag(E{VH +1kV1k}, . . . , E{VH +MkVMk}) = diag(Z1k, . . . , ZMk). For E{Gkl¯Fl,uGH +kl}, we notice that the +(m, m′)-submatrix of E{Gkl¯Fl,uGH +kl} is E{VH +mkHml¯Fl,uHH +m′lVm′k}. +Based on [28], we compute E{VH +mkHml¯Fl,uHH +m′lVm′k} for four possible AP-UE combinations. For +“m ̸= m′, l /∈ Pk”, we have E{VH +mkHml¯Fl,uHH +m′lVm′k} = 0 for the independence between Vmk and +Hml. For “m ̸= m′, l ∈ Pk”, we have E{VH +mkHml¯Fl,uHH +m′lVm′k} = E{VH +mkHml}¯Fl,uE{HH +m′lVm′k} = +Λmkl¯Fl,uΛm′lk, where the (n, n′)-th element of N ×N-dimension complex matrices Λmkl ≜ E{VH +mkHml}, + +29 +Λm′lk ≜ E{HH +m′lVm′k} are [Λmkl]nn′ = E{ˆhH +mk,nˆhml,n′} = tr(Ξn′n +mkl) and [Λm′lk]nn′ = E{ˆhH +m′l,nˆhmk,n′} = +tr(Ξn′n +m′lk) with Ξmkl ≜ E{ˆhmlˆhH +mk} = τpRml˜FH +l,pΨ−1 +mk˜Fk,pRmk, Ξm′lk ≜ E{ˆhm′kˆhH +m′l} = τpRm′k˜FH +k,pΨ−1 +m′k˜Fl,pRm′l. +For “m = m′, l /∈ Pk”, we define Γ(1) +mkl ≜ E{VH +mkHml¯Fl,uHH +mlVmk} ∈ CN×N with the (n, n′)-th element +[Γ(1) +mkl]nn′ = �N +i=1 +�N +i′=1 [¯Fl,u]i′iE{ˆhH +mk,nhml,i′hH +ml,iˆhmk,n′} being +[Γ(1) +mkl]nn′ = +N +� +i=1 +N +� +i′=1 +[¯Fl,u]i′itr(E +� +hml,i′hH +ml,i +� +E{ˆhmk,n′ˆhH +mk,n}) = +N +� +i=1 +N +� +i′=1 +[¯Fl,u]i′itr(Ri′i +ml ˆRn′n +mk) +(47) +since ˆHmk and Hml are independent. Finally, for “m = m′, l ∈ Pk”, ˆHmk and Hml are no longer inde- +pendent. We define Γ(2) +mkl ≜ E{VH +mkHml¯Fl,uHH +mlVmk} ∈ CN×N whose (n, n′)-th element is [Γ(2) +mkl]nn′ = +�N +i=1 +�N +i′=1 [¯Fl,u]i′iE{ˆhH +mk,nhml,i′hH +ml,iˆhmk,n′}. We follow the similar method in [28] and derive +[Γ(2) +kl,m]nn′ = �N +i=1 +�N +i′=1 [¯Fl,u]i′itr(Ri′i +mlPn′n +mkl,(1))+τ 2 +p +�N +q1=1 +�N +q2=1 [¯Fl,u]i′i[tr(˜Pq1n +mkl,(2) ˜Ri′q2 +ml ˜Rq2i +ml ˜Pn′q1 +mkl,(2))].+ +τ 2 +p +�N +q1=1 +�N +q2=1 [¯Fl,u]i′itr(˜Pq1n +mkl,(2) ˜Ri′q2 +ml )tr(˜Pn′q2 +mkl,(2) ˜Rq2i +ml), where Pmkl,(1) ≜ τpSmk(Ψmk−τp˜Fl,pRml˜FH +l,p)SH +mk, +Smk ≜ Rmk˜FH +k,pΨ−1 +mk and Pmkl,(2) ≜ Smk˜Fl,pRml˜FH +l,pSH +mk, respectively. Besides, ˜Rni +ml and ˜Pni +mkl,(2) denote +(n, i)-submatrix of R +1 +2 +ml and P +1 +2 +mkl,(2), respectively. +In summary, combining all the cases, we have E{Gkl¯Fl,uGH +kl} = Tkl,(1) + Tkl,(2) if l ∈ Pk and Tkl,(1) +otherwise, where Tkl,(1) ≜ diag(Γ(1) +kl,1, . . . , Γ(1) +kl,M) ∈ CMN×MN and Tmm′ +kl,(2) = Γ(2) +kl,m − Γ(1) +kl,m if m = m′ +and Λmkl¯Fl,uΛm′lk otherwise. Plugging the derived results into (28) and (31), we can easily compute the +optimal LSFD coefficient matrix and MSE matrix in closed-form as (34). So we have finished the proof +of Theorem 2. For more details on the derived expression, please refer to [28, Appendix D]. +APPENDIX D +PROOF OF THEOREM 1 +When other optimization variables are fixed, we derive the partial derivative of (17) w.r.t F(1) +k,u as +∂f +� +F1,u,(1), . . . , FK,u,(1) +� +∂Fk,u,(1) += +K +� +l=1 +µl,(1) +� +ˆHH +k VlWl,(1)VH +l ˆHk + E +� +˜HH +k VlWl,(1)VH +l +˜Hk +��� V, W +�� ++ λk,(1)IN +− µk,(1) ˆHH +k VH +k Wk,(1). +(48) +By applying the first-order optimality condition and setting +∂f(F1,u,(1),...,FK,u,(1)) +∂Fk,u,(1) += 0, we can easily obtain +the optimal precoding scheme. Besides, λk,(1) and Fk,u,(1) should also satisfy KKT condition as (19). +As for ¯Ckl ≜ E{ ˜HH +k VlWl,(1)VH +l ˜Hk|V, W} ∈ CN×N, by applying Lemma 1, the (i, n)-th element of +¯Ckl is tr( ¯VlE{˜hk,n˜hH +k,i}) where ¯Vl ≜ VlWl,(1)VH +l +and ˜hk,n = [˜hT +1k,n, . . . , ˜hT +Mk,n]T ∈ CML is the n-th +column of ˜Hk. Finally, we derive Ck,ni ≜ E{˜hk,n˜hH +k,i} = diag (Cni +1k, . . . , Cni +Mk) ∈ CML×ML since ˜hmk,n + +30 +and ˜hm′k,n for m ̸= m′ are independent and both have zero mean. So Ck,ni is a block-diagonal matrix +with the square matrices Cni +1k = E{˜h1k,n˜hH +1k,i}, . . . , Cni +Mk = E{˜hMk,n˜hH +Mk,i} on the diagonal. +APPENDIX E +PROOF OF (15) +For the LSFD scheme, the conditional MSE matrix for UE k can be written as (26). Based on [28, +Appendix C], we prove that (28) can also minimize MSEk,(2) = tr +� +Ek,(2) +� +. With (28) implemented, Ek,(2) +is given by (31). Then, by applying Lemma 2, we have +� +Eopt +k,(2) +�−1 += IN + FH +k,u,(2)E +� +GH +kk +� +� K +� +l=1 +E +� +Gkl¯Fl,u,(2)GH +kl +� +− E {Gkk} ¯Fk,u,(2)E +� +GH +kk +� ++ σ2Sk +�−1 +× E {Gkk} Fk,u,(2), +where A ≜ IN, B ≜ −FH +k,u,(2)E{GH +kk}, C ≜ (�K +l=1 E{Gkl¯Fl,u,(2)GH +kl}+σ2Sk)−1 and D ≜ E{Gkk}Fk,u,(2), +respectively. We show the equivalence between SEopt +k,(2) and log2 |(Eopt +k,(2))−1| without a factor (1 − τp/τc). +APPENDIX F +PROOF OF THEOREM 4 +When MR combining Vmk = ˆHmk and the optimal LSFD scheme applied, we can easily compute +E{GH +kk}, Aopt +k , and Eopt +k,(2) in closed-form as Theorem 2. Furthermore, by applying Lemma 1, the (i, n)-th +entry of ¯Tlk = E{GH +lkAlE−1 +l,(2)AH +l Glk} ∈ CN×N can be denoted as tr(¯AlE{glk,ngH +lk,i}), where ¯Al ≜ +AlE−1 +l,(2)AH +l +and glk,n ∈ CMN is the n-th column of Glk. Note that the (m − 1) N + p-th element of +glk,n is ˆhH +ml,phmk,n so the [(m − 1) N + p, (m′ − 1) N + p′]-th (or [o, j]-th briefly) entry of ¯Glk,ni ≜ +E{glk,ngH +lk,i} ∈ CMN×MN can be denoted as E{ˆhH +ml,phmk,nhH +m′k,iˆhm′l,p′}, which can be computed for four +AP-UE combinations as Theorem 2. +For “l /∈ Pk, m ̸= m′”, we have E{ˆhH +ml,phmk,nhH +m′k,iˆhm′l,p′} = 0. For “l ∈ Pk, m ̸= m′”, we have +E{ˆhH +ml,phmk,nhH +mk,iˆhml,p′} = tr(Rni +mk ˆRp′p +ml). For “l /∈ Pk, m = m′”, we have E{ˆhH +ml,phmk,nhH +m′k,iˆhm′l,p′} = +E{ˆhH +ml,phmk,n}E{hH +m′k,iˆhm′l,p′} = tr(Ξnp +mlk)tr(Ξp′i +m′kl), where Ξmlk = τpRmk˜FH +k,pΨ−1 +mk˜Fl,pRml and Ξm′kl = +τpRm′l˜FH +l,pΨ−1 +m′l˜Fk,pRm′k. For “l ∈ Pk, m = m′”, we obtain E{ˆhH +ml,phmk,nhH +mk,iˆhml,p′} = tr(Rni +mkPp′p +mkl,(1))+ +τ 2 +p +�N +q1=1 +�N +q2=1 tr(˜Pq1p +mlk,(2) ˜Rnq2 +mk ˜Rq2i +mk ˜Pp′q1 +mlk,(2)) + τ 2 +p +�N +q1=1 +�N +q2=1 tr(˜Pq1n +mlk,(2) ˜Rnq1 +mk)tr(˜Pp′q2 +mlk,(2) ˜Rq2i +mk), where +Sml = Rml˜FH +l,pΨ−1 +ml, Pmlk,(1) = τpSml(Ψml−τp˜Fk,pRmk˜FH +k,p)SH +ml and Pmlk,(2) = Sml˜Fk,pRmk˜FH +k,pSH +ml with +˜Rni +mk and ˜Pni +mkl,(2) being (n, i)-submatrix of R +1 +2 +mk and P +1 +2 +mkl,(2), respectively. We can compute E{glk,ngH +lk,i}oj +in closed-form as (42) and Fopt +k,u,(2) in closed-form as (43). + +31 +REFERENCES +[1] Z. Wang, J. Zhang, H. Q. Ngo, B. Ai, and M. Debbah, “Iteratively weighted MMSE uplink precoding for cell-free massive MIMO,” +in Proc. IEEE ICC, May 2022, pp. 1–6. +[2] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive MIMO versus small cells,” IEEE Trans. +Wireless Commun., vol. 16, no. 3, pp. 1834–1850, Mar. 2017. +[3] J. Zhang, E. Bj¨ornson, M. Matthaiou, D. W. K. Ng, H. Yang, and D. J. Love, “Prospective multiple antenna technologies for beyond +5G,” IEEE J. Sel. Areas Commun, vol. 38, no. 8, pp. 1637–1660, Jun. 2020. +[4] S. Chen, J. Zhang, J. Zhang, E. Bj¨ornson, and B. Ai, “A survey on user-centric cell-free massive MIMO systems,” Digit. Commun. +Netw., 2021. +[5] E. Bj¨ornson and L. Sanguinetti, “Making cell-free massive MIMO competitive with MMSE processing and centralized implementation,” +IEEE Trans. Wireless Commun., vol. 19, no. 1, pp. 77–90, Jan. 2019. +[6] S. Elhoushy, M. Ibrahim, and W. Hamouda, “Cell-free massive MIMO: A survey,” IEEE Commun. Surveys Tuts., vol. 24, no. 1, pp. +492–523, 1st Quart. 2022. +[7] J. Zhang, S. Chen, Y. Lin, J. Zheng, B. Ai, and L. Hanzo, “Cell-free massive MIMO: A new next-generation paradigm,” IEEE Access, +vol. 7, pp. 99 878–99 888, Jul. 2019. +[8] H. Q. Ngo, L. Tran, T. Q. Duong, M. Matthaiou, and E. G. Larsson, “On the total energy efficiency of cell-free massive MIMO,” IEEE +Trans. Green Commun. Netw, vol. 2, no. 1, pp. 25–39, Mar. 2018. +[9] E. Bj¨ornson and L. Sanguinetti, “Scalable cell-free massive MIMO systems,” IEEE Trans. Commun., vol. 68, no. 7, pp. 4247–4261, +Jul. 2020. +[10] S. Chen, J. Zhang, E. Bj¨ornson, J. Zhang, and B. Ai, “Structured massive access for scalable cell-free massive MIMO systems,” IEEE +J. Sel. Areas Commun, vol. 39, no. 4, pp. 1086–1100, Aug. 2021. +[11] A. Papazafeiropoulos, E. Bj¨ornson, P. Kourtessis, S. Chatzinotas, and J. M. Senior, “Scalable cell-free massive MIMO systems: Impact +of hardware impairments,” IEEE Trans. Veh. Technol., vol. 70, no. 10, pp. 9701–9715, Oct. 2021. +[12] J. Zheng, J. Zhang, E. Bj¨ornson, Z. Li, and B. Ai, “Cell-free massive MIMO-OFDM for high-speed train communications,” IEEE J. +Sel. Areas Commun., vol. 40, no. 10, pp. 2823–2839, Oct. 2022. +[13] M. Attarifar, A. Abbasfar, and A. Lozano, “Subset MMSE receivers for cell-free networks,” IEEE Trans. Wireless Commun., vol. 19, +no. 6, pp. 4183–4194, Jun. 2020. +[14] E. Nayebi, A. Ashikhmin, T. L. Marzetta, and B. D. Rao, “Performance of cell-free massive MIMO systems with MMSE and LSFD +receivers,” in Proc. Asilomar Conf. Signals, Syst. Comput., Nov. 2016, pp. 203–207. +[15] +¨O. ¨Ozdogan, E. Bj¨ornson, and J. Zhang, “Performance of cell-free massive MIMO with Rician fading and phase shifts,” IEEE Trans. +Wireless Commun., vol. 18, no. 11, pp. 5299–5315, Nov. 2019. +[16] Z. Wang, J. Zhang, E. Bj¨ornson, and B. Ai, “Uplink performance of cell-free massive MIMO over spatially correlated Rician fading +channels,” IEEE Commun. Lett., vol. 25, no. 4, pp. 1348–1352, Apr. 2021. +[17] J. Zhang, J. Zhang, E. Bj¨ornson, and B. Ai, “Local partial zero-forcing combining for cell-free massive MIMO systems,” IEEE Trans. +Commun., vol. 69, no. 12, pp. 8459–8473, Dec. 2021. +[18] G. Interdonato, P. Frenger, and E. G. Larsson, “Scalability aspects of cell-free massive MIMO,” in Proc. IEEE ICC, 2019, pp. 1–6. +[19] Z. H. Shaik, E. Bj¨ornson, and E. G. Larsson, “MMSE-optimal sequential processing for cell-free massive MIMO with radio stripes,” +IEEE Trans. Commun., vol. 69, no. 11, pp. 7775–7789, Nov. 2021. + +32 +[20] H. Masoumi and M. J. Emadi, “Performance analysis of cell-free massive MIMO system with limited fronthaul capacity and hardware +impairments,” IEEE Trans. Wireless Commun., vol. 19, no. 2, pp. 1038–1053, Feb. 2020. +[21] M. Bashar, P. Xiao, R. Tafazolli, K. Cumanan, A. G. Burr, and E. Bj¨ornson, “Limited-fronthaul cell-free massive MIMO with local +MMSE receiver under Rician fading and phase shifts,” IEEE Wireless Commun. Lett., vol. 10, no. 9, pp. 1934–1938, Sep. 2021. +[22] T. Van Chien, H. Q. Ngo, S. Chatzinotas, M. Di Renzo, and B. Ottersten, “Reconfigurable intelligent surface-assisted cell-free massive +MIMO systems over spatially-correlated channels,” IEEE Trans. Wireless Commun., vol. 21, no. 7, pp. 5106–5128, Jul. 2022. +[23] E. Shi, J. Zhang, S. Chen, J. Zheng, Y. Zhang, D. W. Kwan Ng, and B. Ai, “Wireless energy transfer in RIS-aided cell-free massive +MIMO systems: Opportunities and challenges,” IEEE Commun. Mag., vol. 60, no. 3, pp. 26–32, Mar. 2022. +[24] T. C. Mai, H. Q. Ngo, and T. Q. Duong, “Cell-free massive MIMO systems with multi-antenna users,” in IEEE GlobalSIP, Nov. 2018, +pp. 828–832. +[25] S. Buzzi, C. D’Andrea, A. Zappone, and C. D’Elia, “User-centric 5G cellular networks: Resource allocation and comparison with the +cell-free massive MIMO approach,” IEEE Trans. Wireless Commun., vol. 19, no. 2, pp. 1250–1264, Feb. 2020. +[26] T. C. Mai, H. Q. Ngo, and T. Q. Duong, “Downlink spectral efficiency of cell-free massive MIMO systems with multi-antenna users,” +IEEE Trans. Commun., vol. 68, no. 8, pp. 4803–4815, Apr. 2020. +[27] M. Zhou, L. Yang, and H. Zhu, “Sum-SE for multigroup multicast cell-free massive MIMO with multi-antenna users and low-resolution +DACs,” IEEE Wireless Commun. Lett., vol. 10, no. 8, pp. 1702–1706, May 2021. +[28] Z. Wang, J. Zhang, B. Ai, C. Yuen, and M. Debbah, “Uplink performance of cell-free massive MIMO with multi-antenna users over +jointly-correlated Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 21, no. 9, pp. 7391–7406, Sep. 2022. +[29] W. Weichselberger, M. Herdin, H. Ozcelik, and E. Bonek, “A stochastic MIMO channel model with joint correlation of both link ends,” +IEEE Trans. Wireless Commun., vol. 5, no. 1, pp. 90–100, Jan. 2006. +[30] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weighted MMSE approach to distributed sum-utility maximization for +a MIMO interfering broadcast channel,” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4331–4340, Apr. 2011. +[31] J. Shin and J. Moon, “Weighted-sum-rate-maximizing linear transceiver filters for the K-user MIMO interference channel,” IEEE Trans. +Commun., vol. 60, no. 10, pp. 2776–2783, Sep. 2012. +[32] S. S. Christensen, R. Agarwal, E. De Carvalho, and J. M. Cioffi, “Weighted sum-rate maximization using weighted MMSE for MIMO- +BC beamforming design,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 4792–4799, 2008. +[33] X. Li, X. Zhang, Y. Zhou, and L. Hanzo, “Optimal massive-MIMO-Aided clustered base-station coordination,” IEEE Trans. Veh. +Technol., vol. 70, no. 3, pp. 2699–2712, Feb. 2021. +[34] H. Q. Ngo, H. Tataria, M. Matthaiou, S. Jin, and E. G. Larsson, “On the performance of cell-free massive MIMO in Ricean fading,” +in Proc. Asilomar Conf. Signals, Syst., Comput., Oct. 2018, pp. 980–984. +[35] E. Bj¨ornson and B. Ottersten, “A framework for training-based estimation in arbitrarily correlated Rician MIMO channels with Rician +disturbance,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1807–1820, Nov. 2010. +[36] E. Bj¨ornson, J. Hoydis, and L. Sanguinetti, “Massive MIMO networks: Spectral, energy, and hardware efficiency,” Foundations and +Trends® in Signal Processing, vol. 11, no. 3-4, pp. 154–655, 2017. +[37] D. Tse and P. Viswanath, Fundamentals of wireless communication. +Cambridge university press, 2005. +[38] M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, M. Debbah, and P. Xiao, “Max-min rate of cell-free massive MIMO uplink with +optimal uniform quantization,” IEEE Trans. Commun., vol. 67, no. 10, pp. 6796–6815, Oct. 2019. +[39] B. Hu, C. Hua, C. Chen, and X. Guan, “Joint beamformer design for wireless fronthaul and access links in C-RANs,” IEEE Trans. +Wireless Commun., vol. 17, no. 5, pp. 2869–2881, May 2018. + +33 +[40] J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?” IEEE +J. Sel. Areas Commun, vol. 31, no. 2, pp. 160–171, Feb. 2013. +[41] S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “Large system analysis of linear precoding in correlated MISO broadcast +channels under limited feedback,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4509–4537, Jul. 2012. +[42] J. Zhang, J. Zhang, D. W. K. Ng, S. Jin, and B. Ai, “Improving sum-rate of cell-free massive MIMO with expanded compute-and- +forward,” IEEE Trans. Signal Process., vol. 70, pp. 202–215, 2021. +[43] A. Tulino, A. Lozano, and S. Verdu, “Impact of antenna correlation on the capacity of multiantenna channels,” IEEE Trans. Inf. Theory, +vol. 51, no. 7, pp. 2491–2509, Jun. 2005. +[44] Y. Wu, S. Ma, and Y. Gu, “A unified framework of non-orthogonal pilot design for multi-cell massive MIMO systems,” IEEE Trans. +Commun., vol. 68, no. 12, pp. 7623–7633, 2020. +[45] A. Hjørungnes, Complex-valued matrix derivatives: with applications in signal processing and communications. Cambridge University +Press, 2011. + diff --git a/DNE0T4oBgHgl3EQfggEA/content/tmp_files/load_file.txt b/DNE0T4oBgHgl3EQfggEA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..afb7c1660241b63047c0c124671444eac65f7f18 --- /dev/null +++ b/DNE0T4oBgHgl3EQfggEA/content/tmp_files/load_file.txt @@ -0,0 +1,1368 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf,len=1367 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='02417v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='IT] 6 Jan 2023 1 Uplink Precoding Design for Cell-Free Massive MIMO with Iteratively Weighted MMSE Zhe Wang, Jiayi Zhang, Senior Member, IEEE, Hien Quoc Ngo, Senior Member, IEEE, Bo Ai, Fellow, IEEE and M´erouane Debbah, Fellow, IEEE Abstract In this paper, we investigate a cell-free massive multiple-input multiple-output system with both access points and user equipments equipped with multiple antennas over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We study the uplink spectral efficiency (SE) for the fully centralized processing scheme and large-scale fading decoding (LSFD) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further improve the SE performance, we design the uplink precoding schemes based on the weighted sum SE maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since the weighted sum SE maximization problem is not jointly over all optimization variables, two efficient uplink precoding schemes based on Iteratively Weighted sum-Minimum Mean Square Error (I-WMMSE) algorithms, which rely on the iterative minimization of weighted MSE, are proposed for two processing schemes investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, with maximum ratio combining applied in the LSFD scheme, we derive novel closed-form achievable SE expressions and optimal precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Numerical results validate the proposed results and show that the I-WMMSE precoding schemes can achieve excellent sum SE performance with a large number of UE antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Index Terms Cell-free massive MIMO, uplink precoding, weighted sum-rate maximization, spectral efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' INTRODUCTION Cell-free massive multiple-input multiple-output (CF mMIMO) has attracted a lot of research interest and is regarded as a promising technology for future wireless communications, for its ability to achieve This article was presented in part at IEEE International Conference on Communications 2022 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang are with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China, and also with the Frontiers Science Center for Smart High-speed Railway System, Beijing Jiaotong University, Beijing 100044, China (e-mail: {zhewang 77, jiayizhang}@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo is with the Institute of Electronics, Communications, and Information Technology, Queen’s University Belfast, BT3 9DT Belfast, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (email: hien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ngo@qub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China, also with the Frontiers Science Center for Smart High-Speed Railway System and the Henan Joint International Research Laboratory of Intelligent Networking and Data Analysis, Zhengzhou University, Zhengzhou 450001, China, and also with the Research Center of Networks and Communications, Peng Cheng Laboratory, Shenzhen 518066, China (e-mail: boai@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah is with the Technology Innovation Institute, Abu Dhabi, United Arab Emirates, and also with CentraleSup´elec, University Paris-Saclay, 91192 Gif-sur-Yvette, France (e-mail: merouane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='debbah@tii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 uniformly high spectral efficiency (SE) [2]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Basically, a large number of access points (APs), arbitrarily distributed in a wide coverage area and connected to one or several central processing units (CPUs), jointly serve all user equipments (UEs) on the same time-frequency resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Compared with the traditional cellular mMIMO system, the CF mMIMO system operates with no cell boundaries and many more APs than UEs [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Relying upon the prominent network topology of CF mMIMO, four uplink (UL) signal processing schemes, distinguished from levels of the mutual cooperation between all APs and the assistance from the CPU, can be implemented as [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Among these schemes, the “Level 4” and “Level 3” are viewed as efficient processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The so-called Level 4 is a fully-centralized processing scheme where all the pilot and data signals received at APs are transmitted to the CPU via the fronthaul links and the CPU performs channel estimation and data detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The similar scheme was also investigated in [11]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The so-called Level 3 stands for a two layer decoding scheme: in the first layer, each AP estimates channels and decodes the UE data locally by applying an arbitrary combining scheme based on the local channel state information (CSI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' in the second layer, all the local estimates of the UE data are gathered at the CPU in which they are linearly weighted by the optimal large-scale fading decoding (LSFD) coefficient to obtain the final decoding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The LSFD scheme has been widely investigated in [14]–[17] since it can make full use of the prominent network topology for CF mMIMO and achieve excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To promote the practical implementation of the CF mMIMO network, a new framework of scalable CF mMIMO system and its respective processing algorithms were proposed in [9] by exploiting the dynamic cooperation cluster (DCC) concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the scalability aspects in a realistic scenario with multiple CPUs were considered in [18], where the data processing, network topology and power control strategies with multiple CPUs were discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the authors of [19] considered the uplink of a radio-strip- based CF mMIMO network architecture with sequential fronthaul links between APs and proposed MMSE- based sequential processing schemes, which significantly reduced the fronthaul requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, when the CF mMIMO network is operated in practice, a more practical capacity-constrained fronthaul network would have a great effect on the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [20] and [21] discussed the uplink performance of a CF mMIMO system with limited capacity fronthaul links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, it is worth noting that the CF mMIMO architecture has been co-designed with another promising future wireless technology: Reconfigurable Intelligent Surface (RIS) [22], [23], which would undoubtedly provide vital tutorials for the future wireless network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3 The vast majority of scientific papers on CF mMIMO focus on the scenario with single-antenna UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, in practice, contemporary UEs with moderate physical sizes have already been equipped with multiple antennas to achieve higher multiplexing gain and boost the system reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [24] investigated the UL performance of a CF mMIMO system with multi-antenna UEs over maximum ratio (MR) combining and zero-forcing (ZF) combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [25] considered a user-centric (UC) approach for CF mMIMO with multi-antenna UEs and proposed power allocation strategies for either sum- rate maximization or minimum-rate maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the authors of [26] analyzed the downlink SE performance for a CF mMIMO system with multi-antenna UEs and computed SE expressions in closed- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, the SE performance for a CF mMIMO system with multi-antenna UEs and low-resolution DACs was investigated in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Nevertheless, these works only investigated a simple distributed processing scheme and are based on the overly idealistic assumption of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=') Rayleigh fading channels, neglecting the spatial correlation that has a significant impact on practical CF mMIMO systems [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [28] considered a CF mMIMO system with multi-antenna UEs over the jointly-correlated Weichselberger model [29] and analyzed four UL processing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As observed in [26], [28], increasing the number of antennas per UE may not always benefit the SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The SE would reach the maximum value with particular number of antennas per UE, then decrease with the increase of number of antennas per UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One main reason for this phenomenon is that the UEs cannot make full use of the benefit of equipping with multiple antennas to achieve higher SE performance without UL precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So it is undoubtedly vital to design the UL precoding scheme to further improve the performance of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, it is worth noting that the design of UL precoding for CF mMIMO has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the traditional mMIMO or MIMO systems, one popular optimization objective for the uplink/downlink precoding design is to maximize the weighted sum rate (WSR) [30]–[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [30] and [32] discussed the equivalence between the WSR maximization problem and the Weighted sum-Minimum Mean Square Error (WMMSE) problem in MIMO systems and proposed an iteratively downlink transceiver design algorithm for the WSR maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the algorithm relies on the iterative minimization of weighted MSE since the WMMSE problem are not jointly convex over all optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the authors of [31] investigated the UL precoding scheme optimization based on [30] under sum-power-constraint or individual-power-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Motivated by the above observations, we investigate a CF mMIMO system with both multi-antenna APs and UEs over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Two pragmatic processing schemes: 1) 4 the fully centralized processing scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) the large-scale fading decoding scheme are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The main contributions are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We design an efficient UL precoding scheme to maximize the WSR for the fully centralized processing scheme based on an iteratively WMMSE (I-WMMSE) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the design of I-WMMSE precoding scheme for the fully centralized processing scheme is implemented at the CPU and based on the instantaneous CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the LSFD processing scheme, we derive a UL precoding scheme for the WSR maximization based on an iteratively WMMSE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The design of I-WMMSE precoding scheme for the LSFD scheme is implemented at the CPU but based only on channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, we compute achievable SE expressions and optimal precoding schemes in novel closed-form for the LSFD scheme with MR combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We analyze the practical implementation and computation complexity for the proposed I-WMMSE precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is found that the proposed I-WMMSE precoding schemes can be guaranteed to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, the proposed UL precoding schemes are efficient to achieve excellent sum SE/rate performance and the average rate benefits from the multiple antennas at the UE-side, which undoubtedly provides vital insights for the practical implementation of multi-antenna UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that this paper differs from the conference version [1] in the following aspects: i) we investigate the fully centralized processing/LSFD schemes and design their respective UL precoding schemes, while only the LSFD scheme was considered in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ii) we provide details for the derivation of the I-WMMSE precoding schemes, which are omitted in [1] due to the lack of space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' iii) we analyze the practical im- plementation and convergence behavior of the proposed precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, numerical results show vital insights for the CF mMIMO system with the proposed UL precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In Section II, we consider a CF mMIMO system with the Weichselberger Rayleigh fading channel, and describe the channel estimation and data detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, Section III introduces the fully centralized processing and LSFD processing schemes, and provides their respective achievable SE expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Novel closed-form SE expressions for the LSFD scheme with MR combining are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, based on the achievable SE expressions, we propose UL I- WMMSE precoding schemes for two processing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, Section IV provides some insights for the practical implementation and computation complexity of proposed I-WMMSE precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In Section V, numerical results and performance analysis for the I-WMMSE precoding schemes are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 CPU Fronthaul Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A cell-free massive MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, the major conclusions and future directions are drawn in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Notation: Lowercase letters x and boldface uppercase letters X denote the column vectors and matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' E {·}, tr {·} and ≜ are the expectation operator, the trace operator, and the definitions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' |·|, ∥·∥ and ∥·∥F are the determinant of a matrix or the absolute value of a number, the Euclidean norm and the Frobenious norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' vec (A) denotes a column vector formed by the stack of the columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The n×n identity matrix is represented by In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The Kronecker products and the element-wise products are denoted by ⊗ and ⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, x ∼ NC (0, R) is a circularly symmetric complex Gaussian distribution vector with correlation matrix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' SYSTEM MODEL In this paper, we investigate a CF mMIMO system consisting of M APs and K UEs, where all APs are connected to one or several CPUs via fronthaul links as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For simplicity, there is only one CPU and all APs serve all UEs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The numbers of antennas per AP and UE are L and N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A standard block fading model is investigated, in which the channel response is constant and frequency flat in a coherence block of τc-length (channel uses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Let τp and τc − τp denote channel uses dedicated for the channel estimation and data transmission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We denote by Hmk ∈ CL×N the channel response between AP m and UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We assume that Hmk for different AP-UE pairs are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Channel Model Based on the jointly-correlated (also known as the Weichselberger model [29]) Rayleigh fading channel2, Hmk is modeled as Hmk = Umk,r � ˜Ωmk ⊙ Hmk,iid � UH mk,t (1) 1As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1, a more practical network topology is with multiple CPUs and dynamic cooperation clusters, where each UE is only served by a cluster of APs and the APs are grouped into cell-centric clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Each cell-centric cluster is connected to a particular CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2Note that the Rayleigh fading channel is a special case of the Rician fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' And the performance gap between the Rician channel and the Rayleigh channel is small [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, the focus of this paper is not on the channel model but on the UL precoding scheme design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So for the simplicity of analysis, we investigate an essential Rayleigh fading channel by assuming there is no line-of-sight (LoS) link between each UE and AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 0000000000006 where Umk,r = [umk,r,1, · · · , umk,r,L] ∈ CL×L and Umk,t = [umk,t,1, · · · , umk,t,N] ∈ CN×N are the eigen- vector matrices of the one-sided correlation matrices Rmk,r ≜ E � HmkHH mk � and Rmk,t ≜ E � HT mkH∗ mk � , and Hmk,iid ∈ CL×N is composed of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' NC (0, 1) random entries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, we denote by Ωmk ≜ ˜Ωmk ⊙ ˜Ωmk ∈ RL×N the “eigenmode coupling matrix” with the (l, n)-th element [Ωmk]ln specifying the average amount of power coupling from umk,r,l to umk,t,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hmk can also be formed as Hmk = [hmk,1, · · · , hmk,N] with hmk,n ∈ CL being the channel between AP m and n-th antenna of UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By stacking the columns of Hmk on each other, we define hmk ≜ vec (Hmk) = [hT mk,1, · · · , hT mk,N]T ∼ NC (0, Rmk), where Rmk ≜ E{hmkhH mk} is the full correlation matrix Rmk = (U∗ mk,t ⊗ Umk,r)diag (vec (Ωmk)) (U∗ mk,t ⊗ Umk,r)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (2) Moreover, note that Rmk can be structured into the block form as [28] with the (n, i)-th submatrix being Rni mk = E{hmk,nhH mk,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the large-scale fading coefficient βmk can be extracted from Rmk as βmk = 1 LN tr (Rmk) = 1 LN ∥Ωmk∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is worth mentioning that the motivations for adopting the Weichselberger channel model are: 1) The Weichselberger model investigated in (1) not only captures the correlation features at both the AP-side and UE-side but models the joint correlation dependence between each AP-UE pair through the coupling matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) The coupling matrix Ωmk reflects the practical spatial arrangement of scattering objects between AP m and UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More significantly, the Weichselberger model can reduce to most channel models of great interest by adjusting the coupling Ωmk to particular formulation, such as the Kronecker model and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Rayleigh fading model [28], [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3) Compared with other stochastic channel models, the Weichselberger model displays significantly less modeling error, which is validated based on the practical measurement in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Channel Estimation For the channel estimation, mutually orthogonal pilot matrices are constructed and each pilot matrix is composed of N mutually orthogonal pilot sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We denote by Φk the pilot matrix assigned to UE k with ΦH k Φl = τpIN, if l = k and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' And Pk is the index subset of UEs using the same pilot matrix as UE k including itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When all UEs transmit their pilot matrices, the received signal at AP m Yp mk ∈ CL×τp is Yp m = �K k=1 HmkFk,pΦT k + Np m, where Fk,p ∈ CN×N is the precoding matrix for UE k under the phase of pilot transmission, Np m ∈ CL×τp is the additive noise at AP m with independent NC(0, σ2) entries and σ2 being the noise power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The pilot transmission should be implemented under the power constraint as tr(Fk,pFH k,p) ⩽ pk, where pk is the maximum transmit power for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To derive sufficient statistics for hmk, AP m projects Yp mk onto Φ∗ k as Yp mk = Yp mΦ∗ k = 7 �K l=1 HmlFl,p � ΦT l Φ∗ k � +Np mΦ∗ k = � l∈Pk τpHmlFl,p + Qp mk, where Qp mk ≜ Np mΦ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, following the standard MMSE estimation steps in [35] and [36], AP m can compute the MMSE estimation of hmk as ˆhmk = vec( ˆHmk) = Rmk˜FH k,pΨ−1 mkyp mk, (3) where ˆHmk is the MMSE estimation of Hmk, ˜Fk,p = FT k,p⊗IL, yp mk ≜ vec (Yp mk) = � l∈Pk τp˜Fl,phml + qp m, qp m = vec (Qp mk) and Ψmk = � l∈Pk τp˜Fl,pRml˜FH l,p + σ2ILN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the estimate ˆhmk and estimation error ˜hmk = hmk − ˆhmk are independent random vectors distributed as ˆhmk ∼ NC(0, ˆRmk) and ˜hmk ∼ NC(0, Cmk), where ˆRmk ≜ τpRmk˜FH k,pΨ−1 mk˜Fk,pRmk and Cmk ≜ Rmk − ˆRmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We can also form ˆRmk and Cmk in the block structure with the (n, i)-th submatrix being ˆRni mk = E{ˆhmk,nˆhH mk,i} and Cni mk = E{˜hmk,n˜hH mk,i}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Data Transmission For the data transmission, all antennas of all UEs simultaneously transmit their data symbols to all APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The received signal ym ∈ CL at AP m is ym = K � k=1 Hmksk + nm, (4) where nm ∼ NC(0, σ2IL) is the independent receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The transmitted signal from UE k sk ∈ CN can be constructed as sk = Fk,uxk, where xk ∼ NC(0, IN) is the data symbol for UE k and Fk,u ∈ CN×N is the precoding matrix for the data transmission which should satisfy the power constraint of UE k as tr(Fk,uFH k,u) ⩽ pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' SPECTRAL EFFICIENCY ANALYSIS AND I-WMMSE PRECODING DESIGN In this section, we investigate two promising signal processing schemes, called “fully centralized pro- cessing” and “LSFD processing”, and analyze their corresponding SE performance and design respective iteratively WMMSE precoding schemes3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Fully Centralized Processing 1) Spectral Efficiency Analysis: For the fully centralized processing scheme, all M APs send all the received pilot signals and data signals to the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Indeed, both the channel estimation and data detection are implemented at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The collective channel hk ∈ CMLN for UE k can be constructed as hk = [vec(H1k)T, · · · , vec(HMk)T]T ∼ NC(0, Rk) with Rk = diag (R1k, · · · , RMk) ∈ CMLN×MLN being the 3We only optimize the precoding matrices for the phase of data transmission Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The optimization of Fk,p is left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Although we do not design Fk,p in this paper, we try to keep the derived equations more generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So a scenario with arbitrary Fk,p instead of limiting Fk,p to a particular form is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is worth noting that all equations in this paper hold for any Fk,p so undoubtedly provide some important guidelines for the investigation of optimization design for Fk,p in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8 whole block-diagonal correlation matrix for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Similar to (3), the CPU can derive the channel estimate for UE k as4 ˆhk ≜ � ˆhT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ˆhT Mk �T ∼ NC � 0, τpRk¯FH k,pΨ−1 k ¯Fk,pRk � where ¯Fk,p = diag(˜Fk,p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ˜Fk,p � �� � M ) and Ψ−1 k = diag(Ψ−1 1k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Ψ−1 Mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The channel estimation error is ˜hk ∼ NC (0, Ck) where Ck ≜ Rk − τpRk¯FH k,pΨ−1 k ¯Fk,pRk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the received data signal at the CPU can be denoted as [yT 1 , · · · , yT M]T � �� � = y = K � k=1 [HT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , HT Mk]T � �� � = Hk Fk,uxk + [nT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , nT M]T � �� � = n , (5) or a compact form as y = �K k=1 HkFk,uxk + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Under the setting of “fully centralized processing”, we assume that UL precoding matrices (Fk,u and Fk,p) are available at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on the collective channel estimates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' the CPU designs an arbitrary receive combining matrix Vk ∈ CLM×N for UE k to detect xk as ˇxk = VH k y = VH k ˆHkFk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uxk + VH k ˜HkFk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uxk + K � l̸=k VH k HlFl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uxl + VH k n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (6) and the conditional MSE matrix for UE k is Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = E{(xk − ˇxk)(xk − ˇxk)H|{ ˆHk},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' {Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u}} = IN − VH k ˆHkFk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u − FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ˆHH k Vk + VH k � K � l=1 � ˆHl¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ˆHH l + C′ l � + σ2IML � Vk (7) where ¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ≜ Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uFH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C′ l ≜ diag (C′ 1l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C′ Ml) ∈ CML×ML and C′ ml = E{ ˜Hml¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ˜HH ml} ∈ CL×L with the (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' q)-th element of C′ ml being [C′ ml]jq = �N p1=1 �N p2=1 �¯Fl � p2p1 [Cp2p1 ml ]jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By implementing the per-user-basis minimum mean-squared error-based successive interference cancel- lation (MMSE-SIC) detector while treating co-user interference as uncorrelated Gaussian noise, we derive the achievable SE for UE k as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' An achievable for UE k under the setting of “fully centralized processing” with the MMSE estimator is SEk,(1) = � 1 − τp τc � E � log2 ���IN + DH k,(1)Σ−1 k,(1)Dk,(1) ��� � , (8) where Dk,(1) ≜ VH k ˆHkFk,u and Σk,(1) ≜ VH k ��K l=1 ˆHl¯Fl,u ˆHH l − ˆHk¯Fk,u ˆHH k + �K l=1 C′ l + σ2IML � Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4Note that the pilot signals received at the APs are first transmitted to the CPU and then the CPU estimates the channels, where τpML complex scalars are sent from the APs to the CPU at each coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Alternatively, all APs can first estimate the channels as (3), and then send their channel estimates to the CPU, where MKLN complex scalars are sent from the APs to the CPU at each coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since the pilot contamination is investigated (τp < KN) in this paper, we consider the first transmission protocol due to its lower fronthaul overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 The expectations are with respect to all sources of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof follows from the similar approach as [28, Corollary 1] and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that Corollary 1 holds for any combining schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One promising combining scheme is the MMSE combining as VMMSE k = � K � l=1 � ˆHl¯Fl,u ˆHH l + C′ l � + σ2IML �−1 ˆHkFk,u, (9) which can minimize the mean-squared error MSEk,(1) = tr(Ek,(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' With the MMSE combining scheme, the conditional MSE matrix in (7) is Eopt k,(1) = IN − FH k,u ˆHH k � K � l=1 � ˆHl¯Fl,u ˆHH l + C′ l � + σ2IML �−1 ˆHkFk,u (10) More importantly, the MMSE combining in (9) can also maximize the achievable SE in (8) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The achievable SE for UE k in (8) can be maximized by the MMSE combining scheme in (9) with the maximum value SEopt k,(1) = � 1 − τp τc � E \uf8f1 \uf8f2 \uf8f3log2 ������ IN + FH k,u ˆHH k � K � l=1 � ˆHl¯Fl,u ˆHH l + C′ l � − ˆHk¯Fk,u ˆHH k + σ2IML �−1 ˆHkFk,u ������ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof can be found in [28, Appendix B] and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) Iteratively WMMSE Precoding Design: In this part, we design the uplink precoding scheme for the “fully centralized processing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One popular weighted sum-rate maximization problem is investigated as5 max {F} K � k=1 µk,(1)SEk,(1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (12) where µk,(1) represents the priority weight of UE k and SEk,(1) is given by (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5The notation F is short for {Fk,u}k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=',K, denoting all variables Fk,u with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Similar definitions are applied for V, A, W, S in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In this section, we denote by Fk,u,(1) and Fk,u,(2) the UL precoding matrix of UE k for the fully centralized processing and LSFD scheme, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10 As [30] and [32], the matrix-weighted sum-MSE minimization problem as min {F,V,W} K � k=1 µk,(1) � tr � Wk,(1)Ek,(1) � − log2 ��Wk,(1) ��� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (13) is equivalent to the weighted sum-rate maximization problem (12), where Wk,(1) ∈ CN×N is the weight matrix for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that (13) is convex over each optimization variable F, V, W but is not jointly convex over all optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Following the method in [30], we can solve (13) by sequentially fixing two of the three optimization variables F, V, W and updating the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Fixing the other variables, the update of Vk is given by the MMSE solution as (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Under the MMSE combining, the MSE matrix is given by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, note that optimal Wk,(1) for (13) is Wopt k,(1) = E−1 k,(1), (14) which can be easily derived through the first order optimality condition for Wk,(1) by fixing F and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When the MMSE combining VMMSE k and Wopt k,(1) for all UEs are implemented in (13), we have tr(Wk,(1)Ek,(1))−log2 ��Wk,(1) �� = tr (IN)−log2 |(Eopt k,(1))−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the matrix-weighted sum-MSE minimization problem in (13) would reduce to the equivalent optimization problem of (12) as6: max {F} K � k=1 µk,(1) log2 ���� � Eopt k,(1) �−1���� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (15) which is a well-known relationship between Eopt k,(1) and SEopt k,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, fixing V and W, the update of Fk,u,(1) for (13) results in the optimization problem as7 6Note that “SE” is equivalent to “rate” except from having one scaling factor (τc − τp)/τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since τc and τp are constants in this paper, so we ignore the difference between SE and rate in the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7It is worth mentioning that the updates of optimization variables are based on the preliminary of fixing the other optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, when updating Fk,u,(1), we should fix the other optimization variables (Vk and Wk,(1)) but not only limited to their respective optimal solutions VMMSE k and Wopt k,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we update Fk,u,(1) based on (16) with generalized Vk and Wk,(1) instead of (15) with optimal VMMSE k and Wopt k,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 11 min {F} K � k=1 µk,(1)tr � Wk,(1) � IN − VH k ˆHkFk,u,(1) � � IN − VH k ˆHkFk,u,(1) �H� + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l̸=k ˆHlFl,u,(1)FH l,u,(1) ˆHH l � Vk � − K � k=1 µk,(1) log2 ��Wk,(1) �� + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l=1 E � ˜HlFl,u,(1)FH l,u,(1) ˜HH l ��� F � + σ2IML � Vk � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (16) which is a convex quadratic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the classic Lagrange multipliers methods and Karush-Kuhn-Tucker (KKT) conditions can be applied to derive an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The Lagrange function of (16) is f � F1,u,(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , FK,u,(1) � = K � k=1 µk,(1)tr � Wk,(1) � IN − VH k ˆHkFk,u,(1) � � IN − VH k ˆHkFk,u,(1) �H� + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l̸=k ˆHlFl,u,(1)FH l,u,(1) ˆHH l � Vk � + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l=1 E � ˜HlFl,u,(1)FH l,u,(1) ˜HH l ��� F � + σ2IML � Vk � + K � k=1 λk,(1) � tr � Fk,u,(1)FH k,u,(1) � − pk � (17) Finally, we derive the optimal precoding scheme as the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By fixing other optimization variables and applying the first-order optimality condition of (17) with respect to each Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' the optimal precoding scheme is given by Fopt k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � K � l=1 µl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � ˆHH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˆHk + E � ˜HH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˜Hk ��� V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W �� + λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)IN �−1 ˆHH k VkWk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � K � l=1 µl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � ˆHH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˆHk + ¯Ckl � + λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)IN �−1 ˆHH k VkWk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (18) where λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) ⩾ 0 is the Lagrangian multiplier and the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' n)-th element of ¯Ckl ≜ E{ ˜HH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˜Hk|V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W} ∈ CN×N is �¯Ckl � in = tr( ¯VlE{˜hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='n˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i}) = tr �¯VlCk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='in � with ¯Vl ≜ VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l and Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ni ≜ E{˜hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='n˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i} = 12 diag (Cni 1k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Cni Mk) ∈ CML×ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' According to the KKT condition, λk,(1) and Fk,u,(1) should also satisfy ��Fk,u,(1) ��2 ⩽ pk, λk,(1) ���Fk,u,(1) ��2 − pk � = 0, λk,(1) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (19) Proof: The proof is given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We denote by Fk,u,(1)(λk,(1)) the right-hand side of (18), when �K l=1 µl,(1)( ˆHH k VlWl,(1)VH l ˆHk + ¯Ckl) is invertible and tr[Fk,u,(1)(0)Fk,u,(1)(0)H] ⩽ pk, then Fopt k,u,(1) = Fk,u,(1) (0), otherwise we have tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] = pk to satisfy (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] is a monotonically decreasing function of λk,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof: Let DΛDH denote the eigendecomposition of �K l=1 µl,(1)( ˆHH k VlWl,(1)VH l ˆHk + ¯Ckl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Fol- lowing the method in [30], we define Φ = µ2 k,(1)DH ˆHH k VkW2 k,(1) ˆHkVH k D and we have tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] = tr �� DΛDH + λk,(1)IN �−1 DΦDH � DΛDH + λk,(1)IN �−1� = tr �� DΛDH + λk,(1)IN �−2 DΦDH� = tr �� Λ + λk,(1)IN �−2� = N � n=1 [Φ]nn � [Λ]nn + λk,(1) �2, (20) so tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] is a monotonically decreasing function of λk,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on Corollary 3, optimum λk,(1) (denoted by λopt k,(1)) can be easily obtained by a one-dimensional (1-D) bisection algorithm so we derive the solution for Fk,u,(1)(λopt k,(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, an iterative opti- mization algorithm for Fk,u,(1), called “iteratively WMMSE (I-WMMSE) algorithm”, is summarized in Algorithm 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The convergence of Algorithm 1 is proven in [30, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the design of Fk,p is a valuable future direction to further improve the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One valuable optimization problem is to minimize the total MSE of the channel estimators of all UEs as min {Fk,p} K � k=1 tr (Ck) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ∥Fk,p∥2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (21) where the optimization goal is only based on the statistical knowledge so Fk,p is also based on the statistical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8To balance the efficiency and the computational complexity of the proposed algorithm, we also include the stopping criterion “R(i) (1) < R(i−1) (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the I-WMMSE precoding scheme is derived at iteration (i − 1), which may achieve higher sum SE than the one at iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 13 Algorithm 1: I-WMMSE Algorithm for the Design of Fk,u,(1) Input: Collective channel estimates ˆHk for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Estimation error covariance matrices Cml for all possible pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' UE weights µk,(1) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Output: Optimal precoding matrices Fk,u,(1) for all UEs (F(i) k,u,(1) for the first or third stopping criterion and F(i−1) k,u,(1) for the second stopping criterion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 Initiation: i = 0, F(0) k,u,(1) and R(0) (1) = �K k=1 µk,(1)SE(0) k,(1) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' maximum iteration number I(1),max and threshold ε(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 repeat 3 i = i + 1 4 Update the MMSE combining scheme V(i) k with F(i−1) k,u,(1) based on (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 Update optimal MSE matrix E(i) k,(1) with F(i−1) l,u,(1) based on (10), and update W(i) k,(1) based on (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6 Update optimal precoding matrix F(i) l,u,(1) with V(i) k and W(i) k,(1) based on (18), where λ(i) k,(1) is found by a bisection algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7 Update sum weighted rate R(i) (1) = �K k=1 µk,(1)SE(i) k,(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8 until ���R(i) (1) − R(i−1) (1) ��� /R(i−1) (1) ⩽ ε(1) or R(i) (1) < R(i−1) (1) or i ⩾ I(1),max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Large-Scale Fading Decoding 1) Spectral Efficiency Analysis: Another promising processing scheme is “large-scale fading decoding”, which is a two-layer decoding scheme to decode the data symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that UL precoding matrices (Fk,u and Fk,p) are assumed to be available at all APs and the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In the first layer, AP m applies an arbitrary combining matrix Vmk ∈ CL×N to derive local detection of xk as ˜xmk = VH mkym = VH mkHmkFk,uxk + K � l=1,l̸=k VH mkHmlFl,uxl + VH mknm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (22) We notice that Vmk is designed based on local channel estimates at AP m and one handy choice is MR combining Vmk = ˆHmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, local MMSE (L-MMSE) combining Vmk = � K � l=1 � ˆHml¯Fl,u ˆHH ml + C′ ml � + σ2IL �−1 ˆHmkFk,u, (23) is also regarded as a promising scheme, since (23) can minimize E{∥ xk − VH mkym ∥2 |{ ˆHmk}, {Fk,u}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In the second layer, the “LSFD” method is implemented at the CPU [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The CPU weights all the local estimates ˜xmk from all APs by the LSFD coefficient matrix as ˆxk = M � m=1 AH mk˜xmk = M � m=1 AH mkVH mkHmkFk,uxk + M � m=1 K � l=1,l̸=k AH mkVH mkHmlFl,uxl+n′ k, (24) where Amk ∈ CN×N is the complex LSFD coefficient matrix for AP m-UE k and n′ k = �M m=1 AH mkVH mknm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 14 Moreover, we can rewrite ˆxk in a more compact form as ˆxk = AH k GkkFk,uxk + K � l=1,l̸=k AH k GklFl,uxl + n′ k = AH k � GkkFk,uxk + K � l=1,l̸=k GklFl,uxl + ˜n′ k � � �� � ˜xk (25) where Ak ≜ [AT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , AT Mk]T ∈ CMN×N, Gkl ≜ [VH 1kH1l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' VH MkHMl] ∈ CMN×N and ˜n′ k = � VH 1kn1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' VH MknM � ∈ CMN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the CPU does not have the knowledge of channel estimates and is only aware of channel statistics [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The conditional MSE matrix for UE k Ek,(2) ≜ E � (xk − ˆxk) (xk − ˆxk)H |Θ � is Ek,(2) = IN − FH k,uE{GH kk}Ak − AH k E{Gkk}Fk,u + AH k � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk � Ak, (26) where Θ denotes all the channel statistics and Sk = diag(E{VH 1kV1k}, · · · , E{VH MkVMk}) ∈ CMN×MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, we apply classical use-and-then-forget bound to obtain the following ergodic achievable SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the “LSFD” scheme, an achievable SE for UE k can be written as SEk,(2) = � 1 − τp τc � log2 ���IN + DH k,(2)Σ−1 k,(2)Dk,(2) ��� , (27) where Σk,(2) = �K l=1 AH k E{Gkl¯Fl,uGH kl}Ak − Dk,(2)DH k,(2) + σ2AH k SkAk and Dk,(2) = AH k E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof follows similar steps as the proof of [28, Corollary 2] and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that Ak can be optimized by the CPU based on channel statistics to maximize the achievable SE in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on the theory of optimal receivers as in [37], we derive the optimal LSFD coefficient matrix, which not only maximizes the achievable SE but minimizes the conditional MSE, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The achievable SE in (27) is maximized by Aopt k = � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk �−1 E{Gkk}Fk,u, (28) leading to the maximum value as SEopt k,(2) = � 1 − τp τc � log2 ������ IN + FH k,uE {Gkk} � K � l=1 E � Gkl¯Fl,uGH kl � − E {Gkk} ¯Fk,uE � GH kk � + σ2Sk �−1 E {Gkk} Fk,u ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (29) 15 Note that the optimal LSFD coefficient matrix in (28) can also minimize the conditional MSE for UE k MSEk,(2) = tr(Ek,(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' If the optimal LSFD coefficient matrix is applied, the MSE matrix for UE k can be written as Eopt k,(2) = IN − FH k,uE � GH kk � � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk �−1 E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (31) Furthermore, if MR combining Vmk = ˆHmk is applied, we derive closed-form SE expressions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For MR combining Vmk = ˆHmk, (27) can be computed in closed-form as SEk,(2),c = � 1 − τp τc � log2 ���IN + DH k,(2),cΣ−1 k,(2),cDk,(2),c ��� , (32) where Σk,(2),c = AH k (�K l=1 Tkl,(1) + � l∈Pk Tkl,(2))Ak − Dk,(2),cDH k,(2),c + σ2AH k Sk,cAk and Dk,(2),c = AH k ZkFk,u, with E{Gkk} = Zk = [ZT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ZT Mk]T and Sk,c = diag(Z1k, · · · , ZMk) with the (n, n′)- th element of Zmk ∈ CN×N being [Zmk]nn′ = tr(ˆRn′n mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) ≜ diag(Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M) ∈ CMN×MN and Tmm′ kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) = Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m − Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m if m = m′ and Λmkl¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uΛm′lk otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' where Tmm′ kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) de- notes (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m′)-submatrix of Tkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ∈ CMN×MN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' the (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' n′)-th element of N × N-dimension complex matrices Λmkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Λm′lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m and Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m are [Λmkl]nn′ = tr(Ξn′n mkl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [Λm′lk]nn′ = tr(Ξn′n m′lk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [Γ(1) mkl]nn′ = �N i=1 �N i′=1 [¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u]i′itr(Ri′i ml ˆRn′n mk) and [Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m]nn′ given by � Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m � nn′ = N � i=1 N � i′=1 �¯Fl � i′i � tr � Ri′i mlPn′n mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � +τ 2 p N � q1=1 N � q2=1 � tr � ˜Pq1n mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Ri′q2 ml ˜Rq2i ml ˜Pn′q1 mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) � + tr � ˜Pq1n mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Ri′q2 ml � tr � ˜Pn′q2 mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rq2i ml ��� (33) with Ξmkl = τpRml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 mk˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRmk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ξm′lk = τpRm′k˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 m′k˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRm′l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Pmkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = τpSmk(Ψmk − τp˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='p)SH mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Smk = Rmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Pmkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) = Smk˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pSH mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ˜Rni ml and ˜Pni mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) being (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' i)- submatrix of R 1 2 ml and P 1 2 mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, the optimal LSFD coefficient matrix in (28) and MSE matrix in (31) can also be computed in closed-form as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Aopt k,c = ��K l=1 Tkl,(1) + � l∈Pk Tkl,(2) + σ2Sk,c �−1 ZkFk,u, Eopt k,(2),c = IN − FH k,uZH k ��K l=1 Tkl,(1) + � l∈Pk Tkl,(2) + σ2Sk,c �−1 ZkFk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (34) Proof: The proof is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 16 2) Iteratively WMMSE Precoding Design: For the LSFD scheme, we also investigate a weighted sum- rate maximization problem as max {F} K � k=1 µk,(2)SEk,(2) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (35) where µk,(2) represents the priority weight of UE k for the “LSFD” scheme and SEk,(2) is given in (27) with arbitrary combining structure in the first decoding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Similarly, the matrix-weighted sum-MSE minimization problem as9 min {F,A,W,G,S} K � k=1 µk,(2) � tr � Wk,(2)Ek,(2) � − log2 ��Wk,(2) ��� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (36) is equivalent to the weighted sum-rate maximization problem (35), where Wk,(2) is the weight matrix for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that (36) is convex over each optimization variable F, A, W, G, S but is not jointly convex over all optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we can solve (36) by sequentially fixing four of the five optimization variables F, A, W, G, S and updating the fifth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='10 The update of Ak and Ek,(2) are given by the optimal LSFD scheme (28) and MSE matrix with optimal LSFD scheme (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that optimal Wk,(2) for (36) is Wopt k,(2) = E−1 k,(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When Aopt k and Wopt k,(2) for all UEs are applied in (36), we notice that (36) becomes to the equivalent optimization problem of (35) as max {F,G,S} K � k=1 µk,(2) log2 ���� � Eopt k,(2) �−1���� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (37) which is a well-known relationship between Eopt k,(2) and SEopt k,(2) and proven in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Last but not least, fixing other variables, the update of Fk,u,(2) for (36) results in the optimization 9The notation G denotes all G-relevant variables, like E{Gkl¯Fl,u,(2)GH kl} and E{Gkk}, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10As for G and S, if L-MMSE combining scheme applied, E {Gkk} and Sk are relevant to Fk,u,(2) so we should also update them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' On the contrary, E{Gkk} and Sk with MR combining structure are irrelevant to F so we only need to update E{Gkl¯Fl,u,(2)GH kl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 17 problem as min {F} K � k=1 µk,(2) � tr � Wk � IN − FH k,u,(2)E � GH kk � Ak � � IN − FH k,u,(2)E � GH kk � Ak �H�� + K � k=1 µk,(2) � tr � Wk,(2)AH k � K � l̸=k E � Gkl¯Fl,u,(2)GH kl � + σ2Sk � Ak �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (38) which is a convex quadratic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Thus, we can also derive the optimal precoding scheme by applying classic Lagrange multipliers methods and KKT conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The Lagrange function of (38) is f � F1,u,(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , FK,u,(2) � = K � k=1 µk,(2) � tr � Wk,(2) � IN − FH k,u,(2)E � GH kk � Ak � � IN − FH k,u,(2)E � GH kk � Ak �H�� + K � k=1 µk,(2) � tr � Wk,(2)AH k � K � l̸=k E � Gkl¯Fl,u,(2)GH kl � + σ2Sk � Ak �� + K � k=1 λk,(2) � tr � Fk,u,(2)FH k,u,(2) � − pk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (39) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By applying the first-order optimality condition of (39) with respect to each Fk,u,(2) and fixing other optimization variables, we obtain the optimal precoding scheme as Fopt k,u,(2) = µk,(2) � K � l=1 µl,(2)E � GH lkAlE−1 l,(2)AH l Glk � + λk,(2)IN �−1 E � GH kk � AkE−1 k,(2), (40) where λk,(2) ⩾ 0 is the Lagrangian multiplier during the phase of “LSFD” scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' According to the KKT condition, λk,(2) and Fk,u,(2) should also satisfy ��Fk,u,(2) ��2 ⩽ pk, λk,(2) ���Fk,u,(2) ��2 − pk � = 0, λk,(2) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (41) Note that when �K l=1 µl,(2)E{GH lkAlE−1 l,(2)AH l Glk} is invertible and tr � Fk,u,(2)(0)Fk,u,(2)(0 �H] ⩽ pk, then Fopt k,u,(2) = Fk,u,(2) (0), otherwise we must have tr[Fk,u,(2)(λk,(2))Fk,u,(2)(λk,(2))H] = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Following the similar method in Corollary 3, we notice that λk,(2) can be easily found by a 1-D bisection algorithm since tr[Fk,u,(2)(λk,(2))Fk,u,(2)(λk,(2))H] is a monotonically decreasing function of λk,(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, if MR combining Vmk = ˆHmk is applied in the first layer, we can compute expectations in (40) in closed-form as following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' With MR combining Vmk = ˆHmk and the optimal LSFD scheme applied, we can compute 18 E{GH kk}, Aopt k , and Eopt k,(2) in closed-form as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' we have ¯Tlk = E{GH lkAlE−1 l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2)AH l Glk} ∈ CN×N where the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' n)-th element of ¯Tlk is tr(¯Al ¯Glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ni) with ¯Al ≜ AlE−1 l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2)AH l and the [(m − 1) N + p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (m′ − 1) N + p′]-th (or [o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' j]-th briefly) entry of ¯Glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ni ≜ E{glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ngH lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i} ∈ CMN×MN being E{glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ngH lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i}oj = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l /∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m ̸= m′ tr(Rni mk ˆRp′p ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l /∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m = m′ tr(Ξnp mlk)tr(Ξp′i m′kl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l ∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m ̸= m′ tr � Rni mkPp′p mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � + τ 2 p �N q1=1 �N q2=1 tr � ˜Pq1p mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rnq2 mk ˜Rq2i mk ˜Pp′q1 mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) � +τ 2 p �N q1=1 �N q2=1 tr � ˜Pq1n mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rnq1 mk � tr � ˜Pp′q2 mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rq2i mk � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l ∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m = m′ (42) where Ξmlk = τpRmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 mk˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRml,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ξm′kl = τpRm′l˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 m′l˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRm′k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sml = Rml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 ml,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Pmlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = τpSml(Ψml − τp˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='p)SH ml and Pmlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) = Sml˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pSH ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Plugging the derived results into (40), we can compute Fopt k,u,(2) in closed-form as Fopt k,u,(2),c = µk,(2) � K � l=1 µl,(2) ¯Tlk + λk,(2)IN �−1 ZH k Aopt k,c Eopt,−1 k,(2),c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (43) Proof: The proof is given in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, an iterative optimization algorithm for Fk,u,(2) is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The con- vergence of Algorithm 2 is proven in [30, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Relying on the iterative minimization of weighted MSE, two efficient uplink I-WMMSE precoding schemes to maximize the weighted sum SE are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding schemes for the “FCP” and “LSFD” schemes are investigated in Algorithm 1 and Algorithm 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the design of I-WMMSE precoding scheme for the FCP/LSFD is based on instantaneous CSI/channel statistics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, we can compute I-WMMSE precoding schemes in novel closed- form only for the LSFD scheme with MR combining based on Theorm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' PRECODING IMPLEMENTATION AND COMPLEXITY ANALYSIS In this section, we discuss the practical implementation and analyze computational complexity for the UL precoding schemes investigated in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Precoding Implementation 1) Precoding Characteristics: As described above, we investigate a standard block fading model, where the channel response is constant and frequency flat in a coherence block, which contains τc channel 19 Algorithm 2: I-WMMSE Algorithm for the Design of Fk,u,(2) Input: Channel statistics Θ for all possible pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' UE weights µk,(2) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Output: Optimal precoding matrices Fk,u,(2) for all UEs (F(i) k,u,(2) for the first or third stopping criterion and F(i−1) k,u,(2) for the second stopping criterion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 Initiation: i = 0, F(0) k,u,(2) and R(0) (2) = �K k=1 µk,(2)SE(0) k,(2) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' maximum iteration number I(2),max and threshold ε(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 repeat 3 i = i + 1 4 Update channel statistics Θ(i), such as E{G(i) kk}, E{G(i) kl ¯F(i−1) l,u (G(i) kl )H} and S(i) k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 Update optimal LSFD matrix A(i) k with F(i−1) l,u,(2) and Θ(i) based on (28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6 Update optimal MSE matrix E(i) k,(2) with F(i−1) l,u,(2), A(i) k and E{G(i) kk} based on (31) and update W(i) k,(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7 Update optimal precoding matrix F(i) k,u,(2) with A(i) k , W(i) k,(2) and Θ(i) based on (40), where λ(i), k,(2) is found by a bisection algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8 Update sum weighted rate R(i) (2) = �K k=1 µk,(2)SE(i) k,(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 until |R(i) (2) − R(i−1) (2) |/R(i−1) (2) ⩽ ε(2) or R(i) (2) < R(i−1) (2) or i ⩾ I(2),max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the “fully centralized processing” scheme, we notice that the I-WMMSE precoding design is implemented at the CPU based on the instantaneous CSI as (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, to guarantee the convergence of Algorithm 1, only MMSE combining as (9) is advocated to detect the UL data since the equivalent relationship between Eopt k,(1) and SEopt k,(1), which only satisfies with MMSE combining, should be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the LSFD scheme, the optimal design of Fopt k,u,(2) as (40) can only be implemented at the CPU, but relies only on channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, L-MMSE or MR combining can be applied at each AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When MR combining is applied, all terms in Algorithm 2 can be computed in closed-form as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) Fronthaul Requirements: For the FCP scheme with the I-WMMSE precoding, in each coherence block, all APs should relay their received signals to the CPU and the CPU requires precoding matrices Fk,u,(1) feedback to all UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' All APs need to send τcML complex scalars (τpML complex scalars for the pilot signals and (τc − τp)ML complex scalars for the received data signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the full correlation matrices {Rmk} are available at the CPU, which contains MKL2N2/2 complex scalars for each realization of the AP/UE locations/statistics11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the CPU transmits optimal precoding matrices to all UEs, which are described by KN2 complex scalars per coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In summary, for the FCP scheme with the I-WMMSE precoding implemented, total τcMLNr + MKL2N2/2 + KN2Nr complex scalars are transmitted via fronthaul links for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For comparison, when 11Note that the channel statistics remain constant for each realization of the AP/UE locations and each realization of the AP/UE locations contains Nr channel realizations (coherence blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 20 the FCP scheme without the I-WMMSE precoding is implemented, all APs should also transmit τcML complex scalars for the received signals to the CPU in each coherence block and MKL2N2/2 complex scalars for {Rmk} to the CPU for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So for the CFP scheme without the I-WMMSE precoding, total τcMLNr + MKL2N2/2 complex scalars are transmitted via fronthaul links for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the LSFD scheme with the I-WMMSE precoding, all APs transmit their local data estimates ˜xmk, described by (τc − τp)MKN complex scalars, to the CPU per coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, E{Gkk} ∈ CMN×N, described by MKN2 complex scalars for each realization of the AP/UE locations, are also required at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for E{Gkl¯Fl,u,(2)GH kl} ∈ CMN×MN, following the formulation method investigated in Appendix F, the optimization of ¯Fl,u,(2) requires the knowledge of � E � vH ml,phmk,nhH m′k,ivm′l,p′�� , described by M2K2N4/2 complex scalars for each realization of the AP/UE locations, where vml,p denotes the p-th column of Vml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the CPU requires optimal precoding matrices Fk,u,(2) feedback to all APs and UEs only for each realization of the AP/UE locations, which are KN2 complex scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the LSFD scheme without the I-WMMSE precoding, local data estimates ˜xmk, described by (τc − τp)MKN complex scalars per coherence block, E{Gkk}, described by MKN2 complex scalars for each realization of the AP/UE locations, and E{GklGH kl} ∈ CMN×MN, described by M2K2N2/2 complex scalars for each realization of the AP/UE locations, are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' That is total (τc −τp)MKNNr + MKN2 + M2K2N2/2 complex scalars transmitted via fronthaul links for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3) Practical Implementation: Note that the basic motivation of the investigated I-WMMSE precoding schemes is to achieve as good the sum uplink SE performance as possible so we ignore some practical issues, which are vital for the realistic implementation of the investigated precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When the precoding schemes are implemented in practice, these realistic issues should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Capacity-constrained fronthaul network As discussed above, the I-WMMSE precoding require more fronthaul requirements than the case without the I-WMMSE precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is quite vital to consider a more practical capacity-constrained fronthaul network [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the wireless fronthaul [39], which is more flexible than the conventional wire fronthaul, would also be regarded as a promising solution to boost the practical implementation of the I-WMMSE precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Scalability aspects with dynamic cooperation clusters When the precoding schemes are implemented in practice, a more realistic network architecture with 21 multiple CPUs and dynamic cooperation clusters should be advocated, where each UE is only served by a cluster of APs (that a is user-centric cluster) and the APs are grouped into cell-centric clusters as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note a user-centric cluster might consist of APs connecting with different CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on the signal processing schemes in [9], [18], the analytical framework in this paper can be implemented in a scalable paradigm where the fronthaul requirements and computational complexity can be relieved with an anticipated modest performance loss compared with canonical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding design with these two practical aspects is left in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To bring valuable technical insights for the study of I-WMMSE precoding schemes with the DCC strategy and the capacity-constrained fronthaul link, we provide two tutorials for the FCP and LSFD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 based on [9], [10], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tutorials to investigate the I-WMMSE precoding scheme with the DCC strategy and the capacity-constrained fronthaul 1: Joint initial access, pilot assignment, and cluster formation for the DCC topology based on a classical algorithm in [9, Sec V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A] or a more efficient algorithm as [10, Algorithm 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2: Each AP transmits the quantized versions of the local detection signals in (22) to the CPU based on Case 2 in [38] called "Quantized Weighted Signal Available at the CPU" as [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (1), generate the DCC based processing scheme for the LSFD (the local combining design, P-LSFD, and achievable SE computation) motivated by [10, Sec Ⅱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (2), formulate the I-WMMSE precoding design optimization problem with a capacity-constrained fronthaul motivated by [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (24)] and [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (26)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5: Obtain the optimal precoding scheme based on potential methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tutorial 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding for the LSFD with the DCC strategy and the capacity-constrained fronthaul 1: Joint initial access, pilot assignment, and cluster formation for the DCC topology based on a classical algorithm in [9, Sec V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A] or a more efficient algorithm as [10, Algorithm 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2: Each AP transmits the quantized versions of its received pilot signals and data signals to the CPU based on Case 1 in [38] called "Quantized Estimate of the Channel and Quantized Signal Available at the CPU" as [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (11) ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (1), generate the DCC based processing scheme for the FCP (the receive combining and achievable SE computation) motivated by [9, Sec V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (2), formulate the I-WMMSE precoding design optimization problem with a capacity-constrained fronthaul motivated by [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (24)] and [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (26)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5: Obtain the optimal precoding scheme based on potential methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tutorial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding for the FCP with the DCC strategy and the capacity-constrained fronthaul Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Two tutorials to investigate the I-WMMSE precoding schemes with the DCC strategy and the capacity-constrained fronthaul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Complexity Analysis In this subsection, we analyze the computational complexity of two precoding schemes investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since the bisection step for λk,{(1),(2)} generally takes few iterations compared with other steps, we ignore bisection steps for λk,{(1),(2)} in the complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the fully centralized processing scheme and each realization of the AP/UE locations, the per-iteration complexity of iterative optimiza- tion is O (M3K2N5Nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the LSFD scheme and each realization of the AP/UE locations, the per- iteration complexity of iterative optimization based on L-MMSE combining with the Monte-Carlo method, MR combining with the Monte-Carlo method and MR combining with the closed-form expressions are O (M2K2N3Nr), O (M2K2N3Nr + M3KN3) and O (M3K2N5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further reduce the computation complexity, it’s quite necessary to apply the asymptotic analysis method [40], [41] to compute the terms, which cannot be computed in closed-form, in approximation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 22 TABLE I COMPARISON OF TWO PRECODING SCHEMES IN THIS PAPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' THE NUMBER OF COMPLEX SCALARS IS COMPUTED FOR EACH REALIZATION OF THE AP/UE LOCATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' THE SUM SE IMPROVEMENT IS COMPUTED WITH M = 20, K = 10, L = 1 AND N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='FCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='LSFD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Instantaneous CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Statistical CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Detection scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='MMSE combining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='L-MMSE/MR combining + Optimal LSFD scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Number of complex scalars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='sent from APs to the CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='with I-WMMSE precoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='τcMLNr + MKL2N 2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(τc − τp)MKNNr + MKN 2 + M 2K2N 4/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Number of complex scalars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='sent from APs to the CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='without I-WMMSE precoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='τcMLNr + MKL2N 2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(τc − τp)MKNNr + MKN 2 + M 2K2N 2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Number of complex scalars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='feedback sent from the CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='KN 2Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='KN 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Per-iteration computational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='complexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 3K2N 5Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='L-MMSE: O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 2K2N 3Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='MR (Monte-Carlo): O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 2K2N 3Nr + M 3KN 3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='MR (Analytical): O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 3K2N 5� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Sum SE improvement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='93% L-MMSE: 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='74% MR: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='13% V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' NUMERICAL RESULTS In this paper, a CF mMIMO system is investigated, where all APs and UEs are uniformly distributed in a 1×1 km2 area with a wrap-around scheme [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The pathloss and shadow fading are modeled similarly as [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In practice, Umk,r, Umk,t and Ωmk are estimated through measurements [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, we generate them randomly in this paper, where the coupling matrix Ωmk consists of one strong transmit eigendirection capturing dominant power [43]12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, we have Fk,p = F(0) k,u,{(1),(2)} = � pk N IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for Algorithm 1 and Algorithm 2, balancing the convergence and accuracy, we assume that I(1),max = I(2),max = 20, ε(1) = ε(2) = 5 × 10−4, and weights for all UEs are equal (µk,(1) = µk,(2) = 1) without losing generality, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, we consider communication with 20 MHz bandwidth and σ2 = −94 dBm noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' All UEs transmit with 200 mW power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Each coherence block contains τc = 200 channel uses and τp = KN/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, a pilot assignment approach similar as that in [28] is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 3 shows the cumulative distribution function (CDF) of the achievable sum SE over different realizations of the AP/UE locations for two processing schemes investigated (we shortly call “fully centralized processing” as “FCP” in the following) over “I-WMMSE precoding” or “w/o precoding”13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that the FCP scheme undoubtedly achieves higher SE than that of the LSFD scheme since the 12In this paper, we choose one eigendirection capturing dominant channel power (randomly accounting for 80% ∼ 95% of the total channel power) and other eigendirections contain the remaining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 13The “w/o precoding” scenario denotes that identity precoding matrices Fk,u,{(1),(2)} = � pk N IN are implemented without optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 23 0 10 20 30 40 50 60 70 80 90 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='8 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' CDF of the sum SE over different processing schemes and precoding schemes with M = 20, K = 10, L = 2, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 2 3 4 5 6 0 20 40 60 80 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sum SE against the number antennas per AP L over different processing schemes and precoding schemes with M = 20, K = 10, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 2 3 4 5 6 1 2 3 4 5 6 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average rate against the number of antennas per UE N over different processing schemes and precoding schemes with M = 20, K = 10, and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 2 3 4 5 6 1 2 3 4 5 6 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average SE with I-WMMSE precoding schemes against the number of antennas per UE N over different τc with M = 20, K = 10, and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' FCP with MMSE combining is a competitive scheme in CF mMIMO [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, the proposed I-WMMSE schemes are efficient to improve the respective achievable sum SE performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='78%, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='54% and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='13% sum SE improvement for the FCP, the LSFD with MR combining and the LSFD with L-MMSE combining, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, for the LSFD scheme with MR combining, markers “◦” generated by analytical results overlap with the curves generated by simulations, respectively, validating our derived closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 4 shows the achievable sum SE as a function of the number of antennas per AP with two processing schemes investigated and different precoding schemes14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that, for the FCP or LSFD with (L-)MMSE combining, the performance gap between the “I-WMMSE” and “w/o precoding” becomes smaller with the increase of L, which implies that (L-)MMSE combining can use all antennas on each 14Note that the achievable sum SE investigated is the average sum SE value taken over many AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 24 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average SE against the number of APs M for the LSFD scheme with K = 10, L = 4, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 3 4 5 6 8 10 12 2 3 4 5 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average SE against the number of antennas per UE N for different channel models with M = 40, K = 8, and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 4 6 8 10 12 14 16 18 25 26 27 28 29 30 31 32 33 34 (a) FCP 2 4 6 8 10 12 14 16 16 18 20 22 24 26 28 (b) LSFD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Convergence examples of the I-WMMSE algorithm for the FCP and LSFD with M = 20, K = 10, L = 2, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' AP to suppress interference and achieve excellent SE performance even without any precoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, the performance gap between the “I-WMMSE” and “w/o precoding” for the LSFD with L-MMSE combining is 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='74% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='17% over L = 1 and L = 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Meanwhile, for the LSFD with MR combining, the performance gap between the “I-WMMSE” and “w/o precoding” becomes large with the increase of L, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='13% and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='48% for L = 1 and L = 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, for the LSFD scheme with MR combining, markers “✷” generated by analytical results overlap with the curves generated by simulations, respectively, validating our derived closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further show the advantage of the proposed I-WMMSE precoding schemes, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 shows the average rate15 as a function of the number of antennas per UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We find that the average rates for all schemes with I-WMMSE precoding schemes grow with N and the average rates for the case without UL precoding may also suffer the degradation with the increase of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The implementation of the I-WMMSE precoding 15Note that one main reason for the phenomenon that additional UE antennas may give rise to the SE degradation is that increasing N will increase the channel estimation overhead and reduce the pre-log factor “(τc − τp) /τc” in all SE expressions [26], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we investigate “the average rate” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5, ignoring the effect of “(τc − τp) /τc”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 25 50 100 150 200 250 300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='8 4 104 50 100 150 200 250 300 0 2 4 6 105 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Total number of complex scalars sent via the fronthaul per channel use for each realization of the AP/UE locations with M = 20, K = 10, L = 2, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' schemes undoubtedly makes UEs benefit from multiple antennas and achieve excellent rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, we observe that the I-WMMSE precoding schemes perform more efficiently with a larger number of UE antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, the average rate improvements achieved by the I-WMMSE precoding for the LSFD with L-MMSE combining are 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='91% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='43% for N = 6 and N = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, the average SE (with scaling factor (τc − τp)/τc) with I-WMMSE precoding implemented may also degrade with the increase of N as the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 in [1] since, with the increase of N, the prerequisite of “mutually orthogonal pilot matrices” still requires huge channel uses for the pilot transmission and the inter-user interference also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the design of non-orthogonal pilot matrices and per-antenna power control scheme are quite necessary, which are regarded as promising ways to reduce the cost of pilot transmission and further improve the SE performance [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 6 discusses the average SE with I-WMMSE precoding schemes against N over different τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 can be viewed as a special case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6 with the coherence block with infinite length τc = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We observe that the average SE with I-WMMSE precoding schemes increases with N over τc = 500 or ∞, which means the SE performance can benefit from having additional UE antennas when the coherence block resource is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 7 investigates the average SE as a function of M for the LSFD scheme over different precoding schemes16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For MR combining, markers “✷” generated by analytical results overlap with the curves generated by simulations, respectively, validating our derived closed-form expressions again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the I-WMMSE algorithm is more efficient to improve the SE performance for MR combining than that of L-MMSE combining for the scenario over large L and M, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='03% and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='21% SE improvement for L- MMSE combining and MR combining with M = 60, respectively, implying that the L-MMSE combining 16The “WMMSE precoding” denotes the precoding schemes generated by the I-WMMSE algorithm with only single iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 26 based on large L and M can achieve excellent SE performance even without any precoding scheme and the proposed I-WMMSE precoding scheme is handy to mitigate the weakness of MR combining17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 8 considers the average SE as a function of N over the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' and the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As observed, the proposed I-WMMSE precoding schemes are more efficient over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='89% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='77% average SE improvement can be achieved when N = 6 over the “Weichselberger” scenario for the LSFD scheme with MR combining and the FCP scheme, respectively, but only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='29% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='63% average SE improvement can be achieved for “I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Rayleigh channel”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5, we notice that the I-WMMSE precoding scheme for the FCP scheme is more efficient in the highly loaded system (the scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5) where the number of total AP-antennas is comparable with the number of total UE-antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 9 illustrates the convergence behavior of the I-WMMSE algorithms for the FCP scheme and the LSFD scheme with L-MMSE/MR combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note the convergence example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 (a) for the FCP is given by a particular channel realization and the convergence example for the LSFD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 (b) is given by a particular realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the algorithms investigated can be guaranteed to converge and are efficient to achieve excellent sum SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 (b) for the LSFD scheme with MR combining validates our derived closed-form expressions in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 10 investigates the total number of complex scalars sent via the fronthaul per channel use against τc for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As observed, total number of complex scalars per channel use for the FCP/LSFD scheme becomes smaller/larger, which can also be easily found from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the LSFD scheme requires more fronthaul signaling than the FCP scheme since APs under the LSFD scheme need to transmit all received data signals to the CPU, which requires a huge fronthaul load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, with the increase of τc, the gap between “I-WMMSE precoding” and “W/O precoding” becomes smaller for either the FCP scheme or the LSFD scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Considering the SE performance improvement of the I-WMMSE precoding, additional fronthaul loads can be acceptable, especially when the coherence resource is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Although the computational complexity of Algorithm 1 for the FCP scheme is much higher than that of Algorithm 2 for the LSFD scheme, the FCP scheme needs much less fronthaul signaling than that of the LSFD scheme and can achieve better SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So two processing schemes and their respective precoding schemes can be chosen based on different requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 17MR combining is a simple combining scheme but cannot efficiently suppress the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 27 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' CONCLUSION We consider a CF mMIMO system with both APs and UEs equipped with multiple antennas over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The FCP scheme and LSFD scheme are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further improve the sum SE performance, efficient UL precoding schemes based on iteratively WMMSE algorithms are investigated to maximize weighted sum SE for the two processing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that we compute achievable SE expressions and optimal precoding schemes in novel closed-form for the LSFD scheme with MR combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Numerical results show that the investigated I-WMMSE precoding schemes are efficient to achieve excellent sum SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, it can be seen that the proposed I-WMMSE precoding schemes are more efficient with a larger number of UE antennas, which means the I-WMMSE precoding schemes can achieve excellent performance even with a large number of UE antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The derived results undoubtedly provides vital insights for the practical implementation of multi- antenna UEs in CF mMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In future work, we will investigate the design of UL precoding scheme for the phase of pilot transmission and consider the practical implementation of the investigated I-WMMSE precoding schemes with capacity-constrained fronthaul network and dynamic cooperation clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the non-orthogonal pilot matrix design will also be considered to further improve the performance for the CF mMIMO system with multi-antenna UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Last but not least, the UL precoding performance over a more practical Rician fading channel with phase-shifts will also be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX A SOME USEFUL LEMMAS Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Let X ∈ CM×N be a random matrix and Y is a deterministic M × M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The (n, i)-th element of E � XHYX � is tr � Y · E � xixH n �� where xi and xn are the i-th and n-th column of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For matrices A ∈ CN1×N1, B ∈ CN1×N2, C ∈ CN2×N2, and D ∈ CN2×N1, we have (A + BCD)−1 = A−1−A−1B � DA−1B + C−1�−1 DA−1, which is a well-known matrix inversion lemma [36, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX B PROOF COROLLARY 5 Since the CPU is only aware of channel statistics, we need to treat E{Gkk}Fk,u as the true deterministic channel and rewrite ˜xk in (25) as ˜xk = E {Gkk} Fk,uxk+(GkkFk,u − E {Gkk} Fk,u) xk + K � l=1,l̸=k GklFl,uxl + n′ k � �� � v where v is a complex circular symmetric noise with an invertible covariance matrix Ξk = E{vvH|Θ} = 28 �K l=1 E{GklFk,uFH k,uGH kl} − E{Gkk}Fk,uFH k,uE{GH kk} + σ2Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Firstly, we whiten the noise as Ξ − 1 2 k ˆxk = Ξ − 1 2 k E {Gkk} Fk,uxk + ˜v, where ˜v ≜ Ξ − 1 2 k v becomes white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Next, we project Ξ − 1 2 k ˆxk in the direction of Ξ − 1 2 k E {Gkk} Fk,u to obtain an effective scalar channel as � Ξ − 1 2 k E {Gkk} Fk,u �H Ξ − 1 2 k ˆxk = (E {Gkk} Fk,u)H Ξ−1 k E {Gkk} Fk,uxk + (E {Gkk} Fk,u)H Ξ−1 k v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (44) Based on theories of optimal receivers [37], we derive optimal LSFD matrix Ak=Ξ−1 k E {Gkk}Fk,u as Ak = � K � l=1 E � GklFk,uFH k,uGH kl � − E {Gkk} Fk,uFH k,uE � GH kk � + σ2Sk �−1 E {Gkk} Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (45) Moreover, based on the the standard results of matrix derivation in [45], we can easily obtain the LSFD matrix minimizing the conditional MSE for UE k MSE(2) k = tr(E(2) k ) as Ak = � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk �−1 E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (46) We notice that the LSFD matrix in (45) is equivalent to the LSFD matrix in (46), except from having another scaling matrix IN − � CHB−1C + IN �−1 CHB−1C on the right side, which would not affect the value of (27), where B = �K l=1 E{GklFk,uFH k,uGH kl} + σ2Sk and C = E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the LSFD matrix in (46) cannot maximize the achievable SE but minimize the MSE for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX C PROOF OT THEOREM 2 In this part, we compute terms of (27) in closed-form for the LSFD scheme with MR combining Vmk = ˆHmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the first term Dk,(2) = AH k E{Gkk}Fk,u, we have E{Gkk} = [E{VH 1kH1k};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' E{VH MkHMk}] = [ZT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ZT Mk]T ≜ Zk, where Zmk = E{VH mkHmk} = E{ ˆHH mk ˆHmk} ∈ CN×N and the (n, n′)-th el- ement of Zmk can be denoted as [Zmk]nn′ = E{ˆhH mk,nˆhmk,n′} = tr(ˆRn′n mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we derive the closed- form for Dk,(2) as Dk,(2),c = AH k ZkFk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the second term Sk ∈ CMN×MN, we have Sk = diag(E{VH 1kV1k}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , E{VH MkVMk}) = diag(Z1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ZMk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For E{Gkl¯Fl,uGH kl}, we notice that the (m, m′)-submatrix of E{Gkl¯Fl,uGH kl} is E{VH mkHml¯Fl,uHH m′lVm′k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on [28], we compute E{VH mkHml¯Fl,uHH m′lVm′k} for four possible AP-UE combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “m ̸= m′, l /∈ Pk”, we have E{VH mkHml¯Fl,uHH m′lVm′k} = 0 for the independence between Vmk and Hml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “m ̸= m′, l ∈ Pk”, we have E{VH mkHml¯Fl,uHH m′lVm′k} = E{VH mkHml}¯Fl,uE{HH m′lVm′k} = Λmkl¯Fl,uΛm′lk, where the (n, n′)-th element of N ×N-dimension complex matrices Λmkl ≜ E{VH mkHml}, 29 Λm′lk ≜ E{HH m′lVm′k} are [Λmkl]nn′ = E{ˆhH mk,nˆhml,n′} = tr(Ξn′n mkl) and [Λm′lk]nn′ = E{ˆhH m′l,nˆhmk,n′} = tr(Ξn′n m′lk) with Ξmkl ≜ E{ˆhmlˆhH mk} = τpRml˜FH l,pΨ−1 mk˜Fk,pRmk, Ξm′lk ≜ E{ˆhm′kˆhH m′l} = τpRm′k˜FH k,pΨ−1 m′k˜Fl,pRm′l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “m = m′, l /∈ Pk”, we define Γ(1) mkl ≜ E{VH mkHml¯Fl,uHH mlVmk} ∈ CN×N with the (n, n′)-th element [Γ(1) mkl]nn′ = �N i=1 �N i′=1 [¯Fl,u]i′iE{ˆhH mk,nhml,i′hH ml,iˆhmk,n′} being [Γ(1) mkl]nn′ = N � i=1 N � i′=1 [¯Fl,u]i′itr(E � hml,i′hH ml,i � E{ˆhmk,n′ˆhH mk,n}) = N � i=1 N � i′=1 [¯Fl,u]i′itr(Ri′i ml ˆRn′n mk) (47) since ˆHmk and Hml are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, for “m = m′, l ∈ Pk”, ˆHmk and Hml are no longer inde- pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We define Γ(2) mkl ≜ E{VH mkHml¯Fl,uHH mlVmk} ∈ CN×N whose (n, n′)-th element is [Γ(2) mkl]nn′ = �N i=1 �N i′=1 [¯Fl,u]i′iE{ˆhH mk,nhml,i′hH ml,iˆhmk,n′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We follow the similar method in [28] and derive [Γ(2) kl,m]nn′ = �N i=1 �N i′=1 [¯Fl,u]i′itr(Ri′i mlPn′n mkl,(1))+τ 2 p �N q1=1 �N q2=1 [¯Fl,u]i′i[tr(˜Pq1n mkl,(2) ˜Ri′q2 ml ˜Rq2i ml ˜Pn′q1 mkl,(2))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='+ τ 2 p �N q1=1 �N q2=1 [¯Fl,u]i′itr(˜Pq1n mkl,(2) ˜Ri′q2 ml )tr(˜Pn′q2 mkl,(2) ˜Rq2i ml), where Pmkl,(1) ≜ τpSmk(Ψmk−τp˜Fl,pRml˜FH l,p)SH mk, Smk ≜ Rmk˜FH k,pΨ−1 mk and Pmkl,(2) ≜ Smk˜Fl,pRml˜FH l,pSH mk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, ˜Rni ml and ˜Pni mkl,(2) denote (n, i)-submatrix of R 1 2 ml and P 1 2 mkl,(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In summary, combining all the cases, we have E{Gkl¯Fl,uGH kl} = Tkl,(1) + Tkl,(2) if l ∈ Pk and Tkl,(1) otherwise, where Tkl,(1) ≜ diag(Γ(1) kl,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Γ(1) kl,M) ∈ CMN×MN and Tmm′ kl,(2) = Γ(2) kl,m − Γ(1) kl,m if m = m′ and Λmkl¯Fl,uΛm′lk otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Plugging the derived results into (28) and (31), we can easily compute the optimal LSFD coefficient matrix and MSE matrix in closed-form as (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we have finished the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For more details on the derived expression, please refer to [28, Appendix D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX D PROOF OF THEOREM 1 When other optimization variables are fixed, we derive the partial derivative of (17) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t F(1) k,u as ∂f � F1,u,(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , FK,u,(1) � ∂Fk,u,(1) = K � l=1 µl,(1) � ˆHH k VlWl,(1)VH l ˆHk + E � ˜HH k VlWl,(1)VH l ˜Hk ��� V, W �� + λk,(1)IN − µk,(1) ˆHH k VH k Wk,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (48) By applying the first-order optimality condition and setting ∂f(F1,u,(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=',FK,u,(1)) ∂Fk,u,(1) = 0, we can easily obtain the optimal precoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, λk,(1) and Fk,u,(1) should also satisfy KKT condition as (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for ¯Ckl ≜ E{ ˜HH k VlWl,(1)VH l ˜Hk|V, W} ∈ CN×N, by applying Lemma 1, the (i, n)-th element of ¯Ckl is tr( ¯VlE{˜hk,n˜hH k,i}) where ¯Vl ≜ VlWl,(1)VH l and ˜hk,n = [˜hT 1k,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ˜hT Mk,n]T ∈ CML is the n-th column of ˜Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, we derive Ck,ni ≜ E{˜hk,n˜hH k,i} = diag (Cni 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Cni Mk) ∈ CML×ML since ˜hmk,n 30 and ˜hm′k,n for m ̸= m′ are independent and both have zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So Ck,ni is a block-diagonal matrix with the square matrices Cni 1k = E{˜h1k,n˜hH 1k,i}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Cni Mk = E{˜hMk,n˜hH Mk,i} on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX E PROOF OF (15) For the LSFD scheme, the conditional MSE matrix for UE k can be written as (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on [28, Appendix C], we prove that (28) can also minimize MSEk,(2) = tr � Ek,(2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' With (28) implemented, Ek,(2) is given by (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, by applying Lemma 2, we have � Eopt k,(2) �−1 = IN + FH k,u,(2)E � GH kk � � K � l=1 E � Gkl¯Fl,u,(2)GH kl � − E {Gkk} ¯Fk,u,(2)E � GH kk � + σ2Sk �−1 × E {Gkk} Fk,u,(2), where A ≜ IN, B ≜ −FH k,u,(2)E{GH kk}, C ≜ (�K l=1 E{Gkl¯Fl,u,(2)GH kl}+σ2Sk)−1 and D ≜ E{Gkk}Fk,u,(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We show the equivalence between SEopt k,(2) and log2 |(Eopt k,(2))−1| without a factor (1 − τp/τc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX F PROOF OF THEOREM 4 When MR combining Vmk = ˆHmk and the optimal LSFD scheme applied, we can easily compute E{GH kk}, Aopt k , and Eopt k,(2) in closed-form as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, by applying Lemma 1, the (i, n)-th entry of ¯Tlk = E{GH lkAlE−1 l,(2)AH l Glk} ∈ CN×N can be denoted as tr(¯AlE{glk,ngH lk,i}), where ¯Al ≜ AlE−1 l,(2)AH l and glk,n ∈ CMN is the n-th column of Glk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the (m − 1) N + p-th element of glk,n is ˆhH ml,phmk,n so the [(m − 1) N + p, (m′ − 1) N + p′]-th (or [o, j]-th briefly) entry of ¯Glk,ni ≜ E{glk,ngH lk,i} ∈ CMN×MN can be denoted as E{ˆhH ml,phmk,nhH m′k,iˆhm′l,p′}, which can be computed for four AP-UE combinations as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l /∈ Pk, m ̸= m′”, we have E{ˆhH ml,phmk,nhH m′k,iˆhm′l,p′} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l ∈ Pk, m ̸= m′”, we have E{ˆhH ml,phmk,nhH mk,iˆhml,p′} = tr(Rni mk ˆRp′p ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l /∈ Pk, m = m′”, we have E{ˆhH ml,phmk,nhH m′k,iˆhm′l,p′} = E{ˆhH ml,phmk,n}E{hH m′k,iˆhm′l,p′} = tr(Ξnp mlk)tr(Ξp′i m′kl), where Ξmlk = τpRmk˜FH k,pΨ−1 mk˜Fl,pRml and Ξm′kl = τpRm′l˜FH l,pΨ−1 m′l˜Fk,pRm′k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l ∈ Pk, m = m′”, we obtain E{ˆhH ml,phmk,nhH mk,iˆhml,p′} = tr(Rni mkPp′p mkl,(1))+ τ 2 p �N q1=1 �N q2=1 tr(˜Pq1p mlk,(2) ˜Rnq2 mk ˜Rq2i mk ˜Pp′q1 mlk,(2)) + τ 2 p �N q1=1 �N q2=1 tr(˜Pq1n mlk,(2) ˜Rnq1 mk)tr(˜Pp′q2 mlk,(2) ˜Rq2i mk), where Sml = Rml˜FH l,pΨ−1 ml, Pmlk,(1) = τpSml(Ψml−τp˜Fk,pRmk˜FH k,p)SH ml and Pmlk,(2) = Sml˜Fk,pRmk˜FH k,pSH ml with ˜Rni mk and ˜Pni mkl,(2) being (n, i)-submatrix of R 1 2 mk and P 1 2 mkl,(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We can compute E{glk,ngH lk,i}oj in closed-form as (42) and Fopt k,u,(2) in closed-form as (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 31 REFERENCES [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah, “Iteratively weighted MMSE uplink precoding for cell-free massive MIMO,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' IEEE ICC, May 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ashikhmin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Larsson, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Marzetta, “Cell-free massive MIMO versus small cells,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1834–1850, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Matthaiou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Yang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Love, “Prospective multiple antenna technologies for beyond 5G,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Areas Commun, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1637–1660, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “A survey on user-centric cell-free massive MIMO systems,” Digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sanguinetti, “Making cell-free massive MIMO competitive with MMSE processing and centralized implementation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 77–90, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Elhoushy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ibrahim, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hamouda, “Cell-free massive MIMO: A survey,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Surveys Tuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 492–523, 1st Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hanzo, “Cell-free massive MIMO: A new next-generation paradigm,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 99 878–99 888, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Duong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Matthaiou, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Larsson, “On the total energy efficiency of cell-free massive MIMO,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Green Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Netw, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 25–39, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sanguinetti, “Scalable cell-free massive MIMO systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4247–4261, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “Structured massive access for scalable cell-free massive MIMO systems,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Areas Commun, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1086–1100, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Papazafeiropoulos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Kourtessis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chatzinotas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Senior, “Scalable cell-free massive MIMO systems: Impact of hardware impairments,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9701–9715, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Li, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “Cell-free massive MIMO-OFDM for high-speed train communications,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Areas Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2823–2839, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Attarifar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Abbasfar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lozano, “Subset MMSE receivers for cell-free networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4183–4194, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Nayebi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ashikhmin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Marzetta, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Rao, “Performance of cell-free massive MIMO systems with MMSE and LSFD receivers,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Asilomar Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Signals, Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 203–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [15] ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ¨Ozdogan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, “Performance of cell-free massive MIMO with Rician fading and phase shifts,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5299–5315, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [16] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “Uplink performance of cell-free massive MIMO over spatially correlated Rician fading channels,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1348–1352, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “Local partial zero-forcing combining for cell-free massive MIMO systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8459–8473, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Interdonato, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Frenger, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Larsson, “Scalability aspects of cell-free massive MIMO,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' IEEE ICC, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [19] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Shaik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Larsson, “MMSE-optimal sequential processing for cell-free massive MIMO with radio stripes,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7775–7789, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 32 [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Masoumi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Emadi, “Performance analysis of cell-free massive MIMO system with limited fronthaul capacity and hardware impairments,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1038–1053, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bashar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Xiao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tafazolli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Cumanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Burr, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, “Limited-fronthaul cell-free massive MIMO with local MMSE receiver under Rician fading and phase shifts,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1934–1938, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Van Chien, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chatzinotas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Di Renzo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ottersten, “Reconfigurable intelligent surface-assisted cell-free massive MIMO systems over spatially-correlated channels,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5106–5128, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Kwan Ng, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “Wireless energy transfer in RIS-aided cell-free massive MIMO systems: Opportunities and challenges,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 26–32, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Mai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Duong, “Cell-free massive MIMO systems with multi-antenna users,” in IEEE GlobalSIP, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 828–832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Buzzi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' D’Andrea, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zappone, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' D’Elia, “User-centric 5G cellular networks: Resource allocation and comparison with the cell-free massive MIMO approach,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1250–1264, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Mai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Duong, “Downlink spectral efficiency of cell-free massive MIMO systems with multi-antenna users,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4803–4815, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Yang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhu, “Sum-SE for multigroup multicast cell-free massive MIMO with multi-antenna users and low-resolution DACs,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1702–1706, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [28] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Yuen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah, “Uplink performance of cell-free massive MIMO with multi-antenna users over jointly-correlated Rayleigh fading channels,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7391–7406, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [29] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Weichselberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Herdin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ozcelik, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bonek, “A stochastic MIMO channel model with joint correlation of both link ends,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 90–100, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [30] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Shi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Razaviyayn, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Luo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' He, “An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO interfering broadcast channel,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4331–4340, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Shin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moon, “Weighted-sum-rate-maximizing linear transceiver filters for the K-user MIMO interference channel,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2776–2783, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Christensen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Agarwal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' De Carvalho, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Cioffi, “Weighted sum-rate maximization using weighted MMSE for MIMO- BC beamforming design,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4792–4799, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [33] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhou, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hanzo, “Optimal massive-MIMO-Aided clustered base-station coordination,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2699–2712, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tataria, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Matthaiou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Jin, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Larsson, “On the performance of cell-free massive MIMO in Ricean fading,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Asilomar Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Signals, Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 980–984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [35] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ottersten, “A framework for training-based estimation in arbitrarily correlated Rician MIMO channels with Rician disturbance,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1807–1820, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bj¨ornson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hoydis, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sanguinetti, “Massive MIMO networks: Spectral, energy, and hardware efficiency,” Foundations and Trends® in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3-4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 154–655, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tse and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Viswanath, Fundamentals of wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Cambridge university press, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Bashar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Cumanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Burr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Xiao, “Max-min rate of cell-free massive MIMO uplink with optimal uniform quantization,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6796–6815, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [39] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hua, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Chen, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Guan, “Joint beamformer design for wireless fronthaul and access links in C-RANs,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2869–2881, May 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 33 [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hoydis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ten Brink, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Areas Commun, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 160–171, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wagner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Couillet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Slock, “Large system analysis of linear precoding in correlated MISO broadcast channels under limited feedback,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4509–4537, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Jin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai, “Improving sum-rate of cell-free massive MIMO with expanded compute-and- forward,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 202–215, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tulino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lozano, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Verdu, “Impact of antenna correlation on the capacity of multiantenna channels,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2491–2509, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [44] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Gu, “A unified framework of non-orthogonal pilot design for multi-cell massive MIMO systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7623–7633, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hjørungnes, Complex-valued matrix derivatives: with applications in signal processing and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Cambridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} diff --git a/EdE1T4oBgHgl3EQfWgTb/vector_store/index.faiss b/EdE1T4oBgHgl3EQfWgTb/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a89844a32de5aaa92f6a755109659532fb852d32 --- /dev/null +++ b/EdE1T4oBgHgl3EQfWgTb/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b71f34899cdd9cc0ce9e23b7bafbcc82aaba807734f7f3105cf6845690f8b482 +size 7667757 diff --git a/EdE5T4oBgHgl3EQfUw9y/content/2301.05546v1.pdf b/EdE5T4oBgHgl3EQfUw9y/content/2301.05546v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..24fbcd76a1db8432610a84924d56e10f447c26fc --- /dev/null +++ b/EdE5T4oBgHgl3EQfUw9y/content/2301.05546v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8059ee327055e8029001a9f59d3bacd32eddd339177eb8d41d36623b6d3a7e08 +size 292322 diff --git a/EdE5T4oBgHgl3EQfUw9y/vector_store/index.faiss b/EdE5T4oBgHgl3EQfUw9y/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3c52100d0f8c4e7f91a435883e6373a28a2c8a63 --- /dev/null +++ b/EdE5T4oBgHgl3EQfUw9y/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a486f3dd10015351b4ecee684b4df7503240db8bccdc92c22bd485e5150246c4 +size 2555949 diff --git a/EdE5T4oBgHgl3EQfUw9y/vector_store/index.pkl b/EdE5T4oBgHgl3EQfUw9y/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..be5e3605607d6dca50dbb82b44219fde026b8413 --- /dev/null +++ b/EdE5T4oBgHgl3EQfUw9y/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:51003110413668df87ae29b5607c6ec3be742f2726a2ca838a995e1db12ed9a0 +size 97360 diff --git a/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf b/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6ba617aa27e15356a3feff19e507e1440deff958 --- /dev/null +++ b/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6763ab96c00b77d85de9cec775c436ac64cedbbd17896770efc263513b90bb8a +size 460145 diff --git a/F9E2T4oBgHgl3EQf-Qn4/vector_store/index.pkl b/F9E2T4oBgHgl3EQf-Qn4/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b2fd088b303e024d2ba6193b02a3f319ba90d7bf --- /dev/null +++ b/F9E2T4oBgHgl3EQf-Qn4/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bfaf1a03d212e0621560769092b0966f697ca53845d336cf04fac17b327eede6 +size 260672 diff --git a/FdA0T4oBgHgl3EQfBP_A/content/tmp_files/2301.01974v1.pdf.txt b/FdA0T4oBgHgl3EQfBP_A/content/tmp_files/2301.01974v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb6a15cb9a6189fd6e62204eb30a3899f3d25dff --- /dev/null +++ b/FdA0T4oBgHgl3EQfBP_A/content/tmp_files/2301.01974v1.pdf.txt @@ -0,0 +1,703 @@ +arXiv:2301.01974v1 [math.FA] 5 Jan 2023 +THE GENERALISED (UNIFORM) MAZUR +INTERSECTION PROPERTY +PRADIPTA BANDYOPADHYAY AND DEEPAK GOTHWAL +Abstract. Given a family C of closed bounded convex sets in a Banach +space X, we say that X has the C-MIP if every C ∈ C is the intersection +of the closed balls containing it. In this paper, we introduce a stronger +version of the C-MIP and show that it is a more satisfactory general- +isation of the MIP inasmuch as one can obtain complete analogues of +various characterisations of the MIP. +We also introduce uniform versions of the (strong) C-MIP and charac- +terise them analogously. Even in this case, the strong C-UMIP appears +to have richer characterisations than the C-UMIP. +1. Introduction +S. Mazur [10] initiated the study of the following intersection property of +balls in normed spaces, now called the Mazur Intersection Property (MIP): +Every closed bounded convex set is the intersection of closed +balls containing it. +J. R. Giles, D. A. Gregory and B. Sims [7] obtained various characteri- +sations of the MIP, most well-known criterion stating that the w*-denting +points of B(X∗) are norm dense in S(X∗). Chen and Lin [4] introduced the +notion of w*-semidenting points to characterise the MIP. The paper [9] is a +good survey of the MIP and related works. +Whitfield and Zizler in [16] introduced a uniform version of the MIP +(UMIP) and obtained characterisations similar to [7] but the characterisa- +tion related to the w*-denting points was missing. Recently, such a charac- +terisation in terms of “uniform w*-semidenting points” has been obtained +by the authors in [3]. +Soon after [7], there appeared several papers dealing with generalisations +of the MIP, basically studying when members of a given family of closed +bounded convex sets are intersection of balls. For example, +2010 Mathematics Subject Classification. Primary 46B20 +To appear in JMAA. +Key words and phrases. Compatible class, (strong) C-MIP, (strong) C-UMIP. +1 + +2 +BANDYOPADHYAY AND GOTHWAL +(a) K = {all compact convex sets in X} [15, 12] +(b) W = {all weakly compact convex sets in X} [17] +(c) F = {all compact convex sets in X with finite affine dimension} [13] +Let C be a family of bounded subsets of X. Let τC be the topology on +X∗ of uniform convergence on the sets of C. Let us say that X has the +C-MIP if every closed convex set C ∈ C is the intersection of the closed balls +containing it. +In [12, 13, 15], the authors characterised the C-MIP in cases (a) and (c) +above in terms of the density in X∗ of the cone of the extreme points of BX∗ +in the topology τC. +The first author studied the C-MIP for a general family C of bounded +sets in [2] and obtained some necessary and some sufficient conditions that +recaptured the results of [12, 13, 15]. Chen and Lin [5] also discussed the +C-MIP. Both the papers observed that if the cone of the “C-denting points” +of B(X∗), as defined in [5], is τC-dense in X∗, then X has the C-MIP. Both +papers proved the converse under additional hypothesis. +In [7], there are several alternative characterisations of the MIP [7, The- +orem 2.1 (ii), (iii) and (iv)]. Analogous characterisations of the UMIP were +also obtained in [15, 3]. [2] is the only attempt so far to obtain such ana- +logues for the C-MIP (See Theorem 2.6 below). +Recently, Cheng and Dong [6] characterised the C-MIP in terms of the +“C-semidenting” points. However, their definition does not appear to be +a natural generalisation of either the w*-semidenting points of [4] or the +C-denting points of [5]. +Vanderwerff [14] introduced the following stronger notion for the families +K and W above, which we will call strong C-MIP: +For every convex C ∈ C and β ≥ 0, C + βB(X) is the inter- +section of closed balls containing it. +For C = K, he showed that it is equivalent to the extreme points of B(X∗) +being w*-dense in S(X∗). +In this paper, we show that the strong C-MIP is actually a more satisfac- +tory generalisation of the MIP for general families. It can be characterised +in terms of a more natural notion of the (strong) C-semidenting points. +Moreover, one can also obtain complete analogue of [7, Theorem 2.1]. +We also introduce the uniform version of the C-MIP and its stronger form +and characterise them analogously. Even in this case, the strong C-UMIP +appears have richer characterisations than the C-UMIP. + +THE GENERALISED MIP & UMIP +3 +In the last section, we summarise the interrelations between various no- +tions discussed in the paper with examples and counterexamples. +2. Preliminaries +Throughout this article, X is a real Banach space. +For x ∈ X and r > 0, we denote by B(x, r) the open ball {y ∈ X : +∥x − y∥ < r} and by B[x, r] the closed ball {y ∈ X : ∥x − y∥ ≤ r}. We +denote by B(X) the closed unit ball {x ∈ X : ∥x∥ ≤ 1} and by S(X) the +unit sphere {x ∈ X : ∥x∥ = 1}. +For a bounded subset A ⊆ X, denote by co(A) (resp. aco(A)) the convex +(resp. absolutely convex) hull of A. Let ∥f∥A := sup{|f(x)| : x ∈ A} for +f ∈ X∗ and diamA(B) = sup{∥f −g∥A : f, g ∈ B} for any non-empty subset +B of X∗. For f ∈ X∗, ε > 0 and A ∈ C, BA(f, ε) := {g ∈ X∗ : ∥f −g∥A < ε}. +For a bounded set C ⊆ X and x ∈ X, let d(x, C) = inf{∥x − z∥ : z ∈ C} +denote the distance function. +For A ⊆ X, the cone generated by A is cone(A) = {λa : λ ≥ 0, a ∈ A}. +Definition 2.1. We say that a class C of bounded subsets of X is a com- +patible class if +(I) C ∈ C, α ∈ R and x ∈ X =⇒ αC + x ∈ C, C ∪ {x} ∈ C; +(II) C ∈ C implies the closed absolutely convex hull of C ∈ C. +(III) C ∈ C and A ⊆ C implies A ∈ C. +(IV) C1, C2 ∈ C implies C1 ∪ C2 ∈ C. +Our definition is slightly different from the ones in [2, 5, 6]. +Definition 2.2. For a compatible class C, X is said to have the C-MIP if +every closed convex C ∈ C is the intersection of closed balls containing it. +Notice that the families K, W and F above are not exactly compati- +ble classes with this definition. One has to consider such sets and subsets +thereof. However, the property C-MIP remains the same. And we should +add the family B of all bounded sets, corresponding to the MIP itself, in our +list of examples of a compatible class. +Throughout the article, C will denote a compatible class of sets. We will +denote by τC, or, simply τ when there is no ambiguity, the topology on X∗ +of uniform convergence on the sets of C, or, in other words, the topology +on X∗ generated by the family of seminorms {∥ · ∥C : C ∈ C}. Since C is a +compatible class, the family {BC(0, ε) : C ∈ C, ε > 0} forms a local base for +τC at 0. For the classes C = F, K, W and B, the topology τC coincides with + +4 +BANDYOPADHYAY AND GOTHWAL +the w*-, bw*-, Mackey and norm topologies, respectively. We note that the +topologies τF (w*-) and τK (bw*-) coincide on bounded sets. +Definition 2.3. +(a) For x ∈ S(X), we denote by D(x) the set {f ∈ +S(X∗) : f(x) = 1}. Any selection of D is called a support mapping. +(b) A w*-slice of B(X∗) determined by x ∈ S(X) is a set of the form +S(B(X∗), x, δ) := {f ∈ B(X∗) : f(x) > 1 − δ} +for some 0 < δ < 1. +(c) For ε, δ > 0, x ∈ S(X) and C ∈ C denote by +d1(C, x, δ) += +sup +0<λ<δ, y∈C +∥x + λy∥ + ∥x − λy∥ − 2 +λ +d2(C, x, δ) += +diamC(S(B(X∗), x, δ)) +d3(C, x, δ) += +diamC(D(S(X) ∩ B(x, δ))) +(d) [2] For ε, δ > 0 and C ∈ C +Mε,δ,C(X) = {x ∈ S(X) : +sup +0<∥y∥<δ,y∈C +∥x + y∥ + ∥x − y∥ − 2 +∥y∥ +< ε}. +In other words, Mε,δ,C(X) = {x ∈ S(X) : d1(C, x, δ) < ε}. Define +Mε,C(X) = +� +δ>0 +Mε,δ,C(X). +(e) Define H = �{D(Mε,C(X)) +τ : C ∈ C, ε > 0}. +Following lemma is an easy adaptation of [3, Lemma 2.10]. Details can +be found in [1, Lemma 2.1]: +Lemma 2.4. For any α, δ > 0, we have, +(i) d2(C, x, α) ≤ d1(C, x, δ) + 2α +δ . +(ii) d3(C, x, δ) ≤ d2(C, x, δ). +(iii) d1(C, x, δ) ≤ d3(C, x, 2δ). +Definition 2.5. +(a) We say that f ∈ S(X∗) is a w*-denting point of +B(X∗) if for every ε > 0, there exists a w*-slice S of B(X∗) such +that f ∈ S and diam(S) < ε. +(b) [4] We say that f ∈ S(X∗) is a w*-semidenting point of B(X∗) +if for every ε > 0, there exists a w*-slice S of B(X∗) such that +S ⊆ B(f, ε). + +THE GENERALISED MIP & UMIP +5 +(c) [5] We say that f ∈ S(X∗) is a C-denting point of B(X∗) if for +each A ∈ C and ε > 0, there exists a w*-slice S of B(X∗) such that +f ∈ S and diamA(S) < ε. +(d) [6] We say that f ∈ S(X∗) is a C-semidenting point of B(X∗) if +for each A ∈ C and ε > 0, there exists a w*-slice S of B(X∗) such +that S ⊆ cone(BA(f, ε)). +Known results on C-MIP [2, 5, 6] are summarised in our notation and +terminology below. +Theorem 2.6. [2, Theorem 1, Corollary 1], [5, Theorem 1.10], [6, Theorem +2.3] Suppose C is a compatible class of bounded sets. Consider the following +statements : +(a) The cone generated by C-denting points of B(X∗) is τ-dense +in X∗. +(b) The cone generated by H is τ-dense in X∗. +(c) If C1, C2 ∈ C are closed convex such that there exists f ∈ X∗ with +sup f(C1) < inf f(C2), then there exist disjoint closed balls B1, B2 +such that Ci ⊆ Bi, i = 1, 2. +(d) X has the C-MIP. +(e) Every f ∈ S(X∗) is a C-semidenting point of B(X∗). +(f) For every norm dense subset A ⊆ S(X) and every support map- +ping φ, the cone generated by φ(A) is τ-dense in X∗. +Then (a) =⇒ (b) =⇒ (c) =⇒ (d) ⇐⇒ (e) =⇒ (f). Moreover, if +A = {x ∈ S(X) : D(x) contains a C-denting point of B(X∗)} +is norm dense in S(X), then all the statements are equivalent. +Remark 2.7. In [5, Theorem 1.10], it is shown that (a) and (d) are equiv- +alent under a weaker assumption that every w*-slice of B(X∗) contains a +C-denting point. +3. Strong C-MIP +Definition 3.1. A Banach space X is said to have the strong C-MIP if for +every closed convex C ∈ C and β ≥ 0, C + βB(X) is the intersection of +closed balls containing it. +Vanderwerff [14] called this K-IP and W-IP for C = K and W, respectively. +Clearly, for C = B, C-MIP and strong C-MIP coincide. See Section 5 for some +examples and counterexamples. + +6 +BANDYOPADHYAY AND GOTHWAL +Definition 3.2. The duality map on X is said to be norm-τ quasicontinuous +if for every f ∈ S(X∗), ε > 0 and C ∈ C, there exist δ > 0 and x ∈ S(X) +such that D(SX ∩ B(x, δ)) ⊆ BC(f, ε). +In our opinion, the following is a more natural generalisation of both +the w*-semidenting points of [4] and the C-denting points of [5] than the +C-semidenting points defined in [6]. +Definition 3.3. We say that f ∈ S(X∗) is a strong C-semidenting point of +B(X∗) if for each A ∈ C and ε > 0, there exists a w*-slice S of B(X∗) such +that S ⊆ BA(f, ε). +We begin by observing that the strong C-semidenting points can be char- +acterised completely analogous to [7, Lemma 2.2]. +Lemma 3.4. For f ∈ S(X∗), the following are equivalent: +(a) f ∈ H, that is, f ∈ � +ε>0 D(Mε,C(X)) +τ for every C ∈ C. +(b) f is a strong C-semidenting point. +(c) For every β +≥ +0 and closed convex C +∈ +C +such that +inf f(C + βB(X)) > 0, there exists a closed ball B[x0, r0] in X con- +taining C + βB(X) with 0 /∈ B[x0, r0]. +(d) For every ε > 0 and C ∈ C, there exist δ > 0 and x ∈ S(X) such +that D(SX ∩ B(x, δ)) ⊆ BC(f, ε). +Proof. (a) +=⇒ +(b): For C ∈ C, let f ∈ � +ε>0 D(Mε,C(X)) +τ. Let ε > 0 +be given. We have f ∈ D(Mε/4,C(X)) +τ. So, there exist δ > 0 and x ∈ +Mε/4,δ,C(X) and g ∈ D(x) such that ∥f − g∥C < ε/4. +Since x ∈ Mε/4,δ,C(X), we have d1(C, x, δ) < ε/4. By Lemma 2.4(i), for +α = εδ/8, +d2(C, x, α) ≤ d1(C, x, δ) + 2α +δ < ε/4 + ε/4 = ε/2. +Hence, diamC(S(B(X∗), x, α) < ε/2. +Since g ∈ S(B(X∗), x, δ), S(B(X∗), x, δ) ⊆ BC(f, ε). +(b) =⇒ (c): Let β ≥ 0 and C ∈ C such that inf f(C + βB(X)) > 0. +Let inf f(C + βB(X)) = α > 0. Now, there exist δ > 0 and x ∈ S(X) +such that +S := S(B(X∗), x, δ) ⊆ BC(f, α). +Choose g ∈ S and c ∈ C. We have, +g(c) = f(c) − ∥f − g∥C > α + β − α = β. + +THE GENERALISED MIP & UMIP +7 +Hence, for y ∈ C + βB(X), g(y) > 0. +So, by [3, Theorem 2.6], there is a closed ball B[x0, r0] containing C + +βB(X) such that 0 /∈ B[x0, r0]. +(c) =⇒ (b): Let C ∈ C and ε > 0 be given. +Let z0 ∈ X be such that ∥z0∥ = 1 + ε/2 and f(z0) > 1 + ε/4. +Let +D = aco(C ∪ {z0}). We have D ∈ C. +Consider A := {x ∈ D : f(x) ≥ ε/4}. We have A ∈ C. +For β = +ε/4 +1+ε/4, f(A + βB(X)) ≥ ε/4 − β > 0. So, there exists a closed +ball B[x0, r0] containing A + βB(X) such that 0 /∈ B[x0, r0]. +So, for r1 = r0 − β and x1 = x0, A ⊆ B[x1, r1] and d(0, B[x1, r1]) > β. +Consider S = {g ∈ B(X∗) : g(x1/∥x1∥) > (r1 + β)/∥x1∥}. +For g ∈ S, g(x1/∥x1∥) > (r1 + β)/∥x1∥. So, g(x1) > r1 + β which implies +inf g(B[x1, r1]) ≥ β. Thus, g(x) > 0 for all x ∈ A. Therefore, applying +Phelps’ Lemma [3, Theorem 2.5] to the semi-norm ∥ · ∥D, +����f − ∥f∥D +∥g∥D +g +���� +D +< ε/2. +Consider y0 = ε +4 +z0 +f(z0). Since f(z0) > ε/4, D is absolutely convex, and +f(y0) = ε/4, y0 ∈ A, and hence, g(y0) > β. +Therefore, g(z0) = f(z0)g(y0) +ε/4 +> (1 + ε/4)β +ε/4 += 1 and so, ∥g∥D ≥ 1. +Since f, g ∈ B(X∗) and D ⊆ (1 + ε/2)B(X), +1 ≤ ∥f∥D, ∥g∥D ≤ 1 + ε/2. +Hence, +��∥f∥D − ∥g∥D +�� < ε/2. +Finally, +∥f − g∥C ≤ ∥f − g∥D ≤ +����f − ∥f∥D +∥g∥D +g +���� +D ++ +��∥f∥D − ∥g∥D +�� < ε. +(b) =⇒ (d): Let f ∈ S(X∗) be a strong C-semidenting point. Let ε > 0 +and C ∈ C be given. There exist δ > 0 and x ∈ S(X) such that +S(B(X∗), x, δ) ⊆ BC(f, ε/4). +So, d2(C, x, δ) = diamC(S(B(X∗), x, δ)) < ε/2 and D(x) ⊆ BC(f, ε/4). +Now, by Lemma 2.4, d3(C, x, δ) = diamC(D(S(X)∩B(x, δ))) < ε/2. Hence, +D(S(X) ∩ B(x, δ)) ⊆ BC(f, ε). + +8 +BANDYOPADHYAY AND GOTHWAL +(d) +=⇒ +(a): To show f ∈ D(Mε,C(X)) +τ for every C ∈ C and ε > 0, +let 0 < η < ε and C1 ∈ C. Enough to show that there exist x ∈ Mε,C and +g ∈ D(x) such that ∥f − g∥C1 < η. +Let C0 = C ∪ C1 ∈ C. By (d), there exist δ > 0 and x ∈ S(X) such that +D(S(X) ∩ B(x, δ)) ⊆ BC0(f, η/2). +So, d3(C0, x, δ) +≤ +η and D(x) +⊆ +BC0(f, η/2). +By Lemma 2.4, +d1(C0, x, δ/2) ≤ d3(C0, x, δ) ≤ η. Hence, x ∈ Mη,δ/2,C0 ⊆ Mε,δ/2,C ⊆ Mε,C +and ∥f − g∥C1 < η for any g ∈ D(x). +□ +Here is our main theorem characterising the strong C-MIP. Again, we +obtain a complete analogue of [7, Theorem 2.1]. +Theorem 3.5. For a Banach space X, the following are equivalent: +(a) X has the strong C-MIP. +(b) Every f ∈ S(X∗) is a strong C-semidenting point. +(c) For every ε > 0 and C ∈ C, D(Mε,C(X)) is τ-dense in S(X∗). +(d) The duality map on X is norm-τ quasicontinuous. +(e) Every support mapping in X maps norm dense sets in S(X) to +τ-dense sets in S(X∗). +Proof. Equivalence of (a), (b), (c) and (d) follows from Lemma 3.4. +(d) =⇒ (e): Let A ⊆ S(X) be norm dense in S(X). Let φ : S(X) → +S(X∗) be a support mapping. Let f ∈ S(X∗), C ∈ C and ε > 0. It is enough +to find x0 ∈ A such that ∥f − φ(x0)∥C < ε. +By (d), there exist δ > 0 and x ∈ S(X) such that D(SX ∩ B(x, δ)) ⊆ +BC(f, ε). Since, A is dense in S(X), there exists x0 ∈ A such that ∥x0−x∥ < +δ. If follows that D(x0) ⊆ BC(f, ε). Therefore, for any support mapping φ, +∥f − φ(x0)∥C < ε. +(e) =⇒ (d): Suppose the duality map is not norm-τ quasicontinuous. +Then there exist f ∈ S(X∗), ε > 0 and C ∈ C, such that for any x ∈ +S(X) and n ∈ N, there exists yn,x ∈ S(X) and gn,x ∈ D(yn,x) such that +∥yn,x − x∥ < 1/n and ∥gn,x − f∥C ≥ ε. +Let φ : S(X) → S(X∗) be a support mapping that maps yn,x to gn,x. Let +A = {yn,x : x ∈ S(X), n ∈ N}. Then A is norm dense in S(X), but φ(A) is +not τ-dense in S(X∗). +□ +Remark 3.6. For C = K or W, Vanderwerff [14] observed that the strong +C-MIP is equivalent to the following condition: +(b′) For every ε > 0 and C ∈ C, the points in S(X∗) that lie in +w*-slices of B(X∗) having C-diameter ≤ ε is τ-dense in S(X∗). + +THE GENERALISED MIP & UMIP +9 +It is easy to see that this condition is equivalent to (b) above and holds +for any compatible family C. Now consider the following condition: +(b′′) The C-denting points of B(X∗) are τ-dense in S(X∗). +Clearly, (b′′) +=⇒ +(b). +Moreover, if every w*-slice of B(X∗) contains a +C-denting point, then both are equivalent. Notice that this condition holds +in any Asplund space. +Since the τF and τK topologies coincide on bounded sets, we obtain +Corollary 3.7. For a Banach space X, the following are equivalent: +(a) X has the strong K-MIP. +(b) X has the strong F-MIP. +(c) The extreme points of B(X∗) are w*-dense in S(X∗). +Though we have given enough evidence that the strong C-semidenting +points are more natural than the C-semidenting points, one question remains +unanswered: Can we characterise C-MIP in terms of strong C-semidenting +points? +So let us consider the condition: +(a′) The cone generated by strong C-semidenting points of B(X∗) is τ- +dense in X∗. +It is clear from Theorem 2.6 that this condition is generally stronger than +the C-MIP and is equivalent to it if every w*-slice of B(X∗) contains a +strong C-semidenting point, a condition slightly weaker than the one that +gives equivalence in Theorem 2.6. +It is well known that a Banach space with a Fr´echet smooth norm has the +MIP. Indeed, this was known to Mazur himself. Notice that a compatible +class C is a bornology as defined in [11, pg. 64]. Using the definition of +C-smoothness discussed in [11, pg. 64], we have the following corollary. +Corollary 3.8. If the norm on X is C-smooth, then X has the strong C- +MIP. +Remark 3.9. For C = K and W, this was observed by Vanderwerff in [14]. + +10 +BANDYOPADHYAY AND GOTHWAL +4. C-UMIP and Strong C-UMIP +In this section, we discuss uniform versions of C-MIP and strong C-MIP. +The following quantitative result is contained in the proof of [3, Theorem +2.6]. We will need it repeatedly. +Lemma 4.1. Let A ⊆ X be such that M = sup{∥x∥ : x ∈ A} and d = +d(0, A) > 0. Suppose there exist x0 ∈ S(X) and 0 < γ < 1 such that +S = {f ∈ B(X∗) : f(x0) > γ} ⊆ {f ∈ B(X∗) : f(x) > 0 for all x ∈ A}. +Then for λ ≥ M/(1 − γ), +A ⊆ B +� +λx0, λ − d(1 − γ) +1 + γ +� +. +We observe that there are three ways of defining uniform versions of the +strong C-MIP ((a), (b) and (c) below) and they are equivalent. +Theorem 4.2. In a Banach space X, the following are equivalent: +(a) For every ε > 0, β ≥ 0, M ≥ 2 and 0 < η < ε, there exists +K > 0 such that for every closed convex C ∈ C with diam(C) ≤ M +and d(0, C + βB(X)) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that +C + βB(X) ⊆ B[z0, r] and d(0, B[z0, r]) > η. +(b) For every ε > 0, M ≥ 2 and 0 < η < ε, there exists K > 0 +such that for every closed convex C ∈ C with diam(C) ≤ M and +d(0, C) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that C ⊆ B[z0, r] +and d(0, B[z0, r]) > η. +(c) For every ε > 0, β ≥ 0 and M ≥ 2, there exist K > 0 and 0 < +η < ε, such that for every closed convex C ∈ C with diam(C) ≤ M +and d(0, C + βB(X)) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that +C + βB(X) ⊆ B[z0, r] and d(0, B[z0, r]) > η. +(d) For every ε > 0, there exists δ > 0 such that for every C ∈ C, +C ⊆ B(X) and f ∈ S(X∗), there is x ∈ S(X) such that +S(B(X∗), x, δ) ⊆ BC(f, ε). +(e) For every ε > 0, there exists δ > 0 such that for every C ∈ C, +C ⊆ B(X) and f ∈ S(X∗), there is x ∈ S(X) such that D(SX ∩ +B(x, δ)) ⊆ BC(f, ε). +(f) For every ε > 0, there exists δ > 0 such that for every C ∈ C, +C ⊆ B(X) and f ∈ S(X∗), there exists x ∈ Mε,δ,C such that D(x) ⊆ +BC(f, ε). + +THE GENERALISED MIP & UMIP +11 +Proof. Notice that for any bounded C ⊆ X and r, β > 0, +C ⊆ B[z0, r] ⇐⇒ C + βB(X) ⊆ B[z0, r + β] +and +d(0, B[z0, r + β]) = d(0, B[z0, r]) − β +provided d(0, B[z0, r]) > β. +Clearly, (a) =⇒ (b) and (a) =⇒ (c). +(b) =⇒ (d): This is a quantitatively more precise version of the proof of +Lemma 3.4 ((c) =⇒ (b)). +Let ε > 0 be given. Let α = +ε +4+ε > 0 and η ∈ (α, ε/4). +Choose K +for ε/4, η and M = 4. That is, for every C ∈ C with d(0, C) ≥ ε/4 and +diam(C) ≤ 4, there exists a closed ball B[z0, r] containing C with r ≤ K +and d(0, B[z0, r]) > η. We will show that δ = η−α +K+η works. +Let C ∈ C such that C ⊆ B(X) and f ∈ S(X∗). As in the proof of +Lemma 3.4 ((c) +=⇒ +(b)), let z0 ∈ X be such that ∥z0∥ = 1 + ε/2 and +f(z0) > 1 + ε/4 and D = aco(C ∪ {z0}). Again, D ∈ C and ∥f∥D ≥ 1. For +A = {y ∈ D : f(y) ≥ ε/4}, A ∈ C, diam (A) ≤ 4 and d(0, A) ≥ ε/4. +By assumption, there exist x0 ∈ X and r > 0 such that A ⊆ B[x0, r], +d(0, B[x0, r]) > η and r ≤ K. Thus, ∥x0∥ > r + η. Now let +S := S +� +B(X∗), x0 +∥x0∥, δ +� += +� +g ∈ B(X∗) : g +� x0 +∥x0∥ +� +> K + α +K + η +� +. +Then, for g ∈ S, we have g(x0/∥x0∥) > (K +α)/(K +η) > (r+α)/(r+η). +So, g(x0) > (r + α)∥x0∥/(r + η). But, ∥x0∥ > r + η. Thus, g(x0) > r + α. +So, inf g(B[x0, r]) > g(x0) − r > α > 0. And hence, again as in Lemma 3.4 +((c) =⇒ (b)), applying Phelps’ Lemma [3, Theorem 2.5] to ∥.∥D, +����f − ∥f∥D +∥g∥D +g +���� +D +< ε/2. +Now, arguing as in Lemma 3.4 ((c) =⇒ (b)), we have that for g ∈ S, +∥f − g∥C ≤ ∥f − g∥D < ε. +(c) =⇒ (d): This is a minor variant of the above proof. +Let ε > 0 be given. Let α = +ε +4+ε > 0 and β ∈ (α, ε/4). Choose K and η +for ε/4−β, β and M = 4. That is, for every C ∈ C with d(0, C +βB(X)) ≥ +ε/4 − β and diam(C) ≤ 4, there exists a closed ball B[z0, r] containing +C + βB(X), d(0, B[z0, r]) > η > 0 and r ≤ K. +We will show that δ = β−α +K+β works. +Let C ∈ C such that C ⊆ B(X) and f ∈ S(X∗). + +12 +BANDYOPADHYAY AND GOTHWAL +As before, let z0 ∈ X be such that ∥z0∥ = 1 + ε/2 and f(z0) > 1 + ε/4. +Let D = aco(C ∪ {z0}) and A = {y ∈ D : f(y) ≥ ε/4}. Then D, A ∈ C, +∥f∥D ≥ 1, diam(A) ≤ 4 and d(0, A + βB(X)) ≥ ε/4 − β. +By assumption, there exist x0 ∈ X and 0 < r ≤ K such that A+βB(X) ⊆ +B[x0, r], and d(0, B[x0, r]) > η. Thus, ∥x0∥ > r + η and A ⊆ B[x0, r − β], +d(0, B[x0, r − β]) ≥ η + β and r − β ≤ K. +Rest of the proof is identical as above and shows that +S +� +B(X∗), x0 +∥x0∥, δ +� +⊆ BC(f, ε). +(d) =⇒ (a): Let ε > 0, β ≥ 0, M ≥ 2 and 0 < η < ε be given. +Let L = M + ε + 2β. Clearly, L > 1. +Let ε1 = ε/L and η1 = η/L. Choose 0 < δ < 1 such that for any C1 ∈ C +with C1 ⊆ B(X) and f ∈ S(X∗), there exists x ∈ S(X) such that +S(B(X∗), x, δ) ⊆ BC1(f, ε1 − η1). +Let K = (L + ε)/δ. +Let C ∈ C with diam(C) ≤ M and d(0, C + βB(X)) ≥ ε. We will show +that there is a closed ball B of radius ≤ K such that C + βB(X) ⊆ B and +d(0, B) > η. +Case I: C + βB(X) \ B[0, L] ̸= ∅. +If z ∈ C + βB(X) \ B[0, M + ε + 2β], then d(0, B[z, M + 2β]) ≥ ε > η, +C + βB(X) ⊆ B[z, M + 2β] and M + 2β ≤ K. +Case II: C + βB(X) ⊆ B[0, L]. +Let �C = +1 +LC and α = β/L. Then, �C ⊆ D := +1 +L(C + βB(X)) = �C + +αB(X) ⊆ B(X) and d(0, D) ≥ ε1. Thus, d(0, �C) ≥ ε1 + α. It suffices to +show that there is a closed ball B of radius ≤ K/L such that D ⊆ B and +d(0, B) > η1. Let D′ = D + η1B(X) = �C + (η1 + α)B(X). +Choose f ∈ S(X∗) such that inf f( �C) ≥ ε1 + α. For this f, there exists +x ∈ S(X) such that +S := S(B(X∗), x, δ) ⊆ B � +C(f, ε1 − η1). +Let h ∈ S. For y ∈ �C, we have, +h(y) ≥ f(y) − ∥f − h∥ � +C > (ε1 + α) − (ε1 − η1) = η1 + α. +That is, h(y) > 0 for all y ∈ D′. So, +S ⊆ {g ∈ X∗ : g(x) > 0 for all x ∈ D′}. + +THE GENERALISED MIP & UMIP +13 +By Lemma 4.1, we have that for λ = (1+ ε1) +δ +D′ ⊆ B +� +λx, λ − d(0, D′)δ +2 − δ +� +. +Hence, +D ⊆ B +� +λx, λ − d(0, D′)δ +2 − δ +− η1 +� +. +Also, +d(0, B +� +λx, λ − d(0, D′)δ +2 − δ +− η1 +� +) ≥ d(0, D′)δ +2 − δ ++ η1 > η1 +and +λ − d(0, D′)δ +2 − δ +− η1 ≤ λ = 1 + ε1 +δ += K/L. +This completes the proof. +Equivalence of (d), (e) and (f) follows from Lemma 2.4. +□ +Definition 4.3. For a compatible class C, X is said to have strong the +C-UMIP if any of the equivalent conditions of Theorem 4.2 is satisfied. +We again observe that +Corollary 4.4. Strong K-UMIP and strong F-UMIP are equivalent. +Now we come to a uniform version of the C-MIP. +Definition 4.5. For a compatible class C, X is said to have the C-UMIP if +for every ε > 0 and M ≥ 2, there exist K > 0 and 0 < η ≤ ε such that for +every closed convex C ∈ C with diam(C) ≤ M and d(0, C) ≥ ε, there exist +z0 ∈ X, 0 < r ≤ K such that C ⊆ B[z0, r] and d(0, B[z0, r]) ≥ η. +It can be similarly characterised in terms of the uniform version of the +semidenting point discussed in [6]. The proof is a slight modification of the +arguments of Theorem 4.2. We omit the details. +Theorem 4.6. For a compatible class C, the following are equivalent: +(a) X has the C-UMIP +(b) For every ε > 0, there exists δ > 0 such that for every C ∈ C, +C ⊆ B(X) and f ∈ S(X∗), there is x ∈ S(X) such that +S(B(X∗), x, δ) ⊆ cone (BC(f, ε)). + +14 +BANDYOPADHYAY AND GOTHWAL +5. Summary +In this section, we summarise the interrelations between various notions +discussed above. We give counterexamples to some of the reverse implica- +tions. As far as we know, the other implications are open. +MIP +⇒ +̸⇐ +strong W-MIP +⇒ +strong K-MIP +⇔ +strong F-MIP +⇕ +⇓ +⇓̸⇑ +⇓ +MIP +⇒ +̸⇐ +W-MIP +⇒ +K-MIP +⇒ +F-MIP +Examples. +• Vanderwerff in [14, Theorem 2.6] shows that there is a Banach space +which has W-MIP but it does not have the strong K-MIP. Thus, the +K-MIP does not imply the strong K-MIP. This also shows that there +exist Banach spaces with the W-MIP but not the MIP. +• Vanderwerff in [14, Theorem 1.1] also shows that every Banach space +can be renormed to have the W-MIP, and hence, the K-MIP. This +again shows that the W-MIP does not imply the MIP. +• In ℓ1, Gˆateaux differentiability and weak Hadamard (that is, W-) +differentiability coincide (see [8]). Now, ℓ1, being a separable space, +has an equivalent norm that is Gˆateaux smooth, and hence, W- +smooth, and therefore, has the strong W-MIP. But, being a separable +non-Asplund space, it has no MIP renorming. +We also have similar interrelations between the uniform versions. In this +case, most of the reverse implications are open. +UMIP +⇒ +strong W-UMIP +⇒ +strong K-UMIP +⇔ +strong F-UMIP +⇕ +⇓ +⇓ +⇓ +UMIP +⇒ +W-UMIP +⇒ +K-UMIP +⇒ +F-UMIP +References +1. Pradipta Bandyopadhyay, The Mazur Intersection Property in Banach Spaces and +Related Topics, Ph. D Thesis submitted to ISI, Calcutta, 1991. +2. Pradipta Bandyopadhyay, The Mazur Intersection Property for Families of Closed +Bounded Convex Sets in Banach Spaces, Colloq. Math. LXIII (1992), 45–56. +3. Pradipta Bandyopadhyay, Jadav Ganesh and Deepak Gothwal, On Uniform Mazur +Intersection Property Studia Math. 260 (2021), no. 3, 273–283. +4. D. Chen and B. -L. Lin, On B-Convex and Mazur Sets of Banach Spaces, Bull. Pol. +Acad. Sci. Math., 43 (1995), 191-198. + +THE GENERALISED MIP & UMIP +15 +5. D. Chen, B. -L. Lin, Ball Separation Properties in Banach Spaces, Rocky Mountain +J. Math. 28 (1998), 835-873 +6. Q. Cheng, Y. Dong, Characterization of Normed Linear Spaces with Generalized +Mazur Intersection Property, Studia Math. 219 (2013), no. 3, 193–200. +7. J. R. Giles, D. A. Gregory, B. Sims, Characterisation of normed linear spaces with +Mazur’s intersection property, Bull. Austral. Math. Soc., 18 (1978), no. 1, 105-123. +8. J. R. Giles, S. Sciffer, On weak Hadamard differentiability of convex functions on +Banach spaces, Bull. Austral. Math. Soc. 54 (1996), no. 1, 155–166. +9. A. S. Granero, M. Jim´enez-Sevilla, J. P. Moreno, Intersections of closed balls and +geometry of Banach spaces, Extracta Math., 19 (2004), 55-92. +10. S. Mazur, ¨Uber Schwache Konvergenz in den Ra¨umen (Lp), Studia Math. 4 (1933), +128–133. +11. R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture +Notes in Mathematics, 1364, Springer-Verlag, Berlin, 1989. +12. A. Sersouri, The Mazur Property for Compact Sets, Pacific J. Math. 133 (1988), 185– +195. +13. A. Sersouri, Mazur’s Intersection Property for Finite Dimensional Sets, Math. Ann. +283 (1989), 165–170. +14. J. Vanderwerff, Mazur Intersection Properties for Compact and Weakly Compact Con- +vex Sets, Canad. Math. Bull. 41 (1998), no. 2, 225–230. +15. J. H. M. Whitfield, V. Zizler, Mazur’s intersection property of balls for compact convex +sets, Bull. Austral. Math. Soc. 35 (1987), no. 2, 267–274. +16. J. H. M. Whitfield, V. Zizler, Uniform Mazur’s Intersection Property of Balls, Canad. +Math. Bull., 30 (1987), 455-460. +17. V. Zizler, Renorming Concerning Mazur’s Intersection Property of Balls for Weakly +Compact Convex Sets, Math. Ann. 276 (1986), 61–66. +(Pradipta Bandyopadhyay) Stat–Math Division, Indian Statistical Institute, +203, B. T. Road, Kolkata 700108, India. +Email address: pradipta@isical.ac.in +(Deepak Gothwal) Stat–Math Division, Indian Statistical Institute, 203, B. T. +Road, Kolkata 700108, India. +Email address: deepakgothwal190496@gmail.com + diff --git a/FdA0T4oBgHgl3EQfBP_A/content/tmp_files/load_file.txt b/FdA0T4oBgHgl3EQfBP_A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed65fcdf685c56447f58eb868f255b0658b52a11 --- /dev/null +++ b/FdA0T4oBgHgl3EQfBP_A/content/tmp_files/load_file.txt @@ -0,0 +1,501 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf,len=500 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='01974v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='FA] 5 Jan 2023 THE GENERALISED (UNIFORM) MAZUR INTERSECTION PROPERTY PRADIPTA BANDYOPADHYAY AND DEEPAK GOTHWAL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Given a family C of closed bounded convex sets in a Banach space X, we say that X has the C-MIP if every C ∈ C is the intersection of the closed balls containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In this paper, we introduce a stronger version of the C-MIP and show that it is a more satisfactory general- isation of the MIP inasmuch as one can obtain complete analogues of various characterisations of the MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We also introduce uniform versions of the (strong) C-MIP and charac- terise them analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Even in this case, the strong C-UMIP appears to have richer characterisations than the C-UMIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Introduction S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Mazur [10] initiated the study of the following intersection property of balls in normed spaces, now called the Mazur Intersection Property (MIP): Every closed bounded convex set is the intersection of closed balls containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Giles, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Gregory and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Sims [7] obtained various characteri- sations of the MIP, most well-known criterion stating that the w*-denting points of B(X∗) are norm dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Chen and Lin [4] introduced the notion of w*-semidenting points to characterise the MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' The paper [9] is a good survey of the MIP and related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Whitfield and Zizler in [16] introduced a uniform version of the MIP (UMIP) and obtained characterisations similar to [7] but the characterisa- tion related to the w*-denting points was missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Recently, such a charac- terisation in terms of “uniform w*-semidenting points” has been obtained by the authors in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Soon after [7], there appeared several papers dealing with generalisations of the MIP, basically studying when members of a given family of closed bounded convex sets are intersection of balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For example, 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Primary 46B20 To appear in JMAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Compatible class, (strong) C-MIP, (strong) C-UMIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 1 2 BANDYOPADHYAY AND GOTHWAL (a) K = {all compact convex sets in X} [15, 12] (b) W = {all weakly compact convex sets in X} [17] (c) F = {all compact convex sets in X with finite affine dimension} [13] Let C be a family of bounded subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let τC be the topology on X∗ of uniform convergence on the sets of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let us say that X has the C-MIP if every closed convex set C ∈ C is the intersection of the closed balls containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In [12, 13, 15], the authors characterised the C-MIP in cases (a) and (c) above in terms of the density in X∗ of the cone of the extreme points of BX∗ in the topology τC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' The first author studied the C-MIP for a general family C of bounded sets in [2] and obtained some necessary and some sufficient conditions that recaptured the results of [12, 13, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Chen and Lin [5] also discussed the C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Both the papers observed that if the cone of the “C-denting points” of B(X∗), as defined in [5], is τC-dense in X∗, then X has the C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Both papers proved the converse under additional hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In [7], there are several alternative characterisations of the MIP [7, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1 (ii), (iii) and (iv)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Analogous characterisations of the UMIP were also obtained in [15, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' [2] is the only attempt so far to obtain such ana- logues for the C-MIP (See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Recently, Cheng and Dong [6] characterised the C-MIP in terms of the “C-semidenting” points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' However, their definition does not appear to be a natural generalisation of either the w*-semidenting points of [4] or the C-denting points of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Vanderwerff [14] introduced the following stronger notion for the families K and W above, which we will call strong C-MIP: For every convex C ∈ C and β ≥ 0, C + βB(X) is the inter- section of closed balls containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For C = K, he showed that it is equivalent to the extreme points of B(X∗) being w*-dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In this paper, we show that the strong C-MIP is actually a more satisfac- tory generalisation of the MIP for general families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' It can be characterised in terms of a more natural notion of the (strong) C-semidenting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Moreover, one can also obtain complete analogue of [7, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We also introduce the uniform version of the C-MIP and its stronger form and characterise them analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Even in this case, the strong C-UMIP appears have richer characterisations than the C-UMIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 3 In the last section, we summarise the interrelations between various no- tions discussed in the paper with examples and counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Preliminaries Throughout this article, X is a real Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For x ∈ X and r > 0, we denote by B(x, r) the open ball {y ∈ X : ∥x − y∥ < r} and by B[x, r] the closed ball {y ∈ X : ∥x − y∥ ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We denote by B(X) the closed unit ball {x ∈ X : ∥x∥ ≤ 1} and by S(X) the unit sphere {x ∈ X : ∥x∥ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a bounded subset A ⊆ X, denote by co(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' aco(A)) the convex (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' absolutely convex) hull of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let ∥f∥A := sup{|f(x)| : x ∈ A} for f ∈ X∗ and diamA(B) = sup{∥f −g∥A : f, g ∈ B} for any non-empty subset B of X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For f ∈ X∗, ε > 0 and A ∈ C, BA(f, ε) := {g ∈ X∗ : ∥f −g∥A < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a bounded set C ⊆ X and x ∈ X, let d(x, C) = inf{∥x − z∥ : z ∈ C} denote the distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For A ⊆ X, the cone generated by A is cone(A) = {λa : λ ≥ 0, a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We say that a class C of bounded subsets of X is a com- patible class if (I) C ∈ C, α ∈ R and x ∈ X =⇒ αC + x ∈ C, C ∪ {x} ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (II) C ∈ C implies the closed absolutely convex hull of C ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (III) C ∈ C and A ⊆ C implies A ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (IV) C1, C2 ∈ C implies C1 ∪ C2 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Our definition is slightly different from the ones in [2, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a compatible class C, X is said to have the C-MIP if every closed convex C ∈ C is the intersection of closed balls containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Notice that the families K, W and F above are not exactly compati- ble classes with this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' One has to consider such sets and subsets thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' However, the property C-MIP remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' And we should add the family B of all bounded sets, corresponding to the MIP itself, in our list of examples of a compatible class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Throughout the article, C will denote a compatible class of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We will denote by τC, or, simply τ when there is no ambiguity, the topology on X∗ of uniform convergence on the sets of C, or, in other words, the topology on X∗ generated by the family of seminorms {∥ · ∥C : C ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since C is a compatible class, the family {BC(0, ε) : C ∈ C, ε > 0} forms a local base for τC at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For the classes C = F, K, W and B, the topology τC coincides with 4 BANDYOPADHYAY AND GOTHWAL the w*-, bw*-, Mackey and norm topologies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We note that the topologies τF (w*-) and τK (bw*-) coincide on bounded sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (a) For x ∈ S(X), we denote by D(x) the set {f ∈ S(X∗) : f(x) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Any selection of D is called a support mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) A w*-slice of B(X∗) determined by x ∈ S(X) is a set of the form S(B(X∗), x, δ) := {f ∈ B(X∗) : f(x) > 1 − δ} for some 0 < δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) For ε, δ > 0, x ∈ S(X) and C ∈ C denote by d1(C, x, δ) = sup 0<λ<δ, y∈C ∥x + λy∥ + ∥x − λy∥ − 2 λ d2(C, x, δ) = diamC(S(B(X∗), x, δ)) d3(C, x, δ) = diamC(D(S(X) ∩ B(x, δ))) (d) [2] For ε, δ > 0 and C ∈ C Mε,δ,C(X) = {x ∈ S(X) : sup 0<∥y∥<δ,y∈C ∥x + y∥ + ∥x − y∥ − 2 ∥y∥ < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In other words, Mε,δ,C(X) = {x ∈ S(X) : d1(C, x, δ) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Define Mε,C(X) = � δ>0 Mε,δ,C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (e) Define H = �{D(Mε,C(X)) τ : C ∈ C, ε > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Following lemma is an easy adaptation of [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Details can be found in [1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For any α, δ > 0, we have, (i) d2(C, x, α) ≤ d1(C, x, δ) + 2α δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (ii) d3(C, x, δ) ≤ d2(C, x, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (iii) d1(C, x, δ) ≤ d3(C, x, 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (a) We say that f ∈ S(X∗) is a w*-denting point of B(X∗) if for every ε > 0, there exists a w*-slice S of B(X∗) such that f ∈ S and diam(S) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) [4] We say that f ∈ S(X∗) is a w*-semidenting point of B(X∗) if for every ε > 0, there exists a w*-slice S of B(X∗) such that S ⊆ B(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 5 (c) [5] We say that f ∈ S(X∗) is a C-denting point of B(X∗) if for each A ∈ C and ε > 0, there exists a w*-slice S of B(X∗) such that f ∈ S and diamA(S) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) [6] We say that f ∈ S(X∗) is a C-semidenting point of B(X∗) if for each A ∈ C and ε > 0, there exists a w*-slice S of B(X∗) such that S ⊆ cone(BA(f, ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Known results on C-MIP [2, 5, 6] are summarised in our notation and terminology below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' [2, Theorem 1, Corollary 1], [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='10], [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='3] Suppose C is a compatible class of bounded sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Consider the following statements : (a) The cone generated by C-denting points of B(X∗) is τ-dense in X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) The cone generated by H is τ-dense in X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) If C1, C2 ∈ C are closed convex such that there exists f ∈ X∗ with sup f(C1) < inf f(C2), then there exist disjoint closed balls B1, B2 such that Ci ⊆ Bi, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) X has the C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (e) Every f ∈ S(X∗) is a C-semidenting point of B(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (f) For every norm dense subset A ⊆ S(X) and every support map- ping φ, the cone generated by φ(A) is τ-dense in X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then (a) =⇒ (b) =⇒ (c) =⇒ (d) ⇐⇒ (e) =⇒ (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Moreover, if A = {x ∈ S(X) : D(x) contains a C-denting point of B(X∗)} is norm dense in S(X), then all the statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='10], it is shown that (a) and (d) are equiv- alent under a weaker assumption that every w*-slice of B(X∗) contains a C-denting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Strong C-MIP Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' A Banach space X is said to have the strong C-MIP if for every closed convex C ∈ C and β ≥ 0, C + βB(X) is the intersection of closed balls containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Vanderwerff [14] called this K-IP and W-IP for C = K and W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Clearly, for C = B, C-MIP and strong C-MIP coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' See Section 5 for some examples and counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 6 BANDYOPADHYAY AND GOTHWAL Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' The duality map on X is said to be norm-τ quasicontinuous if for every f ∈ S(X∗), ε > 0 and C ∈ C, there exist δ > 0 and x ∈ S(X) such that D(SX ∩ B(x, δ)) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In our opinion, the following is a more natural generalisation of both the w*-semidenting points of [4] and the C-denting points of [5] than the C-semidenting points defined in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We say that f ∈ S(X∗) is a strong C-semidenting point of B(X∗) if for each A ∈ C and ε > 0, there exists a w*-slice S of B(X∗) such that S ⊆ BA(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We begin by observing that the strong C-semidenting points can be char- acterised completely analogous to [7, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For f ∈ S(X∗), the following are equivalent: (a) f ∈ H, that is, f ∈ � ε>0 D(Mε,C(X)) τ for every C ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) f is a strong C-semidenting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) For every β ≥ 0 and closed convex C ∈ C such that inf f(C + βB(X)) > 0, there exists a closed ball B[x0, r0] in X con- taining C + βB(X) with 0 /∈ B[x0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) For every ε > 0 and C ∈ C, there exist δ > 0 and x ∈ S(X) such that D(SX ∩ B(x, δ)) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (a) =⇒ (b): For C ∈ C, let f ∈ � ε>0 D(Mε,C(X)) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We have f ∈ D(Mε/4,C(X)) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, there exist δ > 0 and x ∈ Mε/4,δ,C(X) and g ∈ D(x) such that ∥f − g∥C < ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since x ∈ Mε/4,δ,C(X), we have d1(C, x, δ) < ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4(i), for α = εδ/8, d2(C, x, α) ≤ d1(C, x, δ) + 2α δ < ε/4 + ε/4 = ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Hence, diamC(S(B(X∗), x, α) < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since g ∈ S(B(X∗), x, δ), S(B(X∗), x, δ) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) =⇒ (c): Let β ≥ 0 and C ∈ C such that inf f(C + βB(X)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let inf f(C + βB(X)) = α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now, there exist δ > 0 and x ∈ S(X) such that S := S(B(X∗), x, δ) ⊆ BC(f, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Choose g ∈ S and c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We have, g(c) = f(c) − ∥f − g∥C > α + β − α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 7 Hence, for y ∈ C + βB(X), g(y) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, by [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6], there is a closed ball B[x0, r0] containing C + βB(X) such that 0 /∈ B[x0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) =⇒ (b): Let C ∈ C and ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let z0 ∈ X be such that ∥z0∥ = 1 + ε/2 and f(z0) > 1 + ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let D = aco(C ∪ {z0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We have D ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Consider A := {x ∈ D : f(x) ≥ ε/4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We have A ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For β = ε/4 1+ε/4, f(A + βB(X)) ≥ ε/4 − β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, there exists a closed ball B[x0, r0] containing A + βB(X) such that 0 /∈ B[x0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, for r1 = r0 − β and x1 = x0, A ⊆ B[x1, r1] and d(0, B[x1, r1]) > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Consider S = {g ∈ B(X∗) : g(x1/∥x1∥) > (r1 + β)/∥x1∥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For g ∈ S, g(x1/∥x1∥) > (r1 + β)/∥x1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, g(x1) > r1 + β which implies inf g(B[x1, r1]) ≥ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Thus, g(x) > 0 for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Therefore, applying Phelps’ Lemma [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='5] to the semi-norm ∥ · ∥D, ����f − ∥f∥D ∥g∥D g ���� D < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Consider y0 = ε 4 z0 f(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since f(z0) > ε/4, D is absolutely convex, and f(y0) = ε/4, y0 ∈ A, and hence, g(y0) > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Therefore, g(z0) = f(z0)g(y0) ε/4 > (1 + ε/4)β ε/4 = 1 and so, ∥g∥D ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since f, g ∈ B(X∗) and D ⊆ (1 + ε/2)B(X), 1 ≤ ∥f∥D, ∥g∥D ≤ 1 + ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Hence, ��∥f∥D − ∥g∥D �� < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Finally, ∥f − g∥C ≤ ∥f − g∥D ≤ ����f − ∥f∥D ∥g∥D g ���� D + ��∥f∥D − ∥g∥D �� < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) =⇒ (d): Let f ∈ S(X∗) be a strong C-semidenting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let ε > 0 and C ∈ C be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' There exist δ > 0 and x ∈ S(X) such that S(B(X∗), x, δ) ⊆ BC(f, ε/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, d2(C, x, δ) = diamC(S(B(X∗), x, δ)) < ε/2 and D(x) ⊆ BC(f, ε/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4, d3(C, x, δ) = diamC(D(S(X)∩B(x, δ))) < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Hence, D(S(X) ∩ B(x, δ)) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 8 BANDYOPADHYAY AND GOTHWAL (d) =⇒ (a): To show f ∈ D(Mε,C(X)) τ for every C ∈ C and ε > 0, let 0 < η < ε and C1 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Enough to show that there exist x ∈ Mε,C and g ∈ D(x) such that ∥f − g∥C1 < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let C0 = C ∪ C1 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' By (d), there exist δ > 0 and x ∈ S(X) such that D(S(X) ∩ B(x, δ)) ⊆ BC0(f, η/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, d3(C0, x, δ) ≤ η and D(x) ⊆ BC0(f, η/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4, d1(C0, x, δ/2) ≤ d3(C0, x, δ) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Hence, x ∈ Mη,δ/2,C0 ⊆ Mε,δ/2,C ⊆ Mε,C and ∥f − g∥C1 < η for any g ∈ D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' □ Here is our main theorem characterising the strong C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Again, we obtain a complete analogue of [7, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a Banach space X, the following are equivalent: (a) X has the strong C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) Every f ∈ S(X∗) is a strong C-semidenting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) For every ε > 0 and C ∈ C, D(Mε,C(X)) is τ-dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) The duality map on X is norm-τ quasicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (e) Every support mapping in X maps norm dense sets in S(X) to τ-dense sets in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Equivalence of (a), (b), (c) and (d) follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) =⇒ (e): Let A ⊆ S(X) be norm dense in S(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let φ : S(X) → S(X∗) be a support mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let f ∈ S(X∗), C ∈ C and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' It is enough to find x0 ∈ A such that ∥f − φ(x0)∥C < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' By (d), there exist δ > 0 and x ∈ S(X) such that D(SX ∩ B(x, δ)) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since, A is dense in S(X), there exists x0 ∈ A such that ∥x0−x∥ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' If follows that D(x0) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Therefore, for any support mapping φ, ∥f − φ(x0)∥C < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (e) =⇒ (d): Suppose the duality map is not norm-τ quasicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then there exist f ∈ S(X∗), ε > 0 and C ∈ C, such that for any x ∈ S(X) and n ∈ N, there exists yn,x ∈ S(X) and gn,x ∈ D(yn,x) such that ∥yn,x − x∥ < 1/n and ∥gn,x − f∥C ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let φ : S(X) → S(X∗) be a support mapping that maps yn,x to gn,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let A = {yn,x : x ∈ S(X), n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then A is norm dense in S(X), but φ(A) is not τ-dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For C = K or W, Vanderwerff [14] observed that the strong C-MIP is equivalent to the following condition: (b′) For every ε > 0 and C ∈ C, the points in S(X∗) that lie in w*-slices of B(X∗) having C-diameter ≤ ε is τ-dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 9 It is easy to see that this condition is equivalent to (b) above and holds for any compatible family C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now consider the following condition: (b′′) The C-denting points of B(X∗) are τ-dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Clearly, (b′′) =⇒ (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Moreover, if every w*-slice of B(X∗) contains a C-denting point, then both are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Notice that this condition holds in any Asplund space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Since the τF and τK topologies coincide on bounded sets, we obtain Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a Banach space X, the following are equivalent: (a) X has the strong K-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) X has the strong F-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) The extreme points of B(X∗) are w*-dense in S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Though we have given enough evidence that the strong C-semidenting points are more natural than the C-semidenting points, one question remains unanswered: Can we characterise C-MIP in terms of strong C-semidenting points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So let us consider the condition: (a′) The cone generated by strong C-semidenting points of B(X∗) is τ- dense in X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' It is clear from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6 that this condition is generally stronger than the C-MIP and is equivalent to it if every w*-slice of B(X∗) contains a strong C-semidenting point, a condition slightly weaker than the one that gives equivalence in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' It is well known that a Banach space with a Fr´echet smooth norm has the MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Indeed, this was known to Mazur himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Notice that a compatible class C is a bornology as defined in [11, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Using the definition of C-smoothness discussed in [11, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 64], we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' If the norm on X is C-smooth, then X has the strong C- MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For C = K and W, this was observed by Vanderwerff in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 10 BANDYOPADHYAY AND GOTHWAL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' C-UMIP and Strong C-UMIP In this section, we discuss uniform versions of C-MIP and strong C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' The following quantitative result is contained in the proof of [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We will need it repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let A ⊆ X be such that M = sup{∥x∥ : x ∈ A} and d = d(0, A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Suppose there exist x0 ∈ S(X) and 0 < γ < 1 such that S = {f ∈ B(X∗) : f(x0) > γ} ⊆ {f ∈ B(X∗) : f(x) > 0 for all x ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then for λ ≥ M/(1 − γ), A ⊆ B � λx0, λ − d(1 − γ) 1 + γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We observe that there are three ways of defining uniform versions of the strong C-MIP ((a), (b) and (c) below) and they are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In a Banach space X, the following are equivalent: (a) For every ε > 0, β ≥ 0, M ≥ 2 and 0 < η < ε, there exists K > 0 such that for every closed convex C ∈ C with diam(C) ≤ M and d(0, C + βB(X)) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that C + βB(X) ⊆ B[z0, r] and d(0, B[z0, r]) > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) For every ε > 0, M ≥ 2 and 0 < η < ε, there exists K > 0 such that for every closed convex C ∈ C with diam(C) ≤ M and d(0, C) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that C ⊆ B[z0, r] and d(0, B[z0, r]) > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) For every ε > 0, β ≥ 0 and M ≥ 2, there exist K > 0 and 0 < η < ε, such that for every closed convex C ∈ C with diam(C) ≤ M and d(0, C + βB(X)) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that C + βB(X) ⊆ B[z0, r] and d(0, B[z0, r]) > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) For every ε > 0, there exists δ > 0 such that for every C ∈ C, C ⊆ B(X) and f ∈ S(X∗), there is x ∈ S(X) such that S(B(X∗), x, δ) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (e) For every ε > 0, there exists δ > 0 such that for every C ∈ C, C ⊆ B(X) and f ∈ S(X∗), there is x ∈ S(X) such that D(SX ∩ B(x, δ)) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (f) For every ε > 0, there exists δ > 0 such that for every C ∈ C, C ⊆ B(X) and f ∈ S(X∗), there exists x ∈ Mε,δ,C such that D(x) ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Notice that for any bounded C ⊆ X and r, β > 0, C ⊆ B[z0, r] ⇐⇒ C + βB(X) ⊆ B[z0, r + β] and d(0, B[z0, r + β]) = d(0, B[z0, r]) − β provided d(0, B[z0, r]) > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Clearly, (a) =⇒ (b) and (a) =⇒ (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (b) =⇒ (d): This is a quantitatively more precise version of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4 ((c) =⇒ (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let α = ε 4+ε > 0 and η ∈ (α, ε/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Choose K for ε/4, η and M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' That is, for every C ∈ C with d(0, C) ≥ ε/4 and diam(C) ≤ 4, there exists a closed ball B[z0, r] containing C with r ≤ K and d(0, B[z0, r]) > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We will show that δ = η−α K+η works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let C ∈ C such that C ⊆ B(X) and f ∈ S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' As in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4 ((c) =⇒ (b)), let z0 ∈ X be such that ∥z0∥ = 1 + ε/2 and f(z0) > 1 + ε/4 and D = aco(C ∪ {z0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Again, D ∈ C and ∥f∥D ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For A = {y ∈ D : f(y) ≥ ε/4}, A ∈ C, diam (A) ≤ 4 and d(0, A) ≥ ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' By assumption, there exist x0 ∈ X and r > 0 such that A ⊆ B[x0, r], d(0, B[x0, r]) > η and r ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Thus, ∥x0∥ > r + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now let S := S � B(X∗), x0 ∥x0∥, δ � = � g ∈ B(X∗) : g � x0 ∥x0∥ � > K + α K + η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then, for g ∈ S, we have g(x0/∥x0∥) > (K +α)/(K +η) > (r+α)/(r+η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, g(x0) > (r + α)∥x0∥/(r + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' But, ∥x0∥ > r + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Thus, g(x0) > r + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, inf g(B[x0, r]) > g(x0) − r > α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' And hence, again as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4 ((c) =⇒ (b)), applying Phelps’ Lemma [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='5] to ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='∥D, ����f − ∥f∥D ∥g∥D g ���� D < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now, arguing as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4 ((c) =⇒ (b)), we have that for g ∈ S, ∥f − g∥C ≤ ∥f − g∥D < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (c) =⇒ (d): This is a minor variant of the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let α = ε 4+ε > 0 and β ∈ (α, ε/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Choose K and η for ε/4−β, β and M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' That is, for every C ∈ C with d(0, C +βB(X)) ≥ ε/4 − β and diam(C) ≤ 4, there exists a closed ball B[z0, r] containing C + βB(X), d(0, B[z0, r]) > η > 0 and r ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We will show that δ = β−α K+β works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let C ∈ C such that C ⊆ B(X) and f ∈ S(X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 12 BANDYOPADHYAY AND GOTHWAL As before, let z0 ∈ X be such that ∥z0∥ = 1 + ε/2 and f(z0) > 1 + ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let D = aco(C ∪ {z0}) and A = {y ∈ D : f(y) ≥ ε/4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then D, A ∈ C, ∥f∥D ≥ 1, diam(A) ≤ 4 and d(0, A + βB(X)) ≥ ε/4 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' By assumption, there exist x0 ∈ X and 0 < r ≤ K such that A+βB(X) ⊆ B[x0, r], and d(0, B[x0, r]) > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Thus, ∥x0∥ > r + η and A ⊆ B[x0, r − β], d(0, B[x0, r − β]) ≥ η + β and r − β ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Rest of the proof is identical as above and shows that S � B(X∗), x0 ∥x0∥, δ � ⊆ BC(f, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (d) =⇒ (a): Let ε > 0, β ≥ 0, M ≥ 2 and 0 < η < ε be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let L = M + ε + 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Clearly, L > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let ε1 = ε/L and η1 = η/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Choose 0 < δ < 1 such that for any C1 ∈ C with C1 ⊆ B(X) and f ∈ S(X∗), there exists x ∈ S(X) such that S(B(X∗), x, δ) ⊆ BC1(f, ε1 − η1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let K = (L + ε)/δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let C ∈ C with diam(C) ≤ M and d(0, C + βB(X)) ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We will show that there is a closed ball B of radius ≤ K such that C + βB(X) ⊆ B and d(0, B) > η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Case I: C + βB(X) \\ B[0, L] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' If z ∈ C + βB(X) \\ B[0, M + ε + 2β], then d(0, B[z, M + 2β]) ≥ ε > η, C + βB(X) ⊆ B[z, M + 2β] and M + 2β ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Case II: C + βB(X) ⊆ B[0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let �C = 1 LC and α = β/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Then, �C ⊆ D := 1 L(C + βB(X)) = �C + αB(X) ⊆ B(X) and d(0, D) ≥ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Thus, d(0, �C) ≥ ε1 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' It suffices to show that there is a closed ball B of radius ≤ K/L such that D ⊆ B and d(0, B) > η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let D′ = D + η1B(X) = �C + (η1 + α)B(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Choose f ∈ S(X∗) such that inf f( �C) ≥ ε1 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For this f, there exists x ∈ S(X) such that S := S(B(X∗), x, δ) ⊆ B � C(f, ε1 − η1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Let h ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For y ∈ �C, we have, h(y) ≥ f(y) − ∥f − h∥ � C > (ε1 + α) − (ε1 − η1) = η1 + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' That is, h(y) > 0 for all y ∈ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' So, S ⊆ {g ∈ X∗ : g(x) > 0 for all x ∈ D′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 13 By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1, we have that for λ = (1+ ε1) δ D′ ⊆ B � λx, λ − d(0, D′)δ 2 − δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Hence, D ⊆ B � λx, λ − d(0, D′)δ 2 − δ − η1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Also, d(0, B � λx, λ − d(0, D′)δ 2 − δ − η1 � ) ≥ d(0, D′)δ 2 − δ + η1 > η1 and λ − d(0, D′)δ 2 − δ − η1 ≤ λ = 1 + ε1 δ = K/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Equivalence of (d), (e) and (f) follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a compatible class C, X is said to have strong the C-UMIP if any of the equivalent conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We again observe that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Strong K-UMIP and strong F-UMIP are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now we come to a uniform version of the C-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a compatible class C, X is said to have the C-UMIP if for every ε > 0 and M ≥ 2, there exist K > 0 and 0 < η ≤ ε such that for every closed convex C ∈ C with diam(C) ≤ M and d(0, C) ≥ ε, there exist z0 ∈ X, 0 < r ≤ K such that C ⊆ B[z0, r] and d(0, B[z0, r]) ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' It can be similarly characterised in terms of the uniform version of the semidenting point discussed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' The proof is a slight modification of the arguments of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' For a compatible class C, the following are equivalent: (a) X has the C-UMIP (b) For every ε > 0, there exists δ > 0 such that for every C ∈ C, C ⊆ B(X) and f ∈ S(X∗), there is x ∈ S(X) such that S(B(X∗), x, δ) ⊆ cone (BC(f, ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 14 BANDYOPADHYAY AND GOTHWAL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Summary In this section, we summarise the interrelations between various notions discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We give counterexamples to some of the reverse implica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' As far as we know, the other implications are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' MIP ⇒ ̸⇐ strong W-MIP ⇒ strong K-MIP ⇔ strong F-MIP ⇕ ⇓ ⇓̸⇑ ⇓ MIP ⇒ ̸⇐ W-MIP ⇒ K-MIP ⇒ F-MIP Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Vanderwerff in [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='6] shows that there is a Banach space which has W-MIP but it does not have the strong K-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Thus, the K-MIP does not imply the strong K-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' This also shows that there exist Banach spaces with the W-MIP but not the MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Vanderwerff in [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='1] also shows that every Banach space can be renormed to have the W-MIP, and hence, the K-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' This again shows that the W-MIP does not imply the MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In ℓ1, Gˆateaux differentiability and weak Hadamard (that is, W-) differentiability coincide (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Now, ℓ1, being a separable space, has an equivalent norm that is Gˆateaux smooth, and hence, W- smooth, and therefore, has the strong W-MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' But, being a separable non-Asplund space, it has no MIP renorming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' We also have similar interrelations between the uniform versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' In this case, most of the reverse implications are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' UMIP ⇒ strong W-UMIP ⇒ strong K-UMIP ⇔ strong F-UMIP ⇕ ⇓ ⇓ ⇓ UMIP ⇒ W-UMIP ⇒ K-UMIP ⇒ F-UMIP References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Pradipta Bandyopadhyay, The Mazur Intersection Property in Banach Spaces and Related Topics, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' D Thesis submitted to ISI, Calcutta, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Pradipta Bandyopadhyay, The Mazur Intersection Property for Families of Closed Bounded Convex Sets in Banach Spaces, Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' LXIII (1992), 45–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Pradipta Bandyopadhyay, Jadav Ganesh and Deepak Gothwal, On Uniform Mazur Intersection Property Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 260 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 3, 273–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Chen and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' -L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Lin, On B-Convex and Mazur Sets of Banach Spaces, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=', 43 (1995), 191-198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' THE GENERALISED MIP & UMIP 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' -L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Lin, Ball Separation Properties in Banach Spaces, Rocky Mountain J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 28 (1998), 835-873 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Dong, Characterization of Normed Linear Spaces with Generalized Mazur Intersection Property, Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 219 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 3, 193–200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Giles, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Gregory, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Sims, Characterisation of normed linear spaces with Mazur’s intersection property, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=', 18 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 1, 105-123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Giles, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Sciffer, On weak Hadamard differentiability of convex functions on Banach spaces, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 54 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 1, 155–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Granero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Jim´enez-Sevilla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Moreno, Intersections of closed balls and geometry of Banach spaces, Extracta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=', 19 (2004), 55-92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Mazur, ¨Uber Schwache Konvergenz in den Ra¨umen (Lp), Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 4 (1933), 128–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, 1364, Springer-Verlag, Berlin, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Sersouri, The Mazur Property for Compact Sets, Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 133 (1988), 185– 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Sersouri, Mazur’s Intersection Property for Finite Dimensional Sets, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 283 (1989), 165–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Vanderwerff, Mazur Intersection Properties for Compact and Weakly Compact Con- vex Sets, Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 41 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 2, 225–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Whitfield, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Zizler, Mazur’s intersection property of balls for compact convex sets, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 35 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 2, 267–274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Whitfield, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Zizler, Uniform Mazur’s Intersection Property of Balls, Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=', 30 (1987), 455-460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Zizler, Renorming Concerning Mazur’s Intersection Property of Balls for Weakly Compact Convex Sets, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' 276 (1986), 61–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' (Pradipta Bandyopadhyay) Stat–Math Division, Indian Statistical Institute, 203, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Road, Kolkata 700108, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Email address: pradipta@isical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='in (Deepak Gothwal) Stat–Math Division, Indian Statistical Institute, 203, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Road, Kolkata 700108, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content=' Email address: deepakgothwal190496@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdA0T4oBgHgl3EQfBP_A/content/2301.01974v1.pdf'} diff --git a/G9A0T4oBgHgl3EQfBv-A/vector_store/index.faiss b/G9A0T4oBgHgl3EQfBv-A/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4bf465af532b2a3d27ccb4a18e32c7357a237ebd --- /dev/null +++ b/G9A0T4oBgHgl3EQfBv-A/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0753808bf233890665c61adbc80f292e186fa4456f0d01c28d9f1134bd72255a +size 2424877 diff --git a/HdFIT4oBgHgl3EQfXith/content/2301.11244v1.pdf b/HdFIT4oBgHgl3EQfXith/content/2301.11244v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9801af0d6bcd7d1e656819439d969cff4dc47ce8 --- /dev/null +++ b/HdFIT4oBgHgl3EQfXith/content/2301.11244v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a845a25d3f21fd03dfde52ee34e8021b0fdb118bc4fba649a7f1c515cef72999 +size 412255 diff --git a/HdFIT4oBgHgl3EQfXith/vector_store/index.pkl b/HdFIT4oBgHgl3EQfXith/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9a6da0a677fe581f9a131788e6a673ec5504dcdb --- /dev/null +++ b/HdFIT4oBgHgl3EQfXith/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c7fef4f699beecca518cbcbaaa966b966784b478c96675892c93a999a440dc78 +size 201336 diff --git a/I9A0T4oBgHgl3EQfCP_M/content/2301.01988v1.pdf b/I9A0T4oBgHgl3EQfCP_M/content/2301.01988v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..26c27bb9da1c7ee96b52ad1ab850bc6c49151b13 --- /dev/null +++ b/I9A0T4oBgHgl3EQfCP_M/content/2301.01988v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4afd551720837adc7ea41fcd00d27a356d0fdd85c325f0cf5bf6c53c9f80e579 +size 6409740 diff --git a/I9A0T4oBgHgl3EQfCP_M/vector_store/index.faiss b/I9A0T4oBgHgl3EQfCP_M/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5851c057da6a076dfde458afc38a3e5b9fbcdb63 --- /dev/null +++ b/I9A0T4oBgHgl3EQfCP_M/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6b86ccd3824bf4b544626b0806aa9ee8fc40cb53447782974aa12ce7e34d4b2e +size 8650797 diff --git a/I9AyT4oBgHgl3EQfsPkA/content/tmp_files/2301.00571v1.pdf.txt b/I9AyT4oBgHgl3EQfsPkA/content/tmp_files/2301.00571v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d657e777783bb4b3c90bbec58870d55acc35fbf0 --- /dev/null +++ b/I9AyT4oBgHgl3EQfsPkA/content/tmp_files/2301.00571v1.pdf.txt @@ -0,0 +1,1417 @@ +January 3, 2023 +1:32 +Konika˙2 +Modern Physics Letters A +© World Scientific Publishing Company +Modified Hawking temperature and entropy of Kerr-de Sitter black +hole in Lorentz violation theory +Y. Onika Laxmi +Mathematics Department, Manipur University +Canchipur, Manipur, India +T. Ibungochouba Singh +Mathematics Department, Manipur University +Canchipur, Manipur, India +ibungochouba@rediffmail.com +Received (Day Month Year) +Revised (Day Month Year) +In this paper, we discuss the tunneling of scalar particles near the event horizon of +stationary and nonstationary Kerr-de Sitter black hole using Lorentz violation theory in +curved space time. The modified form of Hamilton-Jacobi equation is derived from the +Klein-Gordon equation by applying Lorentz violation theory. The Hawking temperatures +derived from stationary and nonstationary Kerr-de Sitter black holes are modified due to +Lorentz violation theory. It is noted that the change in Bekenstein-Hawking entropy and +modified Hawking temperatures of stationary and nonstationary Kerr-de Sitter black +hole not only depend on the black hole parameters but also on ether like vectors uα. +Keywords: Hawking radiation; Bekenstein-Hawking entropy; Specific Heat; Lorentz vio- +lation theory +PACS Nos.: 04.70.Dy, 04.20.Gz, 03.65.-w. +1. Introduction +Hawking1,2 showed theoretically that a black hole radiates like a black body in +which the temperature of radiation is dependent on the surface gravity of black +hole. The discovery of Hawking radiation leads to black hole thermodynamics.3−5 +Since then, many scientists have proposed different techniques to study the quan- +tum tunneling of black holes. Refs. [6,7] proposed a new method to investigate the +quantum thermal and nonthermal radiations of stationary and nonstationary black +holes by applying the tortoise coordinate transformation, Maxwell’s electromagnetic +field equation, Klein-Gordon equation and Dirac equation. Applying this method, +many interesting results in different black holes have been derived in.8−11 Refs. +[12-14] introduced the semiclassical tunneling technique to investigate the Hawking +radiation of black hole. In this method, the Hawking radiation is taken as a tun- +1 +arXiv:2301.00571v1 [gr-qc] 2 Jan 2023 + +January 3, 2023 +1:32 +Konika˙2 +2 +Authors’ names +neling process near the event horizon of the black hole and the outgoing particle +produces the tunneling barrier of black hole. They obtain well-behaved coordinate +system which has no singularity near the event horizon to derive the emission rate. +Refs. [15-17], as an extension of Parikh and Wilczek approach, studied the Hawking +radiation as a tunneling of charged massive particle at the event horizon of black +hole by developing the relation between phase and group velocity of the tunneling +particle. Applying the Hamilton-Jacobi equation, Feynman prescription and WKB +approximation, Refs. [18] investigated the Hawking radiation at the event horizon +of rotating and nonrotating black holes. They showed that the isotropic coordinate +or invariant coordinate gives the correct Hawking temperature whereas naive coor- +dinate leads to half of correct Hawking temperature. Ref. [19] studied the Hawking +radiation at the event horizon of different black holes by using Dirac equation, +Feynman prescription and Pauli Sigma matrices. In this method, the appropriate +gamma matrices obtained from black hole and suitable wave function are substi- +tuted in Dirac equations, then the action related to Boltzman factor for emission in +accordance with semiclassical approximation is obtained. +Refs. [20-23] discussed the corrected Hawking temperature and entropy of black +holes using first law of black hole thermodynamics and Hamilton-Jacobi equation +beyond the semiclassical approximation in Schwarzschild like coordinate system and +Painleve coordinate system. Kruglov24,25 proposed the quantum tunneling of boson +near the event horizon of black hole using Proca equation, WKB approximation +and Feynman prescription. The emission temperature of Schwarzschild background +geometry which is the same as the Hawking temperature corresponding to scalar +particle is also obtained. Following their work, many interesting results have been +obtained in.26−29 +The study of Lorentz symmetry violation theory has been discussed during the +past decades and many researchers have proposed different gravity models induced +by Lorentz symmetry violation.30−34 Refs. [35-37] proposed the Lorentz symmetry +violation in flat space time using Dirac equation and ether like vectors uα. Since +then, the Lorentz violation has been extended to curved space time by choosing +ether like vectors uα so that it could hold uαuα = constant.38−42 +The paper is outlined as follows : In section 2, the modified Hawking tempera- +ture, heat capacity and change in entropy near the event horizon of Kerr-de Sitter +black hole are derived. In section 3, the modified surface gravity and the Hawking +temperature of nonstationary KdS black hole are discussed using tortoise coordinate +transformation in Lorentz violation theory in curved space time. Some discussions +and conclusions are given in section 4 and 5 respectively. +2. Modified Hawking temperature of Kerr-de Sitter black hole +The Kerr-de Sitter (KdS) solution indicates the space time geometry of a rotating +black hole with non-zero cosmological constant. The vacuum solution of KdS black +hole is well known Boyer-Lindquist coordinates (t, r, θ, φ) with geometrical unit + +January 3, 2023 +1:32 +Konika˙2 +3 +(c = G = 1) which is given by43 +ds2 = −∆ − ∆θa2 sin2 θ +R2Ξ2 +dt2 + R2 +∆θ +dθ2 − 2a[∆θ(r2 + a2) − ∆] sin2 θ +R2Ξ2 +dtdφ ++R2 +∆ dr2 + ∆θ(r2 + a2)2 − ∆a2 sin2 θ +R2Ξ2 +sin2 θdφ2, +(1) +where +R2 = r2 + a2 cos2 θ, +Ξ = 1 + 1 +3Λa2, +∆θ = 1 + 1 +3Λa2 cos2 θ, +∆ = (r2 + a2) +� +1 − 1 +3Λr2� +− 2Mr. +(2) +Eq. (1) represents rotating KdS black hole for cosmological constant Λ. The KdS +black hole is well defined in the region −∞ ≤ t ≤ ∞, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. M +and a are the mass and rotational parameter of KdS black hole. If 1 +Λ ≥ M 2 > a2, +then ∆ = 0 gives four distinct roots i.e. r+, rh, r− and r−− (r+ > rh > r− > +r−−). The biggest root r+ denotes the location of cosmological horizon. rh and +r− represent the location of the event horizon and Cauchy horizon respectively. If +r = 0, θ = π +2 , then the other side of r = 0, r = r−− is taken as another cosmological +horizon.44 According to Ref. [45], the event horizon of KdS black hole r = rh can +be written as +rh = 1 +Ξ +� +1 + 4ΛM 2 +3β2Ξ + ... +� � +M + +� +M 2 − a2Ξ +� +, +(3) +where β = +� +1 − Λ +3 a2. There is a frame dragging effect near the event horizon of +KdS black hole. Let φ = ϕ − Ωt and Ω = − g03 +g33 . Then Eq. (1) reduces to +ds2 = − +∆∆θR2 +Ξ2[∆θ(r2 + a2)2 − ∆a2 sin2 θ]dt2 + R2 +∆ dr2 + R2 +∆θ +dθ2 ++[∆θ(r2 + a2)2 − ∆a2 sin2 θ] sin2 θ +R2Ξ2 +dφ2, +(4) +where the angular velocity at the event horizon of KdS black hole is given as +Ω = +a +r2 +h + a2 . +(5) +According to Refs. [46, 47], the surface gravity of KdS black hole at the event +horizon r = rh is derived as +κ = lim +g00→0 +� +− 1 +2 +� +−g11 +g00 +dg00 +dr +� += (rh − M − 2 +3Λr3 +h − 1 +3Λa2rh) +Ξ(r2 +h + a2) +. +(6) +The Hawking temperature of KdS black hole is connected with surface gravity via +TH = +κ +2π as +TH = 1 +2π +� +(rh − M − 2 +3Λr3 +h − 1 +3Λa2rh) +Ξ(r2 +h + a2) +� +. +(7) + +January 3, 2023 +1:32 +Konika˙2 +4 +Authors’ names +To discuss heat capacity near the event horizon of black hole, the mass parameter +KdS black hole might be obtained from ∆(rh) = 0 as +M = rh +2 + a2 +2rh +− Λr3 +h +6 +− Λa2rh +6 +. +(8) +The heat capacity (Ch) of black hole is defined by +Ch = ∂M +∂TH += +� +∂M +∂rh +�� +∂rh +∂TH +� +. +(9) +The heat capacity near the event horizon of KdS black hole is calculated as +Ch = ∂M +∂TH += +2πΞ(r2 +h + a2)2[3r2 +h − 3a2 − Λr2 +h(3r2 +h + a2)] +3(a4 − r4 +h) + 4a2r2 +h(3 − 2Λr2 +h) − Λr2 +h(a4 + 3r4 +h). +(10) +The modified form of Hamilton-Jacobi equation in Lorentz theory in curved space +time is given by48 +(gµν + λuµuν)∂µI∂νI + m2 = 0. +(11) +where λ and uα are the correction parameter and ether like vectors respectively. As +λ tends to zero in the above equation, the Lorentz violation theory is cancelled and +the original Hamilton-Jacobi equation in curved space time is obtained. The ether +like vectors uα are constant in the flat space time of the canonical coordinate system. +The ether like vectors uα are not constant in curved space time. But we can take +the vectors uα from curved space time that satisfies the condition uαuα = constant. +To investigate the change in entropy of stationary KdS black hole, we can choose +uα from (4) that satisfies uαuα = constant and uα are related to coordinate system +acquired by the space time. The expressions of ut, ur, uθ and uφ are defined by +ut = +ct +√−gtt += +ctΞ +� +∆θ(r2 + a2)2 − ∆a2 sin2 θ +√∆∆θR2 +, +ur = +cr +√grr += cr +� +∆ +R2 , +uθ = +cθ +√gθθ += cθ +� +∆θ +R2 , +uφ = +cφ +√gφφ += +cφΞ +√ +R2 +� +[∆θ(r2 + a2)2 − ∆a2 sin2 θ] sin2 θ +, +(12) +where ct, cr, cθ and cφ are arbitrary constants. uα satisfies the condition +uαuα = −c2 +t + c2 +r + c2 +θ + c2 +φ = constant. +(13) +Using Eqs. (12) and (4) in Eq. (11), the dynamical equation of scalar particle +with mass m in stationary KdS black hole is obtained as +g00(∂I +∂t )2 + g11(∂I +∂r )2 + g22(∂I +∂θ )2 + g33( ∂I +∂φ)2 + λuµuν∂µI∂νI + m2 = 0. (14) + +January 3, 2023 +1:32 +Konika˙2 +5 +It is known that the above equation involves the variables t, r, θ and φ. To separate +the variables on the Hamilton principal functions I, we can choose the action I as +follows +I = −ωt + S(r, θ) + jφ + δ, +(15) +where S(r, θ), ω and j are the generalized momentum, particle energy and angular +momentum along the φ-axis respectively and δ is a complex constant. Using Eq. +(15) in Eq. (14), a quadratic equation in ∂S +∂r is obtained as +A +� +∂S +∂r +�2 ++ B +�∂S +∂r +� ++ C = 0. +(16) +Then the two roots of the above equation are given by +S = +� −B ± +√ +B2 − 4AC +2A +dr, +(17) +where +A = g11 + λurur, +B = 2λuruθ +�∂I +∂θ +� +− 2λutur(ω − jΩ) + 2λuruφj, +C = (g00 + λutut)(ω − jΩ)2 + (g22 + λuθuθ) +�∂I +∂θ +�2 ++ (g33 + λuφuφ)j2 +−2λut(ω − jΩ) +� +uθ +�∂I +∂θ +� ++ uφj +� ++ 2λuθuφj +�∂I +∂θ +� ++ m2. +(18) +Applying residue theorem of complex analysis and Feynman prescription near the +event horizon of KdS black hole, the integration of Eq. (17) can be written as +S± = +iπΞ(r2 +h + a2) +� +λctcr ± +� +1 − λc2 +t + λc2r +� +(w − jΩ) +2(1 + λc2r){(1 − Λ +3 a2)rh − M − 2Λr3 +h +3 +} +, +(19) +where S+ and S− are the outgoing particle and ingoing particle respectively. The +probabilities which cross the black hole near the event horizon are given by +Γemission = exp(−2ImI) = exp[−2(ImS+ + Imδ)] +(20) +and +Γabsorption = exp(−2ImI) = exp[−2(ImS− + Imδ)]. +(21) +There is a 100% chance the ingoing particle to enter the KdS black hole according +to WKB approximation. This indicates that ImS+ = −ImS−. We calculate the +probability of outgoing particle as +Γrate = Γemission +Γabsorption += exp +� +− +2γΞπ(r2 +h + a2)(ω − jΩ) +{(1 − Λ +3 a2)rh − M − 2Λr3 +h +3 +} +� +, +(22) + +January 3, 2023 +1:32 +Konika˙2 +6 +Authors’ names +where γ = +√ +1−λc2 +t +λc2r +1+λc2r +. Eq. (22) is similar to Boltzmann factor according to semi- +classical approximation. The Hawking temperature near the event horizon of KdS +black hole in Lorentz violation theory is given by +T = {(1 − Λ +3 a2)rh − M − 2Λr3 +h +3 +} +2γπΞ(r2 +h + a2) +. +(23) +If λ = 0 in the above equation, the Lorentz violation has been cancelled. In such +case Eq. (23) is consistent with Eq. (7). If γ > 1 or γ < 1, the Hawking temperature +decreases or increases due to the presence of correction term λ and ether like vectors +uα. +The modified heat capacity at the event horizon of KdS black hole is obtained +as +C +′ +h = ∂M +∂T = +2πγΞ(r2 +h + a2)2[3r2 +h − 3a2 − Λr2 +h(3r2 +h + a2)] +3(a4 − r4 +h) + 4a2r2 +h(3 − 2Λr2 +h) − Λr2 +h(a4 + 3r4 +h). +(24) +As γ tends to unity, Eq. (24) is consistent with original heat capacity given in Eq. +(10). If γ > 1 or γ < 1, the heat capacity increases or decreases near the event +horizon of KdS black hole depending upon the choices of correction term λ and +ether like vectors uα. +Using Eqs. (3) and (5) in Eq. (19) for the outgoing particle, we get +ImS = γ +′ +2 +� +πk2 +1k2 +2 +Ξ +β2 +Ξ k1k2 − M − δ +ω + +πΞa2 +β2 +Ξ k1k2 − M − δ +ω − +πΞa +β2 +Ξ k1k2 − M − δ +j +� +, (25) +where +γ +′ = λctcr + +� +1 − λc2 +t + λc2r +1 + λc2r +, +k1 = +� +1 + 4ΛM 2 +3β2Ξ + ... +� +, +k2 = +� +M + +� +M 2 − a2Ξ +� +, +δ = 2Λ +3Ξ3 +� +1 + 4ΛM 2 +3β2Ξ + ... +�3 � +M + +� +M 2 − a2Ξ +�3 +. +(26) +To obtain the biggest value of the integration, we ignore second order terms of +KdS black hole mass parameter at the numerator and denominator in Eq. (25). +Then we find as +ImS = γ +′ +2 +� +� +πk2 +2 +β2 +� +k2 − MΞ +β2 +�ω + +πΞ2a2 +β2 +� +k2 − MΞ +β2 +�ω − +πΞ2a +β2 +� +k2 − MΞ +β2 +�j +� +� . +(27) +Taking the self-gravitational interaction into account, the mass parameter KdS black +hole is allowed to fluctuate. If a black hole emits a particle ω and angular momentum +j, the KdS black hole parameter will be M − ω and J − j respectively. To find the + +January 3, 2023 +1:32 +Konika˙2 +7 +change in Bekenstein-Hawking entropy of KdS black hole, the term (1 − +Ξ +β2 )M is +ignored. Then we get +ImS = γ +′ +2 +�� ω +0 +πk2 +2 +β2√ +M 2 − a2Ξ +dω +′ + +� ω +0 +πΞ2a2 +β2√ +M 2 − a2Ξ +dω +′ − +� j +0 +πΞ2a +β2√ +M 2 − a2Ξ +dj +′� +. +(28) +Changing M by M − ω and j by J − j and putting the value of k2 in the above +equation, we obtain +ImS = γ +′ +2 +� +−π +β2 +� M−ω +M +k2 +2 +� +(M ′ − ω′)2 − a2Ξ +d(M ′ − ω +′) +−πΞ2a2 +β2 +� M−ω +M +1 +� +(M ′ − ω′)2 − a2Ξ +d(M ′ − ω +′) ++πΞ2a +β2 +� J−j +j +1 +� +(M ′ − ω′)2 − a2Ξ +d(J − j′) +� +, +(29) +where J − j = (M − ω)a. The imaginary part of the action finally yields +ImS = −γ +′π +2β2 +� � M−ω +M +2(M ′ − ω′)2 + 2(M ′ − ω′) +� +(M ′ − ω′)2 − a2Ξ +� +(M ′ − ω′)2 − a2Ξ +d(M ′ − ω +′) +− +� M−ω +M +a2Ξ +� +(M ′ − ω′)2 − a2Ξ +d(M ′ − ω +′) +� +. +(30) +Calculating the ω +′ integral, that gives +ImS = −γ +′π +2β2 +� +(M − ω) +� +(M − ω)2 − a2Ξ + (M − ω)2 − M +� +M 2 − a2Ξ − M 2 +� +, += −γ +′π +4β2 +�� +(M − ω) + +� +(M − ω)2 − a2Ξ +�2 +− +� +M + +� +M 2 − a2Ξ +�2� +, += −γ +′π +2 +(r2 +f − r2 +i ). +(31) +Using WKB approximation, the tunneling rate is obtained as +Γ ∼ exp(−2ImS), += exp[γ +′π(r2 +f − r2 +i )], += exp[γ +′∆SBH], +(32) +where γ′∆SBH = γ′(SBH(M − ω) − SBH(M)) is the modified entropy of +KdS black hole in Lorentz violation theory. ri = +1 +√ +2β +� +M + +√ +M 2 − a2Ξ +� +and +rf = +1 +√ +2β +� +(M − ω) + +� +(M − ω)2 − a2Ξ +� +are the locations of horizons before +and after emission of particle. If λ = 0, the Lorentz violation theory has been +cancelled, the original change in Bekenstein-Hawking entropy is obtained. When + +January 3, 2023 +1:32 +Konika˙2 +8 +Authors’ names +γ +′ > 1, the change in Bekenstein-Hawking entropy increases and γ +′ < 1, the change +in Bekenstein-Hawking entropy decreases near the event horizon of KdS black hole. +In above cases, the change in Bekenstein-Hawking entropy depends on correction +parameter λ and ether like vectors uα. +3. Nonstationary rotating KdS black hole +The +metric +of +rotating +nonstationary +KdS +black +hole +in +retarded +time +coordinates10(u, r, θ, φ) is defined by +ds2 = +1 +R2Ξ2 [∆λ − ∆θa2 sin2 θ]du2 − R2 +∆θ +dθ2 + +2a +R2Ξ2 [∆θ(r2 + a2) − ∆λ] sin2 θdudφ ++ 2 +Ξ[du − a sin2 θdφ]dr − +1 +R2Ξ2 [∆θ(r2 + a2)2 − ∆λa2 sin2 θ] sin2 θdφ2, +(33) +where R2, Ξ, ∆θ are given in Eq. (2). The term ∆λ is given by +∆λ = r2 + a2 − 2M(u)r − 1 +3Λr2(r2 + a2), +(34) +where M(u) is the mass of nonstationary KdS black hole. The location of event +horizon of stationary and nonstationary KdS black hole can be obtained from null +surface equation F(u, r, θ, φ) = 0. The expression of null surface equation is given +by +gµν ∂F +∂xµ +∂F +∂xν = 0. +(35) +The location of event horizon of stationary and nonstationary black holes can be +obtained from the null surface equation (35) using generalized tortoise coordinate +transformation. The space time geometry outside the event horizon of nonstationary +black hole can be described by the tortoise coordinate and in such case, r∗ tends to +positive infinity when tending to infinite point and r∗ tends to negative infinity at +the event horizon of black hole. To study the Hawking radiation of nonstationary +black hole, the tortoise coordinate transformation is defined by49−54 +r∗ = r + +1 +2κ(u0, θ0, φ0)lnr − rh(u, θ, φ) +rh(u, θ, φ) +, +u∗ = u − u0, θ∗ = θ − θ0, φ∗ = φ − φ0, +(36) +where u0, θ0 and φ0 are the arbitrary constants under the coordinate transformation. +From the above equation, we get +∂ +∂r = +� +1 + +1 +2κ(r − rh) +� ∂ +∂r∗ +, +∂ +∂u = +∂ +∂u∗ +− +rrh,u +2κrh(r − rh) +∂ +∂r∗ +, +∂ +∂θ = +∂ +∂θ∗ +− +rrh,θ +2κrh(r − rh) +∂ +∂r∗ +, + +January 3, 2023 +1:32 +Konika˙2 +9 +∂ +∂φ = +∂ +∂φ∗ +− +rrh,φ +2κrj(r − rh) +∂ +∂r∗ +, +(37) +where rh,u = ∂rh +∂u , rh,θ = ∂rh +∂θ and rh,φ = ∂rh +∂φ . rh,u represents the evaporation rate +at the event horizon of KdS space time. If ∂rh +∂u > 0, the event horizon of KdS space +time is expanded (absorbing black hole ) and when ∂rh +∂u < 0, the event horizon of +KdS space time is contracted. κ ≡ κ(u0, θ0, φ0) is taken as surface gravity of KdS +space time which depends on retarded time and angular coordinates. Using Eqs. +(33) and (36) in Eq. (35) and taking r → rh, the horizon equation of nonstationary +KdS black hole is derived as +a2Ξ2 sin2 θr2 +h,u +∆θ ++ 2(r2 +h + a2)Ξrh,u + 2aΞ2rh,urh,φ +∆θ ++ 2aΞrh,u ++∆λ(rh) + ∆θr2 +h,θ + +Ξ2r2 +h,φ +∆θ sin2 θ = 0, +(38) +where ∆λ(rh) = r2 +h + a2 − 2M(u)rh − 1 +3Λr2 +h(r2 +h + a2). It is noted that the location +of event horizon of nonstationary black hole varies with retarded time u = t − r∗ +and angular co-ordinates θ, φ. From Eqs. (11) and (33), the dynamical equation of +scalar particle with mass m in curved space time is obtained as +g00�∂I +∂u +�2 ++ 2g01�∂I +∂u +��∂I +∂r +� ++ 2g03�∂S +∂u +�� ∂I +∂φ +� ++ g11�∂I +∂r +�2 ++ 2g13�∂I +∂r +�� ∂I +∂φ +� ++g22�∂I +∂θ +�2 ++ g33� ∂I +∂φ +�2 ++ λuµuν∂µI∂νI + m2 = 0. +(39) +Since ether like vectors are not constant in curved space time, we can choose uα +from nonstationary space time Eq. (33) so that we can make uαuα = constant. +The ether like vectors uα are related to the properties of black hole and system of +coordinate adopted by the metric space. The ether like vectors uα are choosen as +uu = +ku +√guu += +ku +√ +R2Ξ +� +∆λ − ∆θa2 sin2 θ +, +ur = kr√guu +gur += kr +� +∆λ − ∆θa2 sin2 θ +√ +R2 +, +uθ = +kθ +√gθθ += kθ +� +−∆θ +R2 , +uφ = ∆λkφ +√gφφ += +∆λkφ +√ +R2Ξ +� +−[∆θ(r2 + a2)2 − ∆λa2 sin2 θ] sin2 θ +, +(40) +where ku, kr, kθ and kφ are arbitrary constants. Using Eqs. (37) and (40) in Eq. +(39), we get +D +E +� ∂I +∂r∗ +�2 ++ 2 +� ∂I +∂u∗ +�� ∂I +∂r∗ +� ++ 2F +E +� ∂I +∂r∗ +� ++ 2κ(r − rh)G +E = 0, +(41) +where +D = +1 +2κ(r − rh)r2 +h +� +(g00 + λuuuu)r2r2 +h,u − 2rh +� +(g01 + λuuur)rrh,u + (g13 + +January 3, 2023 +1:32 +Konika˙2 +10 +Authors’ names ++λuφur)rrh,φ + λuruθrrh,θ +� +{2k(r − rh) + 1} + 2(g03 + λuuuφ)r2rh,urh,φ ++r2 +h(g11 + λurur){2κ(r − rh) + 1}2 + (g22 + λuθuθ)r2r2 +h,θ ++(g33 + λuφuφ)r2r2 +h,φ + 2λuθr2rh,θ(uurh,u + uφrh,φ) +� +, +E = −(g00 + λuuuu)rrh,u +rh ++ (g01 + λuuur){2k(r − rh) + 1} +−(g03 + λuuuφ)rrh,φ +rh +− λuuuθrrh,θ +rh +, +F = −(g03 + λuuuφ)rrh,upφ +rh ++ {g13pφ + λur(uφpφ + uθpθ)}{2k(r − rh) + 1} +−(g22 + λuθuθ)rrh,θpθ +rh +− (g33 + λuφuφ)rrh,φpφ +rh +− λuuuθrrh,upθ +rh +−λuθuφrrh,φpθ +rh +− λuθuφrrh,θpφ +rh +, +G = g00ω2 − 2g03pφω + g22p2 +θ + g33p2 +φ + λuuuuω2 − 2λuuuφωpφ + λuθuθp2 +θ ++λuφuφp2 +φ − 2λuuuθωpθ + 2λuθuφpθpφ + m2. +(42) +To study the Hawking temperature, the action I in Eq. (15) can be writen as I = +−ωu∗ + I0(r∗, θ∗, φ∗), then we get +∂I +∂u∗ += −ω, +∂I +∂θ∗ += pθ, +∂I +∂φ∗ += pφ, +(43) +where ω is the energy of emitted scalar particle. pθ and pφ are the components of +generalized momenta of scalar particle along the angular coordinates θ and φ respec- +tively. To obtain surface gravity and Hawking temperature at the event horizon of +nonstationary KdS black hole, we assume that the coefficient of +� +∂I +∂r∗ +�2 +approaches +to unity as r → rh, u → u0, θ → θ0 and φ → φ0. From Eq. (41), an infinite limit of +the form 0 +0 is obtained near the event horizon of KdS black hole. Using L’Hopital’s +rule, the surface gravity is obtained as +κ = rh(1 + 2Ξrh,u) − M − 2 +3Λr3 +h − Λ +3 a2rh − r−1 +h {∆λ + Ξ(r2 +h + a2)rh,u + aΞrh,φ} +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z + λY +, +(44) +where Z and Y are given by +Z = 2∆θr2 +h,θ + rh,φ +�4aΞ2rh,u +∆θ ++ aΞ2 +∆θ ++ 2Ξ2rh,φ +∆θ sin2 θ + 2aΞ +� +, +Y = k2 +uρ4rh,uΞ2 +∆θa2 sin2 θ + ρ2kukrΞ − ρ2kukθΞrh,θ +a sin θ +. +(45) +Using Eq. (43) in Eq. (41) and taking r → rh, then we obtain +� +∂I +∂r∗ +�2 ++ 2(ω − ω0) ∂I +∂r∗ += 0. +(46) + +January 3, 2023 +1:32 +Konika˙2 +11 +The value of chemical potential, ω0 near the event horizon of nonstationary KdS +black hole is +w0 = +1 +g01 − g00rh,u − g03rh,φ − λuuuurh,u + λurur − λuuuφrh,φ − λuuuθrh,θ +×[g13pφ − g03rh,upφ − g22rh,θpθ − g33rh,φpφ − λuurh,u(uφpφ + uθpθ) +−λ(uθrh,θ + uφrh,φ)(uθpθ + uφpφ) + λur(uθpθ + uφpφ)]. +(47) +Eq. (47) represents the chemical potential of nonstationary KdS space time due +to tortoise coordinate transformation (36). The chemical potential, w0 depends on +the black hole mass, cosmological constant, generalized momenta of scalar particle, +retarded time, angular coordinates, correction parameter λ and ether like vectors +uα. If λ and uα tend to zero, Eq. (47) is consistent with earlier literatures [10, 46]. +From the solution of Eq. (37), we obtain +∂I +∂r∗ += [2κ(r − rh) + 1](ω − ω0) ± (ω − ω0) +[2κ(r − rh)] +. +(48) +It is observed that Eq. (48) has a singularity near the event horizon of KdS black +hole. Integrating Eq. (48) by applying residue theorem of complex analysis and +Feynman prescription, the imaginary part of the radial action I is derived as +ImI± = π +2κ[(ω − ω0) ± (ω − ω0)], +(49) +where I+ and I− represent the outgoing scalar particle and ingoing scalar particle at +the event horizon of KdS black hole. Taking outgoing and ingoing of scalar particle, +the tunneling probability which crosses at the event horizon of KdS black hole is +calculated as +Γ = Γemission +Γabsorption += exp[−(ω − ω0) +T +]. +(50) +The modified Hawking temperature near the event horizon of nonstationary KdS +black hole due to Lorentz violation theory is given by +T = Th(1 + λH)−1 += Th(1 − λH + λ2H2 − λ3H3 + ...), +(51) +where the value of Th and H are +Th = 1 +2π +� +rh(1 + 2Ξrh,u) − M − 2 +3Λr3 +h − Λ +3 a2rh +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z +− +r−1 +h {∆λ + Ξ(r2 +h + a2)rh,u + aΞrh,φ} +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z +� +(52) +and +H = +k2 +uρ4rh,uΞ2 +∆θa2 sin2 θ + ρ2kukrΞ − ρ2kukθΞrh,θ +a sin θ +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z +. +(53) + +January 3, 2023 +1:32 +Konika˙2 +12 +Authors’ names +From Eqs. (52) and (53), it is known that the Hawking temperature near the +event horizon of KdS black hole is modified due to Lorentz violation theory. The +modified Hawking temperature (T) of nonstationary KdS black hole depends not +only on the mass of the black hole but also on the properties of event horizon, +cosmological constant Λ, retarded time u, correction term λ and on the ether like +vectors uα. If kuρ2rh,uΞ + ∆θa sin θ(a sin θkr − kθrh,θ) = 0, then H −→ 0. In such +case, our result is consistent with the earlier literatures.10,11,46 +4. Discussion +First the line element of stationary KdS black hole is transformed into static form +using frame dragging effect given in Eq. (4). Using modified form of Hamilton-Jacobi +Fig. 1. +Plot of original and modified Hawking temperature with radius of event horizon, rh of +KdS black hole. Here a = 0.1, Λ = 0.6, λ = 0.005, cr = 1.5, ct = 0.4. +equation, Feynman prescription and WKB approximation, the modified Hawking +temperature, heat capacity and change in Bekenstein-Hawking entropy are derived +in Eq. (23), Eq. (24) and Eq. (31) respectively. It is noted that both are dependent +on correction term λ and ether like vectors uα. +If γ > 1, the modified Hawking temperature near the event horizon of KdS black +hole decreases and if γ < 1, the modified Hawking temperature increases in Eq. (23). +If γ = 1, the modified heat capacity (24) approaches to original heat capacity (10). + +0.4 +^ Modified-Hawking +0.2 +口1 +Hawking +0 +-0.2 +-0.4 +-0.6 +-0.8 +-1 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +rnJanuary 3, 2023 +1:32 +Konika˙2 +13 +If γ > 1, the modified heat capacity increases and if γ < 1, the modified heat +capacity is smaller than that of original heat capacity near the event horizon of +stationary KdS black hole. +The change in Bekenstein-Hawking entropy near the event horizon of KdS black +hole increases or decreases if γ′ > 1 or γ′ < 1. When λ = 0 and ct = cr, λ ̸= 0, the +Lorentz violation has been cancelled and the original change in Bekenstein-Hawking +entropy near the event horizon of KdS black hole is recovered. +The Hawking temperature of rotating nonstationary KdS black hole is also inves- +tigated using Klein-Gordon equation, generalized tortoise coordinate transformation +and L’Hopital rule in Lorentz violation theory. According to Damour and Ruffini6 +and Sannan7, the traditional coordinate transformation is given by +r∗ = r + +1 +2κ1 +ln(r − rh(u, θ, φ)), +u∗ = u − u0, θ∗ = θ − θ0, φ∗ = φ − φ0. +(54) +Using Eq. (54) in Eq. (41), the modified surface gravity and Hawking temperature +of nonstationary rotating KdS black hole are +κ1 = +rh(1 + 2Ξrh,u) − M − 2 +3Λr3 +h − Λ +3 a2rh +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z + λY +(55) +and +T1 = 1 +2π +rh(1 + 2Ξrh,u) − M − 2 +3Λr3 +h − Λ +3 a2rh +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z + λY +. +(56) +In Eqs. (51) and (56) , if we put λ = pθ = pφ = 0 and kuρ2rh,uΞ+∆θa sin θ(a sin θkr− +kθrh,θ) = 0, Eqs. (51) and (56) are concordant with the literature.46 Eq. (54) gives +another chemical potential of nonstationary rotating KdS black hole in Lorentz +violation theory as +wp = +1 +g01 − g00rh,u − g03rh,φ − λuuuurh,u + λurur − λuuuφrh,φ − λuuuθrh,θ +×[g13pφ − g03rh,upφ − g22rh,θpθ − g33rh,φpφ − λuurh,u(uφpφ + uθpθ) +−λ(uθrh,θ + uφrh,φ)(uθpθ + uφpφ) + λur(uθpθ + uφpφ)]. +(57) +It is observed that Eq. (57) is consistent with Eq. (47). The surface gravities derived +from Eq. (44) and (55) can be combined as +κ = κ1 + Ξ1, +(58) +where κ and κ1 represent the surface gravities of nonstationary KdS black hole due +to tortoise coordinate transformations (36) and (54). Ξ1 indicates the constant term +due to tortoise coordinate transformation (36) and its expression is +Ξ1 = − +r−1 +h {∆λ + Ξ(r2 +h + a2)rh,u + aΞrh,φ} +{Ξ(r2 +h + a2) + a2Ξ2 sin2 θrh,u +∆θ +}(1 + 2rh,u) + Z + λY +. +(59) + +January 3, 2023 +1:32 +Konika˙2 +14 +Authors’ names +The rate of correction for Hawking temperature near the event horizon of nonsta- +tionary KdS black hole is given by +δ = − r−1 +h {∆λ + Ξ(r2 +h + a2)rh,u + aΞrh,φ} +rh(1 + 2Ξrh,u) − M − 2 +3Λr3 +h − Λ +3 a2rh +. +(60) +It is worth mentioning that the correction rate is independent of correction term λ +and the ether like vectors uα but depends on black hole mass M, rotational param- +eter a, angular coordinate θ, cosmological constant Λ and generalized momenta. For +stationary KdS black hole in the absence of Lorentz violation theory, Eq. (51) and +(56) reduce to the same Hawking temperature as +TH = T1 = 1 +2π +� +rh − M − 2 +3Λr3 +h − Λ +3 a2rh +Ξ(r2 +h + a2) +� +. +(61) +It is noted that for the stationary KdS space time, the different tortoise coordinate +transformations give the same Hawking temperature in the absence of Lorentz vio- +lation theory which is exactly equal to the actual calculation given in Eq. (7). From +Eq. (58), κ1 approaches to zero, the Hawking temperature T does not tend to zero +due to extra term Ξ1. If Ξ1 approaches to zero in Eq. (58), Eq. (44) is consistent with +Eq. (55). From Eqs. (47) and (57), we observe that the chemical potential derived +from different tortoise coordinate transformations are equal near the event horizon +of black hole. It can be concluded that the tortoise coordinate transformation given +in Eq. (36) is more suitable and accurate in the study of modified surface gravity +near the event horizon of nonstationary KdS space time in Lorentz violation theory. +From Eqs. (44) and (55), the different surface gravities are obtained using the dif- +ferent tortoise coordinate transformations near the event horizon of nonstationary +KdS black hole in Lorentz violation theory. +5. Conclusion +In this paper, the tunneling of scalar particle near the event horizon of stationary +KdS black hole is investigated using Klein-Gordon equation in Lorentz violation +theory, Feynman prescription and WKB approximation. Then the corresponding +Hawking temperature, heat capacity and change in Bekenstein-Hawking entropy +near the event horizon are derived. The Hawking temperature, heat capacity and +change in entropy are modified due to presence of correction term λ and ether like +vectors uα. +The modified surface gravities of nonstationary rotating black hole are also stud- +ied using different tortoise coordinate transformations in Lorentz violation theory. +Using null surface equation and tortoise coordinate transformation, the horizon +equation of nonstationary KdS black hole is obtained. The modified surface grav- +ities and the modified Hawking temperatures are derived with the help of event +horizon equation. It is known that the modified Hawking temperature depends not +only on the correction term λ but also on the ether like vectors uα. If we use Eq. + +January 3, 2023 +1:32 +Konika˙2 +15 +(36) in the study of surface gravity and Hawking radiation of black hole, a constant +term Ξ1 is seen to be appeared in the expressions of surface gravity and Hawking +temperature near the event horizon of KdS black hole. If Ξ1 tends to zero, the +two modified surface gravities and Hawking temperatures are equal. If λ and Ξ1 ap- +proach to zero, the original surface gravities near the event horizon of nonstationary +black hole are recovered and are concordant with the earlier literatures.8,9,54 It is +also seen that the correction rate δ of Hawking temperature in Lorentz violation +theory does not depend on correction term λ and the ether like vectors uα. The +different tortoise coordinate transformations yield the same chemical potential at +the event horizon of black hole but the values of surface gravities and Hawking +temperatures are different. This shows that Eq. (36) is more reliable and accurate +in the study of modified surface gravity near the event horizon of nonstationary +of KdS black hole. For the stationary space time, the different methods yield the +same Hawking temperature and the change in Bekenstein-Hawking entropy in the +absence of Lorentz violation theory. +Acknowledgments +The first author acknowledges the DST INSPIRE, New Delhi, India for providing +financial support (Grant No. IF190759). +References +1. S. W. Hawking, Nature 248, 30 (1974). +2. S. W. Hawking, Commun. Math. Phys 43, 199 (1975). +3. J .D. Bekenstein, Phys. Rev. D 7, 2333 (1973). +4. J. D. Bekenstein, Phys. Rev. D 9, 3292 (1974). +5. J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973). +6. T. Damour and R. Ruffini Phys. Rev. D 14, 332 (1976). +7. S. Sannan, Gen. Relativ. Gravit. 20, 239 (1988). +8. S. Q. Wu and X. Cai, Gen. Relativ. Gravit. 33, 1181 (2001). +9. S. Q. Wu and X. Cai, Gen. Relativ. Gravit. 34, 557 (2002). +10. T. S. Ibungochouba, I.M. Ablu and K.S. Yugindro, Int. J. Mod. Phys. D 25, 1650061 +(2016). +11. T. S. Ibungochouba, Adv. High Phys. 248, 30 (1974). +12. P. Kraus and F. Wilczek, Mod. Phys. Lett. A 9, 3713 (1994). +13. P. Kraus and F. Wilczek, Nucl. Phys. B 437, 231 (1995). +14. M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000.) +15. J. Y. Zhang and Z. Zhao, Nucl. Phys. B 725, 173 (2005). +16. J. Y. Zhang and Z. Zhao, Phys. Lett. B 638, 110 (2006). +17. J. Y. Zhang and Z. Zhao, JHEP 10, 055 (2005). +18. M. Angheben, M. Nadalini, L. Vanzo and S. Zerbini, JHEP 05, 014 (2005). +19. R. Kerner and B. Mann, Class. Quantum Gravity 25, 095014 (2008). +20. R. Banerjee and B. R. Majhi, JHEP 06, 095 (2008). +21. R. Banerjee and B. R. Majhi, Phys. Lett. B 674, 218 (2009). +22. R. Banerjee and S. K. Modak, JHEP 05, 063 (2009). +23. B. R. Majhi, Phys. Rev. D 79, 044005 (2009). + +January 3, 2023 +1:32 +Konika˙2 +16 +Authors’ names +24. S. I. Kruglov, Mod. Phys. Lett. A 29, 1450203 (2014). +25. S. I. Kruglov, Int. J. Mod. Phys. A 29, 1450118 (2014). +26. T. S. Ibungochouba, I. M. Ablu and K. S. Yugindro, Astropyhs. Space Sci. 361, 103 +(2016). +27. I. Sakalli and A. Ovgun, J. Exp. Theor. Phys. 121, 404 (2015). +28. I. Sakalli and A. Ovgun, Eur. Phys. J. Plus. 130, 110 (2015). +29. T. S. Ibungochouba, Y. M. Kenedy, I. M. Ablu and K. S. Yugindro, Ind. J. Phys. 94, +2061 (2020). +30. P. Horava, Phys. Rev. D 79, 084008 (2009). +31. T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001). +32. K. Lin, S. Mukohyama, A. Z. Wang and T. Zhu, Phys. Rev. D 89, 084022 (2014). +33. V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004). +34. V. A. Kostelecky and Z. Li, Phys. Rev. D 103, 024059 (2021). +35. R. Bluhm, Phys. Rev. D 91, 065034 (2015). +36. R. Casana, M. M. Ferreira and R. P. M. Moreira, Phys. Rev. D 84, 125014 (2011). +37. Z. E. Liu, X. Tan, J. Zhang and S. Z. Yang, Commun. Theor. Phys. 73, 045402 (2021). +38. J. R. Nascimento, A. Yu. Petrov and C. M. Reyes, Phys. Rev. D 92, 045030 (2015). +39. J. Zhang, M. Liu, Z. Liu, B. Sha, X. Tan and Y. Liu, Gen. Relativ. Gravit. 52, 105 +(2020). +40. Z. E. Liu, Y. Z. Liu, J. Zhang and S. Z. Yang, EPL 134, 50008 (2021). +41. Z. E. Liu, J. Zhang and S. Z. Yang, Results in Physics 29, 104710 (2021). +42. M. Gomes, J. R. Nascimento, A. Yu. Petrov and A. J. da Silva, Phys. Rev. D 81, +045018 (2010). +43. B. Carter, Commun. Math. Phys.17, 233 (1970). +44. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977). +45. M. I. Hossain and M. A. Rahman, arXiv: 1309.0502v1 [gr-qc]. +46. S. Christina and T. S. Ibungochouba, Gen. Relativ. Gravit. 53, 43 (2021). +47. Y. M. Kenedy, T. S. Ibungochouba and I. M. Ablu, Chin. Phys. Lett. 36, 030401 +(2019). +48. Y. L. Onika, T. S. Ibungochouba and I. M. Ablu, arXiv: 124545v2 (2022). +49. J. Yang, Z. Zhao, G. H. Tian and W. B. Liu, Chin. Phys. Lett. 26, 120401 (2009). +50. N. Ibohal and T. S. Ibungochouba, Astropyhs. Space Sci. 333, 175 (2011). +51. X. G. Lan, Int. J. Theor. Phys. 51, 1195 (2012). +52. X. G. Lan, Q.Q. Jiang and L. F. Wei, Eur. Phys. J. C 72, 1983 (2012). +53. T. S. Ibungochouba, Astropyhs. Space Sci. 347, 271 (2013). +54. I. M. Ablu, T. S. Ibungochouba and K. S. Yugindro, Int. J. Mod. Phys. D 23, 1450077 +(2014). + diff --git a/J9AzT4oBgHgl3EQfj_2L/content/tmp_files/2301.01525v1.pdf.txt b/J9AzT4oBgHgl3EQfj_2L/content/tmp_files/2301.01525v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19b1237a99e6afb278cbcaa537558524249cfdcd --- /dev/null +++ b/J9AzT4oBgHgl3EQfj_2L/content/tmp_files/2301.01525v1.pdf.txt @@ -0,0 +1,1060 @@ +ASCENT – A balloon-borne hard X-ray imaging spectroscopy +telescope using transition edge sensor microcalorimeter detectors +F. Kislata, D. Beckerb, D. Bennettc, A. Dasguptaa, J. Fowlerc, C. Fryerd, J. Gardb, E. Gaue, D. +Gurgewf, K. Harmong, T. Hayashif, S. Heatwoleg, M.A. Hossene, H. Krawczynskie,h,i, R.J. +Lanzig, J. Legerea, J.A.B. Matesc, M. McConnella, J. Nagye,h,i, T. Okajimaf, T. Satoj, D. +Schmidtc, S. Spoonera, D. Swetzc, K. Tamuraf, J. Ullomc, J. Weberb, A. Westera, P. Youngk +aUniversity of New Hampshire, Department of Physics & Astronomy and Space Science Center, 8 College Rd., +Durham, NH 03824, USA +bUniversity of Colorado, Department of Physics, 2000 Colorado Ave, Boulder, CO 80309, USA +cNIST Boulder Laboratories, 325 Broadway, Boulder, CO 80305, USA +dLos Alamos National Laboratory, Los Alamos, NM 87545, USA +eWashington University in St. Louis, Physics Department, 1 Brookings Dr., CB 1105, St. Louis, MO 63130, USA +fNASA’s Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA +gNASA Wallops Flight Facility, 32400 Fulton St., Wallops Island, VA 23337, USA +hMcDonnell Center for the Space Sciences at Washington University in St. Louis +iQuantum Sensor Center at Washington University in St. Louis +jRikkyo University, 3-34-1 Nishi Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan +kArizona State University, School of Earth and Space Exploration, Tempe, AZ 85287, USA +Abstract. +Core collapse supernovae are thought to be one of the main sources in the galaxy of elements heavier +than iron. Understanding the origin of the elements is thus tightly linked to our understanding of the explosion mecha- +nism of supernovae and supernova nucleosynthesis. X-ray and gamma-ray observations of young supernova remnants, +combined with improved theoretical modeling, have resulted in enormous improvements in our knowledge of these +events. The isotope 44Ti is one of the most sensitive probes of the innermost regions of the core collapse engine, and +its spatial and velocity distribution are key observables. Hard X-ray imaging spectroscopy with the Nuclear Spectro- +scopic Telescope Array (NuSTAR) has provided new insights into the structure of the supernova remnant Cassiopeia A +(Cas A), establishing the convective nature of the supernova engine. However, many questions about the details of this +engine remain. We present here the concept for a balloon-borne follow-up mission called ASCENT (A SuperConduct- +ing ENergetic x-ray Telescope). ASCENT uses transition edge sensor gamma-ray microcalorimeter detectors with a +demonstrated 55 eV Full Width Half Maximum (FWHM) energy resolution at 97 keV. This 8–16-fold improvement +in energy resolution over NuSTAR will allow high resolution imaging and spectroscopy of the 44Ti emission. This will +allow a detailed reconstruction of gamma-ray line redshifts, widths, and shapes, allowing us to address questions such +as: What is the source of the neutron star “kicks”? What is the dominant production pathway for 44Ti? Is the engine +of Cas A unique? +Keywords: X-ray, spectroscopy, instrumentation, Supernova remnants. +*Fabian Kislat, fabian.kislat@unh.edu +1 Introduction +Core-collapse supernovae (CCSNe) of prior generations of stars are thought to be a major source of +elements heavier than iron in our Solar System.1 Hence, understanding their explosion mechanism +is key to understanding the evolution of our Galaxy eventually supporting life on Earth. In this +paper, we describe a concept for a new balloon-borne high-energy X-ray telescope and potential +future satellite mission that will, among other science goals, provide new experimental insights +into the inner workings of the CCSN engine. +Supernovae as a source of heavy elements are supported both by theoretical considerations +and experimental evidence. Since 56Fe is the nucleus with the lowest mass per nucleon, fusion of +1 +arXiv:2301.01525v1 [astro-ph.IM] 4 Jan 2023 + +heavier elements cannot serve as a source of energy in stellar cores. Instead, heavier elements are +formed via slow (s-process) or rapid neutron capture (r-process). Some of the earliest and strongest +experimental evidence for nucleosynthesis in CCSNe comes from the detection of 847 keV and +1238 keV gamma-rays associated with the decay of 56Co to 56Fe in SN 1987A.2 Direct evidence for +the fact that our Solar System is indeed made of reprocessed stellar ejecta comes from the analysis +of presolar grains in meteoritic material and interplanetary dust, whose isotopic composition is +representative of the seed material of the Solar System.3 In fact, recent simulations show that +15M⊙ CCSNe are capable of producing many of the isotopic anomalies found in certain presolar +SiC grains,4 which have long been argued to condense in supernovae.5 +An important conclusion from the observations of high-energy X-ray and gamma-ray emission +from SN 1987A soon after the explosion was that mixing of material from the different shells of the +progenitor star must occur very early on in the explosion.6 This mixing moves radioactive nickel +outward from the innermost parts of the ejecta, which then drives the X-ray emission. However, +the details of this mixing and the underlying mechanism are still poorly understood and depend +on the local conditions of the early shock, such as peak temperature and density. Due to these +convective instabilities, anisotropies are expected in supernovae and their remnants. +The structure of young supernova remnants (SNR) reflects the conditions of the explosion. +Since many galactic SNR are spatially resolvable in X-rays and gamma-rays, these remnants are +an excellent site to study supernova explosion physics. Regions where explosive Si burning occurs +can be observed via the K line emission from 56Fe, which is a decay product of 56Ni produced +during Si burning with relatively little dependence on local conditions. The observations, however, +come with the caveats that the X-ray emission depends on the heating of the material in the shock, +and that some of the iron may actually be interstellar material swept up in the shock rather than +supernova ejecta. +The production of 44Ti in the same regions, on the other hand, is very sensitive to the local +conditions. This isotope with a half-life of (58.9 ± 0.3) yr7 is in principle observable in galactic +supernova remnants up to a few hundred years old. It decays via 44Ti → 44Sc → 44Ca, emitting +gamma-rays with energies of 1157 keV, 78.32 keV and 67.87 keV with branching ratios between +93 % and 99.9 %.8 These gamma-rays directly trace the distribution of 44Ti decays. Because of +these properties, observations of 44Ti are a particularly powerful tool to test supernova models, +which has been noted as early as 1969.9 +Here, we discuss a concept for a new balloon-borne high-energy X-ray telescope called AS- +CENT (A SuperConducting ENergetic x-ray Telescope), which had been proposed to NASA’s +Astrophysics Pioneers program, and which could form the basis of a future NuSTAR follow-up +mission. ASCENT consists of a novel transition edge sensor (TES) microcalorimeter gamma-ray +detector array in the focal plane of a multi-layer coated Wolter-type focusing X-ray mirror. Transi- +tion edge sensors utilize the rapid change in conductivity with temperature of a superconductor at +its superconducting transition temperature Tc for calorimetric energy measurements.10 A gamma- +ray spectrometer is constructed by coupling the TES to a thick absorbing structure, commonly +made of Sn, which increases the quantum efficiency for the detection of 10–100 keV photons. Re- +cently, an array consisting of 512 detectors with a spectral resolution of 55 eV FWHM at 97 keV +has been demonstrated,11 and individual detectors have achieved a resolution as precise as 22 eV. +Using these detectors, ASCENT’s spectral resolution will be about 15 times better than NuSTAR in +the 60–85 keV energy range (900 eV at 60 keV).12 +2 + +Additionally, ASCENT will use a new Ni/C multilayer structure on its X-ray optics. The energy +bandpass of NuSTAR was limited by the platinum K edge at 78.395 keV of its Pt/C multilayer, +which prevents it from observing the blue-shifted 78 keV 44Ti line. The use of a Ni/C multilayer +will extend ASCENT’s bandpass to 85 keV and beyond. Furthermore, its multilayer structure will +be optimized for the 55–85 keV range, in order to maximize its effective area for observations of +44Ti. While the baseline angular resolution of the ASCENT optics of 2′ will be slightly worse than +NuSTAR, it will still allow resolution of the most prominent 44Ti emission regions in Cas A. +Observations of the supernova remnant Cas A with ASCENT will test if asymmetries of the +ejecta can completely account for compact remnant “kicks”. Furthermore, they will allow us to +determine the dominant pathway for the production of 44Ti. +So far, 44Ti has only been firmly detected from two objects: SN 1987A13,14 and Cas A.15–17 Ten- +tative detections from Vela Jr.18 and Tycho’s SNR19 have so far not been confirmed.20,21 However, +the Compton Spectrometer and Imager (COSI) will map the Galaxy with unprecedented spectral +and spatial resolution and may find additional sources of 44Ti emission.22 While COSI is an ex- +cellent tool to discover 44Ti emission from additional SNR, its spatial and spectral resolution are +not sufficient to map individual objects. A more sensitive space mission based on the ASCENT de- +sign could follow up on these detections and provide detailed maps of additional SNR. This would +answer the question whether features observed in Cas A are universal or whether there is wide +variation in the underlying engine depending on properties of the progenitor star. Furthermore, a +detection of 44Ti from a remnant associated with a type Ia supernova, such as Tycho’s SNR, would +be a major breakthrough. Ordinarily, SNIa are not expected to produce much 44Ti. However, some +models predict a potential detonation below the Chandrasekhar mass limit (see e.g., Woosley and +Weaver, 199423), in which case a large amount of 44Ti may be produced. Such explosions are +thought to be one of the candidates for the origin of Galactic positrons.24 Thus, such an observa- +tion would not only constrain the SNIa explosion mechanism but also provide new insights into +the origin of Galactic positrons. +Launched on a stratospheric balloon from Kiruna, Sweden, ASCENT will float westward at an +altitude of about 125,000 ft to northern Canada. Typical flight times are 5–7 days, allowing for up +to 560 ksec of observation time of Cas A. A Southern Hemisphere flight from McMurdo Station, +Antarctica, will circle the South Pole at least once for a typical flight time of two weeks. Such a +flight will allow deep observations of SN 1987A. +In Section 2 we discuss the scientific questions addressed by the ASCENT balloon mission, +the technical aspects of which we describe in detail in Section 3. In Section 4, we present results +of Geant4 Monte Carlo simulations of the expected performance of ASCENT and the resulting +sensitivity of the instrument to address its science goals. Finally, in Section 5, we summarize the +results and give an outlook towards a space-based mission based on the ASCENT technology. +2 Scientific objectives +ASCENT’s improved spectral resolution will allow it to address some key questions brought up by +the recent NuSTAR observations of Cas A: what is the source of compact remnant "kicks" and what +are the conditions of 44Ti production in the Cas A supernova remnant? +The high yield of 44Ti detected in Cas A15–17,25,26 is seen as strong support for the expected +anisotropies.27 A 2.4 Ms observation with the NuSTAR satellite was used to obtain the first 3-D +map of the 44Ti ejecta,17,28 which confirmed these high yields and, furthermore, found that the 44Ti +3 + +lies in clumpy structures. Because much of the 44Ti was found to lie in unshocked regions, its +observation provides a pristine measurement of the asymmetries in the supernova engine, and this +NuSTAR data decisively showed that Cas A was produced from an explosion with multiple outflows +(as expected from the convective engine) and not a jet. However, the NuSTAR observations left +many questions unanswered and raised a series of new problems with our understanding of Cas A, +which ASCENT will address. +Along with 56Ni, 44Ti is produced in the innermost supernova ejecta. In contrast to 56Ni produc- +tion, 44Ti production is extremely sensitive to the temperature and density evolution of the ejecta29 +and, hence, the nature of the explosion. Within 1–2 years of a supernova explosion, 56Ni decays to +stable iron. This iron produces emission lines when heated by the reverse shock as the supernova +ejecta plows through the circumstellar medium and depends on the distribution of the circumstellar +medium as well as the explosion, making its interpretation complicated. The innermost iron in the +remnant is also difficult to measure accurately, since it is inside of the reverse shock and therefore +cold. Although attempts have been made at detecting this iron in the infrared in Cas A,30 the un- +biased nature of the 44Ti observations and their sensitivity to the explosion characteristics have led +to 44Ti’s important role in shaping our understanding of the core-collapse explosion. +2.1 The core collapse engine of Cas A +A key to understanding the core collapse supernova engine is understanding the production and +mixing of 44Ti during the explosion, which will identify the dominant pathways of 44Ti produc- +tion in supernovae. The processes that produce Ti are quasi-statistical or statistical equilibrium +processes. They produce an equilibrium distribution of nuclei based on the nuclear chemical po- +tentials at a given temperature, density, and electron fraction.29 The local ratio between 44Ti and +56Ni strongly constrains the thermodynamic initial conditions, allowing a fairly precise determi- +nation of the final distribution of all nuclei. The remaining uncertainty can then be attributed to +the thermodynamic history, which determines the details of freezeout. Observations of additional +nuclear species will provide additional constraints and reduce residual uncertainties. +Combining ASCENT’s 44Ti observations with observations of Cas A’s Si and Fe lines from +JAXA’s and NASA’s XRISM mission (to be launched in 2023) will allow us to perform an im- +proved reconstruction of the Si, Fe, and Ti configurations at the current time and at the time of +the supernova explosion. ASCENT will improve the spectral resolution of the 44Ti emission over +NuSTAR, allowing us to construct a more detailed map of the clumpy structures in Cas A, enabling +a more detailed comparison to numerical models and helping to disentangle multiple structures +along a line of sight and the properties of these structures. These comparisons provide direct +constraints on the engine. +NuSTAR’s results indicate that most of the ejecta are moving away from us with a velocity of +1000–5000 km s−1 but that some 44Ti regions (Region 1 in Fig. 1, i.e., Region 20 of Grefenstette +et al., 201728) move towards us with 7500 km s−1. The combination of ASCENT’s 67 eV energy +resolution with its effective area extending to >85 keV will enable an analysis with much smaller +systematic errors than that of the NuSTAR analysis. If confirmed, these high velocity ejecta would +challenge current supernova theory and provide a strong constraint on models. Most current mod- +els that lead to a high yield of 44Ti do not result in velocities >4500 km s−1,31,32 and the highest +velocity found in the models by Vance et al.33 is ∼5500 km s−1. The detailed velocity measurement +can also be used to understand 44Ti production. Nucleosynthesis calculations predict different 44Ti +4 + +1 +2 +Fig 1: X-ray image of emission lines in Cassiopeia A: iron (red), silicon/magnesium (green), +titanium (blue), and continuum emission (yellow).17 Two parts of the remnant of particular interest +are highlighted. Region 1 is the only region in which NuSTAR detected significantly blue-shifted +44Ti ejecta. Region 2 is of interest due to the highly blue-shifted almost pure iron ejecta. Composite +Chandra/NuSTAR image credit NASA/JPL-Caltech/CXC/SAO. +yields depending on the ejecta velocity.33 We can use the ejecta velocities to constrain the trajec- +tories (temperature and density evolution) of the ejecta, allowing us to test both our explosion and +nucleosynthesis models. These ejecta velocities can be tied to more fundamental properties like +the electron fraction and nuclear cross sections. +Because we expect 44Ti to be produced at some level whenever 56Ni is produced, we expect to +see 44Ti lines wherever iron is observed. This raises the question why there are large iron ejecta +with no evident detection of 44Ti in Cas A. The NuSTAR observations could not detect 44Ti in the +iron-rich southeastern region of Cas A (Region 2 in Fig. 1). Is it because that iron is produced +directly (not the decay product of 56Ni), or was there a large amount of 56Ni produced with 44Ti +mass fractions below the NuSTAR detection limit? The NuSTAR upper limit in this region of +the SNR is not very constraining,28 and the iron-rich ejecta are blue-shifted with a velocity up +to 3000 km s−1.34,35 Therefore, ASCENT may be able to detect the 44Ti in this region due to its +sensitivity to the blue-shifted 78 keV line, or significantly improve on the NuSTAR upper limit. +2.2 Compact Remnant Kicks +Observations of pulsar proper motions and the existence of specific peculiar binary systems sug- +gest that momentum is imparted onto compact remnants during their formation (for a review, see +Fryer and Kusenko, 200636). A diverse set of models have been proposed to create these kicks, but +these models can be separated into two categories: asymmetries in the ejecta and asymmetries in +the neutrino emission. Under the convection-enhanced supernova engine paradigm,37 low-mode +convection produces asymmetric explosions with a nonzero net momentum in the ejecta.38 These +asymmetries impart a net momentum to the compact remnant, and a 1 % asymmetry in the ejecta +produces the high observed kick velocities. Although simulations have struggled to produce some +of the highest observed kicks, ejecta asymmetries remains one of the strongest candidate mecha- +nisms for explaining pulsar proper motions. Alternatively, asymmetries in the neutrino emission +5 + +(typically requiring strong magnetic fields – albeit not necessarily strong bipolar magnetic fields) +also carries away a net momentum, imparting an equally strong kick onto the compact remnant.39 +The different mechanisms proposed in the ejecta and neutrino mechanisms make a variety of +predictions on the relation of the compact remnant kicks with relation to angular momentum, dipole +magnetic-field strength, final remnant mass, and the formation of a black hole versus a neutron +star. Many of these predictions are indirect, and it is difficult to place strong constraints on the +mechanism with existing observations. However, NuSTAR observations opened up the potential for +a more direct observational constraint with detailed maps of the 44Ti to compare the asymmetries +in the explosion to the remnant velocities. Because 44Ti is produced in the innermost ejecta, it is an +ideal probe of these explosion asymmetries. However, to truly compare the explosion asymmetries +with the compact remnant kick, we need detailed 3-dimensional ejecta information. Although the +current NuSTAR data hinted at a correlation between the explosion asymmetries and the compact +remnant kick supporting the ejecta kick mechanism,28 the higher-fidelity ASCENT observations +will allow a more quantitative test of the ejecta kick mechanism. +2.3 Validating the convective SN engine +The convective nature of the supernova engine in Cas A has been established quite firmly. However, +there are only two SNR with confirmed detections of 44Ti, and Cas A is the only SNR in which 44Ti +emission has been spatially resolved. This raises the question whether the SN engine of Cas A is +unique, which ASCENT can address through observations of SN 1987A.13 While ASCENT cannot +spatially resolve SN 1987A, a precise measurement of the 44Ti line shapes can be used to quantify +asymmetries in the Ti distribution. +Figure 2 shows the velocity distributions along three lines of sight for two different supernova +explosions. These velocity distributions are derived from 3-dimensional smooth particle hydrody- +namics simulations of asymmetrically-driven supernova explosions.40 One explosion is bimodal +(either produced by a mild “jet” or low-mode convection model with rotation) and the other is +more representative of a low-rotation, low-mode convectively-driven explosion (“Asym”). For the +velocity distributions in this figure, we chose 3 different lines-of-site and measured the velocities of +the ejecta along these lines-of-site (to determine the red- and blue-shifted features). The ASCENT +observations will not only be able to easily differentiate between these models, but also enable us +to further constrain the specific features of the convective engine. +3 Technical implementation +3.1 Overview +The ASCENT experiment (Figure 3) uses a 12 m optical bench with a 45 cm-diameter, F = 12 m +multilayer X-ray mirror at the front end and a cryogenically cooled microcalorimeter detector as- +sembly at the rear end. A balloon gondola holds a two-frame gimbal, pointing the optical bench in +the direction of the observed astrophysical sources with the help of the Wallops Arc Second Pointer +(WASP) system.41 The microcalorimetric detector array is cooled by an Adiabatic Demagnetiza- +tion Refrigerator (ADR) inside a 65 L liquid He dewar. Table 1 summarizes key characteristics +of the ASCENT observatory and its expected performance. In the remainder of this section, we +describe the design of each of the main components. +The telescope will be carried by a 1.1 × 106 m3 He-filled balloon to an altitude of about 38 km. +When launched from Esrange in Kiruna, Sweden, it will partially circle the North Pole, reaching +6 + +NuSTAR +ASCENT +Fig 2: Velocity distribution of the 44Ti ejecta for two different supernova explosions: a bipolar +explosion where the ejecta is fastest along the axis (jet) and an explosion with multiple strong +outflows mimicking the predictions of the convective supernova engine (asym).40 The structure +in the line of sight velocity distribution can be tied to the structure of the supernova engine (i.e., +the bipolar explosion has a very different profile than the asymmetric explosion). The bars at the +top of the graph illustrate the line-of-sight velocity resolution of NuSTAR and ASCENT. Thanks to +ASCENT’s energy resolution, we will be able to measure these differences. +Northern Canada after a typically 5–7-day flight. On this trajectory, the telescope will be able +to continuously observe Cas A with an elevation angle of about 36 − 82°. Additional, longer, +flights from McMurdo Station (Antarctica) will circle the South Pole, enabling deep observations +SN 1987A. +3.2 Focal plane instrumentation +3.2.1 Transition Edge Sensor array +Microcalorimeter technology has shown great promise for transforming X-ray astrophysics. For +example, the Hitomi mission used a 36-pixel Si thermistor microcalorimeter array for its Soft X- +ray Spectrometer (SXS).42 The Athena mission (to be launched in 2032) will use a 4000-pixel +microcalorimeter array for its X-ray Integrated Field Unit (X-IFU).43,44 Over the last 15 years, +transition-edge sensor (TES) microcalorimeter spectrometers have been developed as cutting-edge +tools in the fields of nuclear materials analysis11,45–47 and the X-ray sciences.48–54 +TES microcalorimeters are detectors that measure the energy of individual photons through the +temperature change of a superconducting thin film thermometer (see Fig. 4). The TES thermometer +is coupled to a photon absorber composed of a high-Z element such as bismuth or tin, enabling high +quantum efficiency for x-rays up to 100 keV. The fundamental energy resolution of a calorimeter +is +∆E ∝ +� +kBT 2C, +(1) +7 + +Asym, losl +5.0 +los2 +line of site and velocity bin (M) +los3 +5.5 +Jet,losl +los2 +6.0 +los3 +6.5 +7.5 +8.0 +60- +8.5 +9.0 +-4000 +-3000 +-2000 +-1000 +0 +1000 +2000 +3000 +4000 +Velocity (km/s)Table 1: Key ASCENT payload characteristics and expected performance. Details of the perfor- +mance estimates are provided in Section 4. +Component +Description +Performance +Truss +Carbon fiber tubes and aluminum joints +Focal spot movement <3 mm, +alignment knowledge 0.5 mm (9′′) +Pointing system +Pitch-yaw articulated +Pointing precision 1.0–3.6′′ (3σ) +on source +Star camera +100 mm, f/1.5 short-wave infrared lens +Pointing knowledge <15′′ (3σ) +X-ray mirror +Wolter I, 12 m focal length, diameter +40 cm, 110 Ni/C-coated and 100 Pt/C- +coated shells +Effective area 190 cm2 at 70 keV, +Angular resolution 2′ HPD, Field +of view 5′ FWHM +Cryostat +LHe-backed adiabatic demagnetization +refrigerator +Base temperature 70 mK +Detector +Two-layer gamma-ray TES array, 256 +pixels each, 1.4 × 1.4 × 0.59 mm3 ab- +sorbers (30′′ ×30′′ at 12 m), microwave +multiplexed readout +Bandpass: 2–100 keV, energy res- +olution ∆E(80 keV) += +67 eV +FWHM +Power +Detectors, cryostat, heaters +∼350 W +Mass +Mass under balloon rotator +∼1700 kg +Signal rate +1 Crab source at 45° elevation +0.5 Hz at 60–80 keV +Background rate +BGO shield veto applied +0.02 Hz at 60–80 keV +X-ray optics +Star camera +WASP gimbals +SIP +Battery box and CPU +Truss CPU +Cryostat +ACD +Truss +TDRSS antenna +Fig 3: CAD rendering of the ASCENT telescope. Key components are labeled in the Figure and de- +scribed in Section 3. The gondola and truss design are almost identical to XL-Calibur, maximizing +flight heritage. +8 + +Weak thermal +link +Thermal bath +shunt +R +SQUID +readout +RTES +bias +I +(a) +Time +Temperature +(b) +Temperature +Resistance +Transition +ΔT +ΔR +(c) +Fig 4: (a) Calorimetric spectroscopy of x-rays. An incident photon deposits its energy into a +target with a weak thermal link to a cold isothermal bath. (b) A typical pulse-response curve +with a decay time determined by the properties of the calorimeter element and its coupling to +the isothermal bath. The filtered pulse height is an extremely precise measure of the photon’s +energy. (c) Thermometry is performed with a thin-film superconducting transition-edge sensor. +The extreme precision results from the sharp temperature dependence of the electrical resistance +of the thin film operated close to its superconducting transition temperature. +where T and C are the sensor temperature and heat capacity,10,55 allowing these devices to achieve +extraordinary energy resolution by operating at cryogenic temperatures of about ∼100 mK. +A prototype detector array called “Spectrometer to Leverage Extensive Development of Gamma- +ray TESs for Huge Arrays using Microwave Multiplexed Enabled Readout” (SLEDGEHAMMER, +see Fig. 5 and Mates et al., 201711) has achieved a full-width at half maximum (FWHM) reso- +lution of 55 eV at 97 keV. A resolution as low as 22 eV has been demonstrated with individual +detectors.11,45,56 The 100 keV energy resolution of 55 eV FWHM of SLEDGEHAMMER is 10× +better than that of cryogenically cooled High Purity Germanium spectrometers (HPGe) and ∼20× +better than room-temperature CdZnTe detectors. +ASCENT will use detectors similar to those of the SLEDGEHAMMER hard X-ray/γ-ray spec- +trometer.11 A photograph of a SLEDGEHAMMER detector is shown in Fig. 6. These sensors use +polycrystalline tin absorbers to absorb photons. Tin is chosen because it combines a relatively high +stopping power for γ-rays in the energy range of interest with a low specific heat at cryogenic tem- +peratures. In SLEDGEHAMMER, these absorbers are 1.45 mm × 1.45 mm in area and 0.38 mm +thick. For ASCENT we plan on using 0.59 mm thick absorbers to increase quantum efficiency to +87 % at 68 keV and 75 % at 78 keV. The 55 % increase in absorber volume results in a correspond- +ing increase of the heat capacity, and a 25 % increased energy resolution based on Eq. (1). The +expected energy resolution of the detectors, thus, increases to 68 eV. The absorbers are glued to +epoxy posts, which are connected to the TES element by copper traces of equal length to ensure +a uniform thermal path to the sensor (Fig. 6). The TES element is a 400 µm × 400 µm bilayer of +superconducting material and normal metal, lithographically deposited on a Si3N4 membrane. The +transition temperature Tc is set to ∼120 mK by the superconducting proximity effect in thin-film +bilayers. This Tc includes enough margin above the base temperature of an Adiabatic Demagne- +tization Refrigerator (ADR) to allow for stable operation. Options for bilayer materials include +MoCu, as in SLEDGEHAMMER, as well as MoAu TES using NIST’s patented hasTES process.57 +Current fabrication methods require manual placement of the absorbers on the TES array using +mechanical tweezers. This constrains the minimum size and spacing of the absorbers to dimensions +9 + +Fig 5: Left: Photograph of the fully assembled SLEDGEHAMMER detector package. The package +contains eight TES microcalorimeter chips with 32 sensors each (center), eight microwave mul- +tiplexer chips with 32 channel readout (outer vertical columns), and eight chips each for detector +bias, Nyquist filtering, and signal routing. The TES signals are read out by two pairs of coaxial +cables attached to the box by SMA connectors on the top and bottom of the box, each record- +ing the signals for 128 sensors. ASCENT will use similar architecture to minimize risk, but with +two monolithic detector chips stacked on top of each other, to minimize inter-pixel dead space and +maximize collection efficiency. Right: A combined 153Gd spectrum from 89 active TESs measured +simultaneously using microwave SQUID multiplexing readout. The inset shows a zoomed region +around the 97 keV γ-ray peak (blue) with a Gaussian fit FWHM resolution of 55 eV (red). The +energy resolution achievable with TES microcalorimeters is 15 times better than that achieved by +NuSTAR, achieving 270 km/s accuracy in measurements of the velocity of 44Ti ejecta. Reprinted +from Mates et al. (2017)11 with the permission of AIP Publishing. +Fig 6: (a) Photograph of a TES gamma-ray microcalorimeter pixel before the Sn absorber is at- +tached, showing the Si3N4 membrane (darker area), the Mo-Cu TES in the middle, and 20 SU8 +epoxy posts connected to the TES by the Cu legs. (b) A portion of the detector chip with some of +the Sn absorbers attached. The Sn absorbers are 1.45 mm × 1.45 mm × 0.38 mm thick, placed on +a 1.75 mm pitch. (c) Each ASCENT detector die uses a central array of 256 of these γ-ray sensors +(within the black inner circle) surrounded by an octagonal pattern of eight sets of 32-channel bias +chips (blue), Nyquist filtering chips (red) and microwave SQUID multiplexer chips (purple). For +scale, the outer black circle is 80 mm in diameter, and the overlay indicates a projection of the +Cas A 44Ti distribution measured by NuSTAR. Each die will be fabricated monolithically from a +75 mm Si wafer to minimize space between pixels. Figures (a) and (b) reprinted from Bennett et +al. (2012)45 with the permission of AIP Publishing. +10 + +20 mm +onpicbor +品品 +4(b) +(a) +SU8 posts +MoCu film +1.4 mmclose to those of SLEDGEHAMMER, resulting in a 1.75 mm pixel pitch and an array fill fraction +of about 65 %. At 12 m focal length, this pixel pitch corresponds to an angular separation of 30′′. +The low array fill fraction correspondingly reduces the overall photon collection efficiency of the +array. +To alleviate these issues, the instrument detector package will consist of 512 detectors, in the +form of two dies each containing 256 detectors. The dies will be stacked on top of each other and +offset so that the detectors in the lower die will lie directly underneath the gaps between detectors +in the upper die. This will result in a total detection efficiency for photons striking the array of +80 % at 68 keV and 70 % at 80 keV. The layout and design of each die is conceptually similar to +the proven design of the SLEDGEHAMMER microcalorimeter array. The central array of 256 TES +detectors in each die (Fig. 6(c)) is fabricated from a single 75 mm Si wafer, with wiring to carry the +TES signals to bond pads for connection to the rest of the readout circuitry arranged in an octagon +around the outside of the TES array. +At 12 m focal length, a point spread function with the half power diameter (HPD) of ASCENT +corresponds to 3.5 mm, which is Nyquist sampled by each of the two detector dies. Combining the +two offset detector dies allows a sampling of the PSF with an effective detector pitch of ∼1.2 mm. +The distance between the two dies will be less than 1 cm. Assuming the focal plane of the X-ray +optics is placed directly between the two dies, the HPD of the point spread function will increase +by only ∼2 %. The array diameter corresponds to an angular scale of about 8.8′, significantly larger +than the field of view of the X-ray optics, which eases the requirements on alignment stability as +long as alignment knowledge can be maintained. +3.2.2 Sensor readout +TES arrays use multiplexing to minimize the thermal load and cryogenic complexity of wire con- +nections to room temperature. The development of the Microwave SQUID Multiplexer, which +reads out array of microcalorimeters using microwave techniques,11,58 increases the available mea- +surement bandwidth from ∼30 MHz (the intrinsic limit in previous multiplexing architectures) to +the several GHz of bandwidth available on a single coaxial cable. +In the Microwave SQUID Multiplexer,59 each sensor is coupled to a high-Q, thin-film resonant +circuit by an rf-SQUID that transduces current changes at the sensor to changes in inductive load +on the resonator (Fig. 7). Multiple resonators, each with a unique frequency, are coupled to a +single microwave feedline. A sum of microwave tones (sine waves) is supplied to the feedline, +each tone matched to the frequency of one resonator. Changes in the current through a sensor +will shift the center frequency of its resonator and thus change the amplitude and phase of the +tone that propagates through the feedline. All tones are amplified by a single shared cryogenic +low-noise amplifier before returning to room temperature, where they are analyzed to extract the +detector signals. Signals from different sensors can easily be separated because they appear in +modulation sidebands of their respective tones. The first microcalorimeter array with microwave +readout, SLEDGEHAMMER, demonstrated multiplexing factors of 128 with negligible resolution +degradation, yielding a co-added resolution of 55 eV at the 97 keV gamma-ray peak (Figs. 5 and 6). +ASCENT will use the same 33-resonator microwave SQUID multiplexing chip designs used for +SLEDGEHAMMER, shown in Fig. 7(b). Each resonator has a FWHM bandwidth of ∼300 kHz, +and the resonances are spaced 3 MHz apart. Variations to the design place 32 resonators into each +of eight 125 MHz bands between 5 GHz and 6 GHz, yielding a total density of 256 detectors per +11 + +Fig 7: (a) Circuit schematic showing three channels of a microwave SQUID multiplexing circuit +with TES microcalorimeters. (b) A photograph of a 33-channel microwave SQUID multiplexer +chip used in the SLEDGEHAMMER instrument. (c) A close-up photograph showing quarter-wave +microwave resonators capacitively coupled to a feedline. The resonators are terminated by induc- +tively coupled rf-SQUIDs (left). The microwave SQUID multiplexer takes advantage of the large +bandwidth provided by coaxial cables to significantly reduce the thermal load and design complex- +ity of reading out large-format arrays of TES microcalorimeters, such as those used in ASCENT. +Reprinted from Mates et al. (2017)11 with the permission of AIP Publishing. +GHz of available bandwidth. To read out the 512 detectors used in ASCENT, we will use 2 parallel +pairs of coaxial cables, each reading out a separate set of resonators in the 5–6 GHz range. +The signals will be analyzed by four commercially available ROACH2 Field Programmable +Gate Array (FPGA) systems designed by the CASPER radio astronomy consortium,60 each cover- +ing a bandwidth of 512 MHz using commercially available DAC and ADC daughter boards. +3.2.3 Calibration +Achieving the best possible spectroscopic performance requires constant monitoring of the detector +calibration, in order to be able to correct for calibration drift. ASCENT will carry a calibration +source housed inside a tungsten enclosure outside a dedicated window in the cryostat, illuminating +the detector array from behind. The calibration of gamma-ray TESs drifts on timescales of minutes +to hours. Experience shows that the dominant contribution to short-term drift is correlated with the +baseline and can be corrected for. Therefore, it is necessary to obtain calibration spectra about +once per hour to keep the systematic uncertainty due to residual drift below 10 eV. By collecting +about 100 photons per calibration line, we can keep the statistical uncertainty of the calibration +negligible. Two approaches to calibration are still under consideration: continuous illumination +with a weak source or use of a strong source behind a shutter periodically illuminating the array +for a brief period of time. +The advantage of continuous illumination is that no shutter mechanism is required and that +no artificial dead-time is introduced. This reduces mission complexity and, thus, risk. However, +care must be taken in the selection of calibration isotope or combination of isotopes. Calibration +lines should be close to the energy range of interest, but lines within the energy range of interest, +including escape peaks, will cause unacceptable background. Furthermore, emission lines above +the energy range of interest will cause a background continuum due to Compton scattering in +the detectors. A preliminary Geant461–63 simulation study of calibration spectra obtained with a +selection of viable sources revealed 155Eu with sufficiently strong lines at 60 keV and 86 keV as a +candidate. +12 + +1mmNSA sufficiently strong source behind a shutter will allow us to acquire calibration spectra at +regular intervals over a short period of time. Because calibration data are acquired at known time +intervals during which science data collection will be suspended, it is possible to use calibration +lines in the energy range of interest. This has the advantage of mitigating the impact of non- +linearities in the detector response. The approach also reduces the continuum background due +to high-energy lines from the calibration source, assuming sufficient shielding is possible. The +downside of this approach is that a mechanical shutter is required, adding complexity. While +details still need to be optimized, a preliminary estimate shows that a 1 min calibration window +every hour will be sufficient. A possible calibration isotope is 227Ac which emits a large number of +X-ray and gamma-ray lines in the energy range of interest. This calibration method would result +in an additional dead-time of <2 %. +3.2.4 Anti-coincidence detector +In order to reduce the background, the cryostat section containing the focal plane instrumenta- +tion will almost entirely be enclosed in a ∼2–3 cm-thick active bismuth-germanate (BGO) anti- +coincidence shield. Scintillation light due to particles interacting in the anti-coincidence shield +will be detected by photomultiplier tubes (PMTs) or Silicon Photomultipliers (SiPMs). Signals +in these PMTs or SiPMs will produce a flag vetoing any triggers in the TES detector readout, +significantly reducing the residual background, in addition to the passive shielding provided by +the absorption of particles in the BGO. The design will maximize the solid angle covered by the +active shield. The conservative solution is to place the active shield components at the outside of +the cryostat, avoiding the difficulties associated with bringing the scintillator crystal and detectors +from ambient temperatures and pressure to liquid helium temperatures and near vacuum. In this +case, passive tungsten shielding will be used inside the cryostat for the small solid angle portions +not covered by the active shield outside the cryostat. We are evaluating solutions with active shield- +ing inside the cryostat which would result in a smaller shield and thus reduced cross sections for +interactions with the background, and a reduced shield mass. In both cases, the veto flag will be +fed to the data acquisition and will be digitized along with the TES signals. +3.3 X-ray optics +ASCENT achieves a large effective area in the 65–85 keV energy range using a dedicated multilayer- +coated grazing incidence X-ray mirror. The mirror consists of 213 nested shells in two reflection +stages with a diameter of 40 cm. The innermost 110 shells will be coated with approximately +500 Ni/C layer pairs, while the remaining shells will be coated with roughly 200 Pt/C layer pairs. +Given the focal length of 12 m, the design will limit incidence angles to < 0.23°. Reflectivity over +a broad bandwidth at high X-ray energies is achieved by coating the shells with alternating layers +of high-Z and low-Z material. The ASCENT optics are expected to achieve an angular resolution +of 2′ half-power diameter (HPD). The field of view of 5′ FWHM exceeds the angular size of Cas A +of 4′. +Most X-ray telescopes are designed to achieve a high collection area over a broad energy range. +The broadband design necessitates the deposition of a multilayer stack for soft X-rays on top of +the stack for hard X-rays. However, the thick soft X-ray layers absorb some of the higher-energy +photons. For ASCENT, the multilayer design will be optimized for energies above 60 keV, achiev- +ing there substantially higher reflectivities than a broadband X-ray multilayer coating. Figure 8 +13 + +20 +40 +60 +80 +100 +Energy [keV] +10 +0 +10 +1 +10 +2 +10 +3 +Effective area [cm²] +Pt/C and Ni/C, =4Å +Pt/C contribution +Ni/C contribution +Pt/C and Ni/C, =6Å +Pt/C only, =4Å +NuSTAR (two mirrors) +Fig 8: Comparison of the effective areas of mirror designs for ASCENT and the two NuSTAR +mirrors combined.12 The ASCENT design uses a combination of Ni/C multilayer coatings on the +inner 110 shells and Pt/C on the outer shells. An alternative design using Pt/C layers on all shells +is shown for comparison. The shaded region indicates the expected range of the surface roughness +between σ = 6 Å (worst case) and 4 Å (best case). The dashed and dotted red lines indicate the +contribution of the Pt/C and Ni/C shells in the σ = 4 Å case, respectively. +shows the comparison of the collection areas of the ASCENT and the two NuSTAR mirrors. The +platinum K absorption edge at 78.395 keV limits the effective area at higher energies, preventing a +purely Pt/C multilayer mirror from properly observing the 44Ti line at 78.36 keV, especially when +that line is blueshifted. Therefore, in addition to an optimized multilayer structure, the ASCENT +optics will use Ni/C coatings on the innermost 110 out of 213 nested shells. The Ni/C design +requires about 500 layer pairs, while the Pt/C design requires about 200 layer pairs, which limits +the number of foils that can be fabricated with this method within the project timeline. A surface +roughness of 4–6 Å of the multilayer coatings is expected. The ASCENT mirror achieves collec- +tion areas of >100 cm2 in the 65–80 keV band. Due to the optimization of the layer structure for +44Ti observations, the effective area between ∼30 keV and 55 keV is very small. The low-energy +reflectivity is not due to Bragg reflection on the multilayer, but due to total external reflection on +the surface. The Ni/C multilayer combination has been studied by several groups in the past (e.g., +Spiga et al., 200464) and is considered a top candidate material for future missions, such as HEX-P, +to extend their energy band out to 200 keV.65 +3.4 Cryostat +The ASCENT cryogenic system cools the detector assembly to a nominal base temperature of +70 mK. Our baseline design foresees to use the cryostat architecture of Fig. 9 that uses a commer- +cial adiabatic demagnetization refrigerator (ADR) coupled to a closed-cycle 300 mK refrigerator +and a liquid helium bath. The 65 L tank is designed to cool the detectors for up to 14 days, more +than sufficient for the expect duration of the balloon flights from Sweden to Northern Canada. An +14 + +Fig 9: Conceptual design of the ASCENT cryostat, which provides a 70 mK base temperature for +the detectors. The detectors are housed in a protruding snout to minimize the mass of the shielding. +absolute pressure regulator maintains the tank near atmospheric pressure to provide a 4 K ther- +mal bath, and internal baffles minimize sloshing to prevent resonances with the pointing system. +Counterweights on the pointing system maintain payload balance as the motion of the liquid during +elevation changes and cryogen boil-off shift the center of mass. +The main liquid helium tank is insulated by two vapor-cooled shields. As cold gas boils away +from the liquid helium reservoir, it flows through stainless steel pipes connected to the shields +through low-impedance heat exchangers. Since the helium boil off rate is proportional to the +thermal load, negative feedback enforces temperature stability within the cryostat. G-10 trusses +mechanically support each stage while maintaining sufficient thermal insulation. Liquid nitrogen +cooling was considered as an alternative for the vapor-cooled shields, but would have led to a +heavier and more complex design due to the additional cryogen tank. +A 300 mK temperature stage is provided by a multi-stage closed-cycle He-4/He-3 sorption re- +frigerator coupled to the cryostat’s liquid helium tank. This serves as the launching point for the +ADR, allowing a lower magnetic field strength and thus lower power consumption than launch- +ing directly from 4 K. The 300 mK stage also intercepts the parasitic load from the wiring and +mechanical support structures to reduce the cooling power requirement on the lowest temperature +stage. Similar refrigerators have been successfully used by many different balloon-borne cryogenic +systems (e.g., SPIDER,66 EBEX,67 BLAST-Pol,68 and BOOMERANG69). +The detector assembly is maintained at a base temperature of 70 mK by a commercially avail- +able ADR using a single ferric ammonium alum (FAA) salt pill70 launched from the 300 mK stage +to provide 1 µW of cooling power with 120 mJ cooling capacity. A 3-hour regeneration cycle will +be performed once every 24 hours, providing roughly 90 % observing efficiency. For optimal oper- +ation, the TES detectors and SQUID multiplexer chips must be protected from external magnetic +15 + +Vacuum Vessel +OuterVapor-Cooled Shield +InnerVapor-Cooled Shield +Liquid Helium Tank +4K Shield +He Fridge +ADR +Tungsten Shield +Detector Assembly +Tungsten Collimator +Inner A4K Shield +Outer A4K Shield +BGO Shieldfields. The entire detector package assembly containing both detector dies and all SQUID mul- +tiplexer chips will therefore be enclosed in a two-layer magnetic shield, incorporated within the +cryostat and maintained at cryogenic temperatures. These reduce the magnetic flux in the SQUIDs +due to Earth’s magnetic field by about two orders of magnitude. +As an alternative cooling option, we are currently evaluating the performance of a mini Dilution +Refrigerator.71 First measurements in the lab indicate that the mini Dilution Refrigerator is well +suited for this application, offering continuous cooling to 80 mK temperatures. The cooling power +does not change significantly for elevation changes of ±30°, enabling its use for ASCENT. +3.5 Optical bench and gondola +X-ray optics and cryostat will be supported by a 12 m-long optical bench pointed by NASA’s +Wallops Arc-Second Pointer (WASP) system. The optical bench consists of three sections made +of carbon fiber tubes glued to Aluminum joints. The design results in an extremely stiff truss and +is similar to previous balloon-borne telescopes X-Calibur and XL-Calibur.72,73 For example, the +8 m long truss of X-Calibur achieved a stability of <1.5 mm of the focal point during most of the +flight.74 The X-ray mirror, star tracker, and fiber-optic gyro of the WASP will be mounted to an Al +honeycomb panel at the front end of the truss, and the focal plane instrumentation will be attached +to an Al honeycomb panel at the rear end of the truss. +The stiffness of the optical bench fulfills two requirements. First, the WASP pointing system +requires that the lowest-frequency vibration mode of the pointed body exceeds 10 Hz. Second, we +require a motion of the focal spot <3 mm in order to ensure the entire image is always contained +in the detector array. A focal spot motion of 3 mm corresponds to 50′′ pointing error. In order +to reduce the resulting degradation of the point spread function, ASCENT will use an alignment +monitoring system similar to X-Calibur.74 The system uses an optical camera mounted in the +central bore of the X-ray optics observing a pattern of LEDs mounted to the entrance window +of the detector. On a 12 m truss, it measures the alignment with a precision of 0.15 mm or 2.5′′, +negligible compared to the point spread function of the optics. +An aluminum gondola suspended from the balloon supports the truss pointed by the WASP. +The WASP points the truss in pitch and within a limited yaw range with respect to the gondola. +Coarse pointing in yaw is achieved using a standard NASA balloon rotator coupling the gondola to +the balloon. Absolute pointing information is provided by a star tracker system specially developed +for balloon flight applications. This system achieves a pointing stability of <1′′ and an absolute +pointing accuracy of ∼15′′. +4 Expected performance +During a balloon flight from Kiruna, Sweden, ASCENT will observe Cas A for approximately +500 ksec at an elevation of about 36–82°. We envision multiple northern hemisphere flights in +order to attain longer total observation times. +In order to estimate ASCENT’s sensitivity, we simulated the detector in Geant4 as two stacked +arrays of Sn absorbers with the layout as shown in Fig. 6c and a thickness of 0.59 mm. Input spectra +were folded with the mirror effective area based on the Ni/C multilayer mirror shown in Fig. 8 and +energy-dependent atmospheric absorption at a balloon altitude of 125,000 ft corresponding to an +overburden of 2.9 g cm−2. Photons were distributed according to the mirror point spread function +(PSF) of ASTRO-H HXT,75 which is similar to the expected ASCENT PSF. +16 + +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Energy [keV] +7 +− +10 +6 +− +10 +5 +− +10 +] +-1 + s +-2 + cm +γ + Line Sensitivity [ +σ +3 +500ksec +1Msec +Fig 10: ASCENT narrow line sensitivity as a function of energy, demonstrating ASCENT’s ability +to detect 44Ti in various regions of Cas A with a single 500 ksec flight. The vertical yellow bands +indicate energy ranges of interest and the horizontal black bars are the flux levels measured by +NuSTAR.28 +We estimate the background using measurements made during the Antarctic flight of X-Calibur74 +and taking into account improvements to the anticoincidence shield, which will reduce this back- +ground by a factor of ∼10.73 To account for the difference in detector size we scale with area and +square root of thickness. This results in a background rate of ∼1.7 × 10−6 s−1 keV−1 per TES de- +tector at 68 keV. In the future, this estimate can be refined using data from an upcoming test flight +of a small TES array scheduled for the fall of 2023, as well as with the help of detailed Monte +Carlo simulations.76 +In the sensitivity calculation, we weighted events in each detector by the expected signal-to- +background ratio for a point-like source according to Ref.77 Figure 10 shows the expected narrow +line sensitivity of ASCENT. NuSTAR detected lines with fluxes ranging from 6 × 10−7 cm−2 s−1 +to 1.7 × 10−6 cm−2 s−1. ASCENT’s energy resolution of 67 eV FWHM will allow us to determine +the velocity of 44Ti ejecta with a FWHM accuracy of 270 km s−1, compared to NuSTAR’s FWHM +accuracy of 3600 km s−1. +From each flight we expect highly significant detections of the 44Ti emission from Cas A of 11 σ +and 6.5 σ of the 67.9 keV and 78.4 keV lines, respectively, when summing over all bright spots. +Particularly interesting will be the energy spectrum from Region 1 in Fig. 1, which we expect +to detect with 5 σ at 67.9 keV and with 3 σ at 78.4 keV. The advantage of ASCENT is greatest +in regions where the width of the 44Ti lines is small compared to NuSTAR’s energy resolution. +A single 500 ksec observation with ASCENT will improve measurements in all regions where +NuSTAR detected 44Ti, except two where the lines are broadened and very weak. We estimate that +ASCENT will improve the line centroid and width measurements compared to previous results by +a factor of 2–20 and 2–10, respectively (Fig. 11). These measurements will significantly improve +the 3D localization of the 44Ti ejecta and result in much tighter constraints on the local 56Ni/44Ti +ratio. These improvements in constraining the local ratio will greatly increase our knowledge of +the nuclear production pathways in the supernova explosion. +17 + +66.5 +67 +67.5 +68 +Centroid [keV] +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Width [keV] +67.1 +67.2 +67.3 +67.4 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +ASCENT +NuSTAR +Fig 11: Expected ASCENT results for the measurements of 44Ti emission lines in the 8 44Ti-bright +regions analyzed in Ref.28 compared to the NuSTAR results. Simulations assume the Gaussian line +parameters measured by NuSTAR. In regions where NuSTAR only set an upper limit on the line +width, a smaller value is assumed. The inset shows two lines with assumed widths of 150 eV and +100 eV, respectively. The ASCENT centroid and line width uncertainties are a factor 2–10 smaller +than NuSTAR’s. +5 Summary and Outlook +Core collapse supernovae are considered to be a significant source of mid-Z elements in the Galaxy. +Despite significant theoretical and observational progress in the last few decades, many details of +the explosion are still poorly understood. One observational approach to gaining new insights is to +study the distribution of elements in supernova remnants. The isotope 44Ti is of particular interest +because it is produced in the innermost regions of the supernova engine, and it is radioactive with +a half-life long enough to be observable in historic supernova remnants. +In this paper we present a concept for a balloon-borne hard X-ray telescope called ASCENT +designed to study the nuclear emission lines from Cas A and SN 1987A, as well as potentially +other remnants that may be discovered by COSI. ASCENT uses an array of transition edge sensor +(TES) microcalorimeter detectors as its focal plane instrument, improving spectral resolution by +more than an order of magnitude over existing semiconductor detectors at gamma-ray energies. +Observations with ASCENT will significantly improve 3D maps of 44Ti in Cas A, and can deliver +detailed spectra of 44Ti from SN 1987A. +ASCENT will also demonstrate the viability of hard X-ray TES technology for a future space +mission. The energy resolution of an ASCENT-type mission would benefit all spectral studies +of a NuSTAR follow-up, e.g., broadband observations of Active Galactic Nuclei (AGN) cover- +ing the lines of the soft excess emission, the Fe Kα emission, and the Compton hump emission. +The broadband results obtained for a sample of ∼100 of AGNs can be used to calibrate the spin +measurements of ESA’s ATHENA mission43,44 and the proposed NASA Lynx mission.78–80 The +ASCENT detector technology could be used for example as the focal plane detector of the HEX-P +mission.65 While ASCENT has not been selected when first proposed, the team continues to pursue +the project and is planning to repropose the mission after maturing detector, cryostat, and X-ray +optics technologies. +18 + +Acknowledgments +FK is grateful for support by the Faculty Development Grant program of UNH. The work by CLF +is supported by the US Department of Energy through the Los Alamos National Laboratory. Los +Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear +Security Administration of the U.S. Department of Energy (Contract No. 89233218CNA000001). +HK and JN acknowledge NASA support under grant 80NSSC21K1817. HK acknowledges NASA +support under grants 80NSSC18K0264, 80NSSC22K1291, and NNX16AC42G. +References +1 E. M. Burbidge, G. R. Burbidge, W. A. Fowler, et al., “Synthesis of the Elements in Stars,” +Rev. Mod. Phys. 29, 547–650 (1957). +2 S. M. Matz, G. H. Share, M. D. Leising, et al., “Gamma-ray line emission from SN1987A,” +Nature 331, 416–418 (1988). +3 B.-G. Choi, G. R. Huss, G. J. Wasserburg, et al., “Presolar Corundum and Spinel in Ordinary +Chondrites: Origins from AGB Stars and a Supernova,” Science 282, 1284 (1998). +4 J. Schulte, M. Bose, P. A. Young, et al., “Three-dimensional Supernova Models Provide New +Insights into the Origins of Stardust,” Astrophys. J. 908, 38 (2021). +5 S. Amari, P. Hoppe, E. Zinner, et al., “Interstellar SiC with Unusual Isotopic Compositions: +Grains from a Supernova?,” Astrophys. J. Lett. 394, L43 (1992). +6 M. Herant and W. Benz, “Postexplosion Hydrodynamics of SN 1987A,” Astrophys. J. 387, +294 (1992). +7 I. Ahmad, J. P. Greene, E. F. Moore, et al., “Improved measurement of the Ti44 half-life from +a 14-year long study,” Phys. Rev. C 74, 065803 (2006). +8 J. Chen, B. Singh, and J. A. Cameron, “Nuclear Data Sheets for A = 44,” Nucl. Data Sheets +112, 2357–2495 (2011). +9 D. D. Clayton, S. A. Colgate, and G. J. Fishman, “Gamma-Ray Lines from Young Supernova +Remnants,” Astrophys. J. 155, 75 (1969). +10 J. N. Ullom and D. A. Bennett, “Review of superconducting transition-edge sensors for x-ray +and gamma-ray spectroscopy,” Supercond. Sci. Tech. 28, 084003 (2015). +11 J. A. B. Mates, D. T. Becker, D. A. Bennett, et al., “Simultaneous readout of 128 X-ray and +gamma-ray transition-edge microcalorimeters using microwave SQUID multiplexing,” Appl. +Phys. Lett. 111, 062601 (2017). +12 F. A. Harrison, W. W. Craig, F. E. Christensen, et al., “The Nuclear Spectroscopic Telescope +Array (NuSTAR) High-energy X-Ray Mission,” Astrophys. J. 770, 103 (2013). +13 S. E. Boggs, F. A. Harrison, H. Miyasaka, et al., “44Ti gamma-ray emission lines from +SN1987A reveal an asymmetric explosion,” Science 348, 670–671 (2015). +14 S. A. Grebenev, A. A. Lutovinov, S. S. Tsygankov, et al., “Hard-X-ray emission lines from +the decay of 44Ti in the remnant of supernova 1987A,” Nature 490, 373–375 (2012). +15 A. F. Iyudin, R. Diehl, H. Bloemen, et al., “COMPTEL observations of Ti-44 gamma-ray +line emission from CAS A,” Astron. Astrophys. 284, L1–L4 (1994). +16 J. Vink, J. M. Laming, J. S. Kaastra, et al., “Detection of the 67.9 and 78.4 keV Lines Asso- +ciated with the Radioactive Decay of 44Ti in Cassiopeia A,” Astrophys. J. Lett. 560, L79–L82 +(2001). +19 + +17 B. W. Grefenstette, F. A. Harrison, S. E. Boggs, et al., “Asymmetries in core-collapse super- +novae from maps of radioactive 44Ti in Cassiopeia A,” Nature 506, 339–342 (2014). +18 A. F. Iyudin, V. Schönfelder, K. Bennett, et al., “Emission from 44Ti associated with a previ- +ously unknown Galactic supernova,” Nature 396, 142–144 (1998). +19 E. Troja, A. Segreto, V. La Parola, et al., “Swift/BAT Detection of Hard X-Rays from Tycho’s +Supernova Remnant: Evidence for Titanium-44,” Astrophys. J. Lett. 797, L6 (2014). +20 S. S. Tsygankov, R. A. Krivonos, A. A. Lutovinov, et al., “Galactic survey of 44Ti sources +with the IBIS telescope onboard INTEGRAL,” Mon. Not. Royal Astron. Soc. 458, 3411–3419 +(2016). +21 L. A. Lopez, B. W. Grefenstette, S. P. Reynolds, et al., “A Spatially Resolved Study of +the Synchrotron Emission and Titanium in Tycho’s Supernova Remnant Using NuSTAR,” +Astrophys. J. 814, 132 (2015). +22 J. A. Tomsick, S. E. Boggs, A. Zoglauer, et al., “The Compton Spectrometer and Imager +Project for MeV Astronomy,” in Proc. 37th International Cosmic Ray Conference, (2021). +23 S. E. Woosley and T. A. Weaver, “Sub–Chandrasekhar Mass Models for Type IA Super- +novae,” Astrophys. J. 423, 371 (1994). +24 R. M. Crocker, A. J. Ruiter, I. R. Seitenzahl, et al., “Diffuse Galactic antimatter from faint +thermonuclear supernovae in old stellar populations,” Nature Astronomy 1, 0135 (2017). +25 M. Renaud, J. Vink, A. Decourchelle, et al., “The Signature of 44Ti in Cassiopeia A Revealed +by IBIS/ISGRI on INTEGRAL,” Astrophys. J. Lett. 647, L41–L44 (2006). +26 T. Siegert, R. Diehl, M. G. H. Krause, et al., “Revisiting INTEGRAL/SPI observations of +44Ti from Cassiopeia A,” Astron. Astrophys. 579, A124 (2015). +27 P. A. Young, C. L. Fryer, A. Hungerford, et al., “Constraints on the Progenitor of Cassiopeia +A,” Astrophys. J. 640, 891–900 (2006). +28 B. W. Grefenstette, C. L. Fryer, F. A. Harrison, et al., “The Distribution of Radioactive 44Ti +in Cassiopeia A,” Astrophys. J. 834, 19 (2017). +29 G. Magkotsios, F. X. Timmes, A. L. Hungerford, et al., “Trends in 44Ti and 56Ni from Core- +collapse Supernovae,” Astrophys. J. Suppl. Ser. 191, 66–95 (2010). +30 B.-C. Koo, H.-J. Kim, Y.-H. Lee, et al., “A Deep Near-infrared [Fe II]+[Si I] Emission Line +Image of the Supernova Remnant Cassiopeia A,” Astrophys. J. 866, 139 (2018). +31 N. J. Hammer, H. T. Janka, and E. Müller, “Three-dimensional Simulations of Mixing Insta- +bilities in Supernova Explosions,” Astrophys. J. 714, 1371–1385 (2010). +32 M. Ono, S. Nagataki, H. Ito, et al., “Matter Mixing in Aspherical Core-collapse Supernovae: +A Search for Possible Conditions for Conveying 56Ni into High Velocity Regions,” Astrophys. +J. 773, 161 (2013). +33 G. S. Vance, P. A. Young, C. L. Fryer, et al., “Titanium and Iron in the Cassiopeia A Super- +nova Remnant,” Astrophys. J. 895, 82 (2020). +34 R. Willingale, J. A. M. Bleeker, K. J. van der Heyden, et al., “X-ray spectral imaging and +Doppler mapping of Cassiopeia A,” Astron. Astrophys. 381, 1039–1048 (2002). +35 T. DeLaney, L. Rudnick, M. D. Stage, et al., “The Three-dimensional Structure of Cassiopeia +A,” Astrophys. J. 725, 2038–2058 (2010). +36 C. L. Fryer and A. Kusenko, “Effects of Neutrino-driven Kicks on the Supernova Explosion +Mechanism,” Astrophys. J. Suppl. Ser. 163, 335–343 (2006). +20 + +37 M. Herant, W. Benz, W. R. Hix, et al., “Inside the Supernova: A Powerful Convective En- +gine,” Astrophys. J. 435, 339 (1994). +38 M. Herant, “The convective engine paradigm for the supernova explosion mechanism and its +consequences.,” Phys. Rep. 256, 117–133 (1995). +39 A. Socrates, O. Blaes, A. Hungerford, et al., “The Neutrino Bubble Instability: A Mechanism +for Generating Pulsar Kicks,” Astrophys. J. 632, 531–562 (2005). +40 C. I. Ellinger, P. A. Young, C. L. Fryer, et al., “A Case Study of Small-scale Structure For- +mation in Three-dimensional Supernova Simulations,” Astrophys. J. 755, 160 (2012). +41 D. W. Stuchlik, “The Wallops Arc Second Pointer – A Balloon Borne Fine Pointing System,” +in AIAA Balloon Systems Conference, Proc. AIAA AVIATION Forum (2015). +42 F. Aharonian, H. Akamatsu, F. Akimoto, et al., “The quiescent intracluster medium in the +core of the Perseus cluster,” Nature 535, 117–121 (2016). +43 S. J. Smith, J. S. Adams, S. R. Bandler, et al., “Transition-edge sensor pixel parameter design +of the microcalorimeter array for the x-ray integral field unit on Athena,” in Space Telescopes +and Instrumentation 2016: Ultraviolet to Gamma Ray, Proc. SPIE 9905, 99052H (2016). +44 X. Barcons, D. Barret, A. Decourchelle, et al., “Athena: ESA’s X-ray observatory for the late +2020s,” Astronomische Nachrichten 338, 153–158 (2017). +45 D. A. Bennett, R. D. Horansky, D. R. Schmidt, et al., “A high resolution gamma-ray +spectrometer based on superconducting microcalorimeters,” Rev. Sci. Instrum. 83, 093113– +093113–14 (2012). +46 R. Winkler, A. S. Hoover, M. W. Rabin, et al., “256-pixel microcalorimeter array for high- +resolution γ-ray spectroscopy of mixed-actinide materials,” Nucl. Instrum. Meth. Phys. Res. +A 770, 203–210 (2015). +47 A. S. Hoover, E. M. Bond, M. P. Croce, et al., “Measurement of the 240Pu/239Pu Mass Ratio +Using a Transition-Edge-Sensor Microcalorimeter for Total Decay Energy Spectroscopy,” +Anal. Chem. 87(7), 3996–4000 (2015). +48 J. Uhlig, W. Fullagar, J. N. Ullom, et al., “Table-Top Ultrafast X-Ray Microcalorimeter Spec- +trometry for Molecular Structure,” Phys. Rev. Lett. 110, 138302 (2013). +49 L. Miaja-Avila, G. C. O’Neil, Y. I. Joe, et al., “Ultrafast Time-Resolved Hard X-Ray Emis- +sion Spectroscopy on a Tabletop,” Phys. Rev. X 6, 031047 (2016). +50 M. R. J. Palosaari, M. Käyhkö, K. M. Kinnunen, et al., “Broadband Ultrahigh-Resolution +Spectroscopy of Particle-Induced X Rays: Extending the Limits of Nondestructive Analysis,” +Phys. Rev. Appl. 6, 024002 (2016). +51 S. Okada, D. A. Bennett, C. Curceanu, et al., “First application of superconducting transition- +edge sensor microcalorimeters to hadronic atom X-ray spectroscopy,” Prog. Theor. Exp. Phys. +2016, 091D01 (2016). +52 G. C. O’Neil, L. Miaja-Avila, Y. I. Joe, et al., “Ultrafast Time-Resolved X-ray Absorption +Spectroscopy of Ferrioxalate Photolysis with a Laser Plasma X-ray Source and Microcalori- +meter Array,” J. Phys. Chem. Lett. 8(5), 1099–1104 (2017). +53 W. B. Doriese, P. Abbamonte, B. K. Alpert, et al., “A practical superconducting- +microcalorimeter X-ray spectrometer for beamline and laboratory science,” Rev. Sci. Instrum. +88, 053108 (2017). +21 + +54 C. J. Titus, M. L. Baker, S. J. Lee, et al., “L-edge spectroscopy of dilute, radiation-sensitive +systems using a transition-edge-sensor array,” J. Chem. Phys. 147, 214201 (2017). +55 K. D. Irwin and G. C. Hilton, Transition-Edge Sensors, 63 (2005). +56 M. K. Bacrania, A. S. Hoover, P. J. Karpius, et al., “Large-Area Microcalorimeter Detectors +for Ultra-High-Resolution X-Ray and Gamma-Ray Spectroscopy,” IEEE Trans. Nucl. Sci. +56, 2299–2302 (2009). +57 J. Weber, K. M. Morgan, D. Yan, et al., “Development of a transition-edge sensor bilayer +process providing new modalities for critical temperature control,” Superconductor Science +and Technology (2020). +58 J. A. B. Mates, K. D. Irwin, L. R. Vale, et al., “Flux-Ramp Modulation for SQUID Multi- +plexing,” J. Low Temp. Phys. 167, 707–712 (2012). +59 J. Mates, G. C. Hilton, K. D. Irwin, et al., “Demonstration of a multiplexer of dissipation- +less superconducting quantum interference devices,” Applied Physics Letters 92(2), 023514 +(2008). +60 J. Hickish, Z. Abdurashidova, Z. Ali, et al., “A Decade of Developing Radio-Astronomy +Instrumentation using CASPER Open-Source Technology,” J. Astron. Instrum. 5, 1641001– +12 (2016). +61 S. Agostinelli, J. Allison, K. Amako, et al., “Geant4 – a simulation toolkit,” Nucl. Instrum. +Meth. Phys. Res. A 506(3), 250 – 303 (2003). +62 J. Allison, K. Amako, J. Apostolakis, et al., “Geant4 developments and applications,” IEEE +Trans. Nucl. Sci. 53, 270–278 (2006). +63 J. Allison, K. Amako, J. Apostolakis, et al., “Recent developments in Geant4,” Nucl. Instrum. +Meth. Phys. Res. A 835, 186 – 225 (2016). +64 D. Spiga, G. Pareschi, O. Citterio, et al., “Development of multilayer coatings (Ni/C-Pt/C) +for hard x-ray telescopes by e-beam evaporation with ion assistance,” in UV and Gamma-Ray +Space Telescope Systems, G. Hasinger and M. J. L. Turner, Eds., Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series 5488, 813–819 (2004). +65 K. K. Madsen, F. Harrison, D. Broadway, et al., “Optical instrument design of the high- +energy x-ray probe (HEX-P),” in Space Telescopes and Instrumentation 2018: Ultraviolet +to Gamma Ray, J.-W. A. den Herder, S. Nikzad, and K. Nakazawa, Eds., Society of Photo- +Optical Instrumentation Engineers (SPIE) Conference Series 10699, 106996M (2018). +66 J. E. Gudmundsson, P. A. R. Ade, M. Amiri, et al., “The thermal design, characterization, and +performance of the SPIDER long-duration balloon cryostat,” Cryogenics 72, 65–76 (2015). +67 B. Reichborn-Kjennerud, A. M. Aboobaker, P. Ade, et al., “EBEX: a balloon-borne CMB +polarization experiment,” in Millimeter, Submillimeter, and Far-Infrared Detectors and In- +strumentation for Astronomy V, Society of Photo-Optical Instrumentation Engineers (SPIE) +Conference Series 7741, 77411C (2010). +68 L. M. Fissel, P. A. R. Ade, F. E. Angilè, et al., “The balloon-borne large-aperture submillime- +ter telescope for polarimetry: BLAST-Pol,” in Millimeter, Submillimeter, and Far-Infrared +Detectors and Instrumentation for Astronomy V, Society of Photo-Optical Instrumentation +Engineers (SPIE) Conference Series 7741, 77410E (2010). +69 B. P. Crill, P. A. R. Ade, D. R. Artusa, et al., “BOOMERANG: A Balloon-borne Millimeter- +Wave Telescope and Total Power Receiver for Mapping Anisotropy in the Cosmic Microwave +Background,” The Astrophysical Journal Supplement Series 148, 527–541 (2003). +22 + +70 G. W. Wilson and P. T. Timbie, “Construction techniques for adiabatic demagnetization re- +frigerators using ferric ammonium alum,” Cryogenics 39, 319–322 (1999). +71 S. T. Chase, T. L. R. Brien, S. M. Doyle, et al., “Pre-cooling a 3He/4He dilutor module with +a sealed closed-cycle continuous cooler,” in Materials Science and Engineering Conference +Series, Materials Science and Engineering Conference Series 502, 012134 (2019). +72 F. Kislat, B. Beheshtipour, P. Dowkontt, et al., “Design of the Telescope Truss and Gondola +for the Balloon-Borne X-ray Polarimeter X-Calibur,” J. Astron. Instrum. 6, 1740003 (2017). +73 Q. Abarr, H. Awaki, M. G. Baring, et al., “XL-Calibur - a second-generation balloon-borne +hard X-ray polarimetry mission,” Astropart. Phys. 126, 102529 (2021). +74 Q. Abarr, B. Beheshtipour, M. Beilicke, et al., “Performance of the X-Calibur hard X-ray +polarimetry mission during its 2018/19 long-duration balloon flight,” Astropart. Physics 143, +102749 (2022). +75 H. Matsumoto, H. Awaki, A. Furuzawa, et al., “Ray-tracing simulation and in-orbit perfor- +mance of the ASTRO-H hard x-ray telescope (HXT),” in Space Telescopes and Instrumen- +tation 2016: Ultraviolet to Gamma Ray, J.-W. A. den Herder, T. Takahashi, and M. Bautz, +Eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 9905, +990541 (2016). +76 F. Shirazi, M. Arman Hossen, D. Becker, et al., “The 511-CAM Mission: A Pointed 511 keV +Gamma-Ray Telescope with a Focal Plane Detector Made of Stacked Transition Edge Sensor +Microcalorimeter Arrays,” arXiv e-prints , arXiv:2206.14652 (2022). +77 R. Barlow, “Event Classification Using Weighting Methods,” J. Comp. Phys. 72, 202–219 +(1987). +78 J. A. Gaskin, M. C. Weisskopf, A. Vikhlinin, et al., “The X-ray Surveyor Mission: a concept +study,” in UV, X-Ray, and Gamma-Ray Space Instrumentation for Astronomy XIX, Proc. SPIE +9601, 96010J (2015). +79 S. R. Bandler, J. S. Adams, J. A. Chervenak, et al., “Development of x-ray microcalorimeter +imaging spectrometers for the X-ray Surveyor mission concept,” in Space Telescopes and +Instrumentation 2016: Ultraviolet to Gamma Ray, Proc. SPIE 9905, 99050Q (2016). +80 Physics of the Cosmos Program Office, “Program Annual Technology Report,” (2017). +23 + diff --git a/J9AzT4oBgHgl3EQfj_2L/content/tmp_files/load_file.txt b/J9AzT4oBgHgl3EQfj_2L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1bd26e266f5c955cae61eba72591e9f194922a36 --- /dev/null +++ b/J9AzT4oBgHgl3EQfj_2L/content/tmp_files/load_file.txt @@ -0,0 +1,1212 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf,len=1211 +page_content='ASCENT – A balloon-borne hard X-ray imaging spectroscopy telescope using transition edge sensor microcalorimeter detectors F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Kislata, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Beckerb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bennettc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Dasguptaa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fowlerc, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fryerd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Gardb, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Gaue, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Gurgewf, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Harmong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hayashif, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Heatwoleg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hossene, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Krawczynskie,h,i, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lanzig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Legerea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Matesc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' McConnella, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nagye,h,i, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Okajimaf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Satoj, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Schmidtc, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Spoonera, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Swetzc, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Tamuraf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ullomc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Weberb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Westera, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Youngk aUniversity of New Hampshire, Department of Physics & Astronomy and Space Science Center, 8 College Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Durham, NH 03824, USA bUniversity of Colorado, Department of Physics, 2000 Colorado Ave, Boulder, CO 80309, USA cNIST Boulder Laboratories, 325 Broadway, Boulder, CO 80305, USA dLos Alamos National Laboratory, Los Alamos, NM 87545, USA eWashington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Louis, Physics Department, 1 Brookings Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', CB 1105, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Louis, MO 63130, USA fNASA’s Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA gNASA Wallops Flight Facility, 32400 Fulton St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Wallops Island, VA 23337, USA hMcDonnell Center for the Space Sciences at Washington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Louis iQuantum Sensor Center at Washington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Louis jRikkyo University, 3-34-1 Nishi Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan kArizona State University, School of Earth and Space Exploration, Tempe, AZ 85287, USA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Core collapse supernovae are thought to be one of the main sources in the galaxy of elements heavier than iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Understanding the origin of the elements is thus tightly linked to our understanding of the explosion mecha- nism of supernovae and supernova nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' X-ray and gamma-ray observations of young supernova remnants, combined with improved theoretical modeling, have resulted in enormous improvements in our knowledge of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The isotope 44Ti is one of the most sensitive probes of the innermost regions of the core collapse engine, and its spatial and velocity distribution are key observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hard X-ray imaging spectroscopy with the Nuclear Spectro- scopic Telescope Array (NuSTAR) has provided new insights into the structure of the supernova remnant Cassiopeia A (Cas A), establishing the convective nature of the supernova engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, many questions about the details of this engine remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' We present here the concept for a balloon-borne follow-up mission called ASCENT (A SuperConduct- ing ENergetic x-ray Telescope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT uses transition edge sensor gamma-ray microcalorimeter detectors with a demonstrated 55 eV Full Width Half Maximum (FWHM) energy resolution at 97 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This 8–16-fold improvement in energy resolution over NuSTAR will allow high resolution imaging and spectroscopy of the 44Ti emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This will allow a detailed reconstruction of gamma-ray line redshifts, widths, and shapes, allowing us to address questions such as: What is the source of the neutron star “kicks”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' What is the dominant production pathway for 44Ti?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Is the engine of Cas A unique?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Keywords: X-ray, spectroscopy, instrumentation, Supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fabian Kislat, fabian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='kislat@unh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='edu 1 Introduction Core-collapse supernovae (CCSNe) of prior generations of stars are thought to be a major source of elements heavier than iron in our Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 Hence, understanding their explosion mechanism is key to understanding the evolution of our Galaxy eventually supporting life on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In this paper, we describe a concept for a new balloon-borne high-energy X-ray telescope and potential future satellite mission that will, among other science goals, provide new experimental insights into the inner workings of the CCSN engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Supernovae as a source of heavy elements are supported both by theoretical considerations and experimental evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Since 56Fe is the nucleus with the lowest mass per nucleon, fusion of 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='01525v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='IM] 4 Jan 2023 heavier elements cannot serve as a source of energy in stellar cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instead, heavier elements are formed via slow (s-process) or rapid neutron capture (r-process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Some of the earliest and strongest experimental evidence for nucleosynthesis in CCSNe comes from the detection of 847 keV and 1238 keV gamma-rays associated with the decay of 56Co to 56Fe in SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 Direct evidence for the fact that our Solar System is indeed made of reprocessed stellar ejecta comes from the analysis of presolar grains in meteoritic material and interplanetary dust, whose isotopic composition is representative of the seed material of the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3 In fact, recent simulations show that 15M⊙ CCSNe are capable of producing many of the isotopic anomalies found in certain presolar SiC grains,4 which have long been argued to condense in supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 An important conclusion from the observations of high-energy X-ray and gamma-ray emission from SN 1987A soon after the explosion was that mixing of material from the different shells of the progenitor star must occur very early on in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='6 This mixing moves radioactive nickel outward from the innermost parts of the ejecta, which then drives the X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, the details of this mixing and the underlying mechanism are still poorly understood and depend on the local conditions of the early shock, such as peak temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Due to these convective instabilities, anisotropies are expected in supernovae and their remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The structure of young supernova remnants (SNR) reflects the conditions of the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Since many galactic SNR are spatially resolvable in X-rays and gamma-rays, these remnants are an excellent site to study supernova explosion physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Regions where explosive Si burning occurs can be observed via the K line emission from 56Fe, which is a decay product of 56Ni produced during Si burning with relatively little dependence on local conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The observations, however, come with the caveats that the X-ray emission depends on the heating of the material in the shock, and that some of the iron may actually be interstellar material swept up in the shock rather than supernova ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The production of 44Ti in the same regions, on the other hand, is very sensitive to the local conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This isotope with a half-life of (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3) yr7 is in principle observable in galactic supernova remnants up to a few hundred years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' It decays via 44Ti → 44Sc → 44Ca, emitting gamma-rays with energies of 1157 keV, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='32 keV and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='87 keV with branching ratios between 93 % and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='8 These gamma-rays directly trace the distribution of 44Ti decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Because of these properties, observations of 44Ti are a particularly powerful tool to test supernova models, which has been noted as early as 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 Here, we discuss a concept for a new balloon-borne high-energy X-ray telescope called AS- CENT (A SuperConducting ENergetic x-ray Telescope), which had been proposed to NASA’s Astrophysics Pioneers program, and which could form the basis of a future NuSTAR follow-up mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT consists of a novel transition edge sensor (TES) microcalorimeter gamma-ray detector array in the focal plane of a multi-layer coated Wolter-type focusing X-ray mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Transi- tion edge sensors utilize the rapid change in conductivity with temperature of a superconductor at its superconducting transition temperature Tc for calorimetric energy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='10 A gamma- ray spectrometer is constructed by coupling the TES to a thick absorbing structure, commonly made of Sn, which increases the quantum efficiency for the detection of 10–100 keV photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Re- cently, an array consisting of 512 detectors with a spectral resolution of 55 eV FWHM at 97 keV has been demonstrated,11 and individual detectors have achieved a resolution as precise as 22 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Using these detectors, ASCENT’s spectral resolution will be about 15 times better than NuSTAR in the 60–85 keV energy range (900 eV at 60 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='12 2 Additionally, ASCENT will use a new Ni/C multilayer structure on its X-ray optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The energy bandpass of NuSTAR was limited by the platinum K edge at 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='395 keV of its Pt/C multilayer, which prevents it from observing the blue-shifted 78 keV 44Ti line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The use of a Ni/C multilayer will extend ASCENT’s bandpass to 85 keV and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Furthermore, its multilayer structure will be optimized for the 55–85 keV range, in order to maximize its effective area for observations of 44Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' While the baseline angular resolution of the ASCENT optics of 2′ will be slightly worse than NuSTAR, it will still allow resolution of the most prominent 44Ti emission regions in Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Observations of the supernova remnant Cas A with ASCENT will test if asymmetries of the ejecta can completely account for compact remnant “kicks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Furthermore, they will allow us to determine the dominant pathway for the production of 44Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' So far, 44Ti has only been firmly detected from two objects: SN 1987A13,14 and Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='15–17 Ten- tative detections from Vela Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='18 and Tycho’s SNR19 have so far not been confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='20,21 However, the Compton Spectrometer and Imager (COSI) will map the Galaxy with unprecedented spectral and spatial resolution and may find additional sources of 44Ti emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='22 While COSI is an ex- cellent tool to discover 44Ti emission from additional SNR, its spatial and spectral resolution are not sufficient to map individual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A more sensitive space mission based on the ASCENT de- sign could follow up on these detections and provide detailed maps of additional SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This would answer the question whether features observed in Cas A are universal or whether there is wide variation in the underlying engine depending on properties of the progenitor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Furthermore, a detection of 44Ti from a remnant associated with a type Ia supernova, such as Tycho’s SNR, would be a major breakthrough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ordinarily, SNIa are not expected to produce much 44Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, some models predict a potential detonation below the Chandrasekhar mass limit (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Woosley and Weaver, 199423), in which case a large amount of 44Ti may be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Such explosions are thought to be one of the candidates for the origin of Galactic positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='24 Thus, such an observa- tion would not only constrain the SNIa explosion mechanism but also provide new insights into the origin of Galactic positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Launched on a stratospheric balloon from Kiruna, Sweden, ASCENT will float westward at an altitude of about 125,000 ft to northern Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Typical flight times are 5–7 days, allowing for up to 560 ksec of observation time of Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A Southern Hemisphere flight from McMurdo Station, Antarctica, will circle the South Pole at least once for a typical flight time of two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Such a flight will allow deep observations of SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In Section 2 we discuss the scientific questions addressed by the ASCENT balloon mission, the technical aspects of which we describe in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In Section 4, we present results of Geant4 Monte Carlo simulations of the expected performance of ASCENT and the resulting sensitivity of the instrument to address its science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Finally, in Section 5, we summarize the results and give an outlook towards a space-based mission based on the ASCENT technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 2 Scientific objectives ASCENT’s improved spectral resolution will allow it to address some key questions brought up by the recent NuSTAR observations of Cas A: what is the source of compact remnant "kicks" and what are the conditions of 44Ti production in the Cas A supernova remnant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The high yield of 44Ti detected in Cas A15–17,25,26 is seen as strong support for the expected anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='27 A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 Ms observation with the NuSTAR satellite was used to obtain the first 3-D map of the 44Ti ejecta,17,28 which confirmed these high yields and, furthermore, found that the 44Ti 3 lies in clumpy structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Because much of the 44Ti was found to lie in unshocked regions, its observation provides a pristine measurement of the asymmetries in the supernova engine, and this NuSTAR data decisively showed that Cas A was produced from an explosion with multiple outflows (as expected from the convective engine) and not a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, the NuSTAR observations left many questions unanswered and raised a series of new problems with our understanding of Cas A, which ASCENT will address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Along with 56Ni, 44Ti is produced in the innermost supernova ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In contrast to 56Ni produc- tion, 44Ti production is extremely sensitive to the temperature and density evolution of the ejecta29 and, hence, the nature of the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Within 1–2 years of a supernova explosion, 56Ni decays to stable iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This iron produces emission lines when heated by the reverse shock as the supernova ejecta plows through the circumstellar medium and depends on the distribution of the circumstellar medium as well as the explosion, making its interpretation complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The innermost iron in the remnant is also difficult to measure accurately, since it is inside of the reverse shock and therefore cold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Although attempts have been made at detecting this iron in the infrared in Cas A,30 the un- biased nature of the 44Ti observations and their sensitivity to the explosion characteristics have led to 44Ti’s important role in shaping our understanding of the core-collapse explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 The core collapse engine of Cas A A key to understanding the core collapse supernova engine is understanding the production and mixing of 44Ti during the explosion, which will identify the dominant pathways of 44Ti produc- tion in supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The processes that produce Ti are quasi-statistical or statistical equilibrium processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' They produce an equilibrium distribution of nuclei based on the nuclear chemical po- tentials at a given temperature, density, and electron fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='29 The local ratio between 44Ti and 56Ni strongly constrains the thermodynamic initial conditions, allowing a fairly precise determi- nation of the final distribution of all nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The remaining uncertainty can then be attributed to the thermodynamic history, which determines the details of freezeout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Observations of additional nuclear species will provide additional constraints and reduce residual uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Combining ASCENT’s 44Ti observations with observations of Cas A’s Si and Fe lines from JAXA’s and NASA’s XRISM mission (to be launched in 2023) will allow us to perform an im- proved reconstruction of the Si, Fe, and Ti configurations at the current time and at the time of the supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT will improve the spectral resolution of the 44Ti emission over NuSTAR, allowing us to construct a more detailed map of the clumpy structures in Cas A, enabling a more detailed comparison to numerical models and helping to disentangle multiple structures along a line of sight and the properties of these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These comparisons provide direct constraints on the engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' NuSTAR’s results indicate that most of the ejecta are moving away from us with a velocity of 1000–5000 km s−1 but that some 44Ti regions (Region 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Region 20 of Grefenstette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', 201728) move towards us with 7500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The combination of ASCENT’s 67 eV energy resolution with its effective area extending to >85 keV will enable an analysis with much smaller systematic errors than that of the NuSTAR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' If confirmed, these high velocity ejecta would challenge current supernova theory and provide a strong constraint on models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Most current mod- els that lead to a high yield of 44Ti do not result in velocities >4500 km s−1,31,32 and the highest velocity found in the models by Vance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='33 is ∼5500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The detailed velocity measurement can also be used to understand 44Ti production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nucleosynthesis calculations predict different 44Ti 4 1 2 Fig 1: X-ray image of emission lines in Cassiopeia A: iron (red), silicon/magnesium (green), titanium (blue), and continuum emission (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='17 Two parts of the remnant of particular interest are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Region 1 is the only region in which NuSTAR detected significantly blue-shifted 44Ti ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Region 2 is of interest due to the highly blue-shifted almost pure iron ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Composite Chandra/NuSTAR image credit NASA/JPL-Caltech/CXC/SAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' yields depending on the ejecta velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='33 We can use the ejecta velocities to constrain the trajec- tories (temperature and density evolution) of the ejecta, allowing us to test both our explosion and nucleosynthesis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These ejecta velocities can be tied to more fundamental properties like the electron fraction and nuclear cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Because we expect 44Ti to be produced at some level whenever 56Ni is produced, we expect to see 44Ti lines wherever iron is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This raises the question why there are large iron ejecta with no evident detection of 44Ti in Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The NuSTAR observations could not detect 44Ti in the iron-rich southeastern region of Cas A (Region 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Is it because that iron is produced directly (not the decay product of 56Ni), or was there a large amount of 56Ni produced with 44Ti mass fractions below the NuSTAR detection limit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The NuSTAR upper limit in this region of the SNR is not very constraining,28 and the iron-rich ejecta are blue-shifted with a velocity up to 3000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='34,35 Therefore, ASCENT may be able to detect the 44Ti in this region due to its sensitivity to the blue-shifted 78 keV line, or significantly improve on the NuSTAR upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 Compact Remnant Kicks Observations of pulsar proper motions and the existence of specific peculiar binary systems sug- gest that momentum is imparted onto compact remnants during their formation (for a review, see Fryer and Kusenko, 200636).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A diverse set of models have been proposed to create these kicks, but these models can be separated into two categories: asymmetries in the ejecta and asymmetries in the neutrino emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Under the convection-enhanced supernova engine paradigm,37 low-mode convection produces asymmetric explosions with a nonzero net momentum in the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='38 These asymmetries impart a net momentum to the compact remnant, and a 1 % asymmetry in the ejecta produces the high observed kick velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Although simulations have struggled to produce some of the highest observed kicks, ejecta asymmetries remains one of the strongest candidate mecha- nisms for explaining pulsar proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Alternatively, asymmetries in the neutrino emission 5 (typically requiring strong magnetic fields – albeit not necessarily strong bipolar magnetic fields) also carries away a net momentum, imparting an equally strong kick onto the compact remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='39 The different mechanisms proposed in the ejecta and neutrino mechanisms make a variety of predictions on the relation of the compact remnant kicks with relation to angular momentum, dipole magnetic-field strength, final remnant mass, and the formation of a black hole versus a neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Many of these predictions are indirect, and it is difficult to place strong constraints on the mechanism with existing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, NuSTAR observations opened up the potential for a more direct observational constraint with detailed maps of the 44Ti to compare the asymmetries in the explosion to the remnant velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Because 44Ti is produced in the innermost ejecta, it is an ideal probe of these explosion asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, to truly compare the explosion asymmetries with the compact remnant kick, we need detailed 3-dimensional ejecta information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Although the current NuSTAR data hinted at a correlation between the explosion asymmetries and the compact remnant kick supporting the ejecta kick mechanism,28 the higher-fidelity ASCENT observations will allow a more quantitative test of the ejecta kick mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3 Validating the convective SN engine The convective nature of the supernova engine in Cas A has been established quite firmly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, there are only two SNR with confirmed detections of 44Ti, and Cas A is the only SNR in which 44Ti emission has been spatially resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This raises the question whether the SN engine of Cas A is unique, which ASCENT can address through observations of SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='13 While ASCENT cannot spatially resolve SN 1987A, a precise measurement of the 44Ti line shapes can be used to quantify asymmetries in the Ti distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Figure 2 shows the velocity distributions along three lines of sight for two different supernova explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These velocity distributions are derived from 3-dimensional smooth particle hydrody- namics simulations of asymmetrically-driven supernova explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='40 One explosion is bimodal (either produced by a mild “jet” or low-mode convection model with rotation) and the other is more representative of a low-rotation, low-mode convectively-driven explosion (“Asym”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' For the velocity distributions in this figure, we chose 3 different lines-of-site and measured the velocities of the ejecta along these lines-of-site (to determine the red- and blue-shifted features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The ASCENT observations will not only be able to easily differentiate between these models, but also enable us to further constrain the specific features of the convective engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3 Technical implementation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 Overview The ASCENT experiment (Figure 3) uses a 12 m optical bench with a 45 cm-diameter, F = 12 m multilayer X-ray mirror at the front end and a cryogenically cooled microcalorimeter detector as- sembly at the rear end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A balloon gondola holds a two-frame gimbal, pointing the optical bench in the direction of the observed astrophysical sources with the help of the Wallops Arc Second Pointer (WASP) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='41 The microcalorimetric detector array is cooled by an Adiabatic Demagnetiza- tion Refrigerator (ADR) inside a 65 L liquid He dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Table 1 summarizes key characteristics of the ASCENT observatory and its expected performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In the remainder of this section, we describe the design of each of the main components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The telescope will be carried by a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 × 106 m3 He-filled balloon to an altitude of about 38 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' When launched from Esrange in Kiruna, Sweden, it will partially circle the North Pole, reaching 6 NuSTAR ASCENT Fig 2: Velocity distribution of the 44Ti ejecta for two different supernova explosions: a bipolar explosion where the ejecta is fastest along the axis (jet) and an explosion with multiple strong outflows mimicking the predictions of the convective supernova engine (asym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='40 The structure in the line of sight velocity distribution can be tied to the structure of the supernova engine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', the bipolar explosion has a very different profile than the asymmetric explosion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The bars at the top of the graph illustrate the line-of-sight velocity resolution of NuSTAR and ASCENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Thanks to ASCENT’s energy resolution, we will be able to measure these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Northern Canada after a typically 5–7-day flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' On this trajectory, the telescope will be able to continuously observe Cas A with an elevation angle of about 36 − 82°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Additional, longer, flights from McMurdo Station (Antarctica) will circle the South Pole, enabling deep observations SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 Focal plane instrumentation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 Transition Edge Sensor array Microcalorimeter technology has shown great promise for transforming X-ray astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' For example, the Hitomi mission used a 36-pixel Si thermistor microcalorimeter array for its Soft X- ray Spectrometer (SXS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='42 The Athena mission (to be launched in 2032) will use a 4000-pixel microcalorimeter array for its X-ray Integrated Field Unit (X-IFU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='43,44 Over the last 15 years, transition-edge sensor (TES) microcalorimeter spectrometers have been developed as cutting-edge tools in the fields of nuclear materials analysis11,45–47 and the X-ray sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='48–54 TES microcalorimeters are detectors that measure the energy of individual photons through the temperature change of a superconducting thin film thermometer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The TES thermometer is coupled to a photon absorber composed of a high-Z element such as bismuth or tin, enabling high quantum efficiency for x-rays up to 100 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The fundamental energy resolution of a calorimeter is ∆E ∝ � kBT 2C, (1) 7 Asym, losl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='0 los2 line of site and velocity bin (M) los3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 Jet,losl los2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='0 los3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='0 60- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='0 4000 3000 2000 1000 0 1000 2000 3000 4000 Velocity (km/s)Table 1: Key ASCENT payload characteristics and expected performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Details of the perfor- mance estimates are provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Component Description Performance Truss Carbon fiber tubes and aluminum joints Focal spot movement <3 mm, alignment knowledge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 mm (9′′) Pointing system Pitch-yaw articulated Pointing precision 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='0–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='6′′ (3σ) on source Star camera 100 mm, f/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 short-wave infrared lens Pointing knowledge <15′′ (3σ) X-ray mirror Wolter I, 12 m focal length, diameter 40 cm, 110 Ni/C-coated and 100 Pt/C- coated shells Effective area 190 cm2 at 70 keV, Angular resolution 2′ HPD, Field of view 5′ FWHM Cryostat LHe-backed adiabatic demagnetization refrigerator Base temperature 70 mK Detector Two-layer gamma-ray TES array, 256 pixels each, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='59 mm3 ab- sorbers (30′′ ×30′′ at 12 m), microwave multiplexed readout Bandpass: 2–100 keV, energy res- olution ∆E(80 keV) = 67 eV FWHM Power Detectors, cryostat, heaters ∼350 W Mass Mass under balloon rotator ∼1700 kg Signal rate 1 Crab source at 45° elevation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 Hz at 60–80 keV Background rate BGO shield veto applied 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='02 Hz at 60–80 keV X-ray optics Star camera WASP gimbals SIP Battery box and CPU Truss CPU Cryostat ACD Truss TDRSS antenna Fig 3: CAD rendering of the ASCENT telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Key components are labeled in the Figure and de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The gondola and truss design are almost identical to XL-Calibur, maximizing flight heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 8 Weak thermal link Thermal bath shunt R SQUID readout RTES bias I (a) Time Temperature (b) Temperature Resistance Transition ΔT ΔR (c) Fig 4: (a) Calorimetric spectroscopy of x-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' An incident photon deposits its energy into a target with a weak thermal link to a cold isothermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (b) A typical pulse-response curve with a decay time determined by the properties of the calorimeter element and its coupling to the isothermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The filtered pulse height is an extremely precise measure of the photon’s energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (c) Thermometry is performed with a thin-film superconducting transition-edge sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The extreme precision results from the sharp temperature dependence of the electrical resistance of the thin film operated close to its superconducting transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' where T and C are the sensor temperature and heat capacity,10,55 allowing these devices to achieve extraordinary energy resolution by operating at cryogenic temperatures of about ∼100 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A prototype detector array called “Spectrometer to Leverage Extensive Development of Gamma- ray TESs for Huge Arrays using Microwave Multiplexed Enabled Readout” (SLEDGEHAMMER, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 5 and Mates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', 201711) has achieved a full-width at half maximum (FWHM) reso- lution of 55 eV at 97 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A resolution as low as 22 eV has been demonstrated with individual detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='11,45,56 The 100 keV energy resolution of 55 eV FWHM of SLEDGEHAMMER is 10× better than that of cryogenically cooled High Purity Germanium spectrometers (HPGe) and ∼20× better than room-temperature CdZnTe detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT will use detectors similar to those of the SLEDGEHAMMER hard X-ray/γ-ray spec- trometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='11 A photograph of a SLEDGEHAMMER detector is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These sensors use polycrystalline tin absorbers to absorb photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Tin is chosen because it combines a relatively high stopping power for γ-rays in the energy range of interest with a low specific heat at cryogenic tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In SLEDGEHAMMER, these absorbers are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='45 mm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='45 mm in area and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='38 mm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' For ASCENT we plan on using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='59 mm thick absorbers to increase quantum efficiency to 87 % at 68 keV and 75 % at 78 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The 55 % increase in absorber volume results in a correspond- ing increase of the heat capacity, and a 25 % increased energy resolution based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The expected energy resolution of the detectors, thus, increases to 68 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The absorbers are glued to epoxy posts, which are connected to the TES element by copper traces of equal length to ensure a uniform thermal path to the sensor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The TES element is a 400 µm × 400 µm bilayer of superconducting material and normal metal, lithographically deposited on a Si3N4 membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The transition temperature Tc is set to ∼120 mK by the superconducting proximity effect in thin-film bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This Tc includes enough margin above the base temperature of an Adiabatic Demagne- tization Refrigerator (ADR) to allow for stable operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Options for bilayer materials include MoCu, as in SLEDGEHAMMER, as well as MoAu TES using NIST’s patented hasTES process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='57 Current fabrication methods require manual placement of the absorbers on the TES array using mechanical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This constrains the minimum size and spacing of the absorbers to dimensions 9 Fig 5: Left: Photograph of the fully assembled SLEDGEHAMMER detector package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The package contains eight TES microcalorimeter chips with 32 sensors each (center), eight microwave mul- tiplexer chips with 32 channel readout (outer vertical columns), and eight chips each for detector bias, Nyquist filtering, and signal routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The TES signals are read out by two pairs of coaxial cables attached to the box by SMA connectors on the top and bottom of the box, each record- ing the signals for 128 sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT will use similar architecture to minimize risk, but with two monolithic detector chips stacked on top of each other, to minimize inter-pixel dead space and maximize collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Right: A combined 153Gd spectrum from 89 active TESs measured simultaneously using microwave SQUID multiplexing readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The inset shows a zoomed region around the 97 keV γ-ray peak (blue) with a Gaussian fit FWHM resolution of 55 eV (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The energy resolution achievable with TES microcalorimeters is 15 times better than that achieved by NuSTAR, achieving 270 km/s accuracy in measurements of the velocity of 44Ti ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Reprinted from Mates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (2017)11 with the permission of AIP Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fig 6: (a) Photograph of a TES gamma-ray microcalorimeter pixel before the Sn absorber is at- tached, showing the Si3N4 membrane (darker area), the Mo-Cu TES in the middle, and 20 SU8 epoxy posts connected to the TES by the Cu legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (b) A portion of the detector chip with some of the Sn absorbers attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The Sn absorbers are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='45 mm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='45 mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='38 mm thick, placed on a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='75 mm pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (c) Each ASCENT detector die uses a central array of 256 of these γ-ray sensors (within the black inner circle) surrounded by an octagonal pattern of eight sets of 32-channel bias chips (blue), Nyquist filtering chips (red) and microwave SQUID multiplexer chips (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' For scale, the outer black circle is 80 mm in diameter, and the overlay indicates a projection of the Cas A 44Ti distribution measured by NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Each die will be fabricated monolithically from a 75 mm Si wafer to minimize space between pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Figures (a) and (b) reprinted from Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (2012)45 with the permission of AIP Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 10 20 mm onpicbor 品品 4(b) (a) SU8 posts MoCu film 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 mmclose to those of SLEDGEHAMMER, resulting in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='75 mm pixel pitch and an array fill fraction of about 65 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' At 12 m focal length, this pixel pitch corresponds to an angular separation of 30′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The low array fill fraction correspondingly reduces the overall photon collection efficiency of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' To alleviate these issues, the instrument detector package will consist of 512 detectors, in the form of two dies each containing 256 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The dies will be stacked on top of each other and offset so that the detectors in the lower die will lie directly underneath the gaps between detectors in the upper die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This will result in a total detection efficiency for photons striking the array of 80 % at 68 keV and 70 % at 80 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The layout and design of each die is conceptually similar to the proven design of the SLEDGEHAMMER microcalorimeter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The central array of 256 TES detectors in each die (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6(c)) is fabricated from a single 75 mm Si wafer, with wiring to carry the TES signals to bond pads for connection to the rest of the readout circuitry arranged in an octagon around the outside of the TES array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' At 12 m focal length, a point spread function with the half power diameter (HPD) of ASCENT corresponds to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 mm, which is Nyquist sampled by each of the two detector dies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Combining the two offset detector dies allows a sampling of the PSF with an effective detector pitch of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The distance between the two dies will be less than 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Assuming the focal plane of the X-ray optics is placed directly between the two dies, the HPD of the point spread function will increase by only ∼2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The array diameter corresponds to an angular scale of about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='8′, significantly larger than the field of view of the X-ray optics, which eases the requirements on alignment stability as long as alignment knowledge can be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 Sensor readout TES arrays use multiplexing to minimize the thermal load and cryogenic complexity of wire con- nections to room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The development of the Microwave SQUID Multiplexer, which reads out array of microcalorimeters using microwave techniques,11,58 increases the available mea- surement bandwidth from ∼30 MHz (the intrinsic limit in previous multiplexing architectures) to the several GHz of bandwidth available on a single coaxial cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In the Microwave SQUID Multiplexer,59 each sensor is coupled to a high-Q, thin-film resonant circuit by an rf-SQUID that transduces current changes at the sensor to changes in inductive load on the resonator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Multiple resonators, each with a unique frequency, are coupled to a single microwave feedline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A sum of microwave tones (sine waves) is supplied to the feedline, each tone matched to the frequency of one resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Changes in the current through a sensor will shift the center frequency of its resonator and thus change the amplitude and phase of the tone that propagates through the feedline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' All tones are amplified by a single shared cryogenic low-noise amplifier before returning to room temperature, where they are analyzed to extract the detector signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Signals from different sensors can easily be separated because they appear in modulation sidebands of their respective tones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The first microcalorimeter array with microwave readout, SLEDGEHAMMER, demonstrated multiplexing factors of 128 with negligible resolution degradation, yielding a co-added resolution of 55 eV at the 97 keV gamma-ray peak (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT will use the same 33-resonator microwave SQUID multiplexing chip designs used for SLEDGEHAMMER, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Each resonator has a FWHM bandwidth of ∼300 kHz, and the resonances are spaced 3 MHz apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Variations to the design place 32 resonators into each of eight 125 MHz bands between 5 GHz and 6 GHz, yielding a total density of 256 detectors per 11 Fig 7: (a) Circuit schematic showing three channels of a microwave SQUID multiplexing circuit with TES microcalorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (b) A photograph of a 33-channel microwave SQUID multiplexer chip used in the SLEDGEHAMMER instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (c) A close-up photograph showing quarter-wave microwave resonators capacitively coupled to a feedline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The resonators are terminated by induc- tively coupled rf-SQUIDs (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The microwave SQUID multiplexer takes advantage of the large bandwidth provided by coaxial cables to significantly reduce the thermal load and design complex- ity of reading out large-format arrays of TES microcalorimeters, such as those used in ASCENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Reprinted from Mates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' (2017)11 with the permission of AIP Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' GHz of available bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' To read out the 512 detectors used in ASCENT, we will use 2 parallel pairs of coaxial cables, each reading out a separate set of resonators in the 5–6 GHz range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The signals will be analyzed by four commercially available ROACH2 Field Programmable Gate Array (FPGA) systems designed by the CASPER radio astronomy consortium,60 each cover- ing a bandwidth of 512 MHz using commercially available DAC and ADC daughter boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3 Calibration Achieving the best possible spectroscopic performance requires constant monitoring of the detector calibration, in order to be able to correct for calibration drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT will carry a calibration source housed inside a tungsten enclosure outside a dedicated window in the cryostat, illuminating the detector array from behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The calibration of gamma-ray TESs drifts on timescales of minutes to hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Experience shows that the dominant contribution to short-term drift is correlated with the baseline and can be corrected for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Therefore, it is necessary to obtain calibration spectra about once per hour to keep the systematic uncertainty due to residual drift below 10 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' By collecting about 100 photons per calibration line, we can keep the statistical uncertainty of the calibration negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Two approaches to calibration are still under consideration: continuous illumination with a weak source or use of a strong source behind a shutter periodically illuminating the array for a brief period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The advantage of continuous illumination is that no shutter mechanism is required and that no artificial dead-time is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This reduces mission complexity and, thus, risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, care must be taken in the selection of calibration isotope or combination of isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Calibration lines should be close to the energy range of interest, but lines within the energy range of interest, including escape peaks, will cause unacceptable background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Furthermore, emission lines above the energy range of interest will cause a background continuum due to Compton scattering in the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A preliminary Geant461–63 simulation study of calibration spectra obtained with a selection of viable sources revealed 155Eu with sufficiently strong lines at 60 keV and 86 keV as a candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 12 1mmNSA sufficiently strong source behind a shutter will allow us to acquire calibration spectra at regular intervals over a short period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Because calibration data are acquired at known time intervals during which science data collection will be suspended, it is possible to use calibration lines in the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This has the advantage of mitigating the impact of non- linearities in the detector response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The approach also reduces the continuum background due to high-energy lines from the calibration source, assuming sufficient shielding is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The downside of this approach is that a mechanical shutter is required, adding complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' While details still need to be optimized, a preliminary estimate shows that a 1 min calibration window every hour will be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A possible calibration isotope is 227Ac which emits a large number of X-ray and gamma-ray lines in the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This calibration method would result in an additional dead-time of <2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 Anti-coincidence detector In order to reduce the background, the cryostat section containing the focal plane instrumenta- tion will almost entirely be enclosed in a ∼2–3 cm-thick active bismuth-germanate (BGO) anti- coincidence shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Scintillation light due to particles interacting in the anti-coincidence shield will be detected by photomultiplier tubes (PMTs) or Silicon Photomultipliers (SiPMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Signals in these PMTs or SiPMs will produce a flag vetoing any triggers in the TES detector readout, significantly reducing the residual background, in addition to the passive shielding provided by the absorption of particles in the BGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The design will maximize the solid angle covered by the active shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The conservative solution is to place the active shield components at the outside of the cryostat, avoiding the difficulties associated with bringing the scintillator crystal and detectors from ambient temperatures and pressure to liquid helium temperatures and near vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In this case, passive tungsten shielding will be used inside the cryostat for the small solid angle portions not covered by the active shield outside the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' We are evaluating solutions with active shield- ing inside the cryostat which would result in a smaller shield and thus reduced cross sections for interactions with the background, and a reduced shield mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In both cases, the veto flag will be fed to the data acquisition and will be digitized along with the TES signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3 X-ray optics ASCENT achieves a large effective area in the 65–85 keV energy range using a dedicated multilayer- coated grazing incidence X-ray mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The mirror consists of 213 nested shells in two reflection stages with a diameter of 40 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The innermost 110 shells will be coated with approximately 500 Ni/C layer pairs, while the remaining shells will be coated with roughly 200 Pt/C layer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Given the focal length of 12 m, the design will limit incidence angles to < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='23°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Reflectivity over a broad bandwidth at high X-ray energies is achieved by coating the shells with alternating layers of high-Z and low-Z material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The ASCENT optics are expected to achieve an angular resolution of 2′ half-power diameter (HPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The field of view of 5′ FWHM exceeds the angular size of Cas A of 4′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Most X-ray telescopes are designed to achieve a high collection area over a broad energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The broadband design necessitates the deposition of a multilayer stack for soft X-rays on top of the stack for hard X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' However, the thick soft X-ray layers absorb some of the higher-energy photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' For ASCENT, the multilayer design will be optimized for energies above 60 keV, achiev- ing there substantially higher reflectivities than a broadband X-ray multilayer coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Figure 8 13 20 40 60 80 100 Energy [keV] 10 0 10 1 10 2 10 3 Effective area [cm²] Pt/C and Ni/C, =4Å Pt/C contribution Ni/C contribution Pt/C and Ni/C, =6Å Pt/C only, =4Å NuSTAR (two mirrors) Fig 8: Comparison of the effective areas of mirror designs for ASCENT and the two NuSTAR mirrors combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='12 The ASCENT design uses a combination of Ni/C multilayer coatings on the inner 110 shells and Pt/C on the outer shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' An alternative design using Pt/C layers on all shells is shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The shaded region indicates the expected range of the surface roughness between σ = 6 Å (worst case) and 4 Å (best case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The dashed and dotted red lines indicate the contribution of the Pt/C and Ni/C shells in the σ = 4 Å case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' shows the comparison of the collection areas of the ASCENT and the two NuSTAR mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The platinum K absorption edge at 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='395 keV limits the effective area at higher energies, preventing a purely Pt/C multilayer mirror from properly observing the 44Ti line at 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='36 keV, especially when that line is blueshifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Therefore, in addition to an optimized multilayer structure, the ASCENT optics will use Ni/C coatings on the innermost 110 out of 213 nested shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The Ni/C design requires about 500 layer pairs, while the Pt/C design requires about 200 layer pairs, which limits the number of foils that can be fabricated with this method within the project timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A surface roughness of 4–6 Å of the multilayer coatings is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The ASCENT mirror achieves collec- tion areas of >100 cm2 in the 65–80 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Due to the optimization of the layer structure for 44Ti observations, the effective area between ∼30 keV and 55 keV is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The low-energy reflectivity is not due to Bragg reflection on the multilayer, but due to total external reflection on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The Ni/C multilayer combination has been studied by several groups in the past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Spiga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', 200464) and is considered a top candidate material for future missions, such as HEX-P, to extend their energy band out to 200 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 Cryostat The ASCENT cryogenic system cools the detector assembly to a nominal base temperature of 70 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Our baseline design foresees to use the cryostat architecture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 9 that uses a commer- cial adiabatic demagnetization refrigerator (ADR) coupled to a closed-cycle 300 mK refrigerator and a liquid helium bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The 65 L tank is designed to cool the detectors for up to 14 days, more than sufficient for the expect duration of the balloon flights from Sweden to Northern Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' An 14 Fig 9: Conceptual design of the ASCENT cryostat, which provides a 70 mK base temperature for the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The detectors are housed in a protruding snout to minimize the mass of the shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' absolute pressure regulator maintains the tank near atmospheric pressure to provide a 4 K ther- mal bath, and internal baffles minimize sloshing to prevent resonances with the pointing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Counterweights on the pointing system maintain payload balance as the motion of the liquid during elevation changes and cryogen boil-off shift the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The main liquid helium tank is insulated by two vapor-cooled shields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' As cold gas boils away from the liquid helium reservoir, it flows through stainless steel pipes connected to the shields through low-impedance heat exchangers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Since the helium boil off rate is proportional to the thermal load, negative feedback enforces temperature stability within the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' G-10 trusses mechanically support each stage while maintaining sufficient thermal insulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Liquid nitrogen cooling was considered as an alternative for the vapor-cooled shields, but would have led to a heavier and more complex design due to the additional cryogen tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A 300 mK temperature stage is provided by a multi-stage closed-cycle He-4/He-3 sorption re- frigerator coupled to the cryostat’s liquid helium tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This serves as the launching point for the ADR, allowing a lower magnetic field strength and thus lower power consumption than launch- ing directly from 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The 300 mK stage also intercepts the parasitic load from the wiring and mechanical support structures to reduce the cooling power requirement on the lowest temperature stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Similar refrigerators have been successfully used by many different balloon-borne cryogenic systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', SPIDER,66 EBEX,67 BLAST-Pol,68 and BOOMERANG69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The detector assembly is maintained at a base temperature of 70 mK by a commercially avail- able ADR using a single ferric ammonium alum (FAA) salt pill70 launched from the 300 mK stage to provide 1 µW of cooling power with 120 mJ cooling capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A 3-hour regeneration cycle will be performed once every 24 hours, providing roughly 90 % observing efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' For optimal oper- ation, the TES detectors and SQUID multiplexer chips must be protected from external magnetic 15 Vacuum Vessel OuterVapor-Cooled Shield InnerVapor-Cooled Shield Liquid Helium Tank 4K Shield He Fridge ADR Tungsten Shield Detector Assembly Tungsten Collimator Inner A4K Shield Outer A4K Shield BGO Shieldfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The entire detector package assembly containing both detector dies and all SQUID mul- tiplexer chips will therefore be enclosed in a two-layer magnetic shield, incorporated within the cryostat and maintained at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These reduce the magnetic flux in the SQUIDs due to Earth’s magnetic field by about two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' As an alternative cooling option, we are currently evaluating the performance of a mini Dilution Refrigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='71 First measurements in the lab indicate that the mini Dilution Refrigerator is well suited for this application, offering continuous cooling to 80 mK temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The cooling power does not change significantly for elevation changes of ±30°, enabling its use for ASCENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 Optical bench and gondola X-ray optics and cryostat will be supported by a 12 m-long optical bench pointed by NASA’s Wallops Arc-Second Pointer (WASP) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The optical bench consists of three sections made of carbon fiber tubes glued to Aluminum joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The design results in an extremely stiff truss and is similar to previous balloon-borne telescopes X-Calibur and XL-Calibur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='72,73 For example, the 8 m long truss of X-Calibur achieved a stability of <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 mm of the focal point during most of the flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='74 The X-ray mirror, star tracker, and fiber-optic gyro of the WASP will be mounted to an Al honeycomb panel at the front end of the truss, and the focal plane instrumentation will be attached to an Al honeycomb panel at the rear end of the truss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The stiffness of the optical bench fulfills two requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' First, the WASP pointing system requires that the lowest-frequency vibration mode of the pointed body exceeds 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Second, we require a motion of the focal spot <3 mm in order to ensure the entire image is always contained in the detector array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A focal spot motion of 3 mm corresponds to 50′′ pointing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In order to reduce the resulting degradation of the point spread function, ASCENT will use an alignment monitoring system similar to X-Calibur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='74 The system uses an optical camera mounted in the central bore of the X-ray optics observing a pattern of LEDs mounted to the entrance window of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' On a 12 m truss, it measures the alignment with a precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='15 mm or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5′′, negligible compared to the point spread function of the optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' An aluminum gondola suspended from the balloon supports the truss pointed by the WASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The WASP points the truss in pitch and within a limited yaw range with respect to the gondola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Coarse pointing in yaw is achieved using a standard NASA balloon rotator coupling the gondola to the balloon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Absolute pointing information is provided by a star tracker system specially developed for balloon flight applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This system achieves a pointing stability of <1′′ and an absolute pointing accuracy of ∼15′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 4 Expected performance During a balloon flight from Kiruna, Sweden, ASCENT will observe Cas A for approximately 500 ksec at an elevation of about 36–82°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' We envision multiple northern hemisphere flights in order to attain longer total observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In order to estimate ASCENT’s sensitivity, we simulated the detector in Geant4 as two stacked arrays of Sn absorbers with the layout as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6c and a thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='59 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Input spectra were folded with the mirror effective area based on the Ni/C multilayer mirror shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 8 and energy-dependent atmospheric absorption at a balloon altitude of 125,000 ft corresponding to an overburden of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 g cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Photons were distributed according to the mirror point spread function (PSF) of ASTRO-H HXT,75 which is similar to the expected ASCENT PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 16 50 55 60 65 70 75 80 85 90 95 100 Energy [keV] 7 − 10 6 − 10 5 − 10 ] 1 s 2 cm γ Line Sensitivity [ σ 3 500ksec 1Msec Fig 10: ASCENT narrow line sensitivity as a function of energy, demonstrating ASCENT’s ability to detect 44Ti in various regions of Cas A with a single 500 ksec flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The vertical yellow bands indicate energy ranges of interest and the horizontal black bars are the flux levels measured by NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='28 We estimate the background using measurements made during the Antarctic flight of X-Calibur74 and taking into account improvements to the anticoincidence shield, which will reduce this back- ground by a factor of ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='73 To account for the difference in detector size we scale with area and square root of thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' This results in a background rate of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='7 × 10−6 s−1 keV−1 per TES de- tector at 68 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In the future, this estimate can be refined using data from an upcoming test flight of a small TES array scheduled for the fall of 2023, as well as with the help of detailed Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='76 In the sensitivity calculation, we weighted events in each detector by the expected signal-to- background ratio for a point-like source according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='77 Figure 10 shows the expected narrow line sensitivity of ASCENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' NuSTAR detected lines with fluxes ranging from 6 × 10−7 cm−2 s−1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='7 × 10−6 cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT’s energy resolution of 67 eV FWHM will allow us to determine the velocity of 44Ti ejecta with a FWHM accuracy of 270 km s−1, compared to NuSTAR’s FWHM accuracy of 3600 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' From each flight we expect highly significant detections of the 44Ti emission from Cas A of 11 σ and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 σ of the 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 keV and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 keV lines, respectively, when summing over all bright spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Particularly interesting will be the energy spectrum from Region 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 1, which we expect to detect with 5 σ at 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 keV and with 3 σ at 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The advantage of ASCENT is greatest in regions where the width of the 44Ti lines is small compared to NuSTAR’s energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A single 500 ksec observation with ASCENT will improve measurements in all regions where NuSTAR detected 44Ti, except two where the lines are broadened and very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' We estimate that ASCENT will improve the line centroid and width measurements compared to previous results by a factor of 2–20 and 2–10, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These measurements will significantly improve the 3D localization of the 44Ti ejecta and result in much tighter constraints on the local 56Ni/44Ti ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' These improvements in constraining the local ratio will greatly increase our knowledge of the nuclear production pathways in the supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 17 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 67 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='5 68 Centroid [keV] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 Width [keV] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='3 ASCENT NuSTAR Fig 11: Expected ASCENT results for the measurements of 44Ti emission lines in the 8 44Ti-bright regions analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='28 compared to the NuSTAR results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Simulations assume the Gaussian line parameters measured by NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In regions where NuSTAR only set an upper limit on the line width, a smaller value is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The inset shows two lines with assumed widths of 150 eV and 100 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The ASCENT centroid and line width uncertainties are a factor 2–10 smaller than NuSTAR’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 5 Summary and Outlook Core collapse supernovae are considered to be a significant source of mid-Z elements in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Despite significant theoretical and observational progress in the last few decades, many details of the explosion are still poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' One observational approach to gaining new insights is to study the distribution of elements in supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The isotope 44Ti is of particular interest because it is produced in the innermost regions of the supernova engine, and it is radioactive with a half-life long enough to be observable in historic supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' In this paper we present a concept for a balloon-borne hard X-ray telescope called ASCENT designed to study the nuclear emission lines from Cas A and SN 1987A, as well as potentially other remnants that may be discovered by COSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT uses an array of transition edge sensor (TES) microcalorimeter detectors as its focal plane instrument, improving spectral resolution by more than an order of magnitude over existing semiconductor detectors at gamma-ray energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Observations with ASCENT will significantly improve 3D maps of 44Ti in Cas A, and can deliver detailed spectra of 44Ti from SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' ASCENT will also demonstrate the viability of hard X-ray TES technology for a future space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The energy resolution of an ASCENT-type mission would benefit all spectral studies of a NuSTAR follow-up, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', broadband observations of Active Galactic Nuclei (AGN) cover- ing the lines of the soft excess emission, the Fe Kα emission, and the Compton hump emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The broadband results obtained for a sample of ∼100 of AGNs can be used to calibrate the spin measurements of ESA’s ATHENA mission43,44 and the proposed NASA Lynx mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='78–80 The ASCENT detector technology could be used for example as the focal plane detector of the HEX-P mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='65 While ASCENT has not been selected when first proposed, the team continues to pursue the project and is planning to repropose the mission after maturing detector, cryostat, and X-ray optics technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 18 Acknowledgments FK is grateful for support by the Faculty Development Grant program of UNH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' The work by CLF is supported by the US Department of Energy through the Los Alamos National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Department of Energy (Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 89233218CNA000001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' HK and JN acknowledge NASA support under grant 80NSSC21K1817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' HK acknowledges NASA support under grants 80NSSC18K0264, 80NSSC22K1291, and NNX16AC42G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' References 1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Burbidge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Burbidge, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fowler, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Synthesis of the Elements in Stars,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 29, 547–650 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Matz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Share, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Leising, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Gamma-ray line emission from SN1987A,” Nature 331, 416–418 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Choi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Huss, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Wasserburg, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Presolar Corundum and Spinel in Ordinary Chondrites: Origins from AGB Stars and a Supernova,” Science 282, 1284 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Schulte, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bose, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Young, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Three-dimensional Supernova Models Provide New Insights into the Origins of Stardust,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 908, 38 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 5 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Amari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hoppe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Zinner, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Interstellar SiC with Unusual Isotopic Compositions: Grains from a Supernova?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=',” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 394, L43 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Herant and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Benz, “Postexplosion Hydrodynamics of SN 1987A,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 387, 294 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 7 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ahmad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Greene, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Moore, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Improved measurement of the Ti44 half-life from a 14-year long study,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' C 74, 065803 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Singh, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Cameron, “Nuclear Data Sheets for A = 44,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Data Sheets 112, 2357–2495 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Clayton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Colgate, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fishman, “Gamma-Ray Lines from Young Supernova Remnants,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 155, 75 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ullom and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bennett, “Review of superconducting transition-edge sensors for x-ray and gamma-ray spectroscopy,” Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 28, 084003 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 11 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Mates, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Becker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bennett, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Simultaneous readout of 128 X-ray and gamma-ray transition-edge microcalorimeters using microwave SQUID multiplexing,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 111, 062601 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Harrison, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Craig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Christensen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The Nuclear Spectroscopic Telescope Array (NuSTAR) High-energy X-Ray Mission,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 770, 103 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 13 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Boggs, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Harrison, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Miyasaka, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “44Ti gamma-ray emission lines from SN1987A reveal an asymmetric explosion,” Science 348, 670–671 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Grebenev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lutovinov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Tsygankov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Hard-X-ray emission lines from the decay of 44Ti in the remnant of supernova 1987A,” Nature 490, 373–375 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Iyudin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Diehl, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bloemen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “COMPTEL observations of Ti-44 gamma-ray line emission from CAS A,” Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 284, L1–L4 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Vink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Kaastra, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Detection of the 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='9 and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='4 keV Lines Asso- ciated with the Radioactive Decay of 44Ti in Cassiopeia A,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 560, L79–L82 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 19 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Grefenstette, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Harrison, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Boggs, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Asymmetries in core-collapse super- novae from maps of radioactive 44Ti in Cassiopeia A,” Nature 506, 339–342 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Iyudin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Schönfelder, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bennett, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Emission from 44Ti associated with a previ- ously unknown Galactic supernova,” Nature 396, 142–144 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 19 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Troja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Segreto, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' La Parola, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Swift/BAT Detection of Hard X-Rays from Tycho’s Supernova Remnant: Evidence for Titanium-44,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 797, L6 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Tsygankov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Krivonos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lutovinov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Galactic survey of 44Ti sources with the IBIS telescope onboard INTEGRAL,” Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Royal Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 458, 3411–3419 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 21 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lopez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Grefenstette, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Reynolds, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “A Spatially Resolved Study of the Synchrotron Emission and Titanium in Tycho’s Supernova Remnant Using NuSTAR,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 814, 132 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Tomsick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Boggs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Zoglauer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The Compton Spectrometer and Imager Project for MeV Astronomy,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 37th International Cosmic Ray Conference, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 23 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Woosley and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Weaver, “Sub–Chandrasekhar Mass Models for Type IA Super- novae,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 423, 371 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 24 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Crocker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ruiter, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Seitenzahl, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Diffuse Galactic antimatter from faint thermonuclear supernovae in old stellar populations,” Nature Astronomy 1, 0135 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Renaud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Vink, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Decourchelle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The Signature of 44Ti in Cassiopeia A Revealed by IBIS/ISGRI on INTEGRAL,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 647, L41–L44 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 26 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Siegert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Diehl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Krause, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Revisiting INTEGRAL/SPI observations of 44Ti from Cassiopeia A,” Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 579, A124 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 27 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Young, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fryer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hungerford, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Constraints on the Progenitor of Cassiopeia A,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 640, 891–900 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 28 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Grefenstette, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fryer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Harrison, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The Distribution of Radioactive 44Ti in Cassiopeia A,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 834, 19 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 29 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Magkotsios, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Timmes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hungerford, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Trends in 44Ti and 56Ni from Core- collapse Supernovae,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 191, 66–95 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 30 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Koo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lee, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “A Deep Near-infrared [Fe II]+[Si I] Emission Line Image of the Supernova Remnant Cassiopeia A,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 866, 139 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 31 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hammer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Janka, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Müller, “Three-dimensional Simulations of Mixing Insta- bilities in Supernova Explosions,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 714, 1371–1385 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 32 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ono, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nagataki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ito, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Matter Mixing in Aspherical Core-collapse Supernovae: A Search for Possible Conditions for Conveying 56Ni into High Velocity Regions,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 773, 161 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 33 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Vance, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Young, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fryer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Titanium and Iron in the Cassiopeia A Super- nova Remnant,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 895, 82 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 34 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Willingale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bleeker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' van der Heyden, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “X-ray spectral imaging and Doppler mapping of Cassiopeia A,” Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 381, 1039–1048 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 35 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' DeLaney, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rudnick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Stage, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The Three-dimensional Structure of Cassiopeia A,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 725, 2038–2058 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 36 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fryer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Kusenko, “Effects of Neutrino-driven Kicks on the Supernova Explosion Mechanism,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 163, 335–343 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 20 37 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Herant, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Benz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hix, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Inside the Supernova: A Powerful Convective En- gine,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 435, 339 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 38 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Herant, “The convective engine paradigm for the supernova explosion mechanism and its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=',” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 256, 117–133 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 39 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Socrates, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Blaes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hungerford, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The Neutrino Bubble Instability: A Mechanism for Generating Pulsar Kicks,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 632, 531–562 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 40 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ellinger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Young, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fryer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “A Case Study of Small-scale Structure For- mation in Three-dimensional Supernova Simulations,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 755, 160 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 41 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Stuchlik, “The Wallops Arc Second Pointer – A Balloon Borne Fine Pointing System,” in AIAA Balloon Systems Conference, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' AIAA AVIATION Forum (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 42 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Aharonian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Akamatsu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Akimoto, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The quiescent intracluster medium in the core of the Perseus cluster,” Nature 535, 117–121 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 43 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Adams, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bandler, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Transition-edge sensor pixel parameter design of the microcalorimeter array for the x-ray integral field unit on Athena,” in Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' SPIE 9905, 99052H (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 44 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Barcons, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Barret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Decourchelle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Athena: ESA’s X-ray observatory for the late 2020s,” Astronomische Nachrichten 338, 153–158 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 45 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bennett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Horansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Schmidt, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “A high resolution gamma-ray spectrometer based on superconducting microcalorimeters,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 83, 093113– 093113–14 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 46 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Winkler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hoover, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rabin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “256-pixel microcalorimeter array for high- resolution γ-ray spectroscopy of mixed-actinide materials,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A 770, 203–210 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 47 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hoover, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bond, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Croce, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Measurement of the 240Pu/239Pu Mass Ratio Using a Transition-Edge-Sensor Microcalorimeter for Total Decay Energy Spectroscopy,” Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 87(7), 3996–4000 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 48 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Uhlig, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fullagar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ullom, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Table-Top Ultrafast X-Ray Microcalorimeter Spec- trometry for Molecular Structure,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 110, 138302 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 49 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Miaja-Avila, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' O’Neil, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Joe, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Ultrafast Time-Resolved Hard X-Ray Emis- sion Spectroscopy on a Tabletop,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' X 6, 031047 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 50 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Palosaari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Käyhkö, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Kinnunen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Broadband Ultrahigh-Resolution Spectroscopy of Particle-Induced X Rays: Extending the Limits of Nondestructive Analysis,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6, 024002 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 51 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Okada, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bennett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Curceanu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “First application of superconducting transition- edge sensor microcalorimeters to hadronic atom X-ray spectroscopy,” Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 2016, 091D01 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 52 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' O’Neil, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Miaja-Avila, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Joe, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Ultrafast Time-Resolved X-ray Absorption Spectroscopy of Ferrioxalate Photolysis with a Laser Plasma X-ray Source and Microcalori- meter Array,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 8(5), 1099–1104 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 53 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Doriese, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Abbamonte, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Alpert, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “A practical superconducting- microcalorimeter X-ray spectrometer for beamline and laboratory science,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 88, 053108 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 21 54 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Titus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Baker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Lee, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “L-edge spectroscopy of dilute, radiation-sensitive systems using a transition-edge-sensor array,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 147, 214201 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 55 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Irwin and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hilton, Transition-Edge Sensors, 63 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 56 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bacrania, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hoover, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Karpius, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Large-Area Microcalorimeter Detectors for Ultra-High-Resolution X-Ray and Gamma-Ray Spectroscopy,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 56, 2299–2302 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 57 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Weber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Morgan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Yan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Development of a transition-edge sensor bilayer process providing new modalities for critical temperature control,” Superconductor Science and Technology (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 58 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Mates, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Irwin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Vale, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Flux-Ramp Modulation for SQUID Multi- plexing,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 167, 707–712 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 59 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Mates, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hilton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Irwin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Demonstration of a multiplexer of dissipation- less superconducting quantum interference devices,” Applied Physics Letters 92(2), 023514 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 60 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hickish, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Abdurashidova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ali, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “A Decade of Developing Radio-Astronomy Instrumentation using CASPER Open-Source Technology,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 5, 1641001– 12 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 61 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Agostinelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Allison, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Amako, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Geant4 – a simulation toolkit,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A 506(3), 250 – 303 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 62 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Allison, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Amako, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Apostolakis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Geant4 developments and applications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 53, 270–278 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 63 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Allison, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Amako, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Apostolakis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Recent developments in Geant4,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A 835, 186 – 225 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 64 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Spiga, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Pareschi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Citterio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Development of multilayer coatings (Ni/C-Pt/C) for hard x-ray telescopes by e-beam evaporation with ion assistance,” in UV and Gamma-Ray Space Telescope Systems, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Hasinger and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Turner, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 5488, 813–819 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 65 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Madsen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Harrison, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Broadway, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Optical instrument design of the high- energy x-ray probe (HEX-P),” in Space Telescopes and Instrumentation 2018: Ultraviolet to Gamma Ray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' den Herder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nikzad, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Nakazawa, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Society of Photo- Optical Instrumentation Engineers (SPIE) Conference Series 10699, 106996M (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 66 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Gudmundsson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ade, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Amiri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The thermal design, characterization, and performance of the SPIDER long-duration balloon cryostat,” Cryogenics 72, 65–76 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 67 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Reichborn-Kjennerud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Aboobaker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ade, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “EBEX: a balloon-borne CMB polarization experiment,” in Millimeter, Submillimeter, and Far-Infrared Detectors and In- strumentation for Astronomy V, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 7741, 77411C (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 68 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Fissel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ade, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Angilè, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The balloon-borne large-aperture submillime- ter telescope for polarimetry: BLAST-Pol,” in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy V, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 7741, 77410E (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 69 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Crill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Ade, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Artusa, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “BOOMERANG: A Balloon-borne Millimeter- Wave Telescope and Total Power Receiver for Mapping Anisotropy in the Cosmic Microwave Background,” The Astrophysical Journal Supplement Series 148, 527–541 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 22 70 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Wilson and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Timbie, “Construction techniques for adiabatic demagnetization re- frigerators using ferric ammonium alum,” Cryogenics 39, 319–322 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 71 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Chase, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Brien, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Doyle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Pre-cooling a 3He/4He dilutor module with a sealed closed-cycle continuous cooler,” in Materials Science and Engineering Conference Series, Materials Science and Engineering Conference Series 502, 012134 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 72 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Kislat, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Beheshtipour, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Dowkontt, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Design of the Telescope Truss and Gondola for the Balloon-Borne X-ray Polarimeter X-Calibur,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 6, 1740003 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 73 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Abarr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Awaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Baring, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “XL-Calibur - a second-generation balloon-borne hard X-ray polarimetry mission,” Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 126, 102529 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 74 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Abarr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Beheshtipour, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Beilicke, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Performance of the X-Calibur hard X-ray polarimetry mission during its 2018/19 long-duration balloon flight,” Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Physics 143, 102749 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 75 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Matsumoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Awaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Furuzawa, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Ray-tracing simulation and in-orbit perfor- mance of the ASTRO-H hard x-ray telescope (HXT),” in Space Telescopes and Instrumen- tation 2016: Ultraviolet to Gamma Ray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' den Herder, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Takahashi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bautz, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 9905, 990541 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 76 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Shirazi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Arman Hossen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Becker, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The 511-CAM Mission: A Pointed 511 keV Gamma-Ray Telescope with a Focal Plane Detector Made of Stacked Transition Edge Sensor Microcalorimeter Arrays,” arXiv e-prints , arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content='14652 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 77 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Barlow, “Event Classification Using Weighting Methods,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 72, 202–219 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 78 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Gaskin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Weisskopf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Vikhlinin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “The X-ray Surveyor Mission: a concept study,” in UV, X-Ray, and Gamma-Ray Space Instrumentation for Astronomy XIX, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' SPIE 9601, 96010J (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 79 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Bandler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Adams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' Chervenak, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=', “Development of x-ray microcalorimeter imaging spectrometers for the X-ray Surveyor mission concept,” in Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' SPIE 9905, 99050Q (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 80 Physics of the Cosmos Program Office, “Program Annual Technology Report,” (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfj_2L/content/2301.01525v1.pdf'} diff --git a/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf b/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d84ed9c8ef3b7b5723d9ad058f570204a806a273 --- /dev/null +++ b/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ec8fba981ed4400595c36b16cf0c57dddb99193aea00fcf80b11ea2ca526d584 +size 10947217 diff --git a/KNE4T4oBgHgl3EQf7g74/content/tmp_files/2301.05341v1.pdf.txt b/KNE4T4oBgHgl3EQf7g74/content/tmp_files/2301.05341v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..879511d65b0757247e9c9488e52e84a53384c1af --- /dev/null +++ b/KNE4T4oBgHgl3EQf7g74/content/tmp_files/2301.05341v1.pdf.txt @@ -0,0 +1,1863 @@ +ON A COMPUTABLE SKOROKHOD’S INTEGRAL BASED ESTIMATOR OF THE +DRIFT PARAMETER IN FRACTIONAL SDE +NICOLAS MARIE† +Abstract. This paper deals with a Skorokhod’s integral based least squares type estimator �θN of the +drift parameter θ0 computed from N ∈ N∗ copies X1, . . . , XN of the solution X to dXt = θ0b(Xt)dt + +σdBt, where B is a fractional Brownian motion of Hurst index H ∈ [1/2, 1). On the one hand, a risk +bound is established on �θN when H = 1/2 and X1, . . . , XN are dependent copies of X. On the other +hand, when H > 1/2, Skorokhod’s integral based estimators as �θN cannot be computed directly from +data, but in this paper some convergence results are established on a computable approximation of �θN +when X1, . . . , XN are independent. +Contents +1. +Introduction +1 +2. +Case H = 1/2: risk bound and dependent copies +3 +3. +Case H > 1/2: risk bound and computable estimator +6 +3.1. +Basics on the Skorokhod integral +6 +3.2. +Risk bound on �θN +8 +3.3. +A computable estimator +10 +4. +Numerical experiments +15 +5. +Conclusion and perspectives +16 +References +17 +1. Introduction +Let X = (Xt)t∈[0,T ] be the solution of the differential equation +(1) +Xt = X0 + θ0 +� t +0 +b(Xs)ds + σBt ; t ∈ [0, T], +where T > 0 is fixed, X0 ∈ L2(Ω), B = (Bt)t∈[0,T ] is a fractional Brownian motion of Hurst index +H ∈ [1/2, 1), b ∈ C1(R), b′ is bounded, σ ∈ R∗ and θ0 ∈ R is an unknown parameter to estimate. +The oldest kind of (non)parametric estimators of the drift function is based on the long-time behav- +ior of the solution to Equation (1). For H = 1/2, the reader may refer to the monograph [14] written by +Y. Kutoyants. For H > 1/2, see Kleptsyna & Le Breton [13], Tudor & Viens [22], Hu & Nualart [10], +Neuenkirch & Tindel [19], Hu et al. [11], Marie & Raynaud de Fitte [16], etc. on parametric estimators, +Key words and phrases. Fractional Brownian motion; Least squares estimator; Malliavin calculus; Stochastic differential +equations. +1 +arXiv:2301.05341v1 [math.ST] 13 Jan 2023 + +2 +NICOLAS MARIE† +and see Saussereau [21] and Comte & Marie [4] on nonparametric ones. The stochastic integral involved +in the definition of the estimators studied in [10], [11], [16] and [4] is taken in the sense of Skorokhod. To +be not directly computable from an observation of X is the major drawback of the Skorokhod integral +with respect to X. One of the main purposes of our paper is to bypass this difficulty in another estima- +tion framework because the Skorokhod integral is a nice generalization of Itô’s integral, tailor-made for +advanced statistical investigations. +For H = 1/2, a new kind of estimators of the drift function have been investigated since several years: +those computed from N copies X1, . . . , XN of X observed on [0, T] with T > 0 fixed but N → ∞. +The major part of the literature deals with estimators based on independent copies of X (see Comte & +Genon-Catalot [3], Denis et al. [8], Marie & Rosier [17], etc.), but some recent papers are also devoted +to estimators based on dependent copies (see Della Maestra & Hoffmann [7] and Comte & Marie [6]). +Copies based estimators are well-adapted to some situations difficult to manage with long-time behavior +based estimators: +• Assume that X models the elimination process of a drug administered to one people, and assume +that in a clinical-trial involving N patients, Xi models the elimination process of the same drug +for the i-th patient. Then, X1, . . . , XN are independent copies of X. +• Consider a financial market with N interacting risky assets of same kind and assume that the +i-th asset is modeled by Xi. Here, X1, . . . , XN are not independent, but copies based estimators +of the drift function remain appropriate as mentioned above. +For H > 1/2, Comte & Marie [5] and Marie [15] are the only two references on such estimators up to our +knowledge. +Our paper deals with the least squares type estimator +�θN := +� N +� +i=1 +� T +0 +b(Xi +s)2ds +�−1 � N +� +i=1 +� T +0 +b(Xi +s)δXi +s +� +of +θ0, +where N ∈ N∗, Xi := I(Xi +0, Bi) for every i ∈ {1, . . . , N}, B1, . . . , BN (resp. X1 +0, . . . , XN +0 ) are some +copies of B (resp. independent copies of X0), I(.) is the Itô map for Equation (1), and the stochastic +integral is taken in the sense of Skorokhod. Since �θN is not directly computable when H > 1/2, our +paper also deals with the estimator �θN approximating �θN and defined as a fixed point: +�θN = +1 +NTDN +N +� +i=1 +�� T +0 +b(Xi +s)dXi +s +(2) +−αHσ2 +� T +0 +� t +0 +b′(Xi +t) exp +� +�θN +� t +s +b′(Xi +u)du +� +|t − s|2H−2dsdt +� +, +where αH := H(2H − 1), +DN := +1 +NT +N +� +i=1 +� T +0 +b(Xi +s)2ds +and the stochastic integral in the right-hand side of Equation (2) is taken pathwise (in the sense of +Young). The main purposes of our paper are: + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +3 +(1) To establish a risk bound on �θN when H = 1/2 and X1, . . . , XN are dependent copies of the +solution X to Equation (1) (see Section 2). +(2) To establish some convergence results on the computable estimator �θN when H > 1/2 (see Section +3). In this case, �θN is an auxiliary estimator. +Finally, Section 4 deals with some numerical experiments on �θN. +2. Case H = 1/2: risk bound and dependent copies +Throughout this section, H = 1/2, and then the Skorokhod integral coincides with Itô’s integral on +the space H2 := {U ∈ L2([0, T] × Ω) : U is adapted} (see Nualart [20], Proposition 1.3.11). Moreover, let +us assume that there exists a correlation matrix R such that, for any i, k ∈ {1, . . . , N}, +E(Bi +sBk +t ) = Ri,k(s ∧ t) ; ∀s, t ∈ [0, T]. +This leads (in particular) to d⟨Bi, Bk⟩t = Ri,kdt. +Remark 2.1. From continuous-time observations, to determine the matrix R is not a statistical problem. +Indeed, since σ ̸= 0, for every i, k ∈ {1, . . . , N}, +Ri,k = ⟨Xi, Xk⟩T +σ2T +. +Finally, the probability distribution of Xt, t ∈ (0, T], and the function b need to fulfill the following +assumption. +Assumption 2.2. For every t ∈ (0, T], the probability distribution of Xt has a density ft with respect to +Lebesgue’s measure such that: +(1) The function t �→ ft(x) belongs to L1([0, T]) for every x ∈ R. +(2) The function bα belongs to L2(R, f(x)dx) for every α ∈ R+, where f is the density function +defined by +f(x) := 1 +T +� T +0 +fs(x)ds ; ∀x ∈ R. +Example 2.3. In the two following situations, the probability distribution of Xt, t ∈ (0, T], and the +function b fulfill Assumption 2.2: +(1) Assume that X0(.) = x0 with x0 ∈ R. Since σ ̸= 0 and b′ is bounded, for every t ∈ (0, T], the +probability distribution of Xt has a density ft with respect to Lebesgue’s measure such that, for +every x ∈ R, +(3) +ft(x) ⩽ c0.5t− 1 +2 exp +� +−m0.5 +(x − x0)2 +t +� +where c0.5 and m0.5 are positive constants depending on T but not on t and x (see Menozzi et al. +[18], Theorem 1.2), and then t �→ ft(x) belongs to L1([0, T]). Moreover, since b′ is bounded, still +by Inequality (3), +bα ∈ L2(R, f(x)dx) ; ∀α ∈ R+. +(2) Assume that θ0 > 0 and that b satisfies the dissipativity condition +(4) +∃c > 0 : ∀x ∈ R, b′(x) ⩽ −c. + +4 +NICOLAS MARIE† +Then, Equation (1) has a unique stationary solution X, and the common probability distribution +of the Xt’s has a sub-Gaussian density f0 with respect to Lebesgue’s measure (see Bertin et al. +[2], Remark 1). So, in this situation, f = f0 and +bα ∈ L2(R, f(x)dx) ; ∀α ∈ R+ +because the density function f0 is sub-Gaussian. For instance, (4) is satisfied by the drift function +of the Langevin equation dXt = −θ0Xtdt+σdBt defining the so-called Ornstein-Uhlenbeck process, +and +f0(x) = +� +θ0 +πσ2 exp +� +−θ0x2 +σ2 +� +; ∀x ∈ R. +Notation. The usual norm on L2(R, f(x)dx) is denoted by ∥.∥f. +The following proposition provides a suitable risk bound on the truncated estimator +�θd +N := �θN1DN⩾d +with +d ∈ ∆f = +� +0, +∥b∥2 +f +2 +� +. +Proposition 2.4. Under Assumption 2.2, +E[(�θd +N − θ0)2] ⩽ c2.4 +N +� +1 + |RN| +N +� +where +RN := {(i, k) : i ̸= k and Ri,k ̸= 0} +and +c2.4 := 1 +d2 +� +σ2∥b∥2 +f +T ++ θ2 +0∥b2∥2 +f +� +. +Proof. First of all, since dXi +t = θ0b(Xi +t)dt + σdBi +t for every i ∈ {1, . . . , N}, +�θN = θ0 + UN +DN +with +UN = +σ +NT +N +� +i=1 +� T +0 +b(Xi +s)δBi +s. +On the one hand, by the isometry property of Itô’s integral, +E(U 2 +N) = +σ2 +N 2T 2 +N +� +i=1 +E +� +� +�� T +0 +b(Xi +s)δBi +s +�2� +� ++ +σ2 +N 2T 2 +� +i̸=k +E +��� T +0 +b(Xi +s)δBi +s +� �� T +0 +b(Xk +s )δBk +s +�� += +σ2 +N 2T 2 +� +�N +� T +0 +E(b(Xs)2)ds + +� +i̸=k +Ri,k +� T +0 +E(b(Xi +s)b(Xk +s ))ds +� +� +⩽ +σ2 +NT +� +� +� ∞ +−∞ +b(x)2f(x)dx + +1 +NT +� +i̸=k +|Ri,k| +� T +0 +E(b(Xi +s)2) +1 +2 E(b(Xk +s )2) +1 +2 ds +� +� +⩽ +σ2∥b∥2 +f +NT +� +�1 + 1 +N +� +i̸=k +|Ri,k| +� +� ⩽ +σ2∥b∥2 +f +NT +� +1 + |RN| +N +� +. +(5) +On the other hand, +E(DN) = 1 +T +� T +0 +E(b(Xs)2)ds = ∥b∥2 +f + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +5 +and, since B1, . . . , BN are Gaussian processes, +RN = {(i, k) ∈ {1, . . . , N}2 : i ̸= k and Xi is independent of Xk}. +Then, +E[(DN − ∥b∥2 +f)2] = var(DN) = +1 +N 2T 2 var +� N +� +i=1 +� T +0 +b(Xi +s)2ds +� += 1 +N var +� +1 +T +� T +0 +b(Xs)2ds +� ++ 1 +N 2 +� +(i,k)∈RN +cov +� +1 +T +� T +0 +b(Xi +s)2ds, 1 +T +� T +0 +b(Xk +s )2ds +� +⩽ +∥b2∥2 +f +N ++ |RN| +N 2 var +� +1 +T +� T +0 +b(Xs)2ds +� +⩽ +∥b2∥2 +f +N +� +1 + |RN| +N +� +. +(6) +Note that +E[(�θd +N − θ0)2] = E[(�θN − θ0)21DN⩾d] + θ2 +0P(DN < d) +⩽ 1 +d2 E(U 2 +N) + θ2 +0P +� +|DN − ∥b∥2 +f| > +∥b∥2 +f +2 +� +⩽ 1 +d2 [E(U 2 +N) + θ2 +0E[(DN − ∥b∥2 +f)2]]. +Therefore, by Inequalities (5) and (6), +E[(�θd +N − θ)2] ⩽ +� +σ2∥b∥2 +f +T ++ θ2 +0∥b2∥2 +f +� +1 +d2N +� +1 + |RN| +N +� +. +□ +Remark 2.5. Let us conclude this section with some remarks about Proposition 2.4: +(1) If |RN| ⩽ N, then the rate of convergence of �θd +N remains of order N −1/2 (parametric rate) as +when B1, . . . , BN are independent (case R = 0). +(2) If b(.)2 ⩾ b with b > 0, then +DN = +1 +NT +N +� +i=1 +� T +0 +b(Xi +s)2ds ⩾ b +and +∥b∥2 +f = +� ∞ +−∞ +b(x)2f(x)dx ⩾ b. +So, +�θd +N = �θN +for +d = b/2. +If in addition |b(.)| ⩽ m with m > 0, then +E[(�θN − θ0)2] ⩽ 4 +b2 +�σ2m2 +T ++ θ2 +0m4 +� 1 +N +� +1 + |RN| +N +� +. +(3) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4). Then, Equation (1) has a +unique stationary solution X, and the common probability distribution of the Xt’s has a density +f0 with respect to Lebesgue’s measure as mentioned in Example 2.3.(2). Moreover, assume that +b and f = f0 fulfill Assumption 2.2.(2) as, for instance, when (1) is the Langevin equation. In +practice, f0 is unknown but one may be able to provide a function f0 such that f0(.) ⩾ f0(.) ⩾ 0, + +6 +NICOLAS MARIE† +and then +d ∈ ∆f +when +d ∈ +� +0, 1 +2 +� ∞ +−∞ +b(x)2f0(x)dx +� +. +For instance, assume that b(x) = −x for every x ∈ R, which means that X is the Ornstein- +Uhlenbeck process. +If there exist (known) θmin, θmax > 0 such that θmin ⩽ θ0 ⩽ θmax, then +f0(.) ⩾ f0(.) > 0 with +f0(x) := +� +θmin +πσ2 exp +� +−θmaxx2 +σ2 +� +; ∀x ∈ R. +(4) Assume that X0(.) = x0 with x0 ∈ R. Assume also that θ0 > 0 and that b satisfies the dissipativity +condition (4). From one path of the solution X to Equation (1) observed on R+, which seems to +be a situation only appropriate for long-time behavior based estimators of θ0, one can construct N +independent copies of X|[0,T ]. To that purpose, consider the stopping times τ1, . . . , τN recursively +defined by τ1 = 0 and +τi = inf{t > τi−1 + T : Xt = x0} ; i = 2, . . . , N +with the convention inf(∅) = ∞. Since θ0 > 0 and b fulfills (4), the scale density +s(.) := exp +� +−2θ0 +σ2 +� . +0 +b(x)dx +� +satisfies +� 0 +−∞ +s(x)dx = +� ∞ +0 +s(x)dx = ∞, +and then X is a recurrent Markov process by Khasminskii [12], Example 3.10. +So, for any +i ∈ {1, . . . , N}, P(τi < ∞) = 1 and one can consider the processes +Bi := (Bτi+t − Bτi)t∈[0,T ] +and +Xi := (Xτi+t)t∈[0,T ]. +Since B1, . . . , BN are independent Brownian motions by the strong Markov property, and since +Xi = I(x0, Bi) ; ∀i ∈ {1, . . . , N}, +the processes X1, . . . , XN are independent copies of X|[0,T ]. +3. Case H > 1/2: risk bound and computable estimator +Throughout this section, H > 1/2 and B1, . . . , BN are independent copies of B. The proof of Proposi- +tion 2.4 only relies on the zero mean and on the control of the variance of Itô’s integral. Such properties +for Skorokhod’s integral with respect to the fractional Brownian motion are stated in Subsection 3.1. +Subsection 3.2 deals with a risk bound on �θN. Finally, Subsection 3.3 deals with the existence and some +convergence results on the computable estimator �θN. +3.1. Basics on the Skorokhod integral. Let ⟨., .⟩H be the inner product defined by +⟨h, η⟩H := αH +� T +0 +� T +0 +h(s)η(s)|t − s|2H−2dsdt, + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +7 +and consider the reproducing kernel Hilbert space H = {h : ∥h∥H < ∞} of B, where ∥.∥H is the norm +associated to ⟨., .⟩H. Consider also the isonormal Gaussian process (B(h))h∈H defined by +B(h) := +� . +0 +h(s)dB(s) +which is the Wiener integral of h ∈ H with respect to B. +Definition 3.1. The Malliavin derivative of a smooth functional +F = ϕ(B(h1), . . . , B(hn)) +where n ∈ N∗, ϕ ∈ C∞ +p (Rn; R) (the space of all the smooth functions ϕ : Rn → R such that ϕ and all its +partial derivatives have polynomial growth) and h1, . . . , hn ∈ H, is the H-valued random variable +DF := +n +� +k=1 +∂kϕ(B(h1), . . . , B(hn))hk. +Proposition 3.2. The map D is closable from L2(Ω; R) into L2(Ω; H). Its domain in L2(Ω; R), denoted +by D1,2, is the closure of the smooth functionals space for the norm ∥.∥1,2 defined by +∥F∥2 +1,2 := E(F 2) + E(∥DF∥2 +H). +The Malliavin derivative of F ∈ D1,2 at time s ∈ [0, T] is denoted by DsF. +See Nualart [20], Proposition 1.2.1 for a proof. +Definition 3.3. The adjoint δ of the Malliavin derivative D is the divergence operator. The domain of +δ is denoted by dom(δ), and Y ∈ dom(δ) if and only if there exists a deterministic constant cY > 0 such +that for every F ∈ D1,2, +|E(⟨DF, Y ⟩H)| ⩽ cY E(F 2) +1 +2 . +For any process Y = (Ys)s∈[0,T ] and every t ∈ (0, T], if Y 1[0,t] ∈ dom(δ), then its Skorokhod integral with +respect to B is defined on [0, t] by +� t +0 +YsδBs := δ(Y 1[0,t]), +and its Skorokhod integral with respect to X is defined by +� t +0 +YsδXs := θ0 +� t +0 +Ysb(Xs)ds + σ +� t +0 +YsδBs. +Note that since δ is the adjoint of the Malliavin derivative D, the Skorokhod integral of Y with respect +to B on [0, t] is a centered random variable. Indeed, +(7) +E +�� t +0 +YsδBs +� += E(1 · δ(Y 1[0,t])) = E(⟨D(1), Y 1[0,t]⟩H) = 0. +Let S be the space of the smooth functionals presented in Definition 3.1 and consider D1,2(H), the closure +of +SH := +� +� +� +n +� +j=1 +Fjhj ; h1, . . . , hn ∈ H, F1, . . . , Fn ∈ S +� +� +� +for the norm ∥.∥1,2,H defined by +∥Y ∥2 +1,2,H := E(∥Y ∥2 +H) + E(∥DY ∥2 +H⊗H). + +8 +NICOLAS MARIE† +Consider also the norm ∥.∥H defined by +∥h∥H := +� +αH +� T +0 +� T +0 +|h(s)| · |h(t)| · |t − s|2H−2dsdt +� 1 +2 +, +the Banach space H := {h : ∥h∥H < ∞} and +D1,2(H) := {Y ∈ D1,2(H) : E(∥Y ∥2 +H) + E(∥DY ∥2 +H⊗H) < ∞}. +By Nualart [20], Proposition 1.3.1, +D1,2(H) ⊂ D1,2(H) ⊂ dom(δ). +When H > 1/2, the two following propositions are crucial in order to establish a suitable risk bound on +�θN (see Subsection 3.2) and to compare �θN and �θN (see Subsection 3.3). +Proposition 3.4. For every ϕ ∈ C1(R) of bounded derivative, (ϕ(Xt))t∈[0,T ] belongs to D1,2(H) and +� T +0 +ϕ(Xs)δXs = +� T +0 +ϕ(Xs)dXs − αHσ +� T +0 +� T +0 +Ds[ϕ(Xt)] · |t − s|2H−2dsdt. +Proposition 3.4 is a straightforward consequence of Nualart [20], Proposition 5.2.3. Now, consider +M := sup +x∈R +b′(x). +Proposition 3.5. There exists a constant c3.5 > 0, only depending on H and σ, such that for every +ϕ ∈ C1(R) of bounded derivative, +E +� +� +�� T +0 +ϕ(Xs)δBs +�2� +� ⩽ c3.5mH,M,T +� +� +�� T +0 +E(|ϕ(Xs)| +1 +H )ds +�2H ++ +�� T +0 +E(ϕ′(Xs)2) +1 +2H ds +�2H� +� +with mH,M,T = 1 ∨ mH,M,T and +mH,M,T = +� +− H +M +�2H +1M<0 + T 2H1M=0 + +� H +M +�2H +e2MT 1M>0. +See Hu et al. [11], Proposition 4.4.(2) and Comte & Marie [5], Theorem 2.9 for a proof. +3.2. Risk bound on �θN. In the sequel, as for H = 1/2, the probability distribution of Xt, t ∈ (0, T], +and the function b need to fulfill Assumption 2.2. +Example 3.6. In the two following situations, the probability distribution of Xt, t ∈ (0, T], and the +function b fulfill Assumption 2.2: +(1) Assume that X0(.) = x0 with x0 ∈ R, and that b ∈ C∞ +b (R) (the space of all the smooth functions +ϕ : R → R such that ∥ϕ(k)∥∞ < ∞ for every k ∈ N). As for H = 1/2, for every t ∈ (0, T], the +probability distribution of Xt has a density ft with respect to Lebesgue’s measure such that, for +every x ∈ R, +(8) +ft(x) ⩽ cHt−H exp +� +−mH +(x − x0)2 +t2H +� +where cH and mH are positive constants depending on T but not on t and x (see Baudoin et al. +[1], Theorem 1.5), and then t �→ ft(x) belongs to L1([0, T]). Moreover, since b′ is bounded, still + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +9 +by Inequality (8), +bα ∈ L2(R, f(x)dx) ; ∀α ∈ R+. +(2) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4). Then, Equation (1) has a +unique stationary solution X, and the common probability distribution of the Xt’s has a density +f0 with respect to Lebesgue’s measure (see Hairer [9] and Bertin et al. [2], Proposition 1). It is +plausible that b and f = f0 fulfill Assumption 2.2.(2) but this problem is out of the scope of the +present paper. However, (at least) the fractional Ornstein-Uhlenbeck process fulfills Assumption +2.2 with +f0(x) = +θH +0 +� +2πσ2HΓ(2H) +exp +� +−θ2H +0 +x2 +2σ2HΓ(2H) +� +; ∀x ∈ R. +As in Section 2, the usual norm on L2(R, f(x)dx) is denoted by ∥.∥f. First, note that since B1, . . . , BN +are i.i.d. processes, by the (usual) law of large numbers and Equality (7), +�θN = θ0 + +� N +� +i=1 +� T +0 +b(Xi +s)2ds +�−1 � N +� +i=1 +� T +0 +b(Xi +s)δBi +s +� +(9) +a.s. +−−−−→ +N→∞ θ0 + +1 +∥b∥2 +f +E +� +1 +T +� T +0 +b(Xs)δBs +� += θ0. +Now, the following proposition provides a suitable risk bound on the truncated estimator +�θd +N := �θN1DN⩾d +with +d ∈ ∆f = +� +0, +∥b∥2 +f +2 +� +. +Proposition 3.7. Under Assumption 2.2, +E[(�θd +N − θ0)2] ⩽ c3.7 +N +with +c3.7 = 1 +d2 +� +σ2c3.5mH,M,T +T 2−2H +��� ∞ +−∞ +|b(x)| +1 +H f(x)dx +�2H ++ +� ∞ +−∞ +b′(x)2f(x)dx +� ++ θ2 +0∥b2∥2 +f +� +. +Proof. First of all, since dXi +t = θ0b(Xi +t)dt + σdBi +t for every i ∈ {1, . . . , N}, +�θN = θ0 + UN +DN +with +UN = +σ +NT +N +� +i=1 +� T +0 +b(Xi +s)δBi +s. +Since B1, . . . , BN are independent, +E(U 2 +N) = +σ2 +N 2T 2 +N +� +i=1 +E +� +� +�� T +0 +b(Xi +s)δBi +s +�2� +� +⩽ σ2c3.5mH,M,T +NT 2 +� +� +�� T +0 +E(|b(Xs)| +1 +H )ds +�2H ++ +�� T +0 +E(b′(Xs)2) +1 +2H ds +�2H� +� by Proposition 3.5 +⩽ σ2c3.5mH,M,T +NT 2−2H +��� ∞ +−∞ +|b(x)| +1 +H f(x)dx +�2H ++ +� ∞ +−∞ +b′(x)2f(x)dx +� + +10 +NICOLAS MARIE† +and +E[(DN − ∥b∥2 +f)2] = var(DN) += +1 +N 2T 2 var +� N +� +i=1 +� T +0 +b(Xi +s)2ds +� += 1 +N var +� +1 +T +� T +0 +b(Xs)2ds +� +⩽ +∥b2∥2 +f +N +. +Therefore, as in the proof of Proposition 2.4, +E[(�θd +N − θ0)2] ⩽ 1 +d2 [E(U 2 +N) + θ2 +0E[(DN − ∥b∥2 +f)2]] +⩽ +� +σ2c3.5mH,M,T +T 2−2H +��� ∞ +−∞ +|b(x)| +1 +H f(x)dx +�2H ++ +� ∞ +−∞ +b′(x)2f(x)dx +� ++ θ2 +0∥b2∥2 +f +� +1 +d2N . +□ +Remark 3.8. Let us conclude this section with some remarks about Proposition 3.7: +(1) As for H = 1/2, if b(.)2 ⩾ b with b > 0, then +�θd +N = �θN +for +d = b/2. +If in addition |b(.)| ⩽ m with m > 0, then +E[(�θN − θ0)2] ⩽ 4 +b2 +�σ2c3.5mH,M,T +T 2−2H +(m2 + ∥b′∥2 +∞) + θ2 +0m4 +� 1 +N . +(2) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4). Then, Equation (1) has a +unique stationary solution X, and the common probability distribution of the Xt’s has a density +f0 with respect to Lebesgue’s measure as mentioned in Example 3.6.(2). Moreover, assume that +b and f = f0 fulfill Assumption 2.2.(2) as, for instance, when (1) is the fractional Langevin +equation. In practice, as for H = 1/2, f0 is unknown but one may be able to provide a function +f0 such that f0(.) ⩾ f0(.) ⩾ 0, and then +d ∈ ∆f +when +d ∈ +� +0, 1 +2 +� ∞ +−∞ +b(x)2f0(x)dx +� +. +For instance, assume that b(x) = −x for every x ∈ R, which means that X is the fractional +Ornstein-Uhlenbeck process. If there exist (known) θmin, θmax > 0 such that θmin ⩽ θ0 ⩽ θmax, +then f0(.) ⩾ f0(.) > 0 with +f0(x) := +θH +min +� +2πσ2HΓ(2H) +exp +� +−θ2H +max +x2 +2σ2HΓ(2H) +� +; ∀x ∈ R. +3.3. A computable estimator. When H = 1/2, �θN is computable because as mentioned in Section +2.4, the Skorokhod integral coincides with Itô’s integral on H2. When H > 1/2, the Skorokhod integral +and then �θN are not directly computable. However, this subsection deals with the approximation �θN of +�θN which is computable by solving Equation (2). First of all, let us explain why �θN is defined this way. +For every i ∈ {1, . . . , N}, since +DsXi +t = σ1[0,t](s) exp +� +θ0 +� t +s +b′(Xi +u)du +� +; ∀s, t ∈ [0, T], + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +11 +by Proposition 3.4 and by the chain rule for the Malliavin derivative (see Nualart [20], Proposition 1.2.3), +� T +0 +b(Xi +s)δXi +s = +� T +0 +b(Xi +s)dXi +s − αHσ +� T +0 +� T +0 +b′(Xi +t)(DsXi +t)|t − s|2H−2dsdt += +� T +0 +b(Xi +s)dXi +s − αHσ2 +� T +0 +� t +0 +b′(Xi +t) exp +� +θ0 +� t +s +b′(Xi +u)du +� +|t − s|2H−2dsdt. +Then, since �θN is a converging estimator of θ0 as established in Subsection 3.2, +�θN − IN = ΦN(θ0 − IN) ≈ ΦN(�θN − IN), +where +ΦN(r) := − αHσ2 +NTDN +N +� +i=1 +� T +0 +� t +0 +b′(Xi +t) exp +� +(r + IN) +� t +s +b′(Xi +u)du +� +|t − s|2H−2dsdt +and, by the change of variable formula for Young’s integral, +IN := +1 +NTDN +N +� +i=1 +� T +0 +b(Xi +s)dXi +s += +1 +NTDN +N +� +i=1 +(b(Xi +T ) − b(Xi +0)) +with +b′(.) = b(.). +This legitimates to consider the estimator �θN := IN + RN of θ0, where RN is the fixed point of the map +ΦN. Let us establish that RN exists and is unique under the condition (10) stated below. +Proposition 3.9. Assume that b′(.) ⩽ 0. If +(10) +T 2H MN +DN +⩽ +c +αHσ2∥b′∥2∞ +, +where c is a deterministic constant arbitrarily chosen in (0, 1), +MN := e∥b′∥∞|IN|T +and +αH := +αH +2H(2H + 1), +then ΦN is a contraction from R+ into R+. Moreover, RN exists and is unique. +Proof. Since b′(.) ⩽ 0, ΦN is nonnegative, and in particular ΦN(R+) ⊂ R+. Moreover, by (10), for every +r, ρ ∈ R+, +|ΦN(r) − ΦN(ρ)| ⩽ +αHσ2 +NTDN +N +� +i=1 +� T +0 +� t +0 +|t − s|2H−2|b′(Xi +t)| exp +� +IN +� t +s +b′(Xi +u)du +� +× +����exp +� +r +� t +s +b′(Xi +u)du +� +− exp +� +ρ +� t +s +b′(Xi +u)du +����� dsdt +⩽ +αHσ2 +NTDN +∥b′∥∞MN +N +� +i=1 +� T +0 +� t +0 +|t − s|2H−2 sup +x∈R− +ex +����(r − ρ) +� t +s +b′(Xi +u)du +���� dsdt +⩽ αHσ2∥b′∥2 +∞T 2H MN +DN +|r − ρ| ⩽ c|r − ρ|. +So, ΦN is a contraction from R+ into R+, and then RN exists and is unique by Picard’s fixed point +theorem. +□ + +12 +NICOLAS MARIE† +The following proposition provides a convergence result on the truncated estimator +�θc +N := �θN1ΩN +with +ΩN = +� +T 2H MN +DN +⩽ +c +αHσ2∥b′∥2∞ +� +. +Proposition 3.10. Assume that b′(.) ⩽ 0 and that θ0 > 0. Under Assumption 2.2, if +(11) +T 2H +∥b∥2 +f +exp +� +∥b′∥∞ +∥b∥2 +f +|E(b(XT ) − b(X0))| +� +⩽ +c +αHσ2∥b′∥2∞ +, +then �θc +N → θ0 a.s. when N → ∞. +Proof. First, �θN = IN + ΦN(θ0 − IN) and, on the event ΩN, RN = �θN − IN is the unique fixed point of +the c-contraction ΦN by Proposition 3.9. Then, +|�θN − �θN|1ΩN = |ΦN(RN) − ΦN(θ0 − IN)|1ΩN +⩽ c|RN − (θ0 − IN)|1ΩN ⩽ c|�θN − �θN|1ΩN + c|�θN − θ0|1ΩN . +Since c ∈ (0, 1), +|�θN − �θN|1ΩN ⩽ +c +1 − c|�θN − θ0|1ΩN , +and thus +(12) +|�θc +N − θ0| = |�θN − θ0|1ΩN + |θ0|1Ωc +N ⩽ +1 +1 − c|�θN − θ0| + |θ0|1Ωc +N . +Now, by the (usual) law of large numbers, +DN +a.s. +−−−−→ +N→∞ +1 +T +� T +0 +E(b(Xs)2)ds = ∥b∥2 +f +and +1 +N +N +� +i=1 +(b(Xi +T ) − b(Xi +0)) +a.s. +−−−−→ +N→∞ E(b(XT ) − b(X0)). +Therefore, +MN +DN += +1 +DN +exp +� +∥b′∥∞ +NDN +����� +N +� +i=1 +(b(Xi +T ) − b(Xi +0)) +����� +� +a.s. +−−−−→ +N→∞ +1 +∥b∥2 +f +exp +� +∥b′∥∞ +∥b∥2 +f +|E(b(XT ) − b(X0))| +� +. +This leads to 1Ωc +N → 0 a.s. by (11). In conclusion, by Inequality (12) together with the convergence +result (9), +|�θc +N − θ0| +a.s. +−−−−→ +N→∞ 0. +□ +Remark 3.11. The condition (11) in the statement of Proposition 3.10 is, in fact, a condition on the +time horizon T which can be chosen arbitrarily small in our estimation framework, even when X1, . . . , XN +have been observed on [0, Tmax] with 0 < T ⩽ Tmax. In the two following situations, (11) can be simplified: +(1) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4) (in particular, b′(.) ⩽ 0). +Let f0 be the density with respect to Lebesgue’s measure of the common probability distribution of + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +13 +the Xt’s. Moreover, assume that b and f = f0 fulfill Assumption 2.2.(2). Then, +E(b(XT )) = E(b(X0)) +and, as a consequence, the condition (11) is equivalent to +T ⩽ +� +c +αHσ2∥b′∥2∞ +�� ∞ +−∞ +b(x)2f0(x)dx +�� +1 +2H +. +In practice, as mentioned in Remark 3.8.(2), f0 is unknown but one may be able to provide a +function f0 such that f0(.) ⩾ f0(.) ⩾ 0, and then T fulfills (11) when +T ⩽ +� +c +αHσ2∥b′∥2∞ +�� ∞ +−∞ +b(x)2f0(x)dx +�� +1 +2H +. +For instance, assume that X is the fractional Ornstein-Uhlenbeck process. If there exist (known) +θmin, θmax > 0 such that θmin ⩽ θ0 ⩽ θmax, then T fulfills (11) when +T ⩽ +� +c +αHσ2 +� +θH +min +� +2πσ2HΓ(2H) +� ∞ +−∞ +x2 exp +� +−θ2H +max +x2 +2σ2HΓ(2H) +� +dx +�� +1 +2H +. +(2) Assume that b′(.) ⩽ 0 and that θ0 > 0. Assume also that b ∈ C∞ +b (R). Since b is bounded, b is +Lipschitz continuous, and then +|E(b(XT ) − b(X0))| ⩽ ∥b∥∞E(|XT − X0|) +⩽ ∥b∥∞ +� +θ0 +� T +0 +E(|b(Xs)|)ds + σE(|BT |) +� +⩽ ∥b∥∞(θ0∥b∥∞T + σT H). +In practice, θ0 is unknown but one may be able to provide θmax > 0 such that θ0 ⩽ θmax, leading +to +|E(b(XT ) − b(X0))| ⩽ ∥b∥∞T H +max(θmax∥b∥∞T 1−H +max + σ). +If in addition b(.)2 ⩾ b with b > 0, then ∥b∥2 +f ⩾ b and T fulfills (11) when +T ⩽ +� +cb +αHσ2∥b′∥2∞ +exp +� +−∥b′∥∞ +b +∥b∥∞T H +max(θmax∥b∥∞T 1−H +max + σ) +�� +1 +2H +. +Now, let us establish a risk bound on �θc +N when b(.)2 ⩾ b with b > 0. +Proposition 3.12. Assume that b′(.) ⩽ 0 and that θ0 > 0. Assume also that b(.)2 ⩾ b with b > 0. +Under Assumption 2.2, if +(13) +T 2H +b +exp +�∥b′∥∞ +b +|E(b(XT ) − b(X0))| +� +< +c +αHσ2∥b′∥2∞ +, +then there exists a constant c3.12 > 0, not depending on N, such that +E[(�θc +N − θ0)2] ⩽ c3.12 +N . +Proof. First, as established in the proof of Proposition 3.10, +|�θc +N − θ0| ⩽ +1 +1 − c|�θN − θ0| + |θ0|1Ωc +N . +Moreover, since b(.)2 ⩾ b, +�θN = �θd +N +with +d = b +2 + +14 +NICOLAS MARIE† +by Remark 3.8.(1). Then, by Proposition 3.7, +E[(�θc +N − θ0)2] ⩽ 2(1 − c)−2 c3.7 +N + 2θ2 +0P(Ωc +N). +Now, +P(Ωc +N) = P +� +MN > +c +αHT 2Hσ2∥b′∥2∞ +DN +� +⩽ P +� +MN > +c +αHT 2Hσ2∥b′∥2∞ +d +� +⩽ P +� +1 +N +����� +N +� +i=1 +(b(Xi +T ) − b(Xi +0)) +����� > log +� +c +αHT 2Hσ2∥b′∥2∞ +d +� +d +∥b′∥∞ +� +⩽ P +� +1 +N +����� +N +� +i=1 +[b(Xi +T ) − b(Xi +0) − E(b(Xi +T ) − b(Xi +0))] +����� > u +� +with +u = log +� +c +αHT 2Hσ2∥b′∥2∞ +d +� +d +∥b′∥∞ +− |E(b(XT ) − b(X0))| > 0 +by +(13). +So, by the Bienaymé-Tchebychev inequality, and since X1, . . . , XN are i.i.d. processes, +P(Ωc +N) ⩽ +1 +u2N 2 var +� N +� +i=1 +(b(Xi +T ) − b(Xi +0)) +� +⩽ +1 +u2N E[(b(XT ) − b(X0))2]. +Therefore, +E[(�θc +N − θ0)2] ⩽ +� +2c3.7 +(1 − c)2 + 1 +u2 E[(b(XT ) − b(X0))2] +� 1 +N . +□ +Finally, let us consider the estimator �θc +N,n := �θN,n1ΩN , where �θN,n := IN + RN,n and the sequence +(RN,n)n∈N is defined by +� +RN,0 = 0 +RN,n+1 = ΦN(RN,n) ; n ∈ N . +Proposition 3.13. Let ψ : N → N be a map satisfying +ψ(.) ⩾ −log(m3.13√.) +log(c) +with +m3.13 = +c(1 − c)−1 +2TαH∥b′∥∞ +. +If T satisfies (11), then +�θc +N,ψ(N) +a.s. +−−−−→ +N→∞ θ0. +If b(.)2 ⩾ b with b > 0, and if T satisfies (13), then there exists a constant c3.13 > 0, not depending on +N, such that +E[(�θc +N,ψ(N) − θ0)2] ⩽ c3.13 +N . +Proof. On the event ΩN, note that +|ΦN(0)| ⩽ +αHσ2 +NTDN +N +� +i=1 +� T +0 +� t +0 +|b′(Xi +t)| exp +� +IN +� t +s +b′(Xi +u)du +� +|t − s|2H−2dsdt +⩽ αHσ2 +TDN +∥b′∥∞MN +� T +0 +� t +0 +|t − s|2H−2dsdt ⩽ σ2∥b′∥∞ +2T +T 2H MN +DN +⩽ c1 +with +c1 = +c +2TαH∥b′∥∞ +. + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +15 +Consider n ∈ N∗. Thanks to a well-known consequence of Picard’s fixed point theorem, for every x ∈ R+, +|(ΦN ◦ · · · ◦ ΦN +� +�� +� +n times +)(x) − RN| ⩽ +cn +1 − c|ΦN(x) − x| +on +ΩN. +Then, +|RN,n − RN|1ΩN = |(ΦN ◦ · · · ◦ ΦN)(RN,0) − RN|1ΩN +⩽ +cn +1 − c|ΦN(0)|1ΩN ⩽ c2cn +with +c2 = +c1 +1 − c, +leading to +|�θc +N,n − θ0| ⩽ |RN,n − RN|1ΩN + |�θc +N − θ0| ⩽ c2cn + |�θc +N − θ0|. +If T satisfies (11), then +|�θc +N,ψ(N) − θ0| ⩽ +1 +√ +N ++ |�θc +N − θ0| +a.s. +−−−−→ +N→∞ 0 +by Proposition 3.10. If b(.)2 ⩾ b with b > 0, and if T satisfies (13), then +E[(�θc +N,ψ(N) − θ0)2] ⩽ 2(1 + c3.12) +N +by Proposition 3.12. +□ +4. Numerical experiments +In this section, our computable estimator of θ0 is evaluated on the two following models: +(1) dXt = θ0b1(Xt)dt + dBt with θ0 = 1 and b1(x) := −x. Since b′ +1(x) = −1 < 0, the estimator �θc +N +is well-defined thanks to Proposition 3.9. Moreover, Proposition 3.10 applies to �θc +N for T small +enough. +(2) dXt = θ0b2(Xt)dt+0.25dBt with θ0 = 1 and b2(x) := π+arctan(−x). Since b′ +2(x) = −(1+x2)−1 ∈ +[−1, 0], the estimator �θc +N is well-defined thanks to Proposition 3.9. Since b2(x)2 ⩾ π2/4 > 0, both +Propositions 3.10 and 3.12 apply to �θc +N for T small enough. +For each model, with H = 0.7 and H = 0.9, �θN,30 is computed from N = 1, . . . , 100 paths of the +process X. This experiment is repeated 100 times. The means and the standard deviations of the error +|�θ100,30 − θ0| are stored in Table 1. Moreover, the map N �→ �θN,30 is plotted for 5 datasets generated +Mean error +Error StD. +Model 1 (H = 0.7) +0.0088416 +0.0081571 +Model 1 (H = 0.9) +0.0147383 +0.0108511 +Model 2 (H = 0.7) +0.0169235 +0.0071539 +Model 2 (H = 0.9) +0.0164968 +0.0085905 +Table 1. Means and StD. of the error of �θ100,30 (100 repetitions). +from Model 1 (resp. Model 2) in Figure 1 (resp. Figure 2). + +16 +NICOLAS MARIE† +Figure 1. Plots of N �→ �θN,30 for Model 1 with H = 0.7 (left) and H = 0.9 (right). +Figure 2. Plots of N �→ �θN,30 for Model 2 with H = 0.7 (left) and H = 0.9 (right). +For both H = 0.7 and H = 0.9, the mean error of �θ100,30 is lower for Model 1 than for Model 2. For +Model 1, the mean error is significantly lower when H = 0.7 than when H = 0.9. For Model 2, the mean +error is of same order for both H = 0.7 and H = 0.9. On Figures 1 and 2, for both Model 1 and Model +2, the estimator seems to converge faster to θ0 when H = 0.9 than when H = 0.7. In each situation, +Figures 1 and 2 show that N �→ �θN,30 starts to stabilize from N ≈ 30. +5. Conclusion and perspectives +The main contributions of our paper are: +(1) To provide a risk bound on the truncated least squares estimator �θd +N of the parameter θ0 when +H = 1/2 and X1, . . . , XN are interacting copies of the solution X of Equation (1). Precisely, +Proposition 2.4 says that the rate of convergence of �θd +N remains of order 1/N (parametric), as +when X1, . . . , XN are independent, while |RN| ⩽ N. + +ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE +17 +(2) To provide a computable approximation �θN of �θN when H > 1/2 with some convergence results. +Such approximation of �θN is interesting in practice because it is well-known that the Skorokhod +integral involved in the definition of �θN is not directly computable when H > 1/2. Precisely, an +almost sure convergence result is established in Proposition 3.10 and a risk bound is established +in Proposition 3.12. +In fact, and this will be investigated later, our computable approximation method for �θN could be +extended to the (nonparametric) projection least squares estimator studied in Comte & Marie [5]. For +a regular enough orthonormal family (ϕ1, . . . , ϕm) of L2(R, dx) with m ∈ {1, . . . , N}, [5] deals with the +estimator +�bm(.) := +m +� +j=1 +�θjϕj(.) +with +�θ = �Ψ−1 +m �Zm +of the whole function b0(.) = θ0b(.) in Equation (1), where +�Ψm := +� +1 +NT +N +� +i=1 +� T +0 +ϕj(Xi +s)ϕℓ(Xi +s)ds +� +j,ℓ +and +�Zm := +� +1 +NT +N +� +i=1 +� T +0 +ϕj(Xi +s)δXi +s +� +j +. +By Proposition 3.4 and by the chain rule for the Malliavin derivative, �Zm = γN(b0) with +γN(ψ) := +� +1 +NT +N +� +i=1 +�� T +0 +ϕj(Xi +s)dXi +s − αHσ2 +� T +0 +� t +0 +ϕ′ +j(Xi +t) exp +�� t +s +ψ′(Xi +u)du +� +|t − s|2H−2dsdt +�� +j +. +Then, +�bm = ΓN(b0) ≈ ΓN(�bm) +with +ΓN(ψ)(.) := +m +� +j=1 +[�Ψ−1 +m γN(ψ)]jϕj(.). +This legitimates to consider the estimator �bm(.) := �m +j=1 �θjϕj(.) of b0 such that �bm = ΓN(�bm). +References +[1] F. Baudoin, E. Nualart, C. Ouyang & S. Tindel. On Probability Laws of Solutions to Differential Systems Driven by +a Fractional Brownian Motion. The Annals of Probability 44, 4, 2554-2590, 2016. +[2] K. Bertin, N. Klutchnikoff, F. Panloup & M. Varvenne. Adaptive Estimation of the Stationary Density of a Stochastic +Differential Equation Driven by a Fractional Brownian Motion. Statistical Inference for Stochastic Processes 23, 2, +271-300, 2020. +[3] F. Comte & V. Genon-Catalot. Nonparametric Drift Estimation for I.I.D. Paths of Stochastic Differential Equations. +The Annals of Statistics 48, 6, 3336-3365, 2020. +[4] F. Comte & N. Marie Nonparametric Estimation in Fractional SDE. Statistical Inference for Stochastic Processes 22, +3, 359-382, 2019. +[5] F. Comte & N. Marie Nonparametric Estimation for I.I.D. Paths of Fractional SDE. Statistical Inference for Stochastic +Processes 24, 3, 669-705, 2021. +[6] F. Comte & N. Marie Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions. Preprint, +arXiv: 2210.13173, 2022. +[7] L. Della Maestra & M. Hoffmann. Nonparametric Estimation for Interacting Particle Systems: McKean-Vlasov Models. +Probability Theory and Related Fields 182, 551-613, 2022. +[8] C. Denis, C. Dion & M. Martinez. A Ridge Estimator of the Drift from Discrete Repeated Observations of the Solutions +of a Stochastic Differential Equation. Bernoulli 27, 2675-2713, 2021. +[9] M. Hairer. Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion. The Annals of +Probability 33, 2, 703-758, 2005. + +18 +NICOLAS MARIE† +[10] Y. Hu & D. Nualart. Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes. Statistics and Probability +Letters 80, 1030-1038, 2010. +[11] Y. Hu, D. Nualart & H. Zhou. Drift Parameter Estimation for Nonlinear Stochastic Differential Equations Driven by +Fractional Brownian Motion. Stochastic 91, 8, 1067-1091, 2019. +[12] R. Khasminskii. Stochastic Stability of Differential Equations. Springer, 2012. +[13] M.L. Kleptsyna & A. Le Breton. Some Explicit Statistical Results About Elementary Fractional Type Models. Nonlinear +Analysis 47, 4783-4794, 2001. +[14] Y. Kutoyants. Statistical Inference for Ergodic Diffusion Processes. Springer, 2004. +[15] N. Marie. Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non- +Autonomous Financial Models. Finance and Stochastics 27, 1, 97-126, 2023. +[16] N. Marie & P. Raynaud de Fitte. Almost Periodic and Periodic Solutions of Differential Equations Driven by the +Fractional Brownian Motion with Statistical Application. Stochastics 93, 6, 886-906, 2021. +[17] N. Marie & A. Rosier. Nadaraya-Watson Estimator for I.I.D. Paths of Diffusion Processes. Scandinavian Journal of +Statistics (accepted), 2022. +[18] S. Menozzi, A. Pesce & X. Zhang. Density and Gradient Estimates for Non Degenerate Brownian SDEs with Unbounded +Measurable Drift. Journal of Differential Equations 272, 339-369, 2021. +[19] A. Neuenkirch & S. Tindel. A Least Square-Type Procedure for Parameter Estimation in Stochastic Differential Equa- +tions with Additive Fractional Noise. Statistical Inference for Stochastic Processes 17, 1, 99-120, 2014. +[20] D. Nualart. The Malliavin Calculus and Related Topics. Springer, 2006. +[21] B. Saussereau. Nonparametric Inference for Fractional Diffusions. Bernoulli 20, 2, 878-918, 2014. +[22] C.A. Tudor & F. Viens. Statistical Aspects of the Fractional Stochastic Calculus. The Annals of Statistics 35, 3, +1183-1212, 2007. +†Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France +Email address: nmarie@parisnanterre.fr + diff --git a/KNE4T4oBgHgl3EQf7g74/content/tmp_files/load_file.txt b/KNE4T4oBgHgl3EQf7g74/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9bfc61abc490b3990cfbd1d1bf3213d597e7a90 --- /dev/null +++ b/KNE4T4oBgHgl3EQf7g74/content/tmp_files/load_file.txt @@ -0,0 +1,760 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf,len=759 +page_content='ON A COMPUTABLE SKOROKHOD’S INTEGRAL BASED ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE NICOLAS MARIE† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' This paper deals with a Skorokhod’s integral based least squares type estimator �θN of the drift parameter θ0 computed from N ∈ N∗ copies X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN of the solution X to dXt = θ0b(Xt)dt + σdBt, where B is a fractional Brownian motion of Hurst index H ∈ [1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' On the one hand, a risk bound is established on �θN when H = 1/2 and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are dependent copies of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' On the other hand, when H > 1/2, Skorokhod’s integral based estimators as �θN cannot be computed directly from data, but in this paper some convergence results are established on a computable approximation of �θN when X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Case H = 1/2: risk bound and dependent copies 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Case H > 1/2: risk bound and computable estimator 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Basics on the Skorokhod integral 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Risk bound on �θN 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' A computable estimator 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Numerical experiments 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Conclusion and perspectives 16 References 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Introduction Let X = (Xt)t∈[0,T ] be the solution of the differential equation (1) Xt = X0 + θ0 � t 0 b(Xs)ds + σBt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' t ∈ [0, T], where T > 0 is fixed, X0 ∈ L2(Ω), B = (Bt)t∈[0,T ] is a fractional Brownian motion of Hurst index H ∈ [1/2, 1), b ∈ C1(R), b′ is bounded, σ ∈ R∗ and θ0 ∈ R is an unknown parameter to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The oldest kind of (non)parametric estimators of the drift function is based on the long-time behav- ior of the solution to Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For H = 1/2, the reader may refer to the monograph [14] written by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Kutoyants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For H > 1/2, see Kleptsyna & Le Breton [13], Tudor & Viens [22], Hu & Nualart [10], Neuenkirch & Tindel [19], Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [11], Marie & Raynaud de Fitte [16], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' on parametric estimators, Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Fractional Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Least squares estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Malliavin calculus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='05341v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='ST] 13 Jan 2023 2 NICOLAS MARIE† and see Saussereau [21] and Comte & Marie [4] on nonparametric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The stochastic integral involved in the definition of the estimators studied in [10], [11], [16] and [4] is taken in the sense of Skorokhod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' To be not directly computable from an observation of X is the major drawback of the Skorokhod integral with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' One of the main purposes of our paper is to bypass this difficulty in another estima- tion framework because the Skorokhod integral is a nice generalization of Itô’s integral, tailor-made for advanced statistical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For H = 1/2, a new kind of estimators of the drift function have been investigated since several years: those computed from N copies X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN of X observed on [0, T] with T > 0 fixed but N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The major part of the literature deals with estimators based on independent copies of X (see Comte & Genon-Catalot [3], Denis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [8], Marie & Rosier [17], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ), but some recent papers are also devoted to estimators based on dependent copies (see Della Maestra & Hoffmann [7] and Comte & Marie [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Copies based estimators are well-adapted to some situations difficult to manage with long-time behavior based estimators: Assume that X models the elimination process of a drug administered to one people, and assume that in a clinical-trial involving N patients, Xi models the elimination process of the same drug for the i-th patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are independent copies of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Consider a financial market with N interacting risky assets of same kind and assume that the i-th asset is modeled by Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Here, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are not independent, but copies based estimators of the drift function remain appropriate as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For H > 1/2, Comte & Marie [5] and Marie [15] are the only two references on such estimators up to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Our paper deals with the least squares type estimator �θN := � N � i=1 � T 0 b(Xi s)2ds �−1 � N � i=1 � T 0 b(Xi s)δXi s � of θ0, where N ∈ N∗, Xi := I(Xi 0, Bi) for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' X1 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN 0 ) are some copies of B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' independent copies of X0), I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') is the Itô map for Equation (1), and the stochastic integral is taken in the sense of Skorokhod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since �θN is not directly computable when H > 1/2, our paper also deals with the estimator �θN approximating �θN and defined as a fixed point: �θN = 1 NTDN N � i=1 �� T 0 b(Xi s)dXi s (2) −αHσ2 � T 0 � t 0 b′(Xi t) exp � �θN � t s b′(Xi u)du � |t − s|2H−2dsdt � , where αH := H(2H − 1), DN := 1 NT N � i=1 � T 0 b(Xi s)2ds and the stochastic integral in the right-hand side of Equation (2) is taken pathwise (in the sense of Young).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The main purposes of our paper are: ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 3 (1) To establish a risk bound on �θN when H = 1/2 and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are dependent copies of the solution X to Equation (1) (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) To establish some convergence results on the computable estimator �θN when H > 1/2 (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In this case, �θN is an auxiliary estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Finally, Section 4 deals with some numerical experiments on �θN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Case H = 1/2: risk bound and dependent copies Throughout this section, H = 1/2, and then the Skorokhod integral coincides with Itô’s integral on the space H2 := {U ∈ L2([0, T] × Ω) : U is adapted} (see Nualart [20], Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, let us assume that there exists a correlation matrix R such that, for any i, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, E(Bi sBk t ) = Ri,k(s ∧ t) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀s, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' This leads (in particular) to d⟨Bi, Bk⟩t = Ri,kdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' From continuous-time observations, to determine the matrix R is not a statistical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Indeed, since σ ̸= 0, for every i, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, Ri,k = ⟨Xi, Xk⟩T σ2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Finally, the probability distribution of Xt, t ∈ (0, T], and the function b need to fulfill the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For every t ∈ (0, T], the probability distribution of Xt has a density ft with respect to Lebesgue’s measure such that: (1) The function t �→ ft(x) belongs to L1([0, T]) for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) The function bα belongs to L2(R, f(x)dx) for every α ∈ R+, where f is the density function defined by f(x) := 1 T � T 0 fs(x)ds ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In the two following situations, the probability distribution of Xt, t ∈ (0, T], and the function b fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2: (1) Assume that X0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') = x0 with x0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since σ ̸= 0 and b′ is bounded, for every t ∈ (0, T], the probability distribution of Xt has a density ft with respect to Lebesgue’s measure such that, for every x ∈ R, (3) ft(x) ⩽ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5t− 1 2 exp � −m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5 (x − x0)2 t � where c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5 and m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5 are positive constants depending on T but not on t and x (see Menozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [18], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2), and then t �→ ft(x) belongs to L1([0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, since b′ is bounded, still by Inequality (3), bα ∈ L2(R, f(x)dx) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀α ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4) ∃c > 0 : ∀x ∈ R, b′(x) ⩽ −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 4 NICOLAS MARIE† Then, Equation (1) has a unique stationary solution X, and the common probability distribution of the Xt’s has a sub-Gaussian density f0 with respect to Lebesgue’s measure (see Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [2], Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' So, in this situation, f = f0 and bα ∈ L2(R, f(x)dx) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀α ∈ R+ because the density function f0 is sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For instance, (4) is satisfied by the drift function of the Langevin equation dXt = −θ0Xtdt+σdBt defining the so-called Ornstein-Uhlenbeck process, and f0(x) = � θ0 πσ2 exp � −θ0x2 σ2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The usual norm on L2(R, f(x)dx) is denoted by ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='∥f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The following proposition provides a suitable risk bound on the truncated estimator �θd N := �θN1DN⩾d with d ∈ ∆f = � 0, ∥b∥2 f 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2, E[(�θd N − θ0)2] ⩽ c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 N � 1 + |RN| N � where RN := {(i, k) : i ̸= k and Ri,k ̸= 0} and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 := 1 d2 � σ2∥b∥2 f T + θ2 0∥b2∥2 f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' First of all, since dXi t = θ0b(Xi t)dt + σdBi t for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, �θN = θ0 + UN DN with UN = σ NT N � i=1 � T 0 b(Xi s)δBi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' by the isometry property of Itô’s integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' E(U 2 N) = σ2 N 2T 2 N � i=1 E � � �� T 0 b(Xi s)δBi s �2� � + σ2 N 2T 2 � i̸=k E ��� T 0 b(Xi s)δBi s � �� T 0 b(Xk s )δBk s �� = σ2 N 2T 2 � �N � T 0 E(b(Xs)2)ds + � i̸=k Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='k � T 0 E(b(Xi s)b(Xk s ))ds � � ⩽ σ2 NT � � � ∞ −∞ b(x)2f(x)dx + 1 NT � i̸=k |Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='k| � T 0 E(b(Xi s)2) 1 2 E(b(Xk s )2) 1 2 ds � � ⩽ σ2∥b∥2 f NT � �1 + 1 N � i̸=k |Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='k| � � ⩽ σ2∥b∥2 f NT � 1 + |RN| N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (5) On the other hand, E(DN) = 1 T � T 0 E(b(Xs)2)ds = ∥b∥2 f ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 5 and, since B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN are Gaussian processes, RN = {(i, k) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}2 : i ̸= k and Xi is independent of Xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, E[(DN − ∥b∥2 f)2] = var(DN) = 1 N 2T 2 var � N � i=1 � T 0 b(Xi s)2ds � = 1 N var � 1 T � T 0 b(Xs)2ds � + 1 N 2 � (i,k)∈RN cov � 1 T � T 0 b(Xi s)2ds, 1 T � T 0 b(Xk s )2ds � ⩽ ∥b2∥2 f N + |RN| N 2 var � 1 T � T 0 b(Xs)2ds � ⩽ ∥b2∥2 f N � 1 + |RN| N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (6) Note that E[(�θd N − θ0)2] = E[(�θN − θ0)21DN⩾d] + θ2 0P(DN < d) ⩽ 1 d2 E(U 2 N) + θ2 0P � |DN − ∥b∥2 f| > ∥b∥2 f 2 � ⩽ 1 d2 [E(U 2 N) + θ2 0E[(DN − ∥b∥2 f)2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Therefore, by Inequalities (5) and (6), E[(�θd N − θ)2] ⩽ � σ2∥b∥2 f T + θ2 0∥b2∥2 f � 1 d2N � 1 + |RN| N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let us conclude this section with some remarks about Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4: (1) If |RN| ⩽ N, then the rate of convergence of �θd N remains of order N −1/2 (parametric rate) as when B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN are independent (case R = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) If b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0, then DN = 1 NT N � i=1 � T 0 b(Xi s)2ds ⩾ b and ∥b∥2 f = � ∞ −∞ b(x)2f(x)dx ⩾ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' So, �θd N = �θN for d = b/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If in addition |b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )| ⩽ m with m > 0, then E[(�θN − θ0)2] ⩽ 4 b2 �σ2m2 T + θ2 0m4 � 1 N � 1 + |RN| N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (3) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, Equation (1) has a unique stationary solution X, and the common probability distribution of the Xt’s has a density f0 with respect to Lebesgue’s measure as mentioned in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, assume that b and f = f0 fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) as, for instance, when (1) is the Langevin equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In practice, f0 is unknown but one may be able to provide a function f0 such that f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ 0, 6 NICOLAS MARIE† and then d ∈ ∆f when d ∈ � 0, 1 2 � ∞ −∞ b(x)2f0(x)dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For instance, assume that b(x) = −x for every x ∈ R, which means that X is the Ornstein- Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If there exist (known) θmin, θmax > 0 such that θmin ⩽ θ0 ⩽ θmax, then f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') > 0 with f0(x) := � θmin πσ2 exp � −θmaxx2 σ2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (4) Assume that X0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') = x0 with x0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assume also that θ0 > 0 and that b satisfies the dissipativity condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' From one path of the solution X to Equation (1) observed on R+, which seems to be a situation only appropriate for long-time behavior based estimators of θ0, one can construct N independent copies of X|[0,T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' To that purpose, consider the stopping times τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , τN recursively defined by τ1 = 0 and τi = inf{t > τi−1 + T : Xt = x0} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N with the convention inf(∅) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since θ0 > 0 and b fulfills (4), the scale density s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') := exp � −2θ0 σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 0 b(x)dx � satisfies � 0 −∞ s(x)dx = � ∞ 0 s(x)dx = ∞, and then X is a recurrent Markov process by Khasminskii [12], Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' So, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, P(τi < ∞) = 1 and one can consider the processes Bi := (Bτi+t − Bτi)t∈[0,T ] and Xi := (Xτi+t)t∈[0,T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN are independent Brownian motions by the strong Markov property, and since Xi = I(x0, Bi) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, the processes X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are independent copies of X|[0,T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Case H > 1/2: risk bound and computable estimator Throughout this section, H > 1/2 and B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN are independent copies of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The proof of Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 only relies on the zero mean and on the control of the variance of Itô’s integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Such properties for Skorokhod’s integral with respect to the fractional Brownian motion are stated in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2 deals with a risk bound on �θN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Finally, Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3 deals with the existence and some convergence results on the computable estimator �θN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Basics on the Skorokhod integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='⟩H be the inner product defined by ⟨h, η⟩H := αH � T 0 � T 0 h(s)η(s)|t − s|2H−2dsdt, ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 7 and consider the reproducing kernel Hilbert space H = {h : ∥h∥H < ∞} of B, where ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='∥H is the norm associated to ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Consider also the isonormal Gaussian process (B(h))h∈H defined by B(h) := � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 0 h(s)dB(s) which is the Wiener integral of h ∈ H with respect to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Malliavin derivative of a smooth functional F = ϕ(B(h1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , B(hn)) where n ∈ N∗, ϕ ∈ C∞ p (Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' R) (the space of all the smooth functions ϕ : Rn → R such that ϕ and all its partial derivatives have polynomial growth) and h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , hn ∈ H, is the H-valued random variable DF := n � k=1 ∂kϕ(B(h1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , B(hn))hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The map D is closable from L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' R) into L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Its domain in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' R), denoted by D1,2, is the closure of the smooth functionals space for the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='∥1,2 defined by ∥F∥2 1,2 := E(F 2) + E(∥DF∥2 H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Malliavin derivative of F ∈ D1,2 at time s ∈ [0, T] is denoted by DsF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' See Nualart [20], Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1 for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The adjoint δ of the Malliavin derivative D is the divergence operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The domain of δ is denoted by dom(δ), and Y ∈ dom(δ) if and only if there exists a deterministic constant cY > 0 such that for every F ∈ D1,2, |E(⟨DF, Y ⟩H)| ⩽ cY E(F 2) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For any process Y = (Ys)s∈[0,T ] and every t ∈ (0, T], if Y 1[0,t] ∈ dom(δ), then its Skorokhod integral with respect to B is defined on [0, t] by � t 0 YsδBs := δ(Y 1[0,t]), and its Skorokhod integral with respect to X is defined by � t 0 YsδXs := θ0 � t 0 Ysb(Xs)ds + σ � t 0 YsδBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Note that since δ is the adjoint of the Malliavin derivative D, the Skorokhod integral of Y with respect to B on [0, t] is a centered random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Indeed, (7) E �� t 0 YsδBs � = E(1 · δ(Y 1[0,t])) = E(⟨D(1), Y 1[0,t]⟩H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let S be the space of the smooth functionals presented in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1 and consider D1,2(H), the closure of SH := � � � n � j=1 Fjhj ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , hn ∈ H, F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , Fn ∈ S � � � for the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='∥1,2,H defined by ∥Y ∥2 1,2,H := E(∥Y ∥2 H) + E(∥DY ∥2 H⊗H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 8 NICOLAS MARIE† Consider also the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='∥H defined by ∥h∥H := � αH � T 0 � T 0 |h(s)| · |h(t)| · |t − s|2H−2dsdt � 1 2 , the Banach space H := {h : ∥h∥H < ∞} and D1,2(H) := {Y ∈ D1,2(H) : E(∥Y ∥2 H) + E(∥DY ∥2 H⊗H) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' By Nualart [20], Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='1, D1,2(H) ⊂ D1,2(H) ⊂ dom(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' When H > 1/2, the two following propositions are crucial in order to establish a suitable risk bound on �θN (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2) and to compare �θN and �θN (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For every ϕ ∈ C1(R) of bounded derivative, (ϕ(Xt))t∈[0,T ] belongs to D1,2(H) and � T 0 ϕ(Xs)δXs = � T 0 ϕ(Xs)dXs − αHσ � T 0 � T 0 Ds[ϕ(Xt)] · |t − s|2H−2dsdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 is a straightforward consequence of Nualart [20], Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Now, consider M := sup x∈R b′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' There exists a constant c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5 > 0, only depending on H and σ, such that for every ϕ ∈ C1(R) of bounded derivative, E � � �� T 0 ϕ(Xs)δBs �2� � ⩽ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5mH,M,T � � �� T 0 E(|ϕ(Xs)| 1 H )ds �2H + �� T 0 E(ϕ′(Xs)2) 1 2H ds �2H� � with mH,M,T = 1 ∨ mH,M,T and mH,M,T = � − H M �2H 1M<0 + T 2H1M=0 + � H M �2H e2MT 1M>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' See Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [11], Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) and Comte & Marie [5], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9 for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Risk bound on �θN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In the sequel, as for H = 1/2, the probability distribution of Xt, t ∈ (0, T], and the function b need to fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In the two following situations, the probability distribution of Xt, t ∈ (0, T], and the function b fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2: (1) Assume that X0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') = x0 with x0 ∈ R, and that b ∈ C∞ b (R) (the space of all the smooth functions ϕ : R → R such that ∥ϕ(k)∥∞ < ∞ for every k ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' As for H = 1/2, for every t ∈ (0, T], the probability distribution of Xt has a density ft with respect to Lebesgue’s measure such that, for every x ∈ R, (8) ft(x) ⩽ cHt−H exp � −mH (x − x0)2 t2H � where cH and mH are positive constants depending on T but not on t and x (see Baudoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [1], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5), and then t �→ ft(x) belongs to L1([0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, since b′ is bounded, still ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 9 by Inequality (8), bα ∈ L2(R, f(x)dx) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀α ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, Equation (1) has a unique stationary solution X, and the common probability distribution of the Xt’s has a density f0 with respect to Lebesgue’s measure (see Hairer [9] and Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [2], Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' It is plausible that b and f = f0 fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) but this problem is out of the scope of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' However, (at least) the fractional Ornstein-Uhlenbeck process fulfills Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2 with f0(x) = θH 0 � 2πσ2HΓ(2H) exp � −θ2H 0 x2 2σ2HΓ(2H) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' As in Section 2, the usual norm on L2(R, f(x)dx) is denoted by ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='∥f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' First, note that since B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' processes, by the (usual) law of large numbers and Equality (7), �θN = θ0 + � N � i=1 � T 0 b(Xi s)2ds �−1 � N � i=1 � T 0 b(Xi s)δBi s � (9) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ θ0 + 1 ∥b∥2 f E � 1 T � T 0 b(Xs)δBs � = θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Now, the following proposition provides a suitable risk bound on the truncated estimator �θd N := �θN1DN⩾d with d ∈ ∆f = � 0, ∥b∥2 f 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2, E[(�θd N − θ0)2] ⩽ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 N with c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 = 1 d2 � σ2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5mH,M,T T 2−2H ��� ∞ −∞ |b(x)| 1 H f(x)dx �2H + � ∞ −∞ b′(x)2f(x)dx � + θ2 0∥b2∥2 f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' First of all, since dXi t = θ0b(Xi t)dt + σdBi t for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, �θN = θ0 + UN DN with UN = σ NT N � i=1 � T 0 b(Xi s)δBi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , BN are independent, E(U 2 N) = σ2 N 2T 2 N � i=1 E � � �� T 0 b(Xi s)δBi s �2� � ⩽ σ2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5mH,M,T NT 2 � � �� T 0 E(|b(Xs)| 1 H )ds �2H + �� T 0 E(b′(Xs)2) 1 2H ds �2H� � by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5 ⩽ σ2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5mH,M,T NT 2−2H ��� ∞ −∞ |b(x)| 1 H f(x)dx �2H + � ∞ −∞ b′(x)2f(x)dx � 10 NICOLAS MARIE† and E[(DN − ∥b∥2 f)2] = var(DN) = 1 N 2T 2 var � N � i=1 � T 0 b(Xi s)2ds � = 1 N var � 1 T � T 0 b(Xs)2ds � ⩽ ∥b2∥2 f N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Therefore, as in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4, E[(�θd N − θ0)2] ⩽ 1 d2 [E(U 2 N) + θ2 0E[(DN − ∥b∥2 f)2]] ⩽ � σ2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5mH,M,T T 2−2H ��� ∞ −∞ |b(x)| 1 H f(x)dx �2H + � ∞ −∞ b′(x)2f(x)dx � + θ2 0∥b2∥2 f � 1 d2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let us conclude this section with some remarks about Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7: (1) As for H = 1/2, if b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0, then �θd N = �θN for d = b/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If in addition |b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )| ⩽ m with m > 0, then E[(�θN − θ0)2] ⩽ 4 b2 �σ2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='5mH,M,T T 2−2H (m2 + ∥b′∥2 ∞) + θ2 0m4 � 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, Equation (1) has a unique stationary solution X, and the common probability distribution of the Xt’s has a density f0 with respect to Lebesgue’s measure as mentioned in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, assume that b and f = f0 fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) as, for instance, when (1) is the fractional Langevin equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In practice, as for H = 1/2, f0 is unknown but one may be able to provide a function f0 such that f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ 0, and then d ∈ ∆f when d ∈ � 0, 1 2 � ∞ −∞ b(x)2f0(x)dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For instance, assume that b(x) = −x for every x ∈ R, which means that X is the fractional Ornstein-Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If there exist (known) θmin, θmax > 0 such that θmin ⩽ θ0 ⩽ θmax, then f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') > 0 with f0(x) := θH min � 2πσ2HΓ(2H) exp � −θ2H max x2 2σ2HΓ(2H) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' A computable estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' When H = 1/2, �θN is computable because as mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4, the Skorokhod integral coincides with Itô’s integral on H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' When H > 1/2, the Skorokhod integral and then �θN are not directly computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' However, this subsection deals with the approximation �θN of �θN which is computable by solving Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' First of all, let us explain why �θN is defined this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, since DsXi t = σ1[0,t](s) exp � θ0 � t s b′(Xi u)du � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ∀s, t ∈ [0, T], ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 11 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 and by the chain rule for the Malliavin derivative (see Nualart [20], Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='3), � T 0 b(Xi s)δXi s = � T 0 b(Xi s)dXi s − αHσ � T 0 � T 0 b′(Xi t)(DsXi t)|t − s|2H−2dsdt = � T 0 b(Xi s)dXi s − αHσ2 � T 0 � t 0 b′(Xi t) exp � θ0 � t s b′(Xi u)du � |t − s|2H−2dsdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, since �θN is a converging estimator of θ0 as established in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2, �θN − IN = ΦN(θ0 − IN) ≈ ΦN(�θN − IN), where ΦN(r) := − αHσ2 NTDN N � i=1 � T 0 � t 0 b′(Xi t) exp � (r + IN) � t s b′(Xi u)du � |t − s|2H−2dsdt and, by the change of variable formula for Young’s integral, IN := 1 NTDN N � i=1 � T 0 b(Xi s)dXi s = 1 NTDN N � i=1 (b(Xi T ) − b(Xi 0)) with b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') = b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' This legitimates to consider the estimator �θN := IN + RN of θ0, where RN is the fixed point of the map ΦN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let us establish that RN exists and is unique under the condition (10) stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assume that b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If (10) T 2H MN DN ⩽ c αHσ2∥b′∥2∞ , where c is a deterministic constant arbitrarily chosen in (0, 1), MN := e∥b′∥∞|IN|T and αH := αH 2H(2H + 1), then ΦN is a contraction from R+ into R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, RN exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩽ 0, ΦN is nonnegative, and in particular ΦN(R+) ⊂ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, by (10), for every r, ρ ∈ R+, |ΦN(r) − ΦN(ρ)| ⩽ αHσ2 NTDN N � i=1 � T 0 � t 0 |t − s|2H−2|b′(Xi t)| exp � IN � t s b′(Xi u)du � × ����exp � r � t s b′(Xi u)du � − exp � ρ � t s b′(Xi u)du ����� dsdt ⩽ αHσ2 NTDN ∥b′∥∞MN N � i=1 � T 0 � t 0 |t − s|2H−2 sup x∈R− ex ����(r − ρ) � t s b′(Xi u)du ���� dsdt ⩽ αHσ2∥b′∥2 ∞T 2H MN DN |r − ρ| ⩽ c|r − ρ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' So, ΦN is a contraction from R+ into R+, and then RN exists and is unique by Picard’s fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' □ 12 NICOLAS MARIE† The following proposition provides a convergence result on the truncated estimator �θc N := �θN1ΩN with ΩN = � T 2H MN DN ⩽ c αHσ2∥b′∥2∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assume that b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩽ 0 and that θ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2, if (11) T 2H ∥b∥2 f exp � ∥b′∥∞ ∥b∥2 f |E(b(XT ) − b(X0))| � ⩽ c αHσ2∥b′∥2∞ , then �θc N → θ0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' First, �θN = IN + ΦN(θ0 − IN) and, on the event ΩN, RN = �θN − IN is the unique fixed point of the c-contraction ΦN by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, |�θN − �θN|1ΩN = |ΦN(RN) − ΦN(θ0 − IN)|1ΩN ⩽ c|RN − (θ0 − IN)|1ΩN ⩽ c|�θN − �θN|1ΩN + c|�θN − θ0|1ΩN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since c ∈ (0, 1), |�θN − �θN|1ΩN ⩽ c 1 − c|�θN − θ0|1ΩN , and thus (12) |�θc N − θ0| = |�θN − θ0|1ΩN + |θ0|1Ωc N ⩽ 1 1 − c|�θN − θ0| + |θ0|1Ωc N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Now, by the (usual) law of large numbers, DN a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ 1 T � T 0 E(b(Xs)2)ds = ∥b∥2 f and 1 N N � i=1 (b(Xi T ) − b(Xi 0)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ E(b(XT ) − b(X0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Therefore, MN DN = 1 DN exp � ∥b′∥∞ NDN ����� N � i=1 (b(Xi T ) − b(Xi 0)) ����� � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ 1 ∥b∥2 f exp � ∥b′∥∞ ∥b∥2 f |E(b(XT ) − b(X0))| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' This leads to 1Ωc N → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In conclusion, by Inequality (12) together with the convergence result (9), |�θc N − θ0| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The condition (11) in the statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10 is, in fact, a condition on the time horizon T which can be chosen arbitrarily small in our estimation framework, even when X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN have been observed on [0, Tmax] with 0 < T ⩽ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In the two following situations, (11) can be simplified: (1) Assume that θ0 > 0 and that b satisfies the dissipativity condition (4) (in particular, b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩽ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let f0 be the density with respect to Lebesgue’s measure of the common probability distribution of ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 13 the Xt’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, assume that b and f = f0 fulfill Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, E(b(XT )) = E(b(X0)) and, as a consequence, the condition (11) is equivalent to T ⩽ � c αHσ2∥b′∥2∞ �� ∞ −∞ b(x)2f0(x)dx �� 1 2H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In practice, as mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2), f0 is unknown but one may be able to provide a function f0 such that f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ f0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ 0, and then T fulfills (11) when T ⩽ � c αHσ2∥b′∥2∞ �� ∞ −∞ b(x)2f0(x)dx �� 1 2H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For instance, assume that X is the fractional Ornstein-Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If there exist (known) θmin, θmax > 0 such that θmin ⩽ θ0 ⩽ θmax, then T fulfills (11) when T ⩽ � c αHσ2 � θH min � 2πσ2HΓ(2H) � ∞ −∞ x2 exp � −θ2H max x2 2σ2HΓ(2H) � dx �� 1 2H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) Assume that b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩽ 0 and that θ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assume also that b ∈ C∞ b (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since b is bounded, b is Lipschitz continuous, and then |E(b(XT ) − b(X0))| ⩽ ∥b∥∞E(|XT − X0|) ⩽ ∥b∥∞ � θ0 � T 0 E(|b(Xs)|)ds + σE(|BT |) � ⩽ ∥b∥∞(θ0∥b∥∞T + σT H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In practice, θ0 is unknown but one may be able to provide θmax > 0 such that θ0 ⩽ θmax, leading to |E(b(XT ) − b(X0))| ⩽ ∥b∥∞T H max(θmax∥b∥∞T 1−H max + σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If in addition b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0, then ∥b∥2 f ⩾ b and T fulfills (11) when T ⩽ � cb αHσ2∥b′∥2∞ exp � −∥b′∥∞ b ∥b∥∞T H max(θmax∥b∥∞T 1−H max + σ) �� 1 2H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Now, let us establish a risk bound on �θc N when b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assume that b′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩽ 0 and that θ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Assume also that b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='2, if (13) T 2H b exp �∥b′∥∞ b |E(b(XT ) − b(X0))| � < c αHσ2∥b′∥2∞ , then there exists a constant c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12 > 0, not depending on N, such that E[(�θc N − θ0)2] ⩽ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' First, as established in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10, |�θc N − θ0| ⩽ 1 1 − c|�θN − θ0| + |θ0|1Ωc N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, since b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b, �θN = �θd N with d = b 2 14 NICOLAS MARIE† by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7, E[(�θc N − θ0)2] ⩽ 2(1 − c)−2 c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 N + 2θ2 0P(Ωc N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Now, P(Ωc N) = P � MN > c αHT 2Hσ2∥b′∥2∞ DN � ⩽ P � MN > c αHT 2Hσ2∥b′∥2∞ d � ⩽ P � 1 N ����� N � i=1 (b(Xi T ) − b(Xi 0)) ����� > log � c αHT 2Hσ2∥b′∥2∞ d � d ∥b′∥∞ � ⩽ P � 1 N ����� N � i=1 [b(Xi T ) − b(Xi 0) − E(b(Xi T ) − b(Xi 0))] ����� > u � with u = log � c αHT 2Hσ2∥b′∥2∞ d � d ∥b′∥∞ − |E(b(XT ) − b(X0))| > 0 by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' So, by the Bienaymé-Tchebychev inequality, and since X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' processes, P(Ωc N) ⩽ 1 u2N 2 var � N � i=1 (b(Xi T ) − b(Xi 0)) � ⩽ 1 u2N E[(b(XT ) − b(X0))2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Therefore, E[(�θc N − θ0)2] ⩽ � 2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 (1 − c)2 + 1 u2 E[(b(XT ) − b(X0))2] � 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' □ Finally, let us consider the estimator �θc N,n := �θN,n1ΩN , where �θN,n := IN + RN,n and the sequence (RN,n)n∈N is defined by � RN,0 = 0 RN,n+1 = ΦN(RN,n) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Let ψ : N → N be a map satisfying ψ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') ⩾ −log(m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='13√.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') log(c) with m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='13 = c(1 − c)−1 2TαH∥b′∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If T satisfies (11), then �θc N,ψ(N) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0, and if T satisfies (13), then there exists a constant c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='13 > 0, not depending on N, such that E[(�θc N,ψ(N) − θ0)2] ⩽ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='13 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' On the event ΩN, note that |ΦN(0)| ⩽ αHσ2 NTDN N � i=1 � T 0 � t 0 |b′(Xi t)| exp � IN � t s b′(Xi u)du � |t − s|2H−2dsdt ⩽ αHσ2 TDN ∥b′∥∞MN � T 0 � t 0 |t − s|2H−2dsdt ⩽ σ2∥b′∥∞ 2T T 2H MN DN ⩽ c1 with c1 = c 2TαH∥b′∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 15 Consider n ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Thanks to a well-known consequence of Picard’s fixed point theorem, for every x ∈ R+, |(ΦN ◦ · · · ◦ ΦN � �� � n times )(x) − RN| ⩽ cn 1 − c|ΦN(x) − x| on ΩN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, |RN,n − RN|1ΩN = |(ΦN ◦ · · · ◦ ΦN)(RN,0) − RN|1ΩN ⩽ cn 1 − c|ΦN(0)|1ΩN ⩽ c2cn with c2 = c1 1 − c, leading to |�θc N,n − θ0| ⩽ |RN,n − RN|1ΩN + |�θc N − θ0| ⩽ c2cn + |�θc N − θ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If T satisfies (11), then |�θc N,ψ(N) − θ0| ⩽ 1 √ N + |�θc N − θ0| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' −−−−→ N→∞ 0 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' If b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' )2 ⩾ b with b > 0, and if T satisfies (13), then E[(�θc N,ψ(N) − θ0)2] ⩽ 2(1 + c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12) N by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Numerical experiments In this section, our computable estimator of θ0 is evaluated on the two following models: (1) dXt = θ0b1(Xt)dt + dBt with θ0 = 1 and b1(x) := −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since b′ 1(x) = −1 < 0, the estimator �θc N is well-defined thanks to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10 applies to �θc N for T small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' (2) dXt = θ0b2(Xt)dt+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='25dBt with θ0 = 1 and b2(x) := π+arctan(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since b′ 2(x) = −(1+x2)−1 ∈ [−1, 0], the estimator �θc N is well-defined thanks to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Since b2(x)2 ⩾ π2/4 > 0, both Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12 apply to �θc N for T small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For each model, with H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9, �θN,30 is computed from N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , 100 paths of the process X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' This experiment is repeated 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The means and the standard deviations of the error |�θ100,30 − θ0| are stored in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Moreover, the map N �→ �θN,30 is plotted for 5 datasets generated Mean error Error StD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Model 1 (H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0088416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0081571 Model 1 (H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0147383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0108511 Model 2 (H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0169235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0071539 Model 2 (H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0164968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='0085905 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Means and StD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' of the error of �θ100,30 (100 repetitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' from Model 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Model 2) in Figure 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 16 NICOLAS MARIE† Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Plots of N �→ �θN,30 for Model 1 with H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 (left) and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Plots of N �→ �θN,30 for Model 2 with H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 (left) and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For both H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9, the mean error of �θ100,30 is lower for Model 1 than for Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For Model 1, the mean error is significantly lower when H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 than when H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For Model 2, the mean error is of same order for both H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7 and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' On Figures 1 and 2, for both Model 1 and Model 2, the estimator seems to converge faster to θ0 when H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='9 than when H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In each situation, Figures 1 and 2 show that N �→ �θN,30 starts to stabilize from N ≈ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Conclusion and perspectives The main contributions of our paper are: (1) To provide a risk bound on the truncated least squares estimator �θd N of the parameter θ0 when H = 1/2 and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are interacting copies of the solution X of Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Precisely, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 says that the rate of convergence of �θd N remains of order 1/N (parametric), as when X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , XN are independent, while |RN| ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ON A COMPUTABLE ESTIMATOR OF THE DRIFT PARAMETER IN FRACTIONAL SDE 17 (2) To provide a computable approximation �θN of �θN when H > 1/2 with some convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Such approximation of �θN is interesting in practice because it is well-known that the Skorokhod integral involved in the definition of �θN is not directly computable when H > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Precisely, an almost sure convergence result is established in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='10 and a risk bound is established in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' In fact, and this will be investigated later, our computable approximation method for �θN could be extended to the (nonparametric) projection least squares estimator studied in Comte & Marie [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' For a regular enough orthonormal family (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , ϕm) of L2(R, dx) with m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' , N}, [5] deals with the estimator �bm(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') := m � j=1 �θjϕj(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') with �θ = �Ψ−1 m �Zm of the whole function b0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') = θ0b(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') in Equation (1), where �Ψm := � 1 NT N � i=1 � T 0 ϕj(Xi s)ϕℓ(Xi s)ds � j,ℓ and �Zm := � 1 NT N � i=1 � T 0 ϕj(Xi s)δXi s � j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='4 and by the chain rule for the Malliavin derivative, �Zm = γN(b0) with γN(ψ) := � 1 NT N � i=1 �� T 0 ϕj(Xi s)dXi s − αHσ2 � T 0 � t 0 ϕ′ j(Xi t) exp �� t s ψ′(Xi u)du � |t − s|2H−2dsdt �� j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Then, �bm = ΓN(b0) ≈ ΓN(�bm) with ΓN(ψ)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') := m � j=1 [�Ψ−1 m γN(ψ)]jϕj(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' This legitimates to consider the estimator �bm(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') := �m j=1 �θjϕj(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=') of b0 such that �bm = ΓN(�bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Baudoin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nualart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Ouyang & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Tindel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' On Probability Laws of Solutions to Differential Systems Driven by a Fractional Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Annals of Probability 44, 4, 2554-2590, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Bertin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Klutchnikoff, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Panloup & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Varvenne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Adaptive Estimation of the Stationary Density of a Stochastic Differential Equation Driven by a Fractional Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistical Inference for Stochastic Processes 23, 2, 271-300, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Comte & V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Genon-Catalot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nonparametric Drift Estimation for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Paths of Stochastic Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Annals of Statistics 48, 6, 3336-3365, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Comte & N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Marie Nonparametric Estimation in Fractional SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistical Inference for Stochastic Processes 22, 3, 359-382, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Comte & N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Marie Nonparametric Estimation for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Paths of Fractional SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistical Inference for Stochastic Processes 24, 3, 669-705, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Comte & N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Marie Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Preprint, arXiv: 2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='13173, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Della Maestra & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Hoffmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nonparametric Estimation for Interacting Particle Systems: McKean-Vlasov Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Probability Theory and Related Fields 182, 551-613, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Denis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Dion & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Martinez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' A Ridge Estimator of the Drift from Discrete Repeated Observations of the Solutions of a Stochastic Differential Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Bernoulli 27, 2675-2713, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Hairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Annals of Probability 33, 2, 703-758, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' 18 NICOLAS MARIE† [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Hu & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nualart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistics and Probability Letters 80, 1030-1038, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Hu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nualart & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Drift Parameter Estimation for Nonlinear Stochastic Differential Equations Driven by Fractional Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Stochastic 91, 8, 1067-1091, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Khasminskii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Stochastic Stability of Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Kleptsyna & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Le Breton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Some Explicit Statistical Results About Elementary Fractional Type Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nonlinear Analysis 47, 4783-4794, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Kutoyants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistical Inference for Ergodic Diffusion Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Springer, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Marie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nonparametric Estimation for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Paths of a Martingale Driven Model with Application to Non- Autonomous Financial Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Finance and Stochastics 27, 1, 97-126, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [16] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Marie & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Raynaud de Fitte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Stochastics 93, 6, 886-906, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Marie & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Rosier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nadaraya-Watson Estimator for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Paths of Diffusion Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Scandinavian Journal of Statistics (accepted), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Menozzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Pesce & X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Density and Gradient Estimates for Non Degenerate Brownian SDEs with Unbounded Measurable Drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Journal of Differential Equations 272, 339-369, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Neuenkirch & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Tindel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' A Least Square-Type Procedure for Parameter Estimation in Stochastic Differential Equa- tions with Additive Fractional Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistical Inference for Stochastic Processes 17, 1, 99-120, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nualart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Malliavin Calculus and Related Topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Saussereau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Nonparametric Inference for Fractional Diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Bernoulli 20, 2, 878-918, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Tudor & F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Viens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' Statistical Aspects of the Fractional Stochastic Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' The Annals of Statistics 35, 3, 1183-1212, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content=' †Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France Email address: nmarie@parisnanterre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQf7g74/content/2301.05341v1.pdf'} diff --git a/KNFRT4oBgHgl3EQfDTe1/content/tmp_files/2301.13472v1.pdf.txt b/KNFRT4oBgHgl3EQfDTe1/content/tmp_files/2301.13472v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3aaa9db50bfc1fd9b249b9cfc9519375ed73a41e --- /dev/null +++ b/KNFRT4oBgHgl3EQfDTe1/content/tmp_files/2301.13472v1.pdf.txt @@ -0,0 +1,892 @@ +The Aharonov Casher phase of a bipartite entanglement pair traversing a quantum +square ring +Che-Chun Huang +Department of Physics, National Taiwan University, Taiwan +Seng Ghee Tan∗ +Department of Optoelectric Physics, Chinese Culture University, +55 Hwa-Kang Road, Yang-Ming-Shan, Taipei 11114, Taiwan +Ching-Ray Chang +Quantum Information Center, Chung Yuan Christian University, Taiwan and +Department of Physics, National Taiwan University, Taiwan +We propose in this article a quantum square ring that conveniently generates, annihilates and +distills the Aharonov Casher phase with the aid of entanglement. The non-Abelian phase is carried +by a pair of spin-entangled particles traversing the square ring. At maximal entanglement, dynamic +phases are eliminated from the ring and geometric phases are generated in discrete values. +By +contrast, at partial to no entanglement, both geometric and dynamic phases take on discrete or +locally continuous values depending only on the wavelength and the ring size. We have shown that +entanglement in a non-Abelian system could greatly simplify future experimental efforts revolving +around the studies of geometric phases. +INTRODUCTION +The quantum ring is a useful apparatus to study the +physics of electron phase accumulating and interfering +over the confined trajectories as prescribed by the de- +sign of the ring. Following the successful measurement +of the Aharonov Bohm [1] phase, hot on the heels were a +slew of experiments that had demonstrated the Aharonov +Casher [2] and the Berry-Pancharatnam [3, 4] phases.In +modern context, the Aharonov Casher phase associates +primarily with the spin orbit coupling, particularly in 2D +condensed matter systems. Ring structure carved out of +a 2D spin-orbital semiconductor to enclose a magnetic +field at the center [5–7] was proposed to study the simul- +tainety of the Aharonov Casher and the Aharonov Bohm +effects on interference. Efforts have also been made to +study the parametric effects [8–10] of e.g. the Rashba +constant, and the time-dependent magnetic field. +At +around the same time, the Aharonov Casher effect was +experimentally measured in a number of ring structures +[11, 12]. On a separate study, the Aharonov Casher phase +is also associated with the non-Abelian gauge field for +its spin phase [13–15] and spin force effects [16–19], cat- +egorically reviewed in Ref [20]. +While the spin phase +which comprises the geometric and the dynamic parts +has largely been determined in ring structures, the ex- +act nature of the accumulated phases in these devices +remain ambiguous. The dynamic phase remains an elu- +sive component in most cases, and the process to extract +the geometric phase continues to be complicated. +For +example, in Ref [12], the system is a Rashba 2D that +comprises a hedgehog orientation of the effective mag- +netic fields turned crown-like by a vertical magnetic field. +While the strength of the BP phase is proportional to the +solid angle subtended in the rest frame of the electron, +a dynamic phase proportional to sin θ is also formed in +concomitance. +By applying an in-plane B field, which +modifies the geometric Berry-Pancharatnam, and keeps +the dynamic unaffected to the first order, a distinction +can be made about the two phases. Therefore, isolating +the geometric phase is a complicated effort, the Aharonov +Casher remains largely a total phase for most applica- +tions. +In this article, we propose a quantum square ring +(QSR) that conveniently generates, annihilates or distills +the Aharonov Casher phase with the aid of entanglement +as shown in FIG. 1. The Aharonov Casher phase gener- +ated in this manner comprises the dynamic and the geo- +metric components that can be further separated by tun- +ing the entanglement strength and the device size mea- +sured by the wavelength multiple of a traversing parti- +cle pair. +For example, at maximal entanglement, dy- +namic phases are eliminated from the device and geo- +metric phases are generated in discrete values. Discrete +geometric phases would in turn switch their values on +different ring locations depending on the device size. At +partial to no entanglement, the Aharonov Casher as well +as its dynamic and geometric components can be tuned +according to the quantum ring size to take on discrete val- +ues or vary continuously across the device. The device is +made out of semiconductor or metallic materials that ex- +hibit 2D spin-orbit effects, e.g., the Rashba-Vasko, Dres- +selhaus, or Dresselhaus-Perel effects [21–27]. The spin- +orbit effects will be the source of both the geometric and +the dynamic phases in our system. As the external mag- +netic field is not needed to generate the geometric phase, +nor is it needed to help to eliminate the dynamic phase, +a leaner QSR concept that rules out the Aharonov Bohm +arXiv:2301.13472v1 [quant-ph] 31 Jan 2023 + +2 +FIG. 1: A square quantum ring device that takes a +bipartite spin-pair at the emitter and generates an AC +total phase as well as its separable components of +geometric and dynamics phases. +and the Altshuler–Aronov–Spivak (AAS) effect, and co- +opts only the electrically-controllable Aharonov Casher +is employed in our design. In the absence of strong B or +M field, the adiabatic Berry-Pancharatnam phases in the +QSR [28–31] is also ruled out. Novel to the functioning +of our device though is the entanglement physics [32, 33]. +On the bottom left of the device is an emitter electrode +(FIG. 1) through which an entangled bipartite spin-pair +is injected into the QSR. The top right is the collector +electrode where the injected spin-pair meets again and +carries with it a total phase moderated by the physics of +entanglement and device geometry. Our QSR device is +therefore, by essence a non-adiabatic and a non-Abelian +Aharonov Casher system [34]. The spin-pair’s total phase +is accumulated via spin precession about the spin-orbit +field but under the constant purview of bi-partite entan- +glement, which provides in this paper a viable method +to generate geometric and dynamic phases in a control- +lable manner. As an aside, we note that quantum ring +device has previously been studied for the practical pur- +pose of producing spin entanglement in a controllable +manner. [35, 36] There is, however, no discussion on its +applicability in the context of geometric phases, let alone +any specific discussion on its moderation of the dynamic +phases or its distillation of the geometric phases. +When a pure quantum state |ψ(t)⟩ evolves on the +Hilbert space trajectory in time range Γ +: +t +∈ +[0, τ], +the total phase it accumulates is given by +arg(⟨ψ(0)|ψ(t)⟩). +The dynamic phase can be derived +from D = −i +� τ +0 dt⟨ψ(t)| ˙ +ψ(t)⟩. One can then define the +geometric phase as the result of a total phase minus the +dynamic phase as follows +γ = arg(⟨ψ(0)|ψ(τ)⟩) + i +� τ +0 +dt⟨ψ(t)| ˙ +ψ(t)⟩ +(1) +Consider a QSR ring of size η×η as shown in FIG. 1. The +device comprises 2 paths, each consisting of a horizontal +and a vertical arm. From the material point of view, the +device exhibits the Rashba spin-orbit effect as follows +H = σxky − σykx +(2) +The QSR geometry conspires with the Rashba effect to +generate phase factors for any particle traveling along +path 1 and path 2 as follows +UI = e +iδσx +2 e +−iησy +2 +, UII = e +−iδσy +2 +e +iησx +2 +(3) +Note that δ = ωt for both paths as a result of ωI = +ωII = ω. Therefore, a spin particle traversing the hori- +zontal arm of Path 1 and the vertical arm of Path 2 would +separately accumulate a total phase as denoted by the di- +mensionless η. The phase accumulated over time can be +translated to a phase at an actual location in space de- +pending on the particle velocity (ν) in the actual system +as denoted by ω = kν, where k is the wave-vector. While +η is hence a phase parameter for the first half of either +Path 1 or 2, δ ∈ [0, η] would represent the phase at any +point on the second half of either path. For ease of illus- +trations, we will refer to the η parts of Paths 1 and 2 as, +respectively, η1 and η2. Likewise, the same is prescribed +for δ1 and δ2. The spin-orbit effect when viewed in the +rest frame of the carrier is a form of effective magnetic +field which sets up a perfect environment for spin preces- +sion. As the entangled spin-pair traverses both paths, its +phase evolves as prescribed by the unitary operation of +U ≡ UI +� +UII +(4) +The geometric phase would thus be +γ = arg(⟨ψ(0)|U|ψ(0)⟩) + i +� τ +0 +dt⟨ψ(0)|U† ˙U|ψ(0)⟩ +(5) +Explicitly, the dynamic phase is given by +iU† ˙U = (I +� +(e− iσx +2 η(σy +2 )e +iσx +2 η) +− (e +iσy +2 η(−σx +2 )e− +iσy +2 η) +� +I) +=1 +2 +� +� +� +� +0 +−i cos η − cos η +0 +i cos η +−2 sin η +0 +− cos η +− cos η +0 +2 sin η +−i cos η +0 +− cos η +i cos η +0 +� +� +� +� +(6) +The dynamic phase is expressed in terms of the Pauli +matrices so that the relatable picture of effective mag- +netic fields is not lost. +For calculation though, use is +often made of its 4 by 4 matrix representation. +BIPARTITE ENTANGLED STATES +The initial states of the entangled-spin-pair at the +emitter is then prepared in the Bell-basis of |φ(0)⟩ or + +path 2 +collector +Entangledparticle +source +emitter +path 13 +|ψ(0)⟩ as follows: +|φ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ +|ψ(0)⟩ = ±√p0 |10⟩ + √p1 |01⟩ +� +(7) +where p0 and p1 determine the strength of the entangle- +ment, and p0, p1 ≥ 0, p0 + p1 = 1. +We will now consider the initial states of |φ(0)⟩ = +√p0 |00⟩ ± √p1 |11⟩ to be injected into the QSR through +the emitter. As shown in FIG. 1, spin particles 1 and +2 take to paths of their respective namesakes. The geo- +metric phase is the total Aharonov-Casher phase of the +system minus the dynamic phase as shown below +γ = arg(cos2(δ +2) cos2(η +2) + sin2(δ +2) sin2(η +2)) − D +(8) +The initial states |φ(0)⟩ simply could not generate any +dynamic phase anywhere on the QSR, i.e. D = 0. And, +the argument of the total Aharonov Casher phase factor +consists of parameters that are all real and non-negative. +The geometric phase by virtue of γ ≡ arg(a + ib) − D +vanishes as given by +γ = tan−1 +0 +(a > 0) − D → γ = 0 +(9) +It is clear that the strength of entanglement has no bear- +ing on the geometric and the dynamic phases as p0, p1 +could take on values of the un-entangled states. The re- +sults of zero phases might not seem as trivial though. +It is a testament to the non-Abelian feature of the QSR +device. In fact, the above shows that a bipartite state +composed out of |00⟩ and |11⟩ is ideal for eliminating +both geometric and dynamic phases from the propagat- +ing particles. In terms of applications, this could be a +handy device to remove phases where they are not de- +sired from all the particles. In the following, we provide +an insight of how dynamic phases are removed from the +bipartite state. Let’s examine the dynamic phase by in- +specting the constituent particles of the entangled pair. +Spin 1 travels on arm-η1 as though it is in a superposi- +tion state of |0⟩ and |1⟩ as far as the dynamic phase is +concerned. In either state, its expectation energy is zero +as can be deduced by the circular fashion of its spin rota- +tion about the effective magnetic field of −By. Consider +spin to precess about the effective magnetic field in an +anti-clockwise manner, and that spin 1 to have rotated an +angle θ < π by the end of its journey on arm-η1. Spin 1 +would thus continue on arm-δ1 about +Bx, now inscrib- +ing a conical spin rotation with negative energy for p0, +and positive energy for p1. Likewise for spin 2, a corre- +sponding process happens over arm-η2 about +Bx with +zero energy for both components p0, p1. Spin 2 would +continue its journey on arm-δ2 about −By, inscribing a +conical rotation with positive energy for p0, and negative +energy for p1. +The cone energies on arm-δ cancel one +another identically independent of the strength of p0, p1. +FIG. 2: Schematic illustration of the dynamic phases of +the bipartite spin pair |ψ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ +traversing the QSR. PE, NE, ZE, stand for positive +energy, negative energy, zero energy, respectively. +The effect is thus a complete negation and a net zero of +dynamic phase at all times. +Note that the energy cones are drawn in different sizes +to reflect the energy it carries. This stands against the +reality that spin vector is constant in length. Therefore, +the energy cones are crude illustrations meant only to +provide an intuitive description of the dynamic phases. +The precise description of the cone energies is given by +the expressions below. Equations (10) and (11) describe +the expectation energy for spin particle 1 on arm-δ1. +p0⟨00|(e +iσy +2 η(−σx +2 )e− +iσy +2 η) +� +I)|00⟩ = −p0 +sin η +2 +(10) +p1⟨11|(e +iσy +2 η(−σx +2 )e− +iσy +2 η) +� +I)|11⟩ = p1 +sin η +2 +(11) +Equations (12) and (13) describe the expectation energy +for spin particle 2 travelling on arm-δ2. +p0⟨00|(I +� +(e− iσx +2 η(σy +2 )e +iσx +2 η)|00⟩ = p0 +sin η +2 +(12) +p1⟨11|(I +� +(e− iσx +2 η(σy +2 )e +iσx +2 η)|11⟩ = −p1 +sin η +2 +(13) +The equations above lend clarity and mathematical +credence to our qualitative accounts that the cone ener- +gies on arms-δ cancel one another identically independent +of the strength of p0, p1, resulting in a net zero dynamic +phase at all times. +We will now consider the initial states of |ψ(0)⟩ = +±√p0 |10⟩+√p1 |01⟩ to be injected into the QSR through +the emitter, once again with the spin particles taking to +paths of their respective namesakes. The geometric phase +is given by +γ = arg(cos η + cos δ +2 +∓ √p0p1(sin δ sin η +2 +) ++i(p0 − p1)(sin δ sin η +2 +)) ++2(sin η)(p0 − p1)δ +(14) + +NE P1 +8. +ZE Po +S arms +PE Po +ZE P1 ++Bx +PE P1 +By ++Bx +NE Po +n2 +n arms +y +ZE +Pi +n1 +ZE +x +Po4 +FIG. 3: Schematic illustration of the dynamic phases of +the bipartite spin pair |ψ(0)⟩ = ±√p0 |10⟩ + √p1 |01⟩ +traversing the QSR. PE, NE, ZE, stand for positive +energy, negative energy, zero energy, respectively. +where the dynamic phase is now −2(sin η)(p0 − p1)δ and +the total phase is deduced accordingly. It is clear from +the above that the physics of entanglement has entered +the geometric phase. We will study the dynamic phase +first. At maximum entanglement where p0 = p1 = 1 +2, the +dynamic phase vanishes with the equality of p0 and p1, +leading to a total Aharonov Casher phase that is purely +geometric. Intuitively, at maximal entanglement, the ex- +pected energy of the spin-pair is constantly zero. +On +arm-η1, inital spin |1⟩ or |0⟩ would precess about an ef- +fective −By field in a circular fashion, both with a zero +expectation energy. Likewise on arm-η2, due to entan- +glement, the corresponding spin of |0⟩ or |1⟩ would pre- +cess about an effective +Bx field in a circular fashion, +and once again both with a zero energy. In short, circu- +lar rotation translates to zero expectation of the Zeeman +energy on both arms. Therefore, regardless of entangle- +ment strength, dynamic phase is zero anywhere on arms- +η. The δ sections of the QSR would, however, generate +a dynamic phase at any strength of entanglement other +than the maximum, i.e. when p0 ̸= p1. This is because +the initial states on arms-δ are determined by the dura- +tion of precession on arms-η. On path 1, let spin rotates +an angle θ < π about −By by the end of arm-η1. Spin 1 +would continue on arm-δ1 about +Bx, inscribing a coni- +cal spin rotation with positive energy for component p0, +and negative energy for component p1. Likewise for path +2, a corresponding process that happens over η2 about ++Bx would continue on arm-δ2 about −By, inscribing +once again a conical rotation with positive energy for p0, +and negative energy for p1, as shown in FIG.3 It is clear +that, on arms-δ, component p1 presents a counter effect +of proportion p1 to the energy due to p0. The effect is +thus a complete negation and a net zero dynamic phase +on the equality of p0 = p1. The energy cones are drawn +in different sizes to reflect the energy it carries. +This +clearly stands against the quantum reality that spin vec- +tor is constant in length. Therefore, the energy cones are +crude illustrations meant only to provide an intuitive de- +scription of the dynamic phases. The precise description +of the cone energies is given by the expressions below. +Equations (15) and (16) describe the expectation energy +for spin particle 1 on arm-δ1. +p0⟨10|(e +iσy +2 η(−σx +2 )e− +iσy +2 η) +� +I)|10⟩ = p0 +sin η +2 +(15) +p1⟨01|(e +iσy +2 η(−σx +2 )e− +iσy +2 η) +� +I)|01⟩ = −p1 +sin η +2 +(16) +Equations (17) and (18) describe the expectation energy +for spin particle 2 on arm-δ2. +p0⟨10|(I +� +(e− iσx +2 η(σy +2 )e +iσx +2 η)|10⟩ = p0 +sin η +2 +(17) +p1⟨01|(I +� +(e− iσx +2 η(σy +2 )e +iσx +2 η)|01⟩ = −p1 +sin η +2 +(18) +Spin particles on arms δ1 and δ2 reinforces once another, +the p0 and p1 components become more positive and neg- +ative, respectively. As the equality of the entanglement +strength is crucial for suppressing the dynamic phase on +arms-δ but not on arms-η, a net dynamic phase would ac- +cumulate on arms-δ on the condition of p0 ̸= p1. There is, +however, an exception. If the length of arms-η translate +to a spin rotation of η = nπ, subsequent conical preces- +sion on arms-δ would not have happened. Spin would +simply continue with circular precession and a zero dy- +namic phase throughout. The physics above lends fur- +ther credence to the applicability of the QSR design as +a phase purifier. Tuning the entanglement strength to +p0 = p1 at the source, a maximally-entangled spin-pair +injected at the emitter would propagate with a dynamic +phase suppressed throughout. In the event of p0 ̸= p1 +though, dynamic phase could be suppressed by choosing +the length of arms η = nπ. Having completed our study +of the dynamic phase, we will now examine the geometric +phase, γ. At maximal entanglement, i.e. p0 = p1 = 1 +2, +the imaginary part of γ, denoted by b as shown in Equa- +tion (19) below vanishes. The geometric phase is either 0 +or π depending on the parameters of the real part a(δ, η) +as shown in the denominator of tan γ = +b +a(δ,η), where a +positive denominator corresponds to γ = 0 while a nega- +tive denominator corresponds to γ = π. +γ = arg(1 +2(cos δ + cos η) − 1 +4 sin δη + i0) ≡ arg(a + ib) +(19) +Let us now study in slightly more details the geometric +phase of the spin-pair traversing the δ arms. The crucial +quantity here is range 0 < δ ≤ η. +In the case of no +entanglement, (p0, p1) = (0, 1) or (1, 0) +γ = tan−1 ∓(sin δ sin η) +cos δ + cos η ∓ 2(sin η)δ +(20) + +NE P1 +62 +ZE Po +arms s +PE +Po +ZE P1 ++Bx +PE Po +By +NE P1 ++Bx +ZE P1 +ZE Po +armsn +n2 +y +n1 +x5 +FIG. 4: Schematic illustration of the effect of η on the +geometric phase γ on arms δ. The red and blue lines +represent the paths with π and 0 geometric phases, +respectively. +Dynamic phase is eliminated at arms’ length correspond- +ing to η = nπ. Note that when δ > 0, phase η corre- +sponds to the end location of arms-η. Spin would always +be oriented along the z axis by the time it reaches the end +location. Therefore, advancing on arms-δ, spin would be +precessing in a circular fashion with a net zero expecta- +tion energy, and is thus precluded from generating the +dynamic phase. But at the values of η = nπ, the to- +tal phase alternates between 0 and π on arms-δ. +For +η = 2nπ, the denominator is always positive, and the de- +vice generates a total phase of 0. For η = (2n + 1)π, the +denominator is always negative, and the total phase is π. +Since the dynamic phase is always 0, the total phase at +η = nπ is also the geometric phase. For other values of η, +the total phase takes on continuous values as a function +of η and δ. Analysis above is focused only on the phases +of arms-δ. Phases on arms-η for different η values could, +on the other hand, be found by prescribing δ = 0, details +of which would be discussed later. For illustration, we +refer to FIG.4 for η = π, 2π and observe the geometric +phases on arms-δ. +In the case of partial entanglement, i.e. p0 ̸= p1 +γ = tan−1 +(p0 − p1)(sin δ sin η) +(cos δ + cos η) ∓ √p0p1 sin η sin δ ++ 2(sin η)(p0 − p1)δ +(21) +Like in the above, the dynamic phase can be eliminated +by η = nπ. Once again at these values, the total phase is +discrete and alternates between 0(for η = 2nπ) and π(for +η = (2n + 1)π). As before, the total phase at η = nπ is +also the geometric phase. For other values of η, the total +phase takes on continuous values as a function of η and +δ. Note again that analysis here is focused only on the +phases of arms-δ. +We will now revert to the case of maximum entan- +glement again. +It was known that at maximal entan- +glement, the dynamic phase vanishes and the geometric +phase takes on discrete values of 0 and π. +We would +FIG. 5: +Quantum square rings (QSR) of different sizes +are superimposed for ease of inspection. The red and +blue lines represent the paths with π and 0 geometric +phases, respectively. +now study the exact locations on the QSR where the ge- +ometric phase switches its value. As a matter of fact, the +positions of switching from 0 to π happens on arms-δ. +The exact location can be pinpointed by checking that +the denominator of the geometric phase factor satisfies +2(cos δ + cos η) ∓ (sin δ sin η) > 0 +(22) +The equation above shows that the answer would de- +pend on the length of arms-η, i.e. +the length of the +arms before advancing into arms-δ. For illustration, we +chose arm lengths that correspond to η = +π +2 , π, 3π +2 , 2π +as shown in FIG.5. +The device generates a γ = 0 +on arms η at all times as indicated in blue. +As the +bipartite spin pair advances into arms-δ, the geomet- +ric phase would switch to π on locations as indicated +by the red segments. +At η = 2π though, no switch- +ing is possible and the geometric phase remains 0 at all +times. In the event of a zero denominator, the total phase +arg(⟨ψ(0)|U|ψ(0)⟩) = arg(a+ib) = tan−1 0 +0 is undefined. +The bipartite state at that juncture would have to either +vanish or turn out orthogonal to the initial Bell states. +Last is the particular situation of δ = 0 that cor- +responds to the point where spin-pair starts to take a +right-angle bend into arms-δ. As long as δ = 0, spin- +pair is considered to reside in the η regions of the arms +only. And a quick inspection shows that a(0, η) is posi- +tive throughout, which leads to the conclusion that the +geometric phase on arms-η is 0 throughout, in spite of +the entanglement strength. This is in fact indicated in +FIG.4 and FIG.5 where arms-η are painted blue to indi- +cate a zero geometric phase throughout. This is indeed +the case, barring the issues of singular points correspond- +ing to cos η = −1 which brings upon γ = tan−1 0 +0. At +these points, the geometric phase is undefined. In terms +of spin precession, the singular points correspond to spin +making a rotation of (2n + 1)π. +The odd-pi quantum +states of the spin-pair at this point would then be or- +thogonal to its initial Bell states. In terms of the dynamic + +=0 +(0,4元) +=π +(0,3元) += 0 +(0,2元) +=元 +(8, n)= (0,元) +(8, n) = (0, 元) +(0,2元) +(0,3元) +(0,4元)(0, 2元) +~116° +3元 +0 +2. +(0, 元) +~116° +v64 +(8, n) = (o,2) +~64° +(8, n) = (o,%) +(0, 元) +0,2 +3元 +(0,2元)6 +Non-Abelian: Non-adiabatic +|φ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ +Dynamic phase +Geometric phase +δ = 0 +δ > 0 +δ = 0 +δ > 0 +(p0, p1 = (0, 1)) +(p0, p1 = (1, 0)) +NO ENTG. +0 +0 +0 +0 +p0 = p1 = 1 +2 +MAX. ENTG. +0 +0 +0 +0 +p0 ̸= p1 ̸= 0 +PARTIAL ENTG. +0 +0 +0 +0 +|ψ(0)⟩ = ±√p0 |10⟩ + √p1 |01⟩ +Dynamic phase +2(sin η)(p0 − p1)δ +Geometric phase +δ = 0 +δ > 0 +δ = 0 +δ > 0 +(p0, p1 = (0, 1)) +(p0, p1 = (1, 0)) +NO ENTG. +0 +0(η = 2nπ) +0(η = (2n + 1)π) +Continuous values as +D = −(∓2(sin η)δ) +0 +Discrete 0(η = 2nπ) +Discrete π(η = (2n + 1)π) +Continuous values as +γ = tan−1 ∓(sin δ sin η) +cos δ+cos η +∓2(sin η)δ +p0 = p1 = 1 +2 +MAX. ENTG. +0 +0 +0 +Discrete 0 or π +p0 ̸= p1 ̸= 0 +PARTIAL ENTG. +0 +0(η = 2nπ) +0(η = (2n + 1)π) +Continuous values as +D = −(2(sin η)δ) +0 +Discrete 0(η = 2nπ) +Discrete π(η = (2n + 1)π) +Continuous values +see Equation (21) +Table. I: Analysis of geometric phases for the SQR is +tabulated according to the entanglement strength and +the locations on arms-δ. +phase, δ = 0 suppresses dynamic phases in spite of the +entanglement strength. Table. I provides a summary of +all the analysis that have been carried out for the geo- +metric and dynamic phases corresponding to all the Bell +states spin-pair traversing a non-Abelian QSR device. +CONCLUSION +We have explained in details how a non-Abelian sys- +tem in the form of a QSR could be designed to generate +and purify the Aharonov Casher phases into its geomet- +ric and dynamic components without elaborate experi- +mental set ups. The device requires only an entangled- +particle source to couple to a passive square ring. The +Aharonov Casher phase is generated or annihilated as de- +termined by the choice of the entanglement configuration. +In the correct Bell states, the dynamic phase is eliminated +outright at maximal entanglement. In the case of partial +to no entanglement, dynamic phases are eliminated at +η = nπ. +In all manners of elimination, the Aharonov +Casher phase becomes discrete and fully geometric. This +device could thus be useful for future experimental efforts +to study the physics of discrete geometric phases. The +continuous spectrum of the Aharonov Casher phase re- +mains accessible though at partial to no entanglement, in +which case, the continuous phases are non-geometric. At +maximum entanglement, there is no possibility to access +any continuous form of the geometric phase. In terms of +discrete phases, the manner in which the phase switch +from one discrete value to another varies according to +the entanglement strength. +At partial to no entangle- +ment, switching occurs only at (δ, η) = (0, (2n + 1)π). +By contrast, at maximal entanglement, switching could +take place anywhere on arms-δ, i.e. any value of (δ, η). +In summary, the device has been shown to generate +continuous Aharonov Casher phases, annihilate dynamic +phases, distill discrete geometric phases, and enable dis- +crete phase switching at various locations, all within the +simple construct of a square ring. +ACKNOWLEDGEMENT +We would like to thank the Ministry of Science and +Technology of Taiwan for supporting this work under +Grant. No.: 110-2112-M-034-001-MY3. +∗ Corresponding +author: +csy16@ulive.pccu.edu.tw +; +tansengghee@gmail.com +[1] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). +[2] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 +(1984). +[3] S. Pancharatnam, Proc. Indian Acad. Sci. 44, 247–262 +(1956). +[4] M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984). +[5] J. Nitta, F. E. Meijer, and H. Takayanagi, Applied +Physics Letters 75, 695 (1999). +[6] B. Moln´ar, F. M. Peeters, and P. Vasilopoulos, Phys. Rev. +B 69, 155335 (2004). +[7] D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310 +(2004). +[8] M. B. A. Jalil and S. G. Tan, Journal of Applied Physics +105, 07C709 (2009). +[9] M.-J. Xing, M. B. A. Jalil, S. G. Tan, and Y. Jiang, IEEE +Transactions on Magnetics 46, 1471 (2010). +[10] J. Chen, M. B. Abdul Jalil, and S. Ghee Tan, Journal of +Applied Physics 113, 17C506 (2013). +[11] J. Nitta, T. Bergsten, Y. Kunihashi, and M. Kohda, Jour- +nal of Applied Physics 105, 122402 (2009). +[12] F. Nagasawa, D. Frustaglia, H. Saarikoski, K. Richter, +and J. Nitta, Nature Communications 4, 2526 (2013). +[13] N. Hatano, R. b. o. Shirasaki, and H. Nakamura, Phys. +Rev. A 75, 032107 (2007). +[14] J. Chen, M. B. A. Jalil, and S. G. Tan, Journal of Applied +Physics 109, 07C722 (2011). +[15] S.-H. Chen and C.-R. Chang, Phys. Rev. B 77, 045324 +(2008). +[16] C. S. Ho, M. B. A. Jalil, and S. G. Tan, Europhysics +Letters 108, 27012 (2014). +[17] K. Bliokh and Y. Bliokh, Annals of Physics 319, 13 +(2005). +[18] S.-Q. Shen, Phys. Rev. Lett. 95, 187203 (2005). +[19] S. G. Tan, M. B. A. Jalil, X.-J. Liu, and T. Fujita, Phys. +Rev. B 78, 245321 (2008). +[20] S. G. Tan, S. H. Chen, C. S. Ho, C. C. Huang, M. B. +Jalil, C. R. Chang, and S. Murakami, Physics Reports +882, 1 (2020). +[21] A. V. Vasko, F. T.and Korovin, Journal of Experimental +and Theoretical Physics 96, 102 (2003). +[22] Y. A. Bychkov and E. I. Rashba, JETP Letters 39, 78 +(1984). +[23] E. RASHBA, Sov. Phys.-Solid State 2, 1109 (1960). +[24] M. I. D’Yakonov and V. I. Perel’, Soviet Journal of Exper- +imental and Theoretical Physics Letters 13, 467 (1971). + +7 +[25] S. Datta and B. Das, Applied Physics Letters 56, 665 +(1990). +[26] S. G. T. M. Jalil, Introduction to the Physics of Nano- +electronics (Woodhead publishing, 2012). +[27] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. +Rev. Lett. 78, 1335 (1997). +[28] P. Bruno, V. K. Dugaev, and M. Taillefumier, Phys. Rev. +Lett. 93, 096806 (2004). +[29] T. Fujita, M. B. A. Jalil, S. G. Tan, and S. Murakami, +Journal of Applied Physics 110, 121301 (2011). +[30] S. G. Tan and M. B. A. Jalil, Modern Physics Letters A +31, 1630003 (2016). +[31] J. Anandan, Physics Letters A 133, 171 (1988). +[32] E. +Sj¨oqvist, +International +Journal +of +Quantum +Chemistry +115, +1311 +(2015), +https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.24941. +[33] E. Sj¨oqvist, Phys. Rev. A 62, 022109 (2000). +[34] F. Qu, F. Yang, J. Chen, J. Shen, Y. Ding, J. Lu, Y. Song, +H. Yang, G. Liu, J. Fan, Y. Li, Z. Ji, C. Yang, and L. Lu, +Phys. Rev. Lett. 107, 016802 (2011). +[35] A. Str¨om, H. Johannesson, and P. Recher, Phys. Rev. B +91, 245406 (2015). +[36] J. Dajka, A. Vourdas, S. Zhang, and E. Zipper, Journal +of Physics: Condensed Matter 18, 1367 (2006). + diff --git a/KNFRT4oBgHgl3EQfDTe1/content/tmp_files/load_file.txt b/KNFRT4oBgHgl3EQfDTe1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13b854f849fd05b43a68a5fd147a2409472d316e --- /dev/null +++ b/KNFRT4oBgHgl3EQfDTe1/content/tmp_files/load_file.txt @@ -0,0 +1,469 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf,len=468 +page_content='The Aharonov Casher phase of a bipartite entanglement pair traversing a quantum square ring Che-Chun Huang Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' National Taiwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Taiwan Seng Ghee Tan∗ Department of Optoelectric Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chinese Culture University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 55 Hwa-Kang Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Yang-Ming-Shan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Taipei 11114,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Taiwan Ching-Ray Chang Quantum Information Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chung Yuan Christian University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Taiwan and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' National Taiwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Taiwan We propose in this article a quantum square ring that conveniently generates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' annihilates and distills the Aharonov Casher phase with the aid of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The non-Abelian phase is carried by a pair of spin-entangled particles traversing the square ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At maximal entanglement, dynamic phases are eliminated from the ring and geometric phases are generated in discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' By contrast, at partial to no entanglement, both geometric and dynamic phases take on discrete or locally continuous values depending only on the wavelength and the ring size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' We have shown that entanglement in a non-Abelian system could greatly simplify future experimental efforts revolving around the studies of geometric phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' INTRODUCTION The quantum ring is a useful apparatus to study the physics of electron phase accumulating and interfering over the confined trajectories as prescribed by the de- sign of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Following the successful measurement of the Aharonov Bohm [1] phase, hot on the heels were a slew of experiments that had demonstrated the Aharonov Casher [2] and the Berry-Pancharatnam [3, 4] phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='In modern context, the Aharonov Casher phase associates primarily with the spin orbit coupling, particularly in 2D condensed matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Ring structure carved out of a 2D spin-orbital semiconductor to enclose a magnetic field at the center [5–7] was proposed to study the simul- tainety of the Aharonov Casher and the Aharonov Bohm effects on interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Efforts have also been made to study the parametric effects [8–10] of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' the Rashba constant, and the time-dependent magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At around the same time, the Aharonov Casher effect was experimentally measured in a number of ring structures [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' On a separate study, the Aharonov Casher phase is also associated with the non-Abelian gauge field for its spin phase [13–15] and spin force effects [16–19], cat- egorically reviewed in Ref [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' While the spin phase which comprises the geometric and the dynamic parts has largely been determined in ring structures, the ex- act nature of the accumulated phases in these devices remain ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The dynamic phase remains an elu- sive component in most cases, and the process to extract the geometric phase continues to be complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For example, in Ref [12], the system is a Rashba 2D that comprises a hedgehog orientation of the effective mag- netic fields turned crown-like by a vertical magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' While the strength of the BP phase is proportional to the solid angle subtended in the rest frame of the electron, a dynamic phase proportional to sin θ is also formed in concomitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' By applying an in-plane B field, which modifies the geometric Berry-Pancharatnam, and keeps the dynamic unaffected to the first order, a distinction can be made about the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Therefore, isolating the geometric phase is a complicated effort, the Aharonov Casher remains largely a total phase for most applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In this article, we propose a quantum square ring (QSR) that conveniently generates, annihilates or distills the Aharonov Casher phase with the aid of entanglement as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The Aharonov Casher phase gener- ated in this manner comprises the dynamic and the geo- metric components that can be further separated by tun- ing the entanglement strength and the device size mea- sured by the wavelength multiple of a traversing parti- cle pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For example, at maximal entanglement, dy- namic phases are eliminated from the device and geo- metric phases are generated in discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Discrete geometric phases would in turn switch their values on different ring locations depending on the device size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At partial to no entanglement, the Aharonov Casher as well as its dynamic and geometric components can be tuned according to the quantum ring size to take on discrete val- ues or vary continuously across the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The device is made out of semiconductor or metallic materials that ex- hibit 2D spin-orbit effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=', the Rashba-Vasko, Dres- selhaus, or Dresselhaus-Perel effects [21–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The spin- orbit effects will be the source of both the geometric and the dynamic phases in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As the external mag- netic field is not needed to generate the geometric phase, nor is it needed to help to eliminate the dynamic phase, a leaner QSR concept that rules out the Aharonov Bohm arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='13472v1 [quant-ph] 31 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 1: A square quantum ring device that takes a bipartite spin-pair at the emitter and generates an AC total phase as well as its separable components of geometric and dynamics phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' and the Altshuler–Aronov–Spivak (AAS) effect, and co- opts only the electrically-controllable Aharonov Casher is employed in our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the absence of strong B or M field, the adiabatic Berry-Pancharatnam phases in the QSR [28–31] is also ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Novel to the functioning of our device though is the entanglement physics [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' On the bottom left of the device is an emitter electrode (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 1) through which an entangled bipartite spin-pair is injected into the QSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The top right is the collector electrode where the injected spin-pair meets again and carries with it a total phase moderated by the physics of entanglement and device geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Our QSR device is therefore, by essence a non-adiabatic and a non-Abelian Aharonov Casher system [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The spin-pair’s total phase is accumulated via spin precession about the spin-orbit field but under the constant purview of bi-partite entan- glement, which provides in this paper a viable method to generate geometric and dynamic phases in a control- lable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As an aside, we note that quantum ring device has previously been studied for the practical pur- pose of producing spin entanglement in a controllable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [35, 36] There is, however, no discussion on its applicability in the context of geometric phases, let alone any specific discussion on its moderation of the dynamic phases or its distillation of the geometric phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' When a pure quantum state |ψ(t)⟩ evolves on the Hilbert space trajectory in time range Γ : t ∈ [0, τ], the total phase it accumulates is given by arg(⟨ψ(0)|ψ(t)⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The dynamic phase can be derived from D = −i � τ 0 dt⟨ψ(t)| ˙ ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' One can then define the geometric phase as the result of a total phase minus the dynamic phase as follows γ = arg(⟨ψ(0)|ψ(τ)⟩) + i � τ 0 dt⟨ψ(t)| ˙ ψ(t)⟩ (1) Consider a QSR ring of size η×η as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The device comprises 2 paths, each consisting of a horizontal and a vertical arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' From the material point of view, the device exhibits the Rashba spin-orbit effect as follows H = σxky − σykx (2) The QSR geometry conspires with the Rashba effect to generate phase factors for any particle traveling along path 1 and path 2 as follows UI = e iδσx 2 e −iησy 2 , UII = e −iδσy 2 e iησx 2 (3) Note that δ = ωt for both paths as a result of ωI = ωII = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Therefore, a spin particle traversing the hori- zontal arm of Path 1 and the vertical arm of Path 2 would separately accumulate a total phase as denoted by the di- mensionless η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The phase accumulated over time can be translated to a phase at an actual location in space de- pending on the particle velocity (ν) in the actual system as denoted by ω = kν, where k is the wave-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' While η is hence a phase parameter for the first half of either Path 1 or 2, δ ∈ [0, η] would represent the phase at any point on the second half of either path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For ease of illus- trations, we will refer to the η parts of Paths 1 and 2 as, respectively, η1 and η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Likewise, the same is prescribed for δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The spin-orbit effect when viewed in the rest frame of the carrier is a form of effective magnetic field which sets up a perfect environment for spin preces- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As the entangled spin-pair traverses both paths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' its phase evolves as prescribed by the unitary operation of U ≡ UI � UII (4) The geometric phase would thus be γ = arg(⟨ψ(0)|U|ψ(0)⟩) + i � τ 0 dt⟨ψ(0)|U† ˙U|ψ(0)⟩ (5) Explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' the dynamic phase is given by iU† ˙U = (I � (e− iσx 2 η(σy 2 )e iσx 2 η) − (e iσy 2 η(−σx 2 )e− iσy 2 η) � I) =1 2 � � � � 0 −i cos η − cos η 0 i cos η −2 sin η 0 − cos η − cos η 0 2 sin η −i cos η 0 − cos η i cos η 0 � � � � (6) The dynamic phase is expressed in terms of the Pauli matrices so that the relatable picture of effective mag- netic fields is not lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For calculation though, use is often made of its 4 by 4 matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' BIPARTITE ENTANGLED STATES The initial states of the entangled-spin-pair at the emitter is then prepared in the Bell-basis of |φ(0)⟩ or path 2 collector Entangledparticle source emitter path 13 |ψ(0)⟩ as follows: |φ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ |ψ(0)⟩ = ±√p0 |10⟩ + √p1 |01⟩ � (7) where p0 and p1 determine the strength of the entangle- ment, and p0, p1 ≥ 0, p0 + p1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' We will now consider the initial states of |φ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ to be injected into the QSR through the emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 1, spin particles 1 and 2 take to paths of their respective namesakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The geo- metric phase is the total Aharonov-Casher phase of the system minus the dynamic phase as shown below γ = arg(cos2(δ 2) cos2(η 2) + sin2(δ 2) sin2(η 2)) − D (8) The initial states |φ(0)⟩ simply could not generate any dynamic phase anywhere on the QSR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' And, the argument of the total Aharonov Casher phase factor consists of parameters that are all real and non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The geometric phase by virtue of γ ≡ arg(a + ib) − D vanishes as given by γ = tan−1 0 (a > 0) − D → γ = 0 (9) It is clear that the strength of entanglement has no bear- ing on the geometric and the dynamic phases as p0, p1 could take on values of the un-entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The re- sults of zero phases might not seem as trivial though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' It is a testament to the non-Abelian feature of the QSR device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In fact, the above shows that a bipartite state composed out of |00⟩ and |11⟩ is ideal for eliminating both geometric and dynamic phases from the propagat- ing particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In terms of applications, this could be a handy device to remove phases where they are not de- sired from all the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the following, we provide an insight of how dynamic phases are removed from the bipartite state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Let’s examine the dynamic phase by in- specting the constituent particles of the entangled pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Spin 1 travels on arm-η1 as though it is in a superposi- tion state of |0⟩ and |1⟩ as far as the dynamic phase is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In either state, its expectation energy is zero as can be deduced by the circular fashion of its spin rota- tion about the effective magnetic field of −By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Consider spin to precess about the effective magnetic field in an anti-clockwise manner, and that spin 1 to have rotated an angle θ < π by the end of its journey on arm-η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Spin 1 would thus continue on arm-δ1 about +Bx, now inscrib- ing a conical spin rotation with negative energy for p0, and positive energy for p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Likewise for spin 2, a corre- sponding process happens over arm-η2 about +Bx with zero energy for both components p0, p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Spin 2 would continue its journey on arm-δ2 about −By, inscribing a conical rotation with positive energy for p0, and negative energy for p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The cone energies on arm-δ cancel one another identically independent of the strength of p0, p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 2: Schematic illustration of the dynamic phases of the bipartite spin pair |ψ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ traversing the QSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' PE, NE, ZE, stand for positive energy, negative energy, zero energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The effect is thus a complete negation and a net zero of dynamic phase at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Note that the energy cones are drawn in different sizes to reflect the energy it carries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' This stands against the reality that spin vector is constant in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Therefore, the energy cones are crude illustrations meant only to provide an intuitive description of the dynamic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The precise description of the cone energies is given by the expressions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Equations (10) and (11) describe the expectation energy for spin particle 1 on arm-δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' p0⟨00|(e iσy 2 η(−σx 2 )e− iσy 2 η) � I)|00⟩ = −p0 sin η 2 (10) p1⟨11|(e iσy 2 η(−σx 2 )e− iσy 2 η) � I)|11⟩ = p1 sin η 2 (11) Equations (12) and (13) describe the expectation energy for spin particle 2 travelling on arm-δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' p0⟨00|(I � (e− iσx 2 η(σy 2 )e iσx 2 η)|00⟩ = p0 sin η 2 (12) p1⟨11|(I � (e− iσx 2 η(σy 2 )e iσx 2 η)|11⟩ = −p1 sin η 2 (13) The equations above lend clarity and mathematical credence to our qualitative accounts that the cone ener- gies on arms-δ cancel one another identically independent of the strength of p0, p1, resulting in a net zero dynamic phase at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' We will now consider the initial states of |ψ(0)⟩ = ±√p0 |10⟩+√p1 |01⟩ to be injected into the QSR through the emitter, once again with the spin particles taking to paths of their respective namesakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The geometric phase is given by γ = arg(cos η + cos δ 2 ∓ √p0p1(sin δ sin η 2 ) +i(p0 − p1)(sin δ sin η 2 )) +2(sin η)(p0 − p1)δ (14) NE P1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' ZE Po S arms PE Po ZE P1 +Bx PE P1 By +Bx NE Po n2 n arms y ZE Pi n1 ZE x Po4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 3: Schematic illustration of the dynamic phases of the bipartite spin pair |ψ(0)⟩ = ±√p0 |10⟩ + √p1 |01⟩ traversing the QSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' PE, NE, ZE, stand for positive energy, negative energy, zero energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' where the dynamic phase is now −2(sin η)(p0 − p1)δ and the total phase is deduced accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' It is clear from the above that the physics of entanglement has entered the geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' We will study the dynamic phase first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At maximum entanglement where p0 = p1 = 1 2, the dynamic phase vanishes with the equality of p0 and p1, leading to a total Aharonov Casher phase that is purely geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Intuitively, at maximal entanglement, the ex- pected energy of the spin-pair is constantly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' On arm-η1, inital spin |1⟩ or |0⟩ would precess about an ef- fective −By field in a circular fashion, both with a zero expectation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Likewise on arm-η2, due to entan- glement, the corresponding spin of |0⟩ or |1⟩ would pre- cess about an effective +Bx field in a circular fashion, and once again both with a zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In short, circu- lar rotation translates to zero expectation of the Zeeman energy on both arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Therefore, regardless of entangle- ment strength, dynamic phase is zero anywhere on arms- η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The δ sections of the QSR would, however, generate a dynamic phase at any strength of entanglement other than the maximum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' when p0 ̸= p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' This is because the initial states on arms-δ are determined by the dura- tion of precession on arms-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' On path 1, let spin rotates an angle θ < π about −By by the end of arm-η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Spin 1 would continue on arm-δ1 about +Bx, inscribing a coni- cal spin rotation with positive energy for component p0, and negative energy for component p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Likewise for path 2, a corresponding process that happens over η2 about +Bx would continue on arm-δ2 about −By, inscribing once again a conical rotation with positive energy for p0, and negative energy for p1, as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='3 It is clear that, on arms-δ, component p1 presents a counter effect of proportion p1 to the energy due to p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The effect is thus a complete negation and a net zero dynamic phase on the equality of p0 = p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The energy cones are drawn in different sizes to reflect the energy it carries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' This clearly stands against the quantum reality that spin vec- tor is constant in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Therefore, the energy cones are crude illustrations meant only to provide an intuitive de- scription of the dynamic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The precise description of the cone energies is given by the expressions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Equations (15) and (16) describe the expectation energy for spin particle 1 on arm-δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' p0⟨10|(e iσy 2 η(−σx 2 )e− iσy 2 η) � I)|10⟩ = p0 sin η 2 (15) p1⟨01|(e iσy 2 η(−σx 2 )e− iσy 2 η) � I)|01⟩ = −p1 sin η 2 (16) Equations (17) and (18) describe the expectation energy for spin particle 2 on arm-δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' p0⟨10|(I � (e− iσx 2 η(σy 2 )e iσx 2 η)|10⟩ = p0 sin η 2 (17) p1⟨01|(I � (e− iσx 2 η(σy 2 )e iσx 2 η)|01⟩ = −p1 sin η 2 (18) Spin particles on arms δ1 and δ2 reinforces once another, the p0 and p1 components become more positive and neg- ative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As the equality of the entanglement strength is crucial for suppressing the dynamic phase on arms-δ but not on arms-η, a net dynamic phase would ac- cumulate on arms-δ on the condition of p0 ̸= p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' There is, however, an exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' If the length of arms-η translate to a spin rotation of η = nπ, subsequent conical preces- sion on arms-δ would not have happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Spin would simply continue with circular precession and a zero dy- namic phase throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The physics above lends fur- ther credence to the applicability of the QSR design as a phase purifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tuning the entanglement strength to p0 = p1 at the source, a maximally-entangled spin-pair injected at the emitter would propagate with a dynamic phase suppressed throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the event of p0 ̸= p1 though, dynamic phase could be suppressed by choosing the length of arms η = nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Having completed our study of the dynamic phase, we will now examine the geometric phase, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At maximal entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' p0 = p1 = 1 2, the imaginary part of γ, denoted by b as shown in Equa- tion (19) below vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The geometric phase is either 0 or π depending on the parameters of the real part a(δ, η) as shown in the denominator of tan γ = b a(δ,η), where a positive denominator corresponds to γ = 0 while a nega- tive denominator corresponds to γ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' γ = arg(1 2(cos δ + cos η) − 1 4 sin δη + i0) ≡ arg(a + ib) (19) Let us now study in slightly more details the geometric phase of the spin-pair traversing the δ arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The crucial quantity here is range 0 < δ ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the case of no entanglement, (p0, p1) = (0, 1) or (1, 0) γ = tan−1 ∓(sin δ sin η) cos δ + cos η ∓ 2(sin η)δ (20) NE P1 62 ZE Po arms s PE Po ZE P1 +Bx PE Po By NE P1 +Bx ZE P1 ZE Po armsn n2 y n1 x5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 4: Schematic illustration of the effect of η on the geometric phase γ on arms δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The red and blue lines represent the paths with π and 0 geometric phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Dynamic phase is eliminated at arms’ length correspond- ing to η = nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Note that when δ > 0, phase η corre- sponds to the end location of arms-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Spin would always be oriented along the z axis by the time it reaches the end location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Therefore, advancing on arms-δ, spin would be precessing in a circular fashion with a net zero expecta- tion energy, and is thus precluded from generating the dynamic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' But at the values of η = nπ, the to- tal phase alternates between 0 and π on arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For η = 2nπ, the denominator is always positive, and the de- vice generates a total phase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For η = (2n + 1)π, the denominator is always negative, and the total phase is π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Since the dynamic phase is always 0, the total phase at η = nπ is also the geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For other values of η, the total phase takes on continuous values as a function of η and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Analysis above is focused only on the phases of arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Phases on arms-η for different η values could, on the other hand, be found by prescribing δ = 0, details of which would be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For illustration, we refer to FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='4 for η = π, 2π and observe the geometric phases on arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the case of partial entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' p0 ̸= p1 γ = tan−1 (p0 − p1)(sin δ sin η) (cos δ + cos η) ∓ √p0p1 sin η sin δ + 2(sin η)(p0 − p1)δ (21) Like in the above, the dynamic phase can be eliminated by η = nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Once again at these values, the total phase is discrete and alternates between 0(for η = 2nπ) and π(for η = (2n + 1)π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As before, the total phase at η = nπ is also the geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For other values of η, the total phase takes on continuous values as a function of η and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Note again that analysis here is focused only on the phases of arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' We will now revert to the case of maximum entan- glement again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' It was known that at maximal entan- glement, the dynamic phase vanishes and the geometric phase takes on discrete values of 0 and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' We would FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 5: Quantum square rings (QSR) of different sizes are superimposed for ease of inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The red and blue lines represent the paths with π and 0 geometric phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' now study the exact locations on the QSR where the ge- ometric phase switches its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As a matter of fact, the positions of switching from 0 to π happens on arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The exact location can be pinpointed by checking that the denominator of the geometric phase factor satisfies 2(cos δ + cos η) ∓ (sin δ sin η) > 0 (22) The equation above shows that the answer would de- pend on the length of arms-η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' the length of the arms before advancing into arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' For illustration, we chose arm lengths that correspond to η = π 2 , π, 3π 2 , 2π as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The device generates a γ = 0 on arms η at all times as indicated in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As the bipartite spin pair advances into arms-δ, the geomet- ric phase would switch to π on locations as indicated by the red segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At η = 2π though, no switch- ing is possible and the geometric phase remains 0 at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the event of a zero denominator, the total phase arg(⟨ψ(0)|U|ψ(0)⟩) = arg(a+ib) = tan−1 0 0 is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The bipartite state at that juncture would have to either vanish or turn out orthogonal to the initial Bell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Last is the particular situation of δ = 0 that cor- responds to the point where spin-pair starts to take a right-angle bend into arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' As long as δ = 0, spin- pair is considered to reside in the η regions of the arms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' And a quick inspection shows that a(0, η) is posi- tive throughout, which leads to the conclusion that the geometric phase on arms-η is 0 throughout, in spite of the entanglement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' This is in fact indicated in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='4 and FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='5 where arms-η are painted blue to indi- cate a zero geometric phase throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' This is indeed the case, barring the issues of singular points correspond- ing to cos η = −1 which brings upon γ = tan−1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At these points, the geometric phase is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In terms of spin precession, the singular points correspond to spin making a rotation of (2n + 1)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The odd-pi quantum states of the spin-pair at this point would then be or- thogonal to its initial Bell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In terms of the dynamic =0 (0,4元) =π (0,3元) = 0 (0,2元) =元 (8, n)= (0,元) (8, n) = (0, 元) (0,2元) (0,3元) (0,4元)(0, 2元) ~116° 3元 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' (0, 元) ~116° v64 (8, n) = (o,2) ~64° (8, n) = (o,%) (0, 元) 0,2 3元 (0,2元)6 Non-Abelian: Non-adiabatic |φ(0)⟩ = √p0 |00⟩ ± √p1 |11⟩ Dynamic phase Geometric phase δ = 0 δ > 0 δ = 0 δ > 0 (p0, p1 = (0, 1)) (p0, p1 = (1, 0)) NO ENTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 0 0 0 0 p0 = p1 = 1 2 MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' ENTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 0 0 0 0 p0 ̸= p1 ̸= 0 PARTIAL ENTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 0 0 0 0 |ψ(0)⟩ = ±√p0 |10⟩ + √p1 |01⟩ Dynamic phase 2(sin η)(p0 − p1)δ Geometric phase δ = 0 δ > 0 δ = 0 δ > 0 (p0, p1 = (0, 1)) (p0, p1 = (1, 0)) NO ENTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 0 0(η = 2nπ) 0(η = (2n + 1)π) Continuous values as D = −(∓2(sin η)δ) 0 Discrete 0(η = 2nπ) Discrete π(η = (2n + 1)π) Continuous values as γ = tan−1 ∓(sin δ sin η) cos δ+cos η ∓2(sin η)δ p0 = p1 = 1 2 MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' ENTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 0 0 0 Discrete 0 or π p0 ̸= p1 ̸= 0 PARTIAL ENTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 0 0(η = 2nπ) 0(η = (2n + 1)π) Continuous values as D = −(2(sin η)δ) 0 Discrete 0(η = 2nπ) Discrete π(η = (2n + 1)π) Continuous values see Equation (21) Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' I: Analysis of geometric phases for the SQR is tabulated according to the entanglement strength and the locations on arms-δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' phase, δ = 0 suppresses dynamic phases in spite of the entanglement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' I provides a summary of all the analysis that have been carried out for the geo- metric and dynamic phases corresponding to all the Bell states spin-pair traversing a non-Abelian QSR device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' CONCLUSION We have explained in details how a non-Abelian sys- tem in the form of a QSR could be designed to generate and purify the Aharonov Casher phases into its geomet- ric and dynamic components without elaborate experi- mental set ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The device requires only an entangled- particle source to couple to a passive square ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The Aharonov Casher phase is generated or annihilated as de- termined by the choice of the entanglement configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the correct Bell states, the dynamic phase is eliminated outright at maximal entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In the case of partial to no entanglement, dynamic phases are eliminated at η = nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In all manners of elimination, the Aharonov Casher phase becomes discrete and fully geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' This device could thus be useful for future experimental efforts to study the physics of discrete geometric phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' The continuous spectrum of the Aharonov Casher phase re- mains accessible though at partial to no entanglement, in which case, the continuous phases are non-geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At maximum entanglement, there is no possibility to access any continuous form of the geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In terms of discrete phases, the manner in which the phase switch from one discrete value to another varies according to the entanglement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' At partial to no entangle- ment, switching occurs only at (δ, η) = (0, (2n + 1)π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' By contrast, at maximal entanglement, switching could take place anywhere on arms-δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' any value of (δ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' In summary, the device has been shown to generate continuous Aharonov Casher phases, annihilate dynamic phases, distill discrete geometric phases, and enable dis- crete phase switching at various locations, all within the simple construct of a square ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' ACKNOWLEDGEMENT We would like to thank the Ministry of Science and Technology of Taiwan for supporting this work under Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' : 110-2112-M-034-001-MY3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' ∗ Corresponding author: csy16@ulive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='pccu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='tw ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' tansengghee@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='com [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Aharonov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Bohm, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 115, 485 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Aharonov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Casher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 53, 319 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Pancharatnam, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Indian Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 44, 247–262 (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Berry, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' London, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A 392, 45 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Nitta, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Meijer, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Takayanagi, Applied Physics Letters 75, 695 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Moln´ar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Peeters, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Vasilopoulos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B 69, 155335 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Frustaglia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Richter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B 69, 235310 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, Journal of Applied Physics 105, 07C709 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Xing, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jiang, IEEE Transactions on Magnetics 46, 1471 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Abdul Jalil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Ghee Tan, Journal of Applied Physics 113, 17C506 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Nitta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Bergsten, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Kunihashi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Kohda, Jour- nal of Applied Physics 105, 122402 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Nagasawa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Frustaglia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Saarikoski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Richter, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Nitta, Nature Communications 4, 2526 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Hatano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Shirasaki, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Nakamura, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A 75, 032107 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, Journal of Applied Physics 109, 07C722 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B 77, 045324 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Ho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, Europhysics Letters 108, 27012 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Bliokh and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Bliokh, Annals of Physics 319, 13 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Shen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 95, 187203 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Liu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Fujita, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B 78, 245321 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Ho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Murakami, Physics Reports 882, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Vasko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='and Korovin, Journal of Experimental and Theoretical Physics 96, 102 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Bychkov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rashba, JETP Letters 39, 78 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' RASHBA, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='-Solid State 2, 1109 (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' D’Yakonov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Perel’, Soviet Journal of Exper- imental and Theoretical Physics Letters 13, 467 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 7 [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Datta and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Das, Applied Physics Letters 56, 665 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, Introduction to the Physics of Nano- electronics (Woodhead publishing, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Nitta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Akazaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Takayanagi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Enoki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 78, 1335 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Bruno, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Dugaev, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Taillefumier, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 93, 096806 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Fujita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Murakami, Journal of Applied Physics 110, 121301 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Tan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Jalil, Modern Physics Letters A 31, 1630003 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Anandan, Physics Letters A 133, 171 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [32] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Sj¨oqvist, International Journal of Quantum Chemistry 115, 1311 (2015), https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='1002/qua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content='24941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [33] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Sj¨oqvist, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' A 62, 022109 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Qu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Ji, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Yang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' 107, 016802 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Str¨om, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Johannesson, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Recher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' B 91, 245406 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Dajka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Vourdas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Zhang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} +page_content=' Zipper, Journal of Physics: Condensed Matter 18, 1367 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFRT4oBgHgl3EQfDTe1/content/2301.13472v1.pdf'} diff --git a/KtE4T4oBgHgl3EQfiA3u/content/tmp_files/2301.05131v1.pdf.txt b/KtE4T4oBgHgl3EQfiA3u/content/tmp_files/2301.05131v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..24398dd61372137cb0d91ab2e494dda25dfb2505 --- /dev/null +++ b/KtE4T4oBgHgl3EQfiA3u/content/tmp_files/2301.05131v1.pdf.txt @@ -0,0 +1,2234 @@ +Toward Theoretical Guidance for Two Common +Questions in Practical Cross-Validation based +Hyperparameter Selection +Parikshit Ram +p.ram@acm.org +IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA +Alexander G. Gray +alexander.gray@ibm.com +IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA +Horst C. Samulowitz +samulowitz@us.ibm.com +IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA +Gregory Bramble +bramble@us.ibm.com +IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA +Abstract +We show, to our knowledge, the first theoretical treatments of two common questions +in cross-validation based hyperparameter selection: 1 After selecting the best hyper- +parameter using a held-out set, we train the final model using all of the training data +– since this may or may not improve future generalization error, should one do this? +2 During optimization such as via SGD (stochastic gradient descent), we must set +the optimization tolerance ρ – since it trades off predictive accuracy with computation +cost, how should one set it? Toward these problems, we introduce the hold-in risk (the +error due to not using the whole training data), and the model class mis-specification risk +(the error due to having chosen the wrong model class) in a theoretical view which is +simple, general, and suggests heuristics that can be used when faced with a dataset +instance. In proof-of-concept studies in synthetic data where theoretical quantities can +be controlled, we show that these heuristics can, respectively, 1 always perform at +least as well as always performing retraining or never performing retraining, 2 ei- +ther improve performance or reduce computational overhead by 2× with no loss in +predictive performance. +1 +Introduction +The learning process has various sources of errors. +The first step in (supervised) learning is +the acquisition of (training) data. +Given data, we choose a model or function class F which +corresponds to not just a method (such as Support Vector Machines, Generalized Linear Models, +Neural Networks, Decision trees) but their specific configuration governed by their respective +hyperparameters (such as regularization forms and penalties, trees depth) – these hyperparameters +refer to anything that would affect the predictive performance of the model learned from the +training data. Given our choice of the function class, the learning process searches for the function +that (approximately) minimizes the empirical risk (or some surrogate of it which better represents +the true risk or is easier to optimize). We currently have an understanding of the factors [Vapnik, +2006, Devroye et al., 2013, Bottou and Bousquet, 2008] affecting the excess risk of this chosen +1 +arXiv:2301.05131v1 [cs.LG] 12 Jan 2023 + +function – (i) the choice of the function class and its capacity to model the data generating process, +(ii) the use of an empirical risk estimate instead of the true risk, and (iii) the approximation in the +empirical risk minimization (ERM). +However, in practice, the learning process is not limited to these steps. A significant part of +the whole exercise is the choice of the function class F (method and its specifications). Usually, +we consider a (possibly large) set of function classes and select one of them based on the data- +driven process of model selection or hyperparameter selection. +This search can be done via grid +search (searching over a discretized grid of hyperparameter values) or random search [Bergstra +and Bengio, 2012]. However, AutoML (automated machine learning) has spurred a lot of research +in the area of hyperparameter optimization or HPO [Hutter et al., 2011, Shahriari et al., 2016, Snoek +et al., 2012, Bergstra et al., 2011, 2013]. +The automation allows us to look at even larger sets +of function classes for improved performance while being significantly more efficient than grid +search and more accurate than random search. The problem of HPO has been extended from +machine learning (ML) model configurations to the design of complete ML pipelines known as the +Combined Algorithm Selection and HPO (or CASH) problem, with various schemes that handle +(i) pipelines with fixed architecture [Hutter et al., 2011, Bergstra et al., 2011, Rakotoarison et al., +2019, Liu et al., 2020, Kishimoto et al., 2022], (ii) searching over the pipeline architectures [Katz +et al., 2020, Marinescu et al., 2021], (iii) deployment and fairness constraints [Liu et al., 2020, Ram +et al., 2020], and (iv) operating in the decentralized setting [Zhou et al., 2021, 2022], leading to +multiple open-source tools [Thornton et al., 2012, Kotthoff et al., 2017, Feurer et al., 2015, 2020, +Komer et al., 2014, Bergstra et al., 2015, Baudart et al., 2020, 2021, Hirzel et al., 2022]. +There has also been a significant amount of theoretical work on development of data- +dependent penalties for penalty-based model selection, resulting in guarantees in the form of +“oracle inequalities” – the expected excess risk of the selected model can be shown to be within +a multiplicative and additive factor of the best possible excess risk if an oracle provided us with +the best hyperparameter. This has been widely studied in (binary) classification [Boucheron et al., +2005], (bounded) regression and density estimation [Massart, 2007, Arlot et al., 2010]. However, in +practice, penalty-based model selection is not used for data-driven hyperparameter selection, and +we resort to some form of cross-validation1(CV). These universally applicable CV techniques have +been shown to be theoretically competitive to the penalty-based schemes at the cost of having less +data for the learning since some amount of data is “held-out” from the training data for validation +purposes [Boucheron et al., 2005, Arlot et al., 2010]. We focus on these universal CV based HPO. +Our contributions. +While CV based model selection has been studied theoretically, there are +various questions in practical HPO, which have not been explored in literature. A common prac- +tice is the learning of the final model on the selected hyperparameter with all available data, +reintroducing the held-out data in the training. This is standard practice in many commercial ML +tools since user data is too precious to not include in the training of the deployed model. In this +paper, we provide answers for the following questions2: +[Q1a] Theoretically, when does this practice help in terms of excess risk and by how much? +[Q1b] Is there a practical data-driven way of deciding whether to do this or not? +A common practice is to approximate the ERM with a large tolerance during the HPO and +perform a more accurate ERM during the final model training on the selected HP to reduce the +overall computational costs. We study the following related questions: +1Existing literature terms the single training/validation split as “cross-validation” (see for example [Kearns, 1996, Blum +et al., 1999]) and when there are multiple folds, it is specifically termed as “k-fold cross-validation”. +2We presented a preliminary version of this work at the AutoML@ICML’21 workshop [Ram et al., 2021]. +2 + +[Q2a] How does this ERM approximation in hyperparameter selection affect the excess risk? +[Q2b] Is there a theoretically informed practical way of setting these approximation levels during the +hyperparameter selection to better control the excess risk vs computation tradeoff? +Note that these answers can be utilized both by humans and by automated data science sys- +tems. +Practical motivations. +In applications with large amount of data, the importance of question +Q1a (and consequently Q1b) might appear minimal (the computational aspect of questions Q2a +and Q2b make them critical with large data). However, we believe that even in such cases, the +important signal can still be quite small, and the relevant information in the held-out set can +have large impact (positive or negative) on the final deployed model. For example, in various data +science applications in finance, the class/target of interest is usually very small. In the Home +Credit Default Risk Kaggle Classification Challenge, the training data has only around 8.5% of +some 3 × 105 training examples from the positive class, and in the AllState Claim Prediction +Kaggle Regression Challenge, the training set has less than 1% of some 13 × 106 examples with +nonzero regression targets; rest are zero. Furthermore, there are situations where the population +has groups and the underrepresented minority groups are significantly smaller than the majority +group(s). In this case, the presence or absence of the held-out data can have significant (again +positive or negative) impact on the fairness and accuracy of the deployed model since many of the +fairness metrics quantify the parity in the group specific predictive performance. Finally, “small” +differences in predictive performance can have significant impact depending on problems and +applications – in data science leaderboards such as Kaggle, minor changes in final performance +can lead to significant reordering of the leaderboard (although, relevance of such competitions +and results to practical problems remain an open question). Such scenarios motivate us to study +the question of whether the practice of retraining after reintroducing the data held-out during HPO +is helpful (Q1a). +Empirical motivation. +As evidence of the lack of clarity on this whether question, we consider +HPO for LightGBM [Ke et al., 2017] on 40 OpenML [Vanschoren et al., 2013] data sets (we detail +the evaluation setup in §2.2). Of the 400 different HPO problems we solve, retraining has (i) no +significant effect (positive or negative) in 51/400 (12.75%) cases, (ii) has a positive effect (improving +test error) in 260/400 (65%) cases, and (iii) has a negative effect in 89/400 (22.25%) cases. This +highlights that there is no single universal correct answer to this question. Hence, we study this whether +question rigorously and provide a theoretically motivated data-driven heuristic as one answer. +Outline. +We present our precise problem setting, and existing & novel excess risk decomposi- +tions with empirical support in §2. Then we present and evaluate our theoretical results, tradeoffs +and practical heuristics for HPO in §3. We position our contributions against existing literature in +§4, and conclude in §5. +2 +Decomposing the excess risk +For a particular method (decision trees, linear models, neural networks), let Fλ denote the func- +tion class for some fixed hyperparameter λ ∈ Λ (tree depth, number of trees for tree ensembles; +regularization parameter for linear and nonlinear models) in the space of valid hyperparameters +(HPs) Λ. For any model or function f : X → Y with (x, y), x ∈ X , y ∈ Y generated from a +distribution P over X × Y, and a loss function ℓ : Y × Y, the expected risk E( f ) and the empirical +3 + +Fλ +f ⋆ +¯fλ +ˆfn,λ +˜fn,λ +E +Eapp +Eopt +Eest +(a) Original decomposition +f ⋆ +F¯λ +Fˆλ +E +Emcm +(b) Model mis-specification error +˜gm,ˆλ +˜fn,ˆλ +Ehin +Fˆλ +(c) Hold-in error +Figure 1: Decompositions of excess risk E. Figure 1a shows the decomposition of E incurred by the +approximate empirical risk minimizer ˜fn,λ ∈ Fλ with respect to the Bayes optimal f ⋆. Figure 1b +visualizes the additional excess risk in the form of the model class mis-specification risk Emcm incurred +by selecting the sub-optimal hyperparameter ˆλ in place of ¯λ. Figure 1c visualizes the hold-in risk +Ehin incurred from using ˜gm,ˆλ instead of ˜fn,ˆλ. +risk En( f ) with n samples {(xi, yi)}n +i=1 ∼ Pn of f is given by +E( f ) = E(x,y)∼P [ℓ(y, f (x))] = +� +ℓ(y, f (x))dP, +En( f ) = En [ℓ(y, f (x))] = 1 +n ∑ +n +i=1 ℓ(yi, f (xi)). +(1) +We denote the Bayes optimal model as f ⋆ where, for any (x, y) ∼ P, +f ⋆(x) = arg min +ˆy∈Y +E [ℓ(y, ˆy)|x] . +(2) +We denote with the following: +¯fλ = arg min +f ∈Fλ +E( f ), +ˆfn,λ = arg min +f ∈Fλ +En( f ), +(3) +as the true risk minimizer and the empirical risk minimizer (with n samples) in model class Fλ +respectively. Table 1 defines the various symbols used in the sequel. +When performing ERM over Fλ, the excess risk incurred E = E( ˆfn,λ) − E( f ⋆) decomposes +into two terms: (i) the approximation risk Eapp(λ) = E( ¯fλ) − E( f ⋆), and (ii) the estimation risk +Eest(n, λ) = E( ˆfn,λ) − E( ¯f ). For limited number of samples n, there is a tradeoff between Eapp and +Eest, where a larger function class Fλ usually reduces Eapp(λ) but increases Eest(n, λ) [Vapnik, +2006, Devroye et al., 2013]. Bottou & Bousquet [Bottou and Bousquet, 2008] study the tradeoffs in +a “large-scale” setting where the learning is compute bound (in addition to the limited number +of samples n). Given any computational budget T, they consider the learning setting “small- +scale” when the number of samples n is small enough to allow for the ERM to be performed to +arbitrary precision. In this case, the tradeoff is between the Eapp and Eest terms (as above). They +consider the large scale setting where the ERM needs to be approximated given the computational +budget and discuss the tradeoffs in the excess risk of an approximate empirical risk minimizer +˜fn,λ ∈ Fλ. In addition to Eapp and Eest, they introduce the optimization risk term Eopt – the excess +risk incurred due to approximate ERM – and argue that, in compute-bound large-scale learning, +approximate ERM on all the samples n can achieve better generalization than high precision ERM +on a subsample of size n′ ≤ n. Figure 1a provides a visual representation of this excess risk +decomposition. +4 + +Table 1: Table of symbols +Symbol +Description (1st location in text) +E( f ) +True risk of any model f (1) +En( f ) +Empirical risk of any model f with n samples (1) +Λ +Set of L hyperparameters (HPs) λ, L = |Λ| (§2) +Fλ +Model class for hyperparameter (HP) λ (§2) +f ⋆ +Bayes optimal predictor (2) +¯fλ +True risk minimizer in Fλ (3) +ˆfn,λ +Empirical risk minimizer in Fλ with n samples (3) +˜fn,λ +Approx. empirical risk minimizer Fλ with n samples (5) +¯λ +Oracle hyperparameter (HP) arg minλ∈Λ E( ˆfn,λ) (§2) +ˆλ +Solution to empirical hyperparameter selection (4) +ˆgm,λ +Empirical risk minimizer in Fλ with m samples (4) +˜gm,λ +Approx. empirical risk minimizer in Fλ with m samples (4) +Given a set Λ of L HPs λ ∈ Λ, and n samples from the true distribution, we wish to find +the oracle HP ¯λ such that the (approximate) ERM solution ˜fn,¯λ has the best possible excess risk +– ¯λ = arg minλ∈Λ E( ˜fn,λ). However, in practice, with n iid (independent and identically dis- +tributed) samples from P, we use cross-validation for model selection and solve the following +bilevel problem to pick the HP ˆλ: +ˆλ = arg minλ∈Λ Ev +µ(˜gm,λ) +(outer) +˜gm,λ ∈ {g : Em(g) ≤ Em(ˆgm,λ) + ρin} , (inner) +(4) +where the inner problem is an approximate ERM on Fλ for each λ ∈ Λ with m < n samples at an +approximation tolerance of ρin > 0 producing ˜gm,λ (we use ˆgm,λ to denote the exact ERM solution +in Fλ with m samples), and the outer problem considers an objective Ev +µ(·) which is evaluated +using µ < n samples held-out from the ERM in the inner problem – while Em(·) and Ev +µ(·) might +have the same form, the ·v superscript highlights their difference. Then, a final approximate ERM +on Fˆλ with all n samples to ρout tolerance produces +˜fn,ˆλ ∈ +� +f ∈ Fˆλ : En( f ) ≤ En( ˆfn,ˆλ) + ρout +� +, +(5) +with ˆfn,ˆλ denoting the exact ERM solution in Fˆλ. This embodies the common practice of splitting +the samples into a training and a held-out validation set (of sizes m, µ < n respectively with m + µ ≤ +n). In k-fold cross-validation, the inner ERM is solved k times for each HP λ (on k different sets +of size m = n − n/k each), and the outer optimization averages the objectives from k held-out +sets (of size µ = n/k) across the k learned models. In this paper, we focus on CV with a single +training-validation split, and defer k-fold CV to future work. +At this point, multiple choices have to be made for computational and statistical purposes: +▶ The number of samples m drives the computational cost of solving the inner problem for each +λ ∈ Λ – larger m requires larger compute budget. +▶ The approximation tolerance ρin in the inner ERM also drives the computational cost – smaller +ρin requires larger compute budget. +5 + +▶ The approximation tolerance ρout in the final ERM over Fˆλ drives the computational cost +similar to ρin but to a lesser extent since it is only over a single ˆλ ∈ Λ instead of for each +λ ∈ Λ. For this reason, ρout is usually selected to be smaller3 than ρin. +▶ The function ˜fn,ˆλ is selected over ˜gm,ˆλ for statistical reasons since the former gets more training +data. +Many of these choices are often made ad hoc or via trial and error. To the best of our knowl- +edge, there is no mathematically grounded way of making some of these practical choices. More- +over, it is not clear what is precisely gained by selecting ˜fn,ˆλ over ˜gm,ˆλ. Existing theoretical guar- +antees for CV based model selection focus on the excess risk of ˜gm,ˆλ, while in practical HPO, ˜fn,ˆλ +is deployed, indicating a gap between theory and practice. In this paper, we try to bridge this +gap, and in the process, provide a practical heuristic that allows us to select between ˜gm,ˆλ and ˜fn,ˆλ +in a data-driven manner. Furthermore, ρin and ρout provide a way to control the computation +vs excess risk tradeoff, but it is not clear how to set them to extract computational gains without +significantly increasing the excess risk. We explicitly highlight the role of ρin and ρout in the ex- +cess risk and provide practical heuristics to select ρin and ρout in a data-driven manner to better +control this tradeoff. +2.1 +Novel excess risk decomposition +We first present some intuitive decompositions of the excess risk to understand the different +sources of additional risk (and gains!). After the selection of ˆλ by solving problem (4), existing +literature focuses on the excess risk of ˜gm,ˆλ, yet we are not aware of any decomposition of its +excess risk. We decompose this excess risk as: +E = E(˜gm,ˆλ) − E( f ⋆) += E(˜gm,ˆλ) − E(˜gm,¯λ) +� +�� +� +Emcm ++ E(˜gm,¯λ) − E(ˆgm,¯λ) +� +�� +� +Eopt ++ E(ˆgm,¯λ) − E( ¯f ¯λ) +� +�� +� +Eest ++ E( ¯f ¯λ) − E( f ⋆) +� +�� +� +Eapp +, +(6) +where ¯f ¯λ, ˆgm,¯λ and ˜gm,¯λ are the true risk minimizer, exact ERM solution and approximate ERM +solution respectively in the function class F¯λ corresponding to the oracle HP ¯λ. We introduce +a new term Emcm = E(˜gm,ˆλ) − E(˜gm,¯λ), the model class mis-specification risk. Figure 1b visualizes +this term in the excess risk. This term incorporates the excess risk from selecting a suboptimal +HP (and corresponding model class). However, there is a potential additional excess risk that is +often ignored in literature but is considered crucial in practice – the risk from learning the model +˜gm,ˆλ on m < n samples instead of all n samples, or the hold-in risk, defined as Ehin = E(˜gm,ˆλ) − +E( ˜fn,ˆλ). This term is visualized in Figure 1c. This excess risk term does not appear explicitly +in the risk decomposition (6) for ˜gm,ˆλ but rather is implicitly incorporated in the estimation risk +Eest. +However, when studying the excess risk of ˜fn,ˆλ, the Ehin does explicitly appear in the +decomposition: +E = E( ˜fn,ˆλ) − E( f ⋆) += E( ˜fn,ˆλ) − E(˜gm,ˆλ) +� +�� +� +−Ehin ++ E(˜gm,ˆλ) − E(˜gm,¯λ) +� +�� +� +Emcm ++ +E(˜gm,¯λ) − E( f ⋆) +� +�� +� +Eopt+Eest+Eapp (see (6)) +(7) +This excess risk decomposition for ˜fn,ˆλ is different from previous decompositions in that the +“−Ehin” term in this excess risk decomposition is potentially a risk deficit instead of an additional +3Often, for computational reasons, m might be much less than n, and training the final ˜fn,ˆλ on all n samples to a +tolerance of ρin might be computationally infeasible, making ρout > ρin. +6 + +Table 2: Empirical (relative) estimate of Emcm = E(˜gm,ˆλ) − E(˜gm,¯λ) across 40 OpenML datasets +of varying number of samples n and varying sizes of the held-out validation set µ/n. We report +the percentage of the experiments (for each combination of n and µ/n) where ˆλ ̸= ¯λ. For these +experiments, we also report the (estimate of the) average relative excess risk ˜∆ incurred. +n +µ/n +AuROC +Acc +ˆλ ̸= ¯λ +˜∆ +ˆλ ̸= ¯λ +˜∆ +1000 +0.1 +95 +2.00 +75 +1.67 +0.2 +65 +0.85 +60 +1.24 +0.3 +50 +1.12 +50 +1.43 +5000 +0.1 +65 +0.12 +80 +0.60 +0.2 +70 +0.14 +75 +0.51 +0.3 +70 +0.08 +80 +0.69 +10000 +0.1 +65 +0.11 +75 +0.24 +0.2 +65 +0.10 +85 +0.22 +0.3 +50 +0.01 +90 +0.27 +50000 +0.1 +80 +0.34 +90 +0.25 +0.2 +90 +0.25 +85 +0.16 +0.3 +80 +0.27 +90 +0.18 +risk, highlighting potential risk we can recover from this common practice of training on all the +data with the selected HP ˆλ. These decompositions are intended to explicitly highlight the dif- +ferent sources of risk (and gains) in the practical HPO process, providing some intuition into the +problem. +2.2 +Empirical Validation +To evaluate the practical significance of these newly introduced risk terms Emcm and Ehin, we +consider the HPO problem with LightGBM [Ke et al., 2017] across 40 OpenML binary classification +datasets [Vanschoren et al., 2013]. We consider 10 datasets each with number of rows in the ranges +1000-5000, 5000-10000, 10000-50000 and 50000-100000. For each dataset, we consider 3 different +values of µ/n (the held-out fraction). We perform this exercise with two classification metrics – area +under the ROC curve (AuROC) and balanced accuracy (Acc). We approximate the true risks for +the post-hoc analysis using an additional test set not involved in the HPO. We detail the datasets +and HP search space in Appendix C. +For each HPO experiment (dataset and held-out fraction), we note whether the selected HP ˆλ +matches the oracle HP ¯λ (found post-hoc using the test set), and the (relative) estimate ˜∆ of Emcm. +We report the aggregate findings for each set of size range and held-out fraction in Table 2. The +results indicate that, with the smaller datasets (n ∈ 1000-5000), a higher value of µ/n reduces the +chances of missing the oracle HP ¯λ, but this effect is no longer present with the larger datasets. +As the dataset sizes increase, the chances of missing the oracle HP does increase on aggregate, but +the relative risk ˜∆ decreases from 2% down to around 0.2%. So the Emcm term benefits from larger +data but the effect is still significant. However, there is no explicit indication of how the different +terms such as n, µ play a role. +To understand the impact of Ehin, we further compare the performance of the ˜gm,ˆλ involved +in the HPO to the final retrained ˜fn,ˆλ in the above HPO experiments. We note the percentage +of the time (i) their performances were within a relative difference of 10−5 ( ˜f ≈ ˜g), (ii) ˜fn,ˆλ was +7 + +Table 3: Comparing relative performances of ˜fn,ˆλ and ˜gm,ˆλ in HPO of LightGBM with balanced +accuracy. +n +µ/n +˜f ≈ ˜g +˜f✓ +˜f gain +˜g✓ +˜g gain +1000 +0.1 +40 +45 +1.25 +15 +0.18 +0.2 +25 +60 +1.81 +15 +1.77 +0.3 +30 +50 +2.57 +20 +1.50 +5000 +0.1 +30 +45 +0.51 +25 +0.48 +0.2 +20 +55 +0.47 +25 +0.66 +0.3 +20 +70 +0.82 +10 +0.74 +10000 +0.1 +10 +60 +0.17 +30 +0.12 +0.2 +5 +70 +0.29 +25 +0.09 +0.3 +20 +60 +0.54 +20 +0.03 +50000 +0.1 +15 +55 +0.16 +30 +0.07 +0.2 +15 +70 +0.10 +15 +0.09 +0.3 +0 +80 +0.16 +20 +0.03 +Table 4: Comparing relative performances of ˜fn,ˆλ and ˜gm,ˆλ in HPO of LightGBM with AuROC. +n +µ/n +˜f ≈ ˜g +˜f✓ +˜f gain +˜g✓ +˜g gain +1000 +0.1 +5 +80 +0.79 +15 +0.72 +0.2 +0 +75 +0.42 +25 +2.15 +0.3 +10 +70 +0.34 +20 +0.27 +5000 +0.1 +15 +45 +0.15 +40 +0.15 +0.2 +10 +60 +0.24 +30 +0.06 +0.3 +10 +50 +0.39 +40 +0.05 +10000 +0.1 +5 +75 +0.10 +20 +0.02 +0.2 +0 +90 +0.09 +10 +0.01 +0.3 +0 +100 +0.21 +0 +0.00 +50000 +0.1 +10 +50 +0.15 +40 +0.18 +0.2 +5 +80 +0.18 +15 +0.05 +0.3 +0 +90 +0.35 +10 +0.18 +better ( ˜f✓), and (iii) ˜gm,ˆλ was better (˜g✓). In both cases (ii) and (iii), we noted the average relative +gain the better choice provided (“ ˜f gain” and “˜g gain”). The results are aggregated across dataset +sizes and held-out fraction µ/n in Table 3 for the balanced accuracy metric and in Table 4 for +the AuROC metric. The results indicate that, in most cases, ˜fn,ˆλ is a better choice, justifying the +common practice. However, it also indicates that, in a significant fraction of the cases (around +20% in most but up to 40%), ˜gm,ˆλ appears to be the better choice against common intuition. The +results also indicate that, when ˜fn,ˆλ is the better choice, it also provides higher relative gains over +˜gm,ˆλ on average across most experimental settings. But the average relative gains of ˜gm,ˆλ over ˜fn,ˆλ +are still significant in most cases across both classification metrics. +These results indicate that there is no single best choice and we can obtain improved per- +formance if we are able to make this choice in a more problem-dependent manner. In the next +8 + +section, we theoretically bound the excess risk to explicitly understand the impact of the different +choices in the HPO and leverage these dependencies for improved performance. +3 +Bounding the excess risk +In this section, we bound the excess risks of ˜gm,ˆλ and ˜fn,ˆλ and try to understand any improvement +˜fn,ˆλ might provide and the interaction with the ERM approximation tolerances ρin and ρout based +on our decompositions. We have the following result for ˜gm,ˆλ: +Theorem 3.1. Let P be a distribution over X × Y and let ℓ : Y × Y, Y ⊂ R, be a B-bounded β-Lipschitz +loss. Let Fλ be a class of functions f : X → Y for any HP λ ∈ Λ. Let ˆλ be the solution of (4) over the set +of L HPs Λ with ERM on m < n samples to approximation tolerance ρin ≥ 0, a held-out set of size µ < n, +and m + µ ≤ n. With probability at least 1 − δ for δ ∈ (0, 1), the excess risk E of ˜gm,ˆλ in (6) is bounded +as: +E ≤ minλ∈Λ +� +8 β Rm(Fλ) + Eapp(λ) +� + ρin + 2B +� +log(2(L+1)/δ) (2/ +√ +2m + 1/√ +2µ) , +(8) +where Rm(Fλ) is the Radamacher complexity of Fλ. +The proof is provided in Appendix A.1. The above bound (8) highlights how the different +terms affect the excess risk bounds – we get an excess risk bound within an additive factor of +the class that possesses the minimum combined (scaled) Radamacher complexity (a proxy for +estimation risk Eest [Bartlett and Mendelson, 2002]) and approximation risk Eapp. Note that the +Radamacher complexity is with respect to m < n samples, highlighting the statistical inefficiency +introduced by the held-out data for CV. The result also indicates that a larger held-out set (larger +µ) is preferable. We study this excess risk bound to identify the effect of the choices in HPO (such +as m, µ, ρin); the best choice needs to balance the terms in the bound – making one term, such as +ρin much smaller (say, by an order of magnitude) than the other terms will not improve the excess +risk significantly, but an order of magnitude larger ρin will have significant ill-effects. We have +the following result for ˜fn,ˆλ: 4 +Theorem 3.2. Under the conditions of Theorem 3.1, let ˜fn,ˆλ ∈ Fˆλ be obtained via approximate ERM with +tolerance ρout ≥ 0 over all n samples in (5). Let I ˆλ +n,m(ρin, ρout) := En(˜gm,ˆλ) − En( ˜fn,ˆλ) be the “empirical +risk improvement” from performing the ERM over all n samples. With probability at least 1 − δ for any +δ ∈ (0, 1), the excess risk E of ˜fn,ˆλ in (7) is bounded as: +E ≤ min +λ∈Λ +� +8 β Rm(Fλ) + Eapp(λ) +� + 8 β Rn(Fˆλ) + ρin + B′ (2/ +√ +2n + 2/ +√ +2m + 1/√ +2µ) − ¯I +(9) +where B′ := 2B +� +log(2(L+2)/δ), and +¯I := max +� +I ˆλ +n,m(ρin, ρout), I ˆλ +n,m(0, 0) − ρout − (µ/n)B +� +, +with I ˆλ +n,m(0, 0) = En(ˆgm,ˆλ) − En( ˆfn,ˆλ) denoting the empirical risk improvement if ρin = ρout = 0. +4With further structural assumptions (such as relationships between the variance and expectations of the functions), we +can improve the 1/√µ dependence to 1/µ. See for example, Theorem A.2 in Appendix A. We focus on Theorem 3.1 since +this will subsequently allow us to derive practical heuristics that we cannot with the structural assumptions required to +get the tighter excess risk bounds. +9 + +Table 5: Tradeoffs of the terms in the excess risk (6) & (7) based on the various choices in HPO. +We use Fall = ∪λ∈ΛFλ to denote the union of all function classes. ‘↑’ denotes an increase while +‘↓’ a decrease. +Fall ↑ +|Λ| ↑ +n ↑ +µ ↑ +m ↑ +ρin ↑ +ρout ↑ +Eapp +↓ +↓ +Eest +↑ +↓ +↑ +↓ +Eopt +↑ +Emcm +↑ +↓ +↓ +Ehin +↓ +↑ +↓ +↑ +The proof is provided in Appendix A.2. This result indicates that we are only able to recover +the hold-in risk Ehin in terms of the excess risk if the empirical risk improvement I ˆλ +n,m(ρin, ρout) +is relatively significant. While I ˆλ +n,m(0, 0) is always ≥ 0 by definition of the ERM solution ˆfn,ˆλ, the +quantity I ˆλ +n,m(ρin, ρout) depends more closely to ρout and, we should make ρout small enough +to extract any gain from retraining the final ˜fn,ˆλ on all the n samples – ρout should not exceed +a critical point where the I ˆλ +n,m(ρin, ρout) term is of the same order as the 8βRn(Fˆλ) and the +B′(2/ +√ +2n) terms. We cannot compute this critical point, but we will discuss how we can select ρout +in a data-driven way. Comparing Theorems 3.2 to 3.1 allows us to provide a theoretical answer +to Q1a. In HPO, a critical choice is the value of ρin, properly balancing the computational and +generalization aspects. Theorems 3.1 and 3.2 highlight the roles of these ERM approximation +tolerances ρin and ρout, providing an answer for Q2a. +These results also allow us to conceptually understand the tradeoffs better the different terms +in the excess risk. A visualization of these tradeoffs is presented in Table 5, highlighting the effect +of the individual parameters of the problems (n, m, µ, |Λ|, etc) on the different terms in the excess +risk. For example, HPO over a large set of HPs Λ will potentially reduce the approximation risk +Eapp, but might increase the model class mis-specification risk Emcm. Or increasing the validation +set size µ would reduce Emcm but increasing µ implies reduced m (for fixed n), leading to higher +estimation risk Eest. We believe that understanding these tradeoffs explicitly will allow us to +obtain better generalization with HPO. +3.1 +Data-driven heuristics +In certain situations, ρin and ρout are specified outside our control (for example, with models +learned with techniques other than gradient descent like decision tree), and hence it is not clear +which learned model, ˜gm,ˆλ or ˜fn,ˆλ, is better to deploy. For this, we present a heuristic to make a +data-driven choice between the two. In practical scenarios, the approximation risk Eapp usually +dominates the excess risk. +Hence, if we assume that minλ{8βRm(Fλ) + Eapp(λ)} dominates +8βRn(Fˆλ) (the latter does not have Eapp term and m < n) then we can compare the excess risk +bounds of ˜gm,ˆλ and ˜fn,ˆλ and utilize the following heuristic (note that log 2(L + 1) ≈ log 2(L + 2) +except from really small L), providing a data-driven answer to Q1b: +Heuristic 3.1. Based on the quantities in Theorems 3.1 & 3.2 and δ > 0, we select ˜fn,ˆλ if I ˆλ +n,m > +2B +� +2 log(2(L+2)/δ)/n, else we select ˜gm,ˆλ. +10 + +To answer Q2b, we need a way to make an informed choice in terms of ρin and ρout. Intuitively, +we can compare ρin to the other computable terms in the bounds (8) and (9) – if ρin is of the order +of these terms or larger, the Eopt risk will contribute significantly to the excess risk; if ρin is an +order of magnitude smaller that this term, then the Emcm risk will dominate the Eopt and any +further reduction in ρin is not beneficial. Based on this observation, we propose another heuristic +for HPO with approximate ERM to facilitate the choice of ρin in the inner ERM of the bilevel +problem (4): +Heuristic 3.2. Based on the quantities in Theorem 3.1 and a scaling parameter γ > 0, we set +ρin = γ B +� +2 log(2(L+1)/δ)(2/√m + 1/√µ). +(10) +A larger γ will imply a more computationally efficient HPO, while a smaller γ will improve +excess risk up until a point. As we will demonstrate in our experiments, γ ∼ 0.1 is sufficiently +small5 such that we still gain computational efficiency via the ERM approximation but not see any +adverse effect on the excess risk of ˜gm,ˆλ. +While we have a very precise way of setting ρin given the theoretical result, the choice of ρout +is more involved. We present this in Appendix B. Note that the choice of ρout does not play a +significant role in the computational cost of the HPO compared to ρin since ρin is involved in the +training for each λ ∈ Λ while ρout only influences the final training of ˜fn,ˆλ. For this reason, if +possible, ρout is chosen to be significantly smaller than ρin. Heuristics 3.2 and B.1 (Appendix B) +are our answers to Q2b. +3.2 +Empirical evaluation +We evaluate our heuristics on HPO with neural network configurations. +We consider neural +networks to have better control over ρin in the ERM. We chose a synthetic data distribution to +have control over the experiment and to be able to generate fresh large samples to accurately +estimate the true risks of the different models (as opposed to our results in §2.2 which were +based on the true risk estimated on a limited test set). This is common practice when empirically +studying theoretical bounds on various statistical quantities (see for example [Rodriguez et al., +2009]). This also allows us to perform the empirical evaluation under various setting (such as +different n, µ, Λ). We set the Bayes optimal risk E( f ⋆) = 0. We consider two HPO problems +with grid-search: (a) one with 36 HP configurations (L = |Λ| = 36), and (b) another with 18 HP +configurations (L = |Λ| = 18). The data generation and the HP search spaces are detailed in +Appendix C. We estimate the true risk E( f ) of any model f with a separate large test sample from +the synthetic data distribution. We consider sample sizes n ∈ [29, 214] and different values for +µ/n ∈ [0.1, 0.5] with m set to (n − µ). Each ERM involves 5 restarts and the results are averaged +over 10 trials (corresponding to different samples from the same distribution). We set the failure +probability δ = 0.05. +We first evaluate the practical utility of Heuristic 3.1, which tries to balance the gain from +utilizing the full data for obtaining ˜fn,ˆλ and the associated statistical cost of an additional ERM. +Figure 2 compares this “Choice” based on Heuristic 3.1 (thick translucent red line) to ˜fn,ˆλ & ˜gm,ˆλ +(solid blue ▲ & green ⋆ respectively) for a subset of the combinations of ρin, ρout and µ/n, showing +the excess risk on the vertical axis as the number of samples n is increased. The results indicate +that, depending on Λ, ρin, ρout and µ/n, ˜gm,ˆλ might be preferable to ˜fn,ˆλ and vice versa – one +is not always better than the other (as we also highlighted in §2.2), and always selecting ˜fn,ˆλ (as +5This γ implies that the optimization risk Eopt is an order of magnitude smaller than atleast some other term in the +excess risk. +11 + +(a) HPO with |Λ| = 36 +(b) HPO with |Λ| = 18 +Figure 2: Empirical utility of Heuristic 3.1 for data dependent choice of ˜fn,ˆλ vs ˜gm,ˆλ with a subset +of the varying values of n, µ/n, ρin, ρout. The vertical axis is the excess risk (lower is better) and the +horizontal axis is the total number of available samples n. Note the logscale on both axes. See +Appendix D.1 for results on all combinations. +done in practice) leaves room for improvement. In both types of cases, Heuristic 3.1 is able to +select the better options in many cases – the proposed heuristic provides a data-driven way of selecting +between ˜fn,ˆλ and ˜gm,ˆλ. We provide the full set of results for all considered values of ρin, ρout, µ/n +for both HPO problems in Appendix D.1. There are cases where the heuristic does not make the +right choice, which indicates that there is room for improvement. +To demonstrate the practical utility of the proposed Heuristic 3.2, we continue with the afore- +mentioned HP selection problem over 36 neural network configurations on a synthetic classifica- +tion data. We consider three choices γ ∈ {0.1, 1, 10} in Heuristic 3.2. We set δ = 0.05 and consider +different values of n and µ/n. Figure 3 compares the performance of the HPO with exact ERM +using ˆgm,λ, λ ∈ Λ to the HPO with approximate ERM for different ρin using ˜gm,λ instead. In +Figure 3a, we compare the excess risk incurred from approximate ERM with the data dependent +choice of ρin compared to exact ERM. We see that γ = 0.1 leads to a sufficiently small ρin that +12 + +ny/n0.1 +/n-0.3 +nvfn-0.4 +/n-0.5 +Pin:0.001.Po:0.01 +Pm:0.01.Pout:0.1 +Pin:0.0.Pou:0.0 +Pm:0.01.Pout:0.001 +Excess risk +2 +Choice +E(gn) +2 +E(f,) +21 +212 +214 +212 +214 +212 +224 +2 +22 +214 ++=n/n-0.1 +nv/n=0.2 +/n-0.4 +/n-0.5 +Pin:0.001.Po:0.002 +Pin:0.0.Pou:0.0 +Pm:0.01.Pout:0.1 +Pin:0.01.Pout:0.001 +risk +ExcessI +Choice +★ E(gn) +← E(fn) +21 +272 +214 +212 +224 +210 +212 +224 +211 +272 +214 ++=(a) Excess-risk from Heuristic 3.2. +(b) Speedup from Heuristic 3.2. +Figure 3: Empirical validation of the utility of Heuristic 3.2 for data-dependent choice of ρin +matches the predictive performance of exact ERM. Any smaller approximation ρin would not +improve the excess risk. The results also indicate that γ = 10 leads to a ρin where the optimiza- +tion error dominates the excess risk, implying that ρin should be reduced if possible. Figure 3b +presents the computational speedups obtained for the corresponding data-dependent choices of +ρin – we see that we can get a 2× speedup over exact ERM without any degradation in excess +risk with γ = 0.1 while obtaining around 4 − 8× speedup with slight degradation in performance +with γ = 1. These results provide empirical evidence for the practical utility of the proposed +Heuristic 3.2 obtained from Theorem 3.1 – the proposed heuristic provides a data driven way of setting +the ERM approximation tolerance in HPO. +4 +Related work +While the use of smaller number of samples m < n to select the HP ˆλ is often recognized in +practice, and leads to the final model being learned via ERM on Fˆλ using all n samples, no +theoretical guarantees exist for this process. We explicitly study this situation, introducing the hold-in +risk Ehin, and provide a novel guarantee for such a procedure in Theorem 3.2. Kearns [Kearns, +1996] studies the interaction between the approximation and estimation risks, and under certain +assumptions and restricted class of functions, proposes ways for selecting the sizes of the training +and held-out splits m and µ in an informed manner. To account for the fact that some of the +training data is “wasted” as the held-out set, Blum et al. [Blum et al., 1999] propose two different +ways of retaining the Hoeffding bounds of the error estimate on the held-out set while still being +13 + +nyln=0.1 +nyin=0.2 +nyln=0.3 +nyln=0.4 +nyln=0.5 +2~1 + E(gn.) +H-- E(9n,),Y= 1.0 +E(9n.), Y = 0.1 +H-_E(9n,), Y= 10.0 +Excess risk +2 +210 +212 +214 +226 +212 +214 +22 +212 +214 +23 +212 +224 +22n +212 +214 ++= ++= +n=ny+nt += +=exact ERM +ny/n = 0.1 +ny/n = 0.2 +ny/n = 0.3 +ny/n = 0.4 +ny/n = 0.5 +Exact +H-- Approx, Y= 1.0 +27 +Approx, +: 0.1 +H-- Approx, Y= 10.0 +21 +210 +222 +214 +220 +212 +224 +212 +214 +212 +214 +21 +212 +214 ++= +n=ny+nt ++= +n=ny+nt ++=able to utilize the full training data to train models employed at test time. These techniques are +ways of modifying the standard CV. +In addition to the above, there are various theoretical analyses focusing on various aspects of +the CV process such as obtaining tight variance estimates for the k-fold CV score of any given +HP λ ∈ Λ [Nadeau and Bengio, 1999, 2003, Rodriguez et al., 2009, Markatou et al., 2005]. These +results are complementary to ours and could be used to extend our current results (for the single +training/validation split based CV) to k-fold CV, with the variance estimates for the k-fold CV +metrics involved in the data-driven heuristics. We will pursue this in future research. However, +note that these existing results do not directly help us obtain tighter excess risk bounds or allow +comparison between models ˜gm,ˆλ (used during the HPO) and the final deployed model ˜fn,ˆλ or +provide any intuition regarding the choices for ρin and ρout, which are the main questions (Q1a, +Q1b, Q2a, Q2b) we study. +Finally, as we discussed in §1, HPO has been widely studied over the last decade. However, +the questions we focus on are complementary to any specific HPO scheme. We do not focus on +how the HP was found (with any specific HPO scheme such Bayesian Optimization [Shahriari +et al., 2016]), but rather on (i) the ERM involved in the evaluation of any HP during the HPO, and +(ii) the ERM involved in the final deployment after an HP is selected. Our Heuristic 3.2 allows us +to speed up any HPO scheme without any additional excess risk, and our Heuristics 3.1 and B.1 +allow us to improve the predictive performance of the deployed model for the HP ˆλ selected via +any HPO scheme. “Multi-fidelity” HPO schemes [Li et al., 2018, Jamieson and Talwalkar, 2016, +Sabharwal et al., 2016, Klein et al., 2017, Falkner et al., 2018] significantly improve the compu- +tational efficiency by adaptively setting either the training set size m < n or the optimization +approximation ρin on a per-HP basis instead of using a single value of m or ρin for all λ ∈ Λ. This +is quite different from our proposed Heuristic 3.2 and the excess risk introduced by this adaptive +strategy is not studied to the best of our knowledge. We wish to extend our tradeoff analysis to +multi-fidelity HPO in future work. Meta-learning is another way of improving the efficiency of the +HPO process [Vanschoren, 2018], and has been used in some AutoML toolkits [Feurer et al., 2015, +2020]. Recently, some theoretical guarantees have been established for such meta-learning based +HPO [Ram, 2022], and we also wish to extend our tradeoff analysis to such meta-learning based +HPO. Note that the proposed Heuristics 3.1 and B.1 is still beneficial in both the above situations +(multi-fidelity and meta-learning). +5 +Conclusions +Our contributions focus on aspects of CV based HPO – we explore how to leverage the different +tradeoffs in the excess risk to make various practical decisions in the HPO process in a data-driven +manner. We use the novel excess risk decomposition and theoretical analyses to answer the two +questions in CV-based HPO: (1) When is the process of training the model on all the data after +the HPO beneficial and can we choose between the two in a data driven manner? (Q1a, Q1b) (2) +At what level should we set the tolerance of the optimization involved in model training during +HPO? (Q2a, Q2b). The ideas can be utilized by data science practitioners as well as by automated +data science systems. +References +Sylvain Arlot, Alain Celisse, et al. A survey of cross-validation procedures for model selection. +Statistics surveys, 4:40–79, 2010. +14 + +Peter L Bartlett and Shahar Mendelson. Rademacher and Gaussian complexities: Risk bounds and +structural results. Journal of Machine Learning Research, 3(Nov):463–482, 2002. +Guillaume Baudart, Martin Hirzel, Kiran Kate, Parikshit Ram, and Avraham Shinnar. Lale: Con- +sistent automated machine learning. In 4th KDD Workshop on Automation in Machine Learning +(AutoML@KDD), 2020. +Guillaume Baudart, Martin Hirzel, Kiran Kate, Parikshit Ram, Avi Shinnar, and Jason Tsay. +Pipeline combinators for gradual automl. +Advances in Neural Information Processing Systems, +34:19705–19718, 2021. +J. Bergstra and Y. Bengio. Random search for hyper-parameter optimization. Journal of Machine +Learning Research, 13(1):281–305, 2012. +James Bergstra, R´emi Bardenet, Yoshua Bengio, and Bal´azs K´egl. Algorithms for hyper-parameter +optimization. Advances in neural information processing systems, 24, 2011. +James Bergstra, Daniel Yamins, and David Cox. Making a science of model search: Hyperparam- +eter optimization in hundreds of dimensions for vision architectures. In International conference +on machine learning, pages 115–123. PMLR, 2013. +James Bergstra, Brent Komer, Chris Eliasmith, Dan Yamins, and David D Cox. Hyperopt: a python +library for model selection and hyperparameter optimization. Computational Science & Discovery, +8(1):014008, 2015. +Avrim Blum, Adam Kalai, and John Langford. Beating the hold-out: Bounds for k-fold and pro- +gressive cross-validation. In Proceedings of the twelfth annual conference on Computational learning +theory, pages 203–208, 1999. +L´eon Bottou and Olivier Bousquet. The tradeoffs of large scale learning. In Advances in neural +information processing systems, pages 161–168, 2008. +St´ephane Boucheron, Olivier Bousquet, and G´abor Lugosi. Theory of classification: A survey of +some recent advances. ESAIM: probability and statistics, 9:323–375, 2005. +Luc Devroye, L´aszl´o Gy¨orfi, and G´abor Lugosi. A probabilistic theory of pattern recognition, vol- +ume 31. Springer Science & Business Media, 2013. +Stefan Falkner, Aaron Klein, and Frank Hutter. +BOHB: Robust and efficient hyperparameter +optimization at scale. In International Conference on Machine Learning, pages 1437–1446. PMLR, +2018. +Matthias Feurer, Aaron Klein, Katharina Eggensperger, Jost Springenberg, Manuel Blum, and +Frank Hutter. Efficient and robust automated machine learning. In Advances in Neural Informa- +tion Processing Systems, pages 2962–2970, 2015. +Matthias Feurer, Katharina Eggensperger, Stefan Falkner, Marius Lindauer, and Frank Hutter. +Auto-sklearn 2.0: Hands-free AutoML via meta-learning. arXiv preprint arXiv:2007.04074, 2020. +Isabelle Guyon. Design of experiments of the NIPS 2003 variable selection benchmark. In NIPS +2003 workshop on feature extraction and feature selection, volume 253, 2003. +Martin Hirzel, Kiran Kate, Parikshit Ram, Avraham Shinnar, and Jason Tsay. Gradual automl +using lale. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data +Mining, pages 4794–4795, 2022. +15 + +Frank Hutter, Holger H. Hoos, and Kevin Leyton-Brown. Sequential model-based optimization +for general algorithm configuration. In Proceedings of the 5th International Conference on Learning +and Intelligent Optimization, LION’05, pages 507–523, Berlin, Heidelberg, 2011. Springer-Verlag. +Kevin Jamieson and Ameet Talwalkar. Non-stochastic best arm identification and hyperparameter +optimization. In Artificial Intelligence and Statistics, pages 240–248, 2016. +Michael Katz, Parikshit Ram, Shirin Sohrabi, and Octavian Udrea. Exploring context-free lan- +guages via planning: The case for automating machine learning. In Proceedings of the Interna- +tional Conference on Automated Planning and Scheduling, volume 30, pages 403–411, 2020. +Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, and Tie-Yan +Liu. LightGBM: A highly efficient gradient boosting decision tree. Advances in neural information +processing systems, 30, 2017. +Michael J Kearns. A bound on the error of cross validation using the approximation and estimation +rates, with consequences for the training-test split. In Advances in Neural Information Processing +Systems, pages 183–189, 1996. +Akihiro Kishimoto, Djallel Bouneffouf, Radu Marinescu, Parikshit Ram, Ambrish Rawat, Martin +Wistuba, Paulito Palmes, and Adi Botea. Bandit limited discrepancy search and application +to machine learning pipeline optimization. In Proceedings of the AAAI Conference on Artificial +Intelligence, volume 36, pages 10228–10237, 2022. +Aaron Klein, Stefan Falkner, Simon Bartels, Philipp Hennig, and Frank Hutter. +Fast bayesian +hyperparameter optimization on large datasets. Electronic Journal of Statistics, 11(2):4945–4968, +2017. +Brent Komer, James Bergstra, and Chris Eliasmith. Hyperopt-sklearn: automatic hyperparameter +configuration for scikit-learn. Citeseer, 2014. +Lars Kotthoff, Chris Thornton, Holger H. Hoos, Frank Hutter, and Kevin Leyton-Brown. Auto- +WEKA 2.0: Automatic model selection and hyperparameter optimization in weka. Journal of +Machine Learning Research, 18(1):826–830, January 2017. ISSN 1532-4435. URL http://dl.acm. +org/citation.cfm?id=3122009.3122034. +Lisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar. Hyper- +band: A novel bandit-based approach to hyperparameter optimization. Journal of Machine Learn- +ing Research, 18(185):1–52, 2018. +Sijia Liu, Parikshit Ram, Deepak Vijaykeerthy, Djallel Bouneffouf, Gregory Bramble, Horst Samu- +lowitz, Dakuo Wang, Andrew Conn, and Alexander Gray. +An admm based framework for +automl pipeline configuration. +In Proceedings of the AAAI Conference on Artificial Intelligence, +volume 34, pages 4892–4899, 2020. URL https://arxiv.org/abs/1905.00424v5. +Radu Marinescu, Akihiro Kishimoto, Parikshit Ram, Ambrish Rawat, Martin Wistuba, Paulito P +Palmes, and Adi Botea. Searching for machine learning pipelines using a context-free grammar. +In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 8902–8911, 2021. +Marianthi Markatou, Hong Tian, Shameek Biswas, and George Hripcsak. Analysis of variance of +cross-validation estimators of the generalization error. Journal of Machine Learning Research, 6: +1127–1168, 2005. +Pascal Massart. Concentration inequalities and model selection: Ecole d’Et´e de Probabilit´es de Saint-Flour +XXXIII-2003. Springer, 2007. +16 + +Claude Nadeau and Yoshua Bengio. Inference for the generalization error. Advances in neural +information processing systems, 12, 1999. +Claude Nadeau and Yoshua Bengio. Inference for the generalization error. Machine Learning, 52: +239–281, 2003. +Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor +Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. PyTorch: An imperative style, +high-performance deep learning library. Advances in Neural Information Processing Systems, 32: +8026–8037, 2019. +F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Pretten- +hofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, +and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, +12:2825–2830, 2011. +Herilalaina Rakotoarison, Marc Schoenauer, and Michele Sebag. Automated machine learning +with Monte-Carlo Tree Search. In Proceedings of the Twenty-Eighth International Joint Conference +on Artificial Intelligence, IJCAI-19, pages 3296–3303, 2019. +Parikshit Ram. On the optimality gap of warm-started hyperparameter optimization. In Interna- +tional Conference on Automated Machine Learning, pages 12–1. PMLR, 2022. +Parikshit Ram, Sijia Liu, Deepak Vijaykeerthi, Dakuo Wang, Djallel Bouneffouf, Greg Bramble, +Horst Samulowitz, and Alexander G Gray. Solving constrained CASH problems with ADMM. +In 7th ICML Workshop on Automated Machine Learning (AutoML), 2020. +Parikshit Ram, Alexander G Gray, and Horst Samulowitz. Leveraging theoretical tradeoffs in hy- +perparameter selection for improved empirical performance. In 8th ICML Workshop on Automated +Machine Learning (AutoML), 2021. +Juan D Rodriguez, Aritz Perez, and Jose A Lozano. Sensitivity analysis of k-fold cross validation +in prediction error estimation. IEEE transactions on pattern analysis and machine intelligence, 32(3): +569–575, 2009. +Ashish Sabharwal, Horst Samulowitz, and Gerald Tesauro. Selecting near-optimal learners via +incremental data allocation. In Thirtieth AAAI Conference on Artificial Intelligence, 2016. +B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and N. De Freitas. Taking the human out of the +loop: A review of bayesian optimization. Proceedings of the IEEE, 104(1):148–175, 2016. +Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine +learning algorithms. Advances in neural information processing systems, 25, 2012. +Chris Thornton, Holger H. Hoos, Frank Hutter, and Kevin Leyton-Brown. Auto-WEKA: Auto- +mated selection and hyper-parameter optimization of classification algorithms. +arXiv, 2012. +URL http://arxiv.org/abs/1208.3719. +Joaquin Vanschoren. Meta-learning: A survey. arXiv preprint arXiv:1810.03548, 2018. +Joaquin Vanschoren, Jan N. van Rijn, Bernd Bischl, and Luis Torgo. OpenML: Networked science +in machine learning. SIGKDD Explorations, 15(2):49–60, 2013. doi: 10.1145/2641190.2641198. +URL http://doi.acm.org/10.1145/2641190.2641198. +17 + +Vladimir Vapnik. Estimation of dependences based on empirical data. Springer Science & Business +Media, 2006. +Yi Zhou, Parikshit Ram, Theodoros Salonidis, Nathalie Baracaldo, Horst Samulowitz, and Heiko +Ludwig. +FLoRA: Single-shot hyper-parameter optimization for federated learning. +In 1st +NeurIPS Workshop on New Frontiers in Federated Learning (NFFL 2021), 2021. +Yi Zhou, Parikshit Ram, Theodoros Salonidis, Nathalie Baracaldo, Horst Samulowitz, and Heiko +Ludwig. Single-shot hyper-parameter optimization for federated learning: A general algorithm +& analysis. arXiv preprint arXiv:2202.08338, 2022. +18 + +A +Detailed Proofs +We will make sure of this standard result: +Theorem A.1. ([Bartlett and Mendelson, 2002]) Consider conditions and notations of Theorem 3.1. Then +for any δ > 0, with probability at least 1 − δ, the following is true: +sup +f ∈Fλ +|En( f ) − E( f )| ≤ 4 β Rn(Fλ) + 2 B +� +log(1/δ)/2n. +(11) +where Rn(Fλ) is the Radamacher complexity of Fλ. +A.1 +Proof of Theorem 3.1 +Proof. (Theorem 3.1) By definition of Fˆλ, for any λ ∈ Λ and δ′ > 0, we have the following +relationships: +E(˜gm,ˆλ) +(a) +≤ Ev +µ(˜gm,ˆλ) + B +� +log(1/δ′)/2µ +w.p. ≥ 1 − δ′ +(12) +(b) +≤ Ev +µ(˜gm,λ) + B +� +log(1/δ′)/2µ +(13) +(c) +≤ E(˜gm,λ) + 2B +� +log(1/δ′)/2µ +w.p. ≥ 1 − δ′, +(14) +where (a) and (c) are obtained from an application of Hoeffding’s inequality while (b) is obtained +from the definition of ˆλ in (4) as the minimizer of Ev +µ(˜gm,λ). For any Fλ, λ ∈ Λ, we have the +following: +E(˜gm,λ) +(d) +≤ Em(˜gm,λ) + 4βRm(Fλ) + 2B +� +log(1/δ′)/2m +w.p. ≥ 1 − δ′ +(e) +≤ Em(ˆgm,λ) + ρin + 4βRm(Fλ) + 2B +� +log(1/δ′)/2m +( f ) +≤ Em( ¯fλ) + ρin + 4βRm(Fλ) + 2B +� +log(1/δ′)/2m +(g) +≤ E( ¯fλ) + ρin + 8βRm(Fλ) + 4B +� +log(1/δ′)/2m +w.p. ≥ 1 − δ′ +(h) +≤ E( f ⋆) + Eapp(λ) + ρin + 8βRm(Fλ) + 4B +� +log(1/δ′)/2m, +(15) +where (d) and (g) are obtained by Theorem A.1, (e) is obtained from the problem definition (4) for +the approximate ERM solution, ( f ) is obtained from the definition of ˆgm,λ as the ERM solution for +any class Fλ, and (h) is a simple application of the definition of the approximation risk Eapp(λ) +of any class Fλ. +Since the above holds for any λ ∈ Λ, putting (14) and (15) together using the union bound +with δ′ = δ/(2 + 2|Λ|) = δ/2(L + 1) and minimizing over λ ∈ Λ gives us the desired result in (8) +with a failure probability of at most δ. +A.2 +Proof of Theorem 3.2 +Proof. (Theorem 3.2) By definition of Fˆλ and ˜fn,ˆλ ∈ Fˆλ, we have the following with an application +of Theorem A.1 w.p. ≥ 1 − δ′: +E( ˜fn,ˆλ) ≤ En( ˜fn,ˆλ) + 4βRn(Fˆλ) + 2B +� +log(1/δ′)/2n +(16) +19 + +Now from the problem definition and the definition of I ˆλ +n,m: +En( ˜fn,ˆλ) = En(˜gm,ˆλ) − I ˆλ +n,m(ρin, ρout) +(17) +However, we have an alternate relationship as follows: +En( ˜fn,ˆλ) +(a) +≤ En( ˆfn,ˆλ) + ρout +(18) +(b) += En(ˆgm,ˆλ) − I ˆλ +n,m(0, 0) + ρout +(19) +(c) +≤ En(˜gm,ˆλ) + µ +n B − I ˆλ +n,m(0, 0) + ρout, +(20) +where (a) is from the definition of the approximate ERM solution ˜fn,ˆλ and the exact ERM solution +ˆfn,ˆλ, (b) is obtained from the definition of I ˆλ +n,m(0, 0) in the statement of the theorem, and (c) is +from the definition of ˆgm,ˆλ as the minimizer of Em(·) in Fˆλ and the fact that, for any f, En( f ) = +(m/n)Em( f ) + (µ/n)Eµ( f ) and the loss ℓ is bounded by B. +Putting (17) and (20) gives us: +En( ˜fn,ˆλ) ≤ En(˜gm,ˆλ) − max +� +I ˆλ +n,m(ρin, ρout), +I ˆλ +n,m(0, 0) − µ +n B − ρout +� +� +�� +� +:= ¯I in Theorem 3.2 +. +Using above in (16), we get the following relationships: +E( ˜fn,ˆλ) ≤ En(˜gm,ˆλ) − ¯I + 4βRn(Fˆλ) + 2B +� +log(1/δ′)/2n +w. p. ≥ 1 − δ′ +≤ E(˜gm,ˆλ) − ¯I + 8βRn(Fˆλ) + 4B +� +log(1/δ′)/2n +w. p. ≥ 1 − 2δ′, +where the last inequality is an application of Theorem A.1 on ˜gn,ˆλ ∈ Fˆλ. Combining (21) with +(14) (which holds w.p. ≥ 1 − 2δ′) and (15) (which holds w.p. ≥ 1 − 2δ′ for each λ ∈ Λ) and +applying the union bound over all λ ∈ Λ, we have +E( ˜fn,ˆλ) − E(˜gm,ˆλ) +≤ − ¯I + 8βRn(Fˆλ) + 4B +� +log(1/δ′)/2n += − ¯I + 8βRn(Fˆλ) + B′(2/ +√ +2n) +w.p. ≥ 1 − 2δ′, +(21) +We also know that for all λ ̸= ˆλ ∈ Λ, +E(˜gm,ˆλ) − E(˜gm,λ) ≤ 2B +� +log(1/δ′)/2µ = B′(1/√ +2µ) +w.p. ≥ 1 − 2δ′, +E(˜gm,λ) − E( f ⋆) ≤ min +λ∈Λ [Eapp(λ) + ρin + 8βRm(Fλ)] + 4B +� +log(1/δ′) +2m += min +λ∈Λ [Eapp(λ) + ρin + 8βRm(Fλ)] + B′ +� +2 +√ +2m +� +w.p. ≥ 1 − 2Lδ′ where we replace 2B +� +log(1/δ′) with B′. +Putting the above together, we get the upperbound for E = E( ˜fn,ˆλ) − E( f ⋆) in the statement +of the claim with probability at least 1 − (2 + 2L + 2)δ′ = 1 − 2(L + 2)δ′ = 1 − δ by setting +δ′ = δ/(2(L + 2)). +20 + +A.3 +Tighter version of Theorem 3.1. +Theorem A.2 (Adapted from Boucheron et al. [2005], Thm 8.16). Consider the conditions and no- +tations of Theorem 3.1. If we further assume that there exists a non-decreasing function w : R+ → +R+ with w(x)/√x non-increasing for x ∈ R+ such that, for any function f, +� +Var [| f − f ⋆|] ≤ +w (E( f ) − E( f ⋆)), then the excess risk of ˜gm,ˆλ can be bounded from above as: +E ≤ min +θ∈(0,1) (1 + θ) +� +min +λ∈Λ +� +8 · β · Rn(Fλ) + Eapp(λ) +� + ρin ++2B · +� +2 log((2L + 1)/δ) +m ++ 4 · B · log 2L + 1 +δ +�τ∗(µ) +θ ++ 2 +3µ +�� +, +where τ∗(n) = min{x > 0 : w(x) = x√n} for any n > 0. +Proof. For any λ ∈ Λ and some δ′ > 0, we have the following from Berstein’s inequality: +E(˜gm,λ) − E( f ⋆) ≤ Ev +µ(˜gm,λ) − Ev +µ( f ⋆) + w(E(˜gm,λ) − E( f ⋆)) +� +2 log(1/δ′) +µ ++ 4 log(1/δ′) +3µ +. +(22) +with probability at least 1 − δ′. +Let ¯λ = arg minλ∈Λ E(˜gm,λ). Then again applying Berstein’s inequality with δ′ > 0, we have +E( f ⋆) − E(˜gm,¯λ) ≤ Ev +µ( f ⋆) − Ev +µ(˜gm,¯λ) + w(E(˜gm,¯λ) − E( f ⋆)) +� +2 log(1/δ′) +µ ++ 4 log(1/δ′) +3µ +. +(23) +with probability at least 1 − δ′. +Combining the above two inequalities over all λ ∈ Λ, using the definition of ˜gm,ˆλ as the +minimizer of Ev +µ(·), and the non-decreasing nature of w(·) gives us the following with probability +at least 1 − (L + 1)δ′: +E(˜gm,ˆλ) − E(˜gm,¯λ) ≤ 2w(E(˜gm,ˆλ) − E( f ⋆)) +� +2 log(1/δ′) +µ ++ 8 log(1/δ′) +3µ +. +(24) +Given τ∗(µ) as defined in the statement of the claim, w(τ∗(µ)) = τ∗(µ)√µ. +Now, either +E(˜gm,ˆλ) − E( f ⋆) < τ∗(µ). Or E(˜gm,ˆλ) − E( f ⋆) ≥ τ∗(µ), which gives us the following: +w(E(˜gm,ˆλ) − E( f ⋆)) +� +E(˜gm,ˆλ) − E( f ⋆) +(a) +≤ w(τ∗(µ)) +� +τ∗(µ) +(b) += +� +τ∗(µ)√µ, +(25) +where (a) comes from the assumptions that w(x)/√x is non-increasing in x ∈ R+ and (b) comes +from the definition of τ∗(µ). Using the above in (24) gives us: +E(˜gm,ˆλ) − E(˜gm,¯λ) ≤ 2 +� +E(˜gm,ˆλ) − E( f ⋆) +� +2 log(1/δ′) +µ +� +τ∗(µ) + 8 log(1/δ′) +3µ +(26) +≤ θ +2 +� +E(˜gm,ˆλ) − E( f ⋆) +� ++ 8 +2θ log(1/δ′)τ∗(µ) + 8 log(1/δ′) +3µ +. +(27) +where we utilize the fact that the arithmetic mean is greater than or equal to the geometric mean +for some θ ∈ (0, 1). +21 + +Then we can get, w.p. ≥ 1 − (L + 1)δ′: +� +E(˜gm,ˆλ) − E( f ⋆) +� +(1 − θ/2) ≤ E(˜gm,¯λ) − E( f ⋆) + 4 log(1/δ′) +�τ∗(µ) +θ ++ 2 +3µ +� +. +(28) +Now we have the following by definition of ¯λ and ˜gm,¯λ ∈ F¯λ: +E(˜gm,¯λ) − E( f ⋆) = min +λ∈Λ (E(˜gm,λ) − E( f ⋆)) +(29) +(c) +≤ min +λ∈Λ +� +E( ¯fλ) − E( f ⋆) + 2 sup +f ∈Fλ +|E( f ) − Em( f )| + ρin +� +(30) +(d) +≤ min +λ∈Λ +� +Eapp(λ) + 8 · β · Rn(Fλ) + 2B · +� +2 log(1/δ′) +m ++ ρin +� +w.p. ≥ 1 − Lδ′ +(31) +where (d) is obtained from the application of Theorem A.1 on each λ ∈ Λ, and (c) is obtained as +follows: +E(˜gm,λ) ≤ Em(˜gm,λ) + sup +f ∈Fλ +|E( f ) − Em( f )| +(32) +≤ Em(ˆgm,λ) + ρin + sup +f ∈Fλ +|E( f ) − Em( f )| +(33) +≤ Em( ¯fλ) + ρin + sup +f ∈Fλ +|E( f ) − Em( f )| +(34) +≤ E( ¯fλ) + ρin + 2 sup +f ∈Fλ +|E( f ) − Em( f )|. +(35) +Combining (28) and (31), setting δ′ = δ/(2L + 1) and noting that (1 − θ/2)−1 ≤ (1 + θ) for +θ ∈ (0, 1) gives us the statement of the claim. +B +Data-driven heuristic for ρout +While we have a very precise way of setting ρin given the theoretical result, the choice of ρout is +somewhat more involved. Based on Theorem 3.2, we can utilize the following heuristic to set ρout: +Heuristic B.1. Assuming that ρout can be iteratively reduced during the approximate ERM for ˜fn,ˆλ, based +on the terms in Theorem 3.2 and a given ρin, we propose the following iterative scheme to set ρout with +scaling parameters ν ∈ (0, 1), γ > 0: For T > 0, we iteratively reduce ρout as +ρ(T+1) +out +← ν · ρ(T) +out +if +Γ(ρ(T−1) +out +) − Γ(ρ(T) +out) > γ · κ +exit approx. ERM +otherwise, +(36) +where Γ(ρout) := I ˆλ +n,m(ρin, ρout), ρ(0) +out ← ρin, and κ := ρin + B +� +2 log(2(L+2)/δ)(2/√n + 2/√m + +1/√µ). +This heuristic leverages the fact that the empirical risk improvement I ˆλ +n,m(ρin, ρout) will in- +crease as ρout is reduced up until a point, and the excess risk of ˜fn,ˆλ is closely tied to this empirical +risk improvement – more improvement implies better excess risk. Heuristic B.1 tries to balance +any increase in this empirical risk improvement with the other (computable) terms, denoted as κ, +in the excess risk bound in (9) – we stop reducing ρout when the increase in the empirical risk +22 + +improvement Γ(ρ(T−1) +out +) − Γ(ρ(T) +out) is an order of magnitude below κ, at which point, other terms +dominate the excess risk. Heuristics 3.2 and B.1 are our answers to Q2b. +Heuristic B.1 just presents a way to identify when ρout is sufficiently small in terms of statistical +performance while being able to gain computationally when we are able to approximate the ERM +in an iterative manner and progressively decrease ρout. +C +Empirical evaluation details +Synthetic +data +generation. +The +binary +classification +data +is +generated +using +the +make classification function [Guyon, 2003] in scikit-learn [Pedregosa et al., 2011]. +We +ensure that the classes are not overlapping and there is no label noise, ensuring that the Bayes +optimal risk E( f ⋆) = 0. +Neural network HP search space. +We consider two different Λs for a fully connected neural +network, with (i) depth ∈ {1, 2, 3}, (ii) number of neurons in each layer ∈ {10, 100}, (iii) initial +SGD learning rate ∈ {0.01, 0.1}, (iv) SGD batch size ∈ {8, 32, 128}. The implementation is in +PyTorch [Paszke et al., 2019]. For one problem we consider 36 configurations (that is L = |Λ| = +36), and for another, we consider 18 configurations (that is L = |Λ| = 18). +LightGBM HP search space. +We use the following search space for theLGBMClassifier from +LightGBM [Ke et al., 2017] with RandomizedSearchCV from scikit-learn: (i) learning rate ∈ +[0.01, 0.4], (ii) number of trees ∈ [100, 5000], (iii) number of leaves per tree ∈ [6, 50], (iv) minimum +samples in a child node ∈ [100, 500], (v) minimum weight in a child node ∈ [10−5, 104], (vi) sub- +sampling rate ∈ [.1, .9], (vii) maximum depth per tree ∈ [−1, 7], (viii) column sub-sampling rate +per tree ∈ [.4, .7], (ix) α regularization ∈ [0, 100], (x) λ regularization ∈ [0, 100]. +OpenML data. +The data sets used in this experiment are listed in Table 6 with their OpenML +names and IDs. For each data set, we utilize 3 different values of the train-validation split ratio +µ/n ∈ {0.1, 0.2, 0.3}. +D +Extended empirical evaluation +D.1 +Further evaluation of Heuristic 3.1 +In Figure 2, we demonstrated the empirical utiity of Heuristic 3.1 for a particular choice of ρin = +ρout = 0 implying we leverage exact ERM in the inner level of the HPO problem (to obtain +ˆgm,λ, λ ∈ Λ) and in the final training of the model on the selected HP ˆλ to obtain ˆfn,ˆλ ∈ Fˆλ. In +this subsection, we evaluate Heuristic 3.1 for other pre-set values of ρin and ρout. We start with +trying ρin ∈ {0.0001, 0.001, 0.01, 0.1} and then setting ρout relative to ρin. +We present the following results for the search space with 36 configuration in Figure 4: (a) Fig- +ure 4a for ρout = ρin/10, (b) Figure 4b for ρout = ρin, (c) Figure 4c for ρout = 2 · ρin, (d) Figure 4d +for ρout = 10ρin. We present the following results for the search space with 18 configuration in +Figure 5: (a) Figure 5a for ρout = ρin/10, (b) Figure 5b for ρout = ρin, (c) Figure 5c for ρout = 2 · ρin, +(d) Figure 5d for ρout = 10ρin. +For the cases where ρout ≤ ρin (Figures 4a and 4b), the excess risk of ˜fn,ˆλ is usually better +than the excess risk of ˜gm,ˆλ, and Heuristic 3.1’s “Choice” makes the right choice when there +is significant difference between the performance of the two candidates. For the cases where +ρout > ρin (Figures 4b and 4c), there are some situations where ˜gm,ˆλ has a (significantly) better +23 + +Table 6: Data set name & OpenML ID of all the data sets used with 10 data sets each in the data +size range. +# samples +Name (OpenML ID) +1000-4999 +kr-vs-kp (3) +credit-g (31) +sick (38) +spambase (44) +scene (312) +yeast-ml8 (316) +fri-c3-1000-25 (715) +fri-c4-1000-100 (718) +abalone (720) +fri-c4-1000-25 (723) +5000-9999 +mushroom (24) +bank8FM (725) +cpu-small (735) +puma32H (752) +cpu-act (761) +delta-ailerons (803) +kin8nm (807) +puma8NH (816) +delta-elevators (819) +bank32nh (833) +10000-49999 +BNG(tic-tac-toe) (137) +electricity (151) +adult (179) +BNG(breast-w) (251) +mammography (310) +webdata-wXa (350) +pol (722) +2dplanes (727) +ailerons (734) +house-16H (821) +50000-99999 +vehicle-sensIT (357) +KDDCup09-app (1111) +KDDCup09-churn (1112) +KDDCup09-upsell (1114) +vehicleNorm (1242) +higgs (23512) +numerai28.6 (23517) +Run-or-walk-info (40922) +APSFailure (41138) +kick (41162) +excess risk over ˜fn,ˆλ. In these cases the Heuristic 3.1 “Choice” is able to make the right choice – +see for example the last row in Figure 4c. +D.2 +Evaluation of Heuristic B.1 +Leveraging the data-dependent selection of ρout with Heuristic B.1, we evaluate the ex- +cess risk incurred by approximating the ERM over Fˆλ with all n samples to ρout toler- +ance. +We present the results for the different values of γ in Heuristic B.1 from the set +{0.001, 0.005, 0.001, 0.005, 0.01, 0.05, 0.01} in Figure 6. The excess risks incurred and the speedups +gained from using ˜fn,ˆλ inplace of ˆfn,ˆλ is visualized in Figure 6 – the solid line corresponds to ˆfn,ˆλ +while the dash-dotted lines correspond to ˜fn,ˆλ for different values of γ. And the results corre- +sponding to the excess risk in Figure 6a indicate that, for γ up to 0.005, the increase in excess-risk +is quite small. The speedups obtained for the different choices of γ and corresponding data- +dependent ρout in Figure 6b. It can be seen that we can get up to 2× speedup over exact ERM +without losing much in terms of the excess risk (see for γ up to 0.005); for larger values of γ we +can get up to 4× speedup if we are ready to incur some additional excess risk. +24 + +(a) ρout = ρin/10 +(b) ρout = ρin +(c) ρout = 2 · ρin +(d) ρout = 10 · ρin +Figure 4: Excess-risk of data-dependent choice between ˜fn,ˆλ and ˜gn,ˆλ based on Heuristic 3.1 for +ρin ∈ {0.0001, 0.001, 0.01, 0.1} and search space with |Λ| = 36. Please magnify to view in detail. +25 + +/n0.1 +/n=0.2 +/n=0.4 +/n0.5 +Pm:0.0,Pou:0.0 +Pin:0.0,Pout:0.0 +Pin:0.0,Pout:0.0 +Pm:0.0,Pot:0.0 +Pin:0.0,Pout:0.0 +23 +/n0.1 +/n-0.2 +nv/n-0.3 +nvfn=0.4 +/n-0.5 +Pin:0.0001.Pout:1e-05 +Pin:0.0001.Pout:1e-05 +Pn:0.0001.Pout:1e-05 +Pn:0.0001.Pot:1e-05 +Pn:0.0001.Pout:1e-05 +2-3 +/n0.1 +/n 0.2 +n/n-0.3 +/m- 0.4 +/n- 0.5 +Pin:0.001.Pot:0.0001 +Pin:0.001.Pot:0.0001 +Pm:0.001.Po:0.0001 +Pin:0.001.Pat:0.0001 +Pm:0.001.Po:0.0001 +2 +Excess risk +2 +nvfn-0.1 +v/n-0.2 +nv/n-0.3 +nvfn=0.4 +nvfn-0.5 +Pm:0.01.Pout:0.001 +Pin:0.01.Pout:0.001 +Pm:0.01.Pout:0.001 +Pin:0.01.Pout:0.001 +Pin:0.01.Pout:0.001 +2-2 +23 +Choice ++ +E(gn.) +2 +E(f,) +211 +212 +214 +21 +21 +214 +2 +212 +214 +21 +212 +214 +20 +222 +214 ++=/n-0.1 +/n0.2 +nvfn0.4 +fn-0.5 +Pm:0.0,Pou:0.0 +Pin:0.0,Pou:0.0 +Pin:0.0.Paut:0.0 +Pm:0.0.Pout:0.0 +Pin:0.0.Pot:0.0 +2~ +2~ +/n=0.1 +/n=0.2 +/n-0.4 +/n=0.5 +Pin:0.0001.Pout:0.0001Pm:0.0001,Pout:0.0001Pin:0.0001.Pout:0.0001Pm:0.0001.Pout:0.0001Pin:0.0001,Pout:0.0001 +2-3 +2 +n/n-0.1 +/n 0.2 +n0.3 +yfn-0.4 +n0.5 +Pin:0.001.Pott:0.001 +Pm:0.001.Po:0.001 +Pm:0.001.Po:0.001 +Pn:0.001.Pot:0.001 +Pin:0.001.Pott:0.001 +Excess risk +2 +/n-0.1 +/n-0.2 +nv/n-0.3 +nv/n-0.4 +/n=0.5 +Pin:0.01.Pout:0.01 +Pin:0.01.Pout:0.01 +Pin:0.01.Pout:0.01 +Pin:0.01.Pout:0.01 +Pm:0.01.Pout:0.01 +2-3 +Choice +E(gn.) +214 +E(f,) +21 +212 +214 +2 +212 +214 +2 +212 +214 +20 +212 +214 +210 +212 +214 +=/n-0.1 +/n0.2 +nvfn0.4 +fn-0.5 +Pm:0.0,Pou:0.0 +Pin:0.0,Pout:0.0 +Pin:0.0.Paut:0.0 +Pm:0.0.Pout:0.0 +Pin:0.0.Pot:0.0 +2-2 +2~3 +/n=0.1 +/n=0.2 +/n-0.4 +/n=0.5 +Pin:0.0001.Pout:0.0002Pm:0.0001,Pout:0.0002Pin:0.0001.Pout:0.0002Pm:0.0001.Pout:0.0002pin:0.0001.Pout:0.0002 +23 +2 +n/n-0.1 +n/n-0.2 +n0.3 +n/n-0.4 +n/n-0.5 +Pin:0.001.Pa:0.002 +Pimn:0.001,Po:0.002 +Pimn:0.001,Pot:0.002 +Pin:0.001,Pa:0.002 +Pin:0.001,Pa:0.002 +Excess risk +2~4 +/n-0.1 +nv/n0.2 +nv/n-0.3 +nv/n-0.4 +/n=0.5 +Pm:0.01.Pout:0.02 +Pin:0.01.Pout:0.02 +Pin:0.01,Pout:0.02 +Pm:0.01.Pout:0.02 +Pin:0.01.Pout:0.02 +2~ +Choice +E(gn.) +E(f,) +21 +212 +214 +2 +212 +214 +2 +232 +214 +20 +212 +214 +210 +2 +214 +=/n0.1 +/n=0.2 +/n=0.4 +/n0.5 +Pm:0.0,Pou:0.0 +Pin:0.0,Pout:0.0 +Pin:0.0,Pout:0.0 +Pm:0.0,Pot:0.0 +Pm:0.0,Pout:0.0 +23 +/n0.1 +/n-0.2 +/n-0.3 +nvfn=0.4 +/n0.5 +Pin:0.0001,Pout:0.001 +Pin:0.0001.Pot:0.001 +Pin:0.0001.Pout:0.001 +Pn:0.0001.Pot:0.001 +Pn:0.0001.Pout:0.001 +2-3 +fn0.1 +/m- 0.2 +n/n-0.3 +/m- 0.4 +nv/n-0.5 +Pm:0.001.Po:0.01 +Pin:0.001.Pot:0.01 +Pm:0.001.Pou:0.01 +Pin:0.001.Pat:0.01 +Pm:0.001.Pou:0.01 +2 +Excess risk +2 +nvfn-0.1 +v/n-0.2 +nv/n-0.3 +nvfn=0.4 +nv/n-0.5 +Pin:0.01.Pourt:0.1 +Pin:0.01.Pourt:0.1 +Pin:0.01.Pout:0.1 +Pin:0.01.Pout:0.1 +Pin:0.01.Pourt:0.1 +22 +Choice +E(gn.) +22 +E(f,) +2h +272 +214 +2 +2 +224 +22 +212 +214 +2 +212 +214 +20 +212 +214 ++=U(a) ρout = ρin/10 +(b) ρout = ρin +(c) ρout = 2 · ρin +(d) ρout = 10 · ρin +Figure 5: Excess-risk of data-dependent choice between ˜fn,ˆλ and ˜gn,ˆλ based on Heuristic 3.1 for +ρin ∈ {0.0001, 0.001, 0.01, 0.1} and search space with |Λ| = 18. Please magnify to view in detail. +26 + +/n-0.1 +v/n-0.2 +nvfn-0.3 +m-0.4 +/n0.5 +Pm:0.0,Pout:0.0 +Pm:0.0,Pout:0.0 +Pm:0.0,Pou:0.0 +Pin:0.0,Pou:0.0 +Pin:0.0,Pou:0.0 +nfn-0.1 +-0.2 +/n=0.3 +-0.4 +/n=0.5 +Pn:0.0001,Pout:0.0001 Pin:0.0001,Pout:0.0001 Pin:0.0001,Pout:0.0001 Pin:0.0001,Pout:0.0001 Pin:0.0001,Pout:0.0001 +/n- 0.1 +/m- 0.2 +/m- 0.4 +n0.5 +Pin:0.001.Pot:0.001 +Pin:0.001,Pa:0.001 +Pm:0.001.Pot:0.001 +Pm:0.001,Po:0.001 +Pm:0.001.Pout:0.001 +risk +2 +Excess +nfn=0.1 +/n=0.2 +nv/n=0.3 +nv/n0.4 +/n=0.5 +Pin:0.01.Pout:0.01 +Pn:0.01.Pout:0.01 +Pn:0.01.Pout:0.01 +Pn:0.01.Pout:0.01 +Pin:0.01.Pout:0.01 +2 +Choice +★ E(gn) +E(f,) +214 +2 +214 +21 +21 +214 +214 +2 +212 +214 +=/n-0.1 +v/n-0.2 +nvfn-0.3 +m-0.4 +/n0.5 +Pm:0.0,Pout:0.0 +Pm:0.0,Pout:0.0 +Pin:0.0,Pout:0.0 +Pin:0.0,Pou:0.0 +Pin:0.0,Pou:0.0 +nfn-0.1 +-0.2 +/n=0.3 +-0.4 +/n=0.5 +Pn:0.0001,Pout:0.0002 Pin:0.0001,Pout:0.0002 Pin:0.0001,Pout:0.0002 Pin:0.0001,Pout:0.0002 Pin:0.0001,Pout:0.0002 +/n- 0.1 +/m- 0.2 +/m- 0.4 +n0.5 +Pin:0.001.Pot:0.002 +Pin:0.001.Pa:0.002 +Pim:0.001,Po:0.002 +Pm:0.001.Pot:0.002 +Pm:0.001.Pot:0.002 +risk +Excess +2 +nfn=0.1 +/n=0.2 +/n=0.3 +nv/n0.4 +/n=0.5 +Pm:0.01.Pout:0.02 +Pin:0.01.Pout:0.02 +Pin:0.01.Pout:0.02 +Pin:0.01.Pout:0.02 +Pin:0.01.Pout:0.02 +Choice +E(f,) +21 +214 +2 +214 +21 +21 +214 +214 +21 +212 +214 +=/n0.1 +/n=0.2 +/n=0.4 +/n=0.5 +Pm:0.0,Pout:0.0 +Pin:0.0,Pout:0.0 +Pin:0.0,Pout:0.0 +Pm:0.0,Pot:0.0 +Pin:0.0,Pout:0.0 +n/n-0.1 +/n=0.2 +nv/n-0.3 +/n=0.4 +/n-0.5 +Pn:0.0001,Pout:0.001 +Pin:0.0001,Pot:0.001 +Pin:0.0001.Pout:0.001 +Pin:0.0001.Pout:0.001 +Pin:0.0001.Pout:0.001 +n0.1 +n/n- 0.2 +n0.3 +/m- 0.4 +/n0.5 +Pm:0.001.Po:0.01 +Pin:0.001.Pott:0.01 +Pm:0.001.Pou:0.01 +Pin:0.001.Pat:0.01 +Pm:0.001.Pou:0.01 +risk +2 +Excess +nvfn-0.1 +v/n-0.2 +nv/n-0.3 +nv/n-0.4 +/n=0.5 +Pin:0.01.Pourt:0.1 +Pin:0.01.Pourt:0.1 +Pin:0.01.Pout:0.1 +Pin:0.01.Pout:0.1 +Pin:0.01.Pout:0.1 +Choice +★ E(gn.) +E(f,) +21 +214 +21 +212 +214 +21h +21 +21 +21 +212 +21 +212 +214 ++=/n0.1 +/n=0.2 +/n=0.4 +/n=0.5 +Pm:0.0,Pout:0.0 +Pm:0.0,Pout:0.0 +Pin:0.0,Pout:0.0 +Pm:0.0,Pout:0.0 +Pm:0.0,Pout:0.0 +n/n-0.1 +/n=0.2 +nv/n-0.3 +/n=0.4 +/n=0.5 +Pin:0.0001.Pout:1e-05 +Pin:0.0001,Pout:1e-05 +Pn:0.0001.Pout:1e-05 +Pin:0.0001.Pout:1e-05 +Qn:0.0001.2cut:1e-05 +/n0.1 +/n- 0.2 +/n- 0.3 +/m- 0.4 +n0.5 +Pin:0.001.Po:0.0001 +Pm:0.001.Po:0.0001 +Pn:0.001.Pa:0.0001 +Pin:0.001.Pot:0.0001 +Pm:0.001.Po:0.0001 +risk +Excess +2 +nvfn-0.1 +/n= 0.2 +nv/n-0.3 +/n=0.4 +v/n=0.5 +Pm:0.01.Pout:0.001 +Pin:0.01.Pout:0.001 +Pm:0.01.Pout:0.001 +Pin:0.01.Pout:0.001 +Pm:0.01.Pout:0.001 +Choice +★ E(gn) +E(f,) +214 +212 +214 +21 +21 +21 +21 +214 +2 +212 +214 +=(a) Excess-risk of ˜f vs. ˆf. +(b) Speedup over exact ERM +Figure 6: Empirical validation of the data-dependent choice of ρout in Heuristic B.1. Please mag- +nify to view in detail. +27 + +nyln=0.1 +nyin=0.2 +nyln=0.3 +nyln=0.4 +nyln=0.5 ++ E(fn) +H-- E(f,),Y=0.0050 +2-2 +H--E(f),= 0.0001 +H--E(ff),Y=0.0100 +Excess risk +→-E(f).= 0.0005 +E(f,), Y= 0.0500 +H-- E(f,),Y= 0.0010 +E(f),Y=0.1000 +22 +212 +214 +23 +212 +224 +21 +22 +214 +23 +212 +224 +22 +222 +214 += += += += +=ERM +nyin=0.1 +nyin=0.2 +nuln=0.3 +ny/n = 0.4 +ny/n = 0.5 +Exact +→-Approx, Y= 0.0050 +23 +exact +Approx, = 0.0001 +Approx, = 0.0100 +Approx, Y= 0.0005 +Approx, Y= 0.0500 +Approx,y=0.0010 +Approx,Y= 0.1000 +over +2 +2 +2 +212 +214 +21 +222 +214 +21 +212 +214 +211 +212 +224 +22 +212 +214 ++= ++= +n= ny + nt += += \ No newline at end of file diff --git a/KtE4T4oBgHgl3EQfiA3u/content/tmp_files/load_file.txt b/KtE4T4oBgHgl3EQfiA3u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..756318e88e84c35f162ee542c37b2b4dac8d63a0 --- /dev/null +++ b/KtE4T4oBgHgl3EQfiA3u/content/tmp_files/load_file.txt @@ -0,0 +1,1588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf,len=1587 +page_content='Toward Theoretical Guidance for Two Common Questions in Practical Cross-Validation based Hyperparameter Selection Parikshit Ram p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ram@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='org IBM Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Watson Research Center, Yorktown Heights, NY, USA Alexander G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Gray alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='gray@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='com IBM Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Watson Research Center, Yorktown Heights, NY, USA Horst C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Samulowitz samulowitz@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='com IBM Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Watson Research Center, Yorktown Heights, NY, USA Gregory Bramble bramble@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='com IBM Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Watson Research Center, Yorktown Heights, NY, USA Abstract We show, to our knowledge, the first theoretical treatments of two common questions in cross-validation based hyperparameter selection: 1 After selecting the best hyper- parameter using a held-out set, we train the final model using all of the training data – since this may or may not improve future generalization error, should one do this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 2 During optimization such as via SGD (stochastic gradient descent), we must set the optimization tolerance ρ – since it trades off predictive accuracy with computation cost, how should one set it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Toward these problems, we introduce the hold-in risk (the error due to not using the whole training data), and the model class mis-specification risk (the error due to having chosen the wrong model class) in a theoretical view which is simple, general, and suggests heuristics that can be used when faced with a dataset instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In proof-of-concept studies in synthetic data where theoretical quantities can be controlled, we show that these heuristics can, respectively, 1 always perform at least as well as always performing retraining or never performing retraining, 2 ei- ther improve performance or reduce computational overhead by 2× with no loss in predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 1 Introduction The learning process has various sources of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The first step in (supervised) learning is the acquisition of (training) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Given data, we choose a model or function class F which corresponds to not just a method (such as Support Vector Machines, Generalized Linear Models, Neural Networks, Decision trees) but their specific configuration governed by their respective hyperparameters (such as regularization forms and penalties, trees depth) – these hyperparameters refer to anything that would affect the predictive performance of the model learned from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Given our choice of the function class, the learning process searches for the function that (approximately) minimizes the empirical risk (or some surrogate of it which better represents the true risk or is easier to optimize).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We currently have an understanding of the factors [Vapnik, 2006, Devroye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2013, Bottou and Bousquet, 2008] affecting the excess risk of this chosen 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='05131v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='LG] 12 Jan 2023 function – (i) the choice of the function class and its capacity to model the data generating process, (ii) the use of an empirical risk estimate instead of the true risk, and (iii) the approximation in the empirical risk minimization (ERM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, in practice, the learning process is not limited to these steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A significant part of the whole exercise is the choice of the function class F (method and its specifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Usually, we consider a (possibly large) set of function classes and select one of them based on the data- driven process of model selection or hyperparameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This search can be done via grid search (searching over a discretized grid of hyperparameter values) or random search [Bergstra and Bengio, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, AutoML (automated machine learning) has spurred a lot of research in the area of hyperparameter optimization or HPO [Hutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2011, Shahriari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2016, Snoek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2012, Bergstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2011, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The automation allows us to look at even larger sets of function classes for improved performance while being significantly more efficient than grid search and more accurate than random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The problem of HPO has been extended from machine learning (ML) model configurations to the design of complete ML pipelines known as the Combined Algorithm Selection and HPO (or CASH) problem, with various schemes that handle (i) pipelines with fixed architecture [Hutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2011, Bergstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2011, Rakotoarison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2019, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2020, Kishimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2022], (ii) searching over the pipeline architectures [Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2020, Marinescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2021], (iii) deployment and fairness constraints [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2020, Ram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2020], and (iv) operating in the decentralized setting [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2021, 2022], leading to multiple open-source tools [Thornton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2012, Kotthoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2017, Feurer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2015, 2020, Komer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2014, Bergstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2015, Baudart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2020, 2021, Hirzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' There has also been a significant amount of theoretical work on development of data- dependent penalties for penalty-based model selection, resulting in guarantees in the form of “oracle inequalities” – the expected excess risk of the selected model can be shown to be within a multiplicative and additive factor of the best possible excess risk if an oracle provided us with the best hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This has been widely studied in (binary) classification [Boucheron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2005], (bounded) regression and density estimation [Massart, 2007, Arlot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, in practice, penalty-based model selection is not used for data-driven hyperparameter selection, and we resort to some form of cross-validation1(CV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These universally applicable CV techniques have been shown to be theoretically competitive to the penalty-based schemes at the cost of having less data for the learning since some amount of data is “held-out” from the training data for validation purposes [Boucheron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2005, Arlot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We focus on these universal CV based HPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' While CV based model selection has been studied theoretically, there are various questions in practical HPO, which have not been explored in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A common prac- tice is the learning of the final model on the selected hyperparameter with all available data, reintroducing the held-out data in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This is standard practice in many commercial ML tools since user data is too precious to not include in the training of the deployed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In this paper, we provide answers for the following questions2: [Q1a] Theoretically, when does this practice help in terms of excess risk and by how much?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' [Q1b] Is there a practical data-driven way of deciding whether to do this or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A common practice is to approximate the ERM with a large tolerance during the HPO and perform a more accurate ERM during the final model training on the selected HP to reduce the overall computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We study the following related questions: 1Existing literature terms the single training/validation split as “cross-validation” (see for example [Kearns, 1996, Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 1999]) and when there are multiple folds, it is specifically termed as “k-fold cross-validation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 2We presented a preliminary version of this work at the AutoML@ICML’21 workshop [Ram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 2 [Q2a] How does this ERM approximation in hyperparameter selection affect the excess risk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' [Q2b] Is there a theoretically informed practical way of setting these approximation levels during the hyperparameter selection to better control the excess risk vs computation tradeoff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Note that these answers can be utilized both by humans and by automated data science sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Practical motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In applications with large amount of data, the importance of question Q1a (and consequently Q1b) might appear minimal (the computational aspect of questions Q2a and Q2b make them critical with large data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, we believe that even in such cases, the important signal can still be quite small, and the relevant information in the held-out set can have large impact (positive or negative) on the final deployed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For example, in various data science applications in finance, the class/target of interest is usually very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In the Home Credit Default Risk Kaggle Classification Challenge, the training data has only around 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5% of some 3 × 105 training examples from the positive class, and in the AllState Claim Prediction Kaggle Regression Challenge, the training set has less than 1% of some 13 × 106 examples with nonzero regression targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' rest are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Furthermore, there are situations where the population has groups and the underrepresented minority groups are significantly smaller than the majority group(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In this case, the presence or absence of the held-out data can have significant (again positive or negative) impact on the fairness and accuracy of the deployed model since many of the fairness metrics quantify the parity in the group specific predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Finally, “small” differences in predictive performance can have significant impact depending on problems and applications – in data science leaderboards such as Kaggle, minor changes in final performance can lead to significant reordering of the leaderboard (although, relevance of such competitions and results to practical problems remain an open question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Such scenarios motivate us to study the question of whether the practice of retraining after reintroducing the data held-out during HPO is helpful (Q1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Empirical motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' As evidence of the lack of clarity on this whether question, we consider HPO for LightGBM [Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2017] on 40 OpenML [Vanschoren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2013] data sets (we detail the evaluation setup in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Of the 400 different HPO problems we solve, retraining has (i) no significant effect (positive or negative) in 51/400 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='75%) cases, (ii) has a positive effect (improving test error) in 260/400 (65%) cases, and (iii) has a negative effect in 89/400 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='25%) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This highlights that there is no single universal correct answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hence, we study this whether question rigorously and provide a theoretically motivated data-driven heuristic as one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We present our precise problem setting, and existing & novel excess risk decomposi- tions with empirical support in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Then we present and evaluate our theoretical results, tradeoffs and practical heuristics for HPO in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We position our contributions against existing literature in §4, and conclude in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 2 Decomposing the excess risk For a particular method (decision trees, linear models, neural networks), let Fλ denote the func- tion class for some fixed hyperparameter λ ∈ Λ (tree depth, number of trees for tree ensembles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' regularization parameter for linear and nonlinear models) in the space of valid hyperparameters (HPs) Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For any model or function f : X → Y with (x, y), x ∈ X , y ∈ Y generated from a distribution P over X × Y, and a loss function ℓ : Y × Y, the expected risk E( f ) and the empirical 3 Fλ f ⋆ ¯fλ ˆfn,λ ˜fn,λ E Eapp Eopt Eest (a) Original decomposition f ⋆ F¯λ Fˆλ E Emcm (b) Model mis-specification error ˜gm,ˆλ ˜fn,ˆλ Ehin Fˆλ (c) Hold-in error Figure 1: Decompositions of excess risk E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 1a shows the decomposition of E incurred by the approximate empirical risk minimizer ˜fn,λ ∈ Fλ with respect to the Bayes optimal f ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 1b visualizes the additional excess risk in the form of the model class mis-specification risk Emcm incurred by selecting the sub-optimal hyperparameter ˆλ in place of ¯λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 1c visualizes the hold-in risk Ehin incurred from using ˜gm,ˆλ instead of ˜fn,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' risk En( f ) with n samples {(xi, yi)}n i=1 ∼ Pn of f is given by E( f ) = E(x,y)∼P [ℓ(y, f (x))] = � ℓ(y, f (x))dP, En( f ) = En [ℓ(y, f (x))] = 1 n ∑ n i=1 ℓ(yi, f (xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (1) We denote the Bayes optimal model as f ⋆ where, for any (x, y) ∼ P, f ⋆(x) = arg min ˆy∈Y E [ℓ(y, ˆy)|x] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (2) We denote with the following: ¯fλ = arg min f ∈Fλ E( f ), ˆfn,λ = arg min f ∈Fλ En( f ), (3) as the true risk minimizer and the empirical risk minimizer (with n samples) in model class Fλ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Table 1 defines the various symbols used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' When performing ERM over Fλ, the excess risk incurred E = E( ˆfn,λ) − E( f ⋆) decomposes into two terms: (i) the approximation risk Eapp(λ) = E( ¯fλ) − E( f ⋆), and (ii) the estimation risk Eest(n, λ) = E( ˆfn,λ) − E( ¯f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For limited number of samples n, there is a tradeoff between Eapp and Eest, where a larger function class Fλ usually reduces Eapp(λ) but increases Eest(n, λ) [Vapnik, 2006, Devroye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Bottou & Bousquet [Bottou and Bousquet, 2008] study the tradeoffs in a “large-scale” setting where the learning is compute bound (in addition to the limited number of samples n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Given any computational budget T, they consider the learning setting “small- scale” when the number of samples n is small enough to allow for the ERM to be performed to arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In this case, the tradeoff is between the Eapp and Eest terms (as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' They consider the large scale setting where the ERM needs to be approximated given the computational budget and discuss the tradeoffs in the excess risk of an approximate empirical risk minimizer ˜fn,λ ∈ Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In addition to Eapp and Eest, they introduce the optimization risk term Eopt – the excess risk incurred due to approximate ERM – and argue that, in compute-bound large-scale learning, approximate ERM on all the samples n can achieve better generalization than high precision ERM on a subsample of size n′ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 1a provides a visual representation of this excess risk decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 4 Table 1: Table of symbols Symbol Description (1st location in text) E( f ) True risk of any model f (1) En( f ) Empirical risk of any model f with n samples (1) Λ Set of L hyperparameters (HPs) λ, L = |Λ| (§2) Fλ Model class for hyperparameter (HP) λ (§2) f ⋆ Bayes optimal predictor (2) ¯fλ True risk minimizer in Fλ (3) ˆfn,λ Empirical risk minimizer in Fλ with n samples (3) ˜fn,λ Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' empirical risk minimizer Fλ with n samples (5) ¯λ Oracle hyperparameter (HP) arg minλ∈Λ E( ˆfn,λ) (§2) ˆλ Solution to empirical hyperparameter selection (4) ˆgm,λ Empirical risk minimizer in Fλ with m samples (4) ˜gm,λ Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' empirical risk minimizer in Fλ with m samples (4) Given a set Λ of L HPs λ ∈ Λ, and n samples from the true distribution, we wish to find the oracle HP ¯λ such that the (approximate) ERM solution ˜fn,¯λ has the best possible excess risk – ¯λ = arg minλ∈Λ E( ˜fn,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' in practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' with n iid (independent and identically dis- tributed) samples from P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' we use cross-validation for model selection and solve the following bilevel problem to pick the HP ˆλ: ˆλ = arg minλ∈Λ Ev µ(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='λ) (outer) ˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='λ ∈ {g : Em(g) ≤ Em(ˆgm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='λ) + ρin} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (inner) (4) where the inner problem is an approximate ERM on Fλ for each λ ∈ Λ with m < n samples at an approximation tolerance of ρin > 0 producing ˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='λ (we use ˆgm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='λ to denote the exact ERM solution in Fλ with m samples),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' and the outer problem considers an objective Ev µ(·) which is evaluated using µ < n samples held-out from the ERM in the inner problem – while Em(·) and Ev µ(·) might have the same form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' the ·v superscript highlights their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Then, a final approximate ERM on Fˆλ with all n samples to ρout tolerance produces ˜fn,ˆλ ∈ � f ∈ Fˆλ : En( f ) ≤ En( ˆfn,ˆλ) + ρout � , (5) with ˆfn,ˆλ denoting the exact ERM solution in Fˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This embodies the common practice of splitting the samples into a training and a held-out validation set (of sizes m, µ < n respectively with m + µ ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In k-fold cross-validation, the inner ERM is solved k times for each HP λ (on k different sets of size m = n − n/k each), and the outer optimization averages the objectives from k held-out sets (of size µ = n/k) across the k learned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In this paper, we focus on CV with a single training-validation split, and defer k-fold CV to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' At this point, multiple choices have to be made for computational and statistical purposes: ▶ The number of samples m drives the computational cost of solving the inner problem for each λ ∈ Λ – larger m requires larger compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ▶ The approximation tolerance ρin in the inner ERM also drives the computational cost – smaller ρin requires larger compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 5 ▶ The approximation tolerance ρout in the final ERM over Fˆλ drives the computational cost similar to ρin but to a lesser extent since it is only over a single ˆλ ∈ Λ instead of for each λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For this reason, ρout is usually selected to be smaller3 than ρin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ▶ The function ˜fn,ˆλ is selected over ˜gm,ˆλ for statistical reasons since the former gets more training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Many of these choices are often made ad hoc or via trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' To the best of our knowl- edge, there is no mathematically grounded way of making some of these practical choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' More- over, it is not clear what is precisely gained by selecting ˜fn,ˆλ over ˜gm,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Existing theoretical guar- antees for CV based model selection focus on the excess risk of ˜gm,ˆλ, while in practical HPO, ˜fn,ˆλ is deployed, indicating a gap between theory and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In this paper, we try to bridge this gap, and in the process, provide a practical heuristic that allows us to select between ˜gm,ˆλ and ˜fn,ˆλ in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Furthermore, ρin and ρout provide a way to control the computation vs excess risk tradeoff, but it is not clear how to set them to extract computational gains without significantly increasing the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We explicitly highlight the role of ρin and ρout in the ex- cess risk and provide practical heuristics to select ρin and ρout in a data-driven manner to better control this tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Novel excess risk decomposition We first present some intuitive decompositions of the excess risk to understand the different sources of additional risk (and gains!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' After the selection of ˆλ by solving problem (4), existing literature focuses on the excess risk of ˜gm,ˆλ, yet we are not aware of any decomposition of its excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We decompose this excess risk as: E = E(˜gm,ˆλ) − E( f ⋆) = E(˜gm,ˆλ) − E(˜gm,¯λ) � �� � Emcm + E(˜gm,¯λ) − E(ˆgm,¯λ) � �� � Eopt + E(ˆgm,¯λ) − E( ¯f ¯λ) � �� � Eest + E( ¯f ¯λ) − E( f ⋆) � �� � Eapp , (6) where ¯f ¯λ, ˆgm,¯λ and ˜gm,¯λ are the true risk minimizer, exact ERM solution and approximate ERM solution respectively in the function class F¯λ corresponding to the oracle HP ¯λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We introduce a new term Emcm = E(˜gm,ˆλ) − E(˜gm,¯λ), the model class mis-specification risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 1b visualizes this term in the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This term incorporates the excess risk from selecting a suboptimal HP (and corresponding model class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, there is a potential additional excess risk that is often ignored in literature but is considered crucial in practice – the risk from learning the model ˜gm,ˆλ on m < n samples instead of all n samples, or the hold-in risk, defined as Ehin = E(˜gm,ˆλ) − E( ˜fn,ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This term is visualized in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This excess risk term does not appear explicitly in the risk decomposition (6) for ˜gm,ˆλ but rather is implicitly incorporated in the estimation risk Eest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' when studying the excess risk of ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' the Ehin does explicitly appear in the decomposition: E = E( ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) − E( f ⋆) = E( ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) − E(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) � �� � −Ehin + E(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) − E(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='¯λ) � �� � Emcm + E(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='¯λ) − E( f ⋆) � �� � Eopt+Eest+Eapp (see (6)) (7) This excess risk decomposition for ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ is different from previous decompositions in that the “−Ehin” term in this excess risk decomposition is potentially a risk deficit instead of an additional 3Often,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' for computational reasons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' m might be much less than n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' and training the final ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ on all n samples to a tolerance of ρin might be computationally infeasible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' making ρout > ρin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 6 Table 2: Empirical (relative) estimate of Emcm = E(˜gm,ˆλ) − E(˜gm,¯λ) across 40 OpenML datasets of varying number of samples n and varying sizes of the held-out validation set µ/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We report the percentage of the experiments (for each combination of n and µ/n) where ˆλ ̸= ¯λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For these experiments, we also report the (estimate of the) average relative excess risk ˜∆ incurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' n µ/n AuROC Acc ˆλ ̸= ¯λ ˜∆ ˆλ ̸= ¯λ ˜∆ 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='00 75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='85 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='12 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='43 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='12 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='14 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='08 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='69 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='11 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='10 85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='27 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='34 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='25 85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='27 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='18 risk, highlighting potential risk we can recover from this common practice of training on all the data with the selected HP ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These decompositions are intended to explicitly highlight the dif- ferent sources of risk (and gains) in the practical HPO process, providing some intuition into the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 Empirical Validation To evaluate the practical significance of these newly introduced risk terms Emcm and Ehin, we consider the HPO problem with LightGBM [Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2017] across 40 OpenML binary classification datasets [Vanschoren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We consider 10 datasets each with number of rows in the ranges 1000-5000, 5000-10000, 10000-50000 and 50000-100000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For each dataset, we consider 3 different values of µ/n (the held-out fraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We perform this exercise with two classification metrics – area under the ROC curve (AuROC) and balanced accuracy (Acc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We approximate the true risks for the post-hoc analysis using an additional test set not involved in the HPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We detail the datasets and HP search space in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For each HPO experiment (dataset and held-out fraction), we note whether the selected HP ˆλ matches the oracle HP ¯λ (found post-hoc using the test set), and the (relative) estimate ˜∆ of Emcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We report the aggregate findings for each set of size range and held-out fraction in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The results indicate that, with the smaller datasets (n ∈ 1000-5000), a higher value of µ/n reduces the chances of missing the oracle HP ¯λ, but this effect is no longer present with the larger datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' As the dataset sizes increase, the chances of missing the oracle HP does increase on aggregate, but the relative risk ˜∆ decreases from 2% down to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' So the Emcm term benefits from larger data but the effect is still significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, there is no explicit indication of how the different terms such as n, µ play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' To understand the impact of Ehin, we further compare the performance of the ˜gm,ˆλ involved in the HPO to the final retrained ˜fn,ˆλ in the above HPO experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We note the percentage of the time (i) their performances were within a relative difference of 10−5 ( ˜f ≈ ˜g), (ii) ˜fn,ˆλ was 7 Table 3: Comparing relative performances of ˜fn,ˆλ and ˜gm,ˆλ in HPO of LightGBM with balanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' n µ/n ˜f ≈ ˜g ˜f✓ ˜f gain ˜g✓ ˜g gain 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 40 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='25 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 25 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='81 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 30 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='57 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='50 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 30 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='51 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 20 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='47 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 20 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='82 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='74 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 10 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='17 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 5 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='29 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 20 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='54 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='03 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 15 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='16 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 15 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 0 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='16 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='03 Table 4: Comparing relative performances of ˜fn,ˆλ and ˜gm,ˆλ in HPO of LightGBM with AuROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' n µ/n ˜f ≈ ˜g ˜f✓ ˜f gain ˜g✓ ˜g gain 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 5 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='79 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 0 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='42 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 10 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='34 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='27 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 15 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='15 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 10 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='24 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 10 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='39 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='05 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 5 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 0 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='09 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='21 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='00 50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 10 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='15 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 5 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='18 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 0 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='35 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='18 better ( ˜f✓), and (iii) ˜gm,ˆλ was better (˜g✓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In both cases (ii) and (iii), we noted the average relative gain the better choice provided (“ ˜f gain” and “˜g gain”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The results are aggregated across dataset sizes and held-out fraction µ/n in Table 3 for the balanced accuracy metric and in Table 4 for the AuROC metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The results indicate that, in most cases, ˜fn,ˆλ is a better choice, justifying the common practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, it also indicates that, in a significant fraction of the cases (around 20% in most but up to 40%), ˜gm,ˆλ appears to be the better choice against common intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The results also indicate that, when ˜fn,ˆλ is the better choice, it also provides higher relative gains over ˜gm,ˆλ on average across most experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' But the average relative gains of ˜gm,ˆλ over ˜fn,ˆλ are still significant in most cases across both classification metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These results indicate that there is no single best choice and we can obtain improved per- formance if we are able to make this choice in a more problem-dependent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In the next 8 section, we theoretically bound the excess risk to explicitly understand the impact of the different choices in the HPO and leverage these dependencies for improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 3 Bounding the excess risk In this section, we bound the excess risks of ˜gm,ˆλ and ˜fn,ˆλ and try to understand any improvement ˜fn,ˆλ might provide and the interaction with the ERM approximation tolerances ρin and ρout based on our decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We have the following result for ˜gm,ˆλ: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Let P be a distribution over X × Y and let ℓ : Y × Y, Y ⊂ R, be a B-bounded β-Lipschitz loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Let Fλ be a class of functions f : X → Y for any HP λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Let ˆλ be the solution of (4) over the set of L HPs Λ with ERM on m < n samples to approximation tolerance ρin ≥ 0, a held-out set of size µ < n, and m + µ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' With probability at least 1 − δ for δ ∈ (0, 1), the excess risk E of ˜gm,ˆλ in (6) is bounded as: E ≤ minλ∈Λ � 8 β Rm(Fλ) + Eapp(λ) � + ρin + 2B � log(2(L+1)/δ) (2/ √ 2m + 1/√ 2µ) , (8) where Rm(Fλ) is the Radamacher complexity of Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The above bound (8) highlights how the different terms affect the excess risk bounds – we get an excess risk bound within an additive factor of the class that possesses the minimum combined (scaled) Radamacher complexity (a proxy for estimation risk Eest [Bartlett and Mendelson, 2002]) and approximation risk Eapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Note that the Radamacher complexity is with respect to m < n samples, highlighting the statistical inefficiency introduced by the held-out data for CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The result also indicates that a larger held-out set (larger µ) is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We study this excess risk bound to identify the effect of the choices in HPO (such as m, µ, ρin);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' the best choice needs to balance the terms in the bound – making one term, such as ρin much smaller (say, by an order of magnitude) than the other terms will not improve the excess risk significantly, but an order of magnitude larger ρin will have significant ill-effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We have the following result for ˜fn,ˆλ: 4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, let ˜fn,ˆλ ∈ Fˆλ be obtained via approximate ERM with tolerance ρout ≥ 0 over all n samples in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Let I ˆλ n,m(ρin, ρout) := En(˜gm,ˆλ) − En( ˜fn,ˆλ) be the “empirical risk improvement” from performing the ERM over all n samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' With probability at least 1 − δ for any δ ∈ (0, 1), the excess risk E of ˜fn,ˆλ in (7) is bounded as: E ≤ min λ∈Λ � 8 β Rm(Fλ) + Eapp(λ) � + 8 β Rn(Fˆλ) + ρin + B′ (2/ √ 2n + 2/ √ 2m + 1/√ 2µ) − ¯I (9) where B′ := 2B � log(2(L+2)/δ), and ¯I := max � I ˆλ n,m(ρin, ρout), I ˆλ n,m(0, 0) − ρout − (µ/n)B � , with I ˆλ n,m(0, 0) = En(ˆgm,ˆλ) − En( ˆfn,ˆλ) denoting the empirical risk improvement if ρin = ρout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 4With further structural assumptions (such as relationships between the variance and expectations of the functions), we can improve the 1/√µ dependence to 1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' See for example, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We focus on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 since this will subsequently allow us to derive practical heuristics that we cannot with the structural assumptions required to get the tighter excess risk bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 9 Table 5: Tradeoffs of the terms in the excess risk (6) & (7) based on the various choices in HPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We use Fall = ∪λ∈ΛFλ to denote the union of all function classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ‘↑’ denotes an increase while ‘↓’ a decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Fall ↑ |Λ| ↑ n ↑ µ ↑ m ↑ ρin ↑ ρout ↑ Eapp ↓ ↓ Eest ↑ ↓ ↑ ↓ Eopt ↑ Emcm ↑ ↓ ↓ Ehin ↓ ↑ ↓ ↑ The proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This result indicates that we are only able to recover the hold-in risk Ehin in terms of the excess risk if the empirical risk improvement I ˆλ n,m(ρin, ρout) is relatively significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' While I ˆλ n,m(0, 0) is always ≥ 0 by definition of the ERM solution ˆfn,ˆλ, the quantity I ˆλ n,m(ρin, ρout) depends more closely to ρout and, we should make ρout small enough to extract any gain from retraining the final ˜fn,ˆλ on all the n samples – ρout should not exceed a critical point where the I ˆλ n,m(ρin, ρout) term is of the same order as the 8βRn(Fˆλ) and the B′(2/ √ 2n) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We cannot compute this critical point, but we will discuss how we can select ρout in a data-driven way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Comparing Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 allows us to provide a theoretical answer to Q1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In HPO, a critical choice is the value of ρin, properly balancing the computational and generalization aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 highlight the roles of these ERM approximation tolerances ρin and ρout, providing an answer for Q2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These results also allow us to conceptually understand the tradeoffs better the different terms in the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A visualization of these tradeoffs is presented in Table 5, highlighting the effect of the individual parameters of the problems (n, m, µ, |Λ|, etc) on the different terms in the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For example, HPO over a large set of HPs Λ will potentially reduce the approximation risk Eapp, but might increase the model class mis-specification risk Emcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Or increasing the validation set size µ would reduce Emcm but increasing µ implies reduced m (for fixed n), leading to higher estimation risk Eest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We believe that understanding these tradeoffs explicitly will allow us to obtain better generalization with HPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Data-driven heuristics In certain situations, ρin and ρout are specified outside our control (for example, with models learned with techniques other than gradient descent like decision tree), and hence it is not clear which learned model, ˜gm,ˆλ or ˜fn,ˆλ, is better to deploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For this, we present a heuristic to make a data-driven choice between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In practical scenarios, the approximation risk Eapp usually dominates the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hence, if we assume that minλ{8βRm(Fλ) + Eapp(λ)} dominates 8βRn(Fˆλ) (the latter does not have Eapp term and m < n) then we can compare the excess risk bounds of ˜gm,ˆλ and ˜fn,ˆλ and utilize the following heuristic (note that log 2(L + 1) ≈ log 2(L + 2) except from really small L), providing a data-driven answer to Q1b: Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Based on the quantities in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 and δ > 0, we select ˜fn,ˆλ if I ˆλ n,m > 2B � 2 log(2(L+2)/δ)/n, else we select ˜gm,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 10 To answer Q2b, we need a way to make an informed choice in terms of ρin and ρout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Intuitively, we can compare ρin to the other computable terms in the bounds (8) and (9) – if ρin is of the order of these terms or larger, the Eopt risk will contribute significantly to the excess risk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' if ρin is an order of magnitude smaller that this term, then the Emcm risk will dominate the Eopt and any further reduction in ρin is not beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Based on this observation, we propose another heuristic for HPO with approximate ERM to facilitate the choice of ρin in the inner ERM of the bilevel problem (4): Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Based on the quantities in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 and a scaling parameter γ > 0, we set ρin = γ B � 2 log(2(L+1)/δ)(2/√m + 1/√µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (10) A larger γ will imply a more computationally efficient HPO, while a smaller γ will improve excess risk up until a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' As we will demonstrate in our experiments, γ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 is sufficiently small5 such that we still gain computational efficiency via the ERM approximation but not see any adverse effect on the excess risk of ˜gm,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' While we have a very precise way of setting ρin given the theoretical result, the choice of ρout is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We present this in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Note that the choice of ρout does not play a significant role in the computational cost of the HPO compared to ρin since ρin is involved in the training for each λ ∈ Λ while ρout only influences the final training of ˜fn,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For this reason, if possible, ρout is chosen to be significantly smaller than ρin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Heuristics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 (Appendix B) are our answers to Q2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 Empirical evaluation We evaluate our heuristics on HPO with neural network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We consider neural networks to have better control over ρin in the ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We chose a synthetic data distribution to have control over the experiment and to be able to generate fresh large samples to accurately estimate the true risks of the different models (as opposed to our results in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 which were based on the true risk estimated on a limited test set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This is common practice when empirically studying theoretical bounds on various statistical quantities (see for example [Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This also allows us to perform the empirical evaluation under various setting (such as different n, µ, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We set the Bayes optimal risk E( f ⋆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We consider two HPO problems with grid-search: (a) one with 36 HP configurations (L = |Λ| = 36), and (b) another with 18 HP configurations (L = |Λ| = 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The data generation and the HP search spaces are detailed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We estimate the true risk E( f ) of any model f with a separate large test sample from the synthetic data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We consider sample sizes n ∈ [29, 214] and different values for µ/n ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5] with m set to (n − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Each ERM involves 5 restarts and the results are averaged over 10 trials (corresponding to different samples from the same distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We set the failure probability δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We first evaluate the practical utility of Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, which tries to balance the gain from utilizing the full data for obtaining ˜fn,ˆλ and the associated statistical cost of an additional ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 2 compares this “Choice” based on Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 (thick translucent red line) to ˜fn,ˆλ & ˜gm,ˆλ (solid blue ▲ & green ⋆ respectively) for a subset of the combinations of ρin, ρout and µ/n, showing the excess risk on the vertical axis as the number of samples n is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The results indicate that, depending on Λ, ρin, ρout and µ/n, ˜gm,ˆλ might be preferable to ˜fn,ˆλ and vice versa – one is not always better than the other (as we also highlighted in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2), and always selecting ˜fn,ˆλ (as 5This γ implies that the optimization risk Eopt is an order of magnitude smaller than atleast some other term in the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 11 (a) HPO with |Λ| = 36 (b) HPO with |Λ| = 18 Figure 2: Empirical utility of Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 for data dependent choice of ˜fn,ˆλ vs ˜gm,ˆλ with a subset of the varying values of n, µ/n, ρin, ρout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The vertical axis is the excess risk (lower is better) and the horizontal axis is the total number of available samples n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Note the logscale on both axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 for results on all combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' done in practice) leaves room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In both types of cases, Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 is able to select the better options in many cases – the proposed heuristic provides a data-driven way of selecting between ˜fn,ˆλ and ˜gm,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We provide the full set of results for all considered values of ρin, ρout, µ/n for both HPO problems in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' There are cases where the heuristic does not make the right choice, which indicates that there is room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' To demonstrate the practical utility of the proposed Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2, we continue with the afore- mentioned HP selection problem over 36 neural network configurations on a synthetic classifica- tion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We consider three choices γ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, 1, 10} in Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We set δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='05 and consider different values of n and µ/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 3 compares the performance of the HPO with exact ERM using ˆgm,λ, λ ∈ Λ to the HPO with approximate ERM for different ρin using ˜gm,λ instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Figure 3a, we compare the excess risk incurred from approximate ERM with the data dependent choice of ρin compared to exact ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We see that γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 leads to a sufficiently small ρin that 12 ny/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Excess risk 2 Choice E(gn) 2 E(f,) 21 212 214 212 214 212 224 2 22 214 +=n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 nv/n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 risk ExcessI Choice ★ E(gn) ← E(fn) 21 272 214 212 224 210 212 224 211 272 214 +=(a) Excess-risk from Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (b) Speedup from Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 3: Empirical validation of the utility of Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 for data-dependent choice of ρin matches the predictive performance of exact ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Any smaller approximation ρin would not improve the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The results also indicate that γ = 10 leads to a ρin where the optimiza- tion error dominates the excess risk, implying that ρin should be reduced if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Figure 3b presents the computational speedups obtained for the corresponding data-dependent choices of ρin – we see that we can get a 2× speedup over exact ERM without any degradation in excess risk with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 while obtaining around 4 − 8× speedup with slight degradation in performance with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These results provide empirical evidence for the practical utility of the proposed Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 obtained from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 – the proposed heuristic provides a data driven way of setting the ERM approximation tolerance in HPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 4 Related work While the use of smaller number of samples m < n to select the HP ˆλ is often recognized in practice, and leads to the final model being learned via ERM on Fˆλ using all n samples, no theoretical guarantees exist for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We explicitly study this situation, introducing the hold-in risk Ehin, and provide a novel guarantee for such a procedure in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Kearns [Kearns, 1996] studies the interaction between the approximation and estimation risks, and under certain assumptions and restricted class of functions, proposes ways for selecting the sizes of the training and held-out splits m and µ in an informed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' To account for the fact that some of the training data is “wasted” as the held-out set, Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' [Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 1999] propose two different ways of retaining the Hoeffding bounds of the error estimate on the held-out set while still being 13 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 nyin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 2~1 E(gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=') H-- E(9n,),Y= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 E(9n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ), Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 H-_E(9n,), Y= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Excess risk 2 210 212 214 226 212 214 22 212 214 23 212 224 22n 212 214 += += n=ny+nt = =exact ERM ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Exact H-- Approx, Y= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 27 Approx, : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 H-- Approx, Y= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 21 210 222 214 220 212 224 212 214 212 214 21 212 214 += n=ny+nt += n=ny+nt +=able to utilize the full training data to train models employed at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These techniques are ways of modifying the standard CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In addition to the above, there are various theoretical analyses focusing on various aspects of the CV process such as obtaining tight variance estimates for the k-fold CV score of any given HP λ ∈ Λ [Nadeau and Bengio, 1999, 2003, Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2009, Markatou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' These results are complementary to ours and could be used to extend our current results (for the single training/validation split based CV) to k-fold CV, with the variance estimates for the k-fold CV metrics involved in the data-driven heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We will pursue this in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, note that these existing results do not directly help us obtain tighter excess risk bounds or allow comparison between models ˜gm,ˆλ (used during the HPO) and the final deployed model ˜fn,ˆλ or provide any intuition regarding the choices for ρin and ρout, which are the main questions (Q1a, Q1b, Q2a, Q2b) we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Finally, as we discussed in §1, HPO has been widely studied over the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' However, the questions we focus on are complementary to any specific HPO scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We do not focus on how the HP was found (with any specific HPO scheme such Bayesian Optimization [Shahriari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2016]), but rather on (i) the ERM involved in the evaluation of any HP during the HPO, and (ii) the ERM involved in the final deployment after an HP is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Our Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 allows us to speed up any HPO scheme without any additional excess risk, and our Heuristics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 allow us to improve the predictive performance of the deployed model for the HP ˆλ selected via any HPO scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' “Multi-fidelity” HPO schemes [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2018, Jamieson and Talwalkar, 2016, Sabharwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2016, Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2017, Falkner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2018] significantly improve the compu- tational efficiency by adaptively setting either the training set size m < n or the optimization approximation ρin on a per-HP basis instead of using a single value of m or ρin for all λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This is quite different from our proposed Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 and the excess risk introduced by this adaptive strategy is not studied to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We wish to extend our tradeoff analysis to multi-fidelity HPO in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Meta-learning is another way of improving the efficiency of the HPO process [Vanschoren, 2018], and has been used in some AutoML toolkits [Feurer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2015, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Recently, some theoretical guarantees have been established for such meta-learning based HPO [Ram, 2022], and we also wish to extend our tradeoff analysis to such meta-learning based HPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Note that the proposed Heuristics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 is still beneficial in both the above situations (multi-fidelity and meta-learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 5 Conclusions Our contributions focus on aspects of CV based HPO – we explore how to leverage the different tradeoffs in the excess risk to make various practical decisions in the HPO process in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We use the novel excess risk decomposition and theoretical analyses to answer the two questions in CV-based HPO: (1) When is the process of training the model on all the data after the HPO beneficial and can we choose between the two in a data driven manner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (Q1a, Q1b) (2) At what level should we set the tolerance of the optimization involved in model training during HPO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (Q2a, Q2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The ideas can be utilized by data science practitioners as well as by automated data science systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' References Sylvain Arlot, Alain Celisse, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A survey of cross-validation procedures for model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Statistics surveys, 4:40–79, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 14 Peter L Bartlett and Shahar Mendelson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Rademacher and Gaussian complexities: Risk bounds and structural results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Journal of Machine Learning Research, 3(Nov):463–482, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Guillaume Baudart, Martin Hirzel, Kiran Kate, Parikshit Ram, and Avraham Shinnar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Lale: Con- sistent automated machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In 4th KDD Workshop on Automation in Machine Learning (AutoML@KDD), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Guillaume Baudart, Martin Hirzel, Kiran Kate, Parikshit Ram, Avi Shinnar, and Jason Tsay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Pipeline combinators for gradual automl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:19705–19718, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Bergstra and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Random search for hyper-parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Journal of Machine Learning Research, 13(1):281–305, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' James Bergstra, R´emi Bardenet, Yoshua Bengio, and Bal´azs K´egl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Algorithms for hyper-parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Advances in neural information processing systems, 24, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' James Bergstra, Daniel Yamins, and David Cox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Making a science of model search: Hyperparam- eter optimization in hundreds of dimensions for vision architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In International conference on machine learning, pages 115–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' James Bergstra, Brent Komer, Chris Eliasmith, Dan Yamins, and David D Cox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hyperopt: a python library for model selection and hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Computational Science & Discovery, 8(1):014008, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Avrim Blum, Adam Kalai, and John Langford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Beating the hold-out: Bounds for k-fold and pro- gressive cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the twelfth annual conference on Computational learning theory, pages 203–208, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' L´eon Bottou and Olivier Bousquet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The tradeoffs of large scale learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Advances in neural information processing systems, pages 161–168, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' St´ephane Boucheron, Olivier Bousquet, and G´abor Lugosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Theory of classification: A survey of some recent advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ESAIM: probability and statistics, 9:323–375, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Luc Devroye, L´aszl´o Gy¨orfi, and G´abor Lugosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A probabilistic theory of pattern recognition, vol- ume 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Stefan Falkner, Aaron Klein, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' BOHB: Robust and efficient hyperparameter optimization at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1437–1446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Matthias Feurer, Aaron Klein, Katharina Eggensperger, Jost Springenberg, Manuel Blum, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Efficient and robust automated machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Advances in Neural Informa- tion Processing Systems, pages 2962–2970, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Matthias Feurer, Katharina Eggensperger, Stefan Falkner, Marius Lindauer, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Auto-sklearn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0: Hands-free AutoML via meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='04074, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Isabelle Guyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Design of experiments of the NIPS 2003 variable selection benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In NIPS 2003 workshop on feature extraction and feature selection, volume 253, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Martin Hirzel, Kiran Kate, Parikshit Ram, Avraham Shinnar, and Jason Tsay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Gradual automl using lale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 4794–4795, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 15 Frank Hutter, Holger H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hoos, and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Sequential model-based optimization for general algorithm configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the 5th International Conference on Learning and Intelligent Optimization, LION’05, pages 507–523, Berlin, Heidelberg, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Kevin Jamieson and Ameet Talwalkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Non-stochastic best arm identification and hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Artificial Intelligence and Statistics, pages 240–248, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Michael Katz, Parikshit Ram, Shirin Sohrabi, and Octavian Udrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Exploring context-free lan- guages via planning: The case for automating machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the Interna- tional Conference on Automated Planning and Scheduling, volume 30, pages 403–411, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' LightGBM: A highly efficient gradient boosting decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Michael J Kearns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A bound on the error of cross validation using the approximation and estimation rates, with consequences for the training-test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 183–189, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Akihiro Kishimoto, Djallel Bouneffouf, Radu Marinescu, Parikshit Ram, Ambrish Rawat, Martin Wistuba, Paulito Palmes, and Adi Botea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Bandit limited discrepancy search and application to machine learning pipeline optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 10228–10237, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Aaron Klein, Stefan Falkner, Simon Bartels, Philipp Hennig, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Fast bayesian hyperparameter optimization on large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Electronic Journal of Statistics, 11(2):4945–4968, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Brent Komer, James Bergstra, and Chris Eliasmith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hyperopt-sklearn: automatic hyperparameter configuration for scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Citeseer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Lars Kotthoff, Chris Thornton, Holger H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hoos, Frank Hutter, and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Auto- WEKA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0: Automatic model selection and hyperparameter optimization in weka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Journal of Machine Learning Research, 18(1):826–830, January 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ISSN 1532-4435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' URL http://dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' org/citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='id=3122009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3122034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Lisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hyper- band: A novel bandit-based approach to hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Journal of Machine Learn- ing Research, 18(185):1–52, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Sijia Liu, Parikshit Ram, Deepak Vijaykeerthy, Djallel Bouneffouf, Gregory Bramble, Horst Samu- lowitz, Dakuo Wang, Andrew Conn, and Alexander Gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' An admm based framework for automl pipeline configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 4892–4899, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='org/abs/1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='00424v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Radu Marinescu, Akihiro Kishimoto, Parikshit Ram, Ambrish Rawat, Martin Wistuba, Paulito P Palmes, and Adi Botea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Searching for machine learning pipelines using a context-free grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 8902–8911, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Marianthi Markatou, Hong Tian, Shameek Biswas, and George Hripcsak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Analysis of variance of cross-validation estimators of the generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Journal of Machine Learning Research, 6: 1127–1168, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Pascal Massart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Concentration inequalities and model selection: Ecole d’Et´e de Probabilit´es de Saint-Flour XXXIII-2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Springer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 16 Claude Nadeau and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Inference for the generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Advances in neural information processing systems, 12, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Claude Nadeau and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Inference for the generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Machine Learning, 52: 239–281, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' PyTorch: An imperative style, high-performance deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32: 8026–8037, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Pedregosa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Varoquaux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Gramfort, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Michel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Thirion, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Grisel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Blondel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Pretten- hofer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Weiss, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Dubourg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Vanderplas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Passos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Cournapeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Perrot, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Duchesnay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Scikit-learn: Machine learning in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Journal of Machine Learning Research, 12:2825–2830, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Herilalaina Rakotoarison, Marc Schoenauer, and Michele Sebag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Automated machine learning with Monte-Carlo Tree Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19, pages 3296–3303, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Parikshit Ram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' On the optimality gap of warm-started hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Interna- tional Conference on Automated Machine Learning, pages 12–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Parikshit Ram, Sijia Liu, Deepak Vijaykeerthi, Dakuo Wang, Djallel Bouneffouf, Greg Bramble, Horst Samulowitz, and Alexander G Gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Solving constrained CASH problems with ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In 7th ICML Workshop on Automated Machine Learning (AutoML), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Parikshit Ram, Alexander G Gray, and Horst Samulowitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Leveraging theoretical tradeoffs in hy- perparameter selection for improved empirical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In 8th ICML Workshop on Automated Machine Learning (AutoML), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Juan D Rodriguez, Aritz Perez, and Jose A Lozano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Sensitivity analysis of k-fold cross validation in prediction error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 32(3): 569–575, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Ashish Sabharwal, Horst Samulowitz, and Gerald Tesauro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Selecting near-optimal learners via incremental data allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In Thirtieth AAAI Conference on Artificial Intelligence, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Shahriari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Swersky, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Adams, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' De Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Taking the human out of the loop: A review of bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Proceedings of the IEEE, 104(1):148–175, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Jasper Snoek, Hugo Larochelle, and Ryan P Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Practical bayesian optimization of machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Advances in neural information processing systems, 25, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Chris Thornton, Holger H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Hoos, Frank Hutter, and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Auto-WEKA: Auto- mated selection and hyper-parameter optimization of classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' arXiv, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='org/abs/1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Joaquin Vanschoren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Meta-learning: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='03548, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Joaquin Vanschoren, Jan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' van Rijn, Bernd Bischl, and Luis Torgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' OpenML: Networked science in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' SIGKDD Explorations, 15(2):49–60, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1145/2641190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2641198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' URL http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1145/2641190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2641198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 17 Vladimir Vapnik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Estimation of dependences based on empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Springer Science & Business Media, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Yi Zhou, Parikshit Ram, Theodoros Salonidis, Nathalie Baracaldo, Horst Samulowitz, and Heiko Ludwig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' FLoRA: Single-shot hyper-parameter optimization for federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In 1st NeurIPS Workshop on New Frontiers in Federated Learning (NFFL 2021), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Yi Zhou, Parikshit Ram, Theodoros Salonidis, Nathalie Baracaldo, Horst Samulowitz, and Heiko Ludwig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Single-shot hyper-parameter optimization for federated learning: A general algorithm & analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='08338, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 18 A Detailed Proofs We will make sure of this standard result: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ([Bartlett and Mendelson, 2002]) Consider conditions and notations of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Then for any δ > 0, with probability at least 1 − δ, the following is true: sup f ∈Fλ |En( f ) − E( f )| ≤ 4 β Rn(Fλ) + 2 B � log(1/δ)/2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (11) where Rn(Fλ) is the Radamacher complexity of Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1) By definition of Fˆλ, for any λ ∈ Λ and δ′ > 0, we have the following relationships: E(˜gm,ˆλ) (a) ≤ Ev µ(˜gm,ˆλ) + B � log(1/δ′)/2µ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − δ′ (12) (b) ≤ Ev µ(˜gm,λ) + B � log(1/δ′)/2µ (13) (c) ≤ E(˜gm,λ) + 2B � log(1/δ′)/2µ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − δ′, (14) where (a) and (c) are obtained from an application of Hoeffding’s inequality while (b) is obtained from the definition of ˆλ in (4) as the minimizer of Ev µ(˜gm,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For any Fλ, λ ∈ Λ, we have the following: E(˜gm,λ) (d) ≤ Em(˜gm,λ) + 4βRm(Fλ) + 2B � log(1/δ′)/2m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − δ′ (e) ≤ Em(ˆgm,λ) + ρin + 4βRm(Fλ) + 2B � log(1/δ′)/2m ( f ) ≤ Em( ¯fλ) + ρin + 4βRm(Fλ) + 2B � log(1/δ′)/2m (g) ≤ E( ¯fλ) + ρin + 8βRm(Fλ) + 4B � log(1/δ′)/2m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − δ′ (h) ≤ E( f ⋆) + Eapp(λ) + ρin + 8βRm(Fλ) + 4B � log(1/δ′)/2m, (15) where (d) and (g) are obtained by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, (e) is obtained from the problem definition (4) for the approximate ERM solution, ( f ) is obtained from the definition of ˆgm,λ as the ERM solution for any class Fλ, and (h) is a simple application of the definition of the approximation risk Eapp(λ) of any class Fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Since the above holds for any λ ∈ Λ, putting (14) and (15) together using the union bound with δ′ = δ/(2 + 2|Λ|) = δ/2(L + 1) and minimizing over λ ∈ Λ gives us the desired result in (8) with a failure probability of at most δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2) By definition of Fˆλ and ˜fn,ˆλ ∈ Fˆλ, we have the following with an application of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − δ′: E( ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) ≤ En( ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) + 4βRn(Fˆλ) + 2B � log(1/δ′)/2n (16) 19 Now from the problem definition and the definition of I ˆλ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='m: En( ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) = En(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) − I ˆλ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='m(ρin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ρout) (17) However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' we have an alternate relationship as follows: En( ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) (a) ≤ En( ˆfn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) + ρout (18) (b) = En(ˆgm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) − I ˆλ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='m(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 0) + ρout (19) (c) ≤ En(˜gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ) + µ n B − I ˆλ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='m(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 0) + ρout,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (20) where (a) is from the definition of the approximate ERM solution ˜fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ and the exact ERM solution ˆfn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (b) is obtained from the definition of I ˆλ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='m(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 0) in the statement of the theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' and (c) is from the definition of ˆgm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ˆλ as the minimizer of Em(·) in Fˆλ and the fact that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' for any f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' En( f ) = (m/n)Em( f ) + (µ/n)Eµ( f ) and the loss ℓ is bounded by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Putting (17) and (20) gives us: En( ˜fn,ˆλ) ≤ En(˜gm,ˆλ) − max � I ˆλ n,m(ρin, ρout), I ˆλ n,m(0, 0) − µ n B − ρout � � �� � := ¯I in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Using above in (16), we get the following relationships: E( ˜fn,ˆλ) ≤ En(˜gm,ˆλ) − ¯I + 4βRn(Fˆλ) + 2B � log(1/δ′)/2n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − δ′ ≤ E(˜gm,ˆλ) − ¯I + 8βRn(Fˆλ) + 4B � log(1/δ′)/2n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − 2δ′, where the last inequality is an application of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 on ˜gn,ˆλ ∈ Fˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Combining (21) with (14) (which holds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − 2δ′) and (15) (which holds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − 2δ′ for each λ ∈ Λ) and applying the union bound over all λ ∈ Λ, we have E( ˜fn,ˆλ) − E(˜gm,ˆλ) ≤ − ¯I + 8βRn(Fˆλ) + 4B � log(1/δ′)/2n = − ¯I + 8βRn(Fˆλ) + B′(2/ √ 2n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − 2δ′, (21) We also know that for all λ ̸= ˆλ ∈ Λ, E(˜gm,ˆλ) − E(˜gm,λ) ≤ 2B � log(1/δ′)/2µ = B′(1/√ 2µ) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − 2δ′, E(˜gm,λ) − E( f ⋆) ≤ min λ∈Λ [Eapp(λ) + ρin + 8βRm(Fλ)] + 4B � log(1/δ′) 2m = min λ∈Λ [Eapp(λ) + ρin + 8βRm(Fλ)] + B′ � 2 √ 2m � w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − 2Lδ′ where we replace 2B � log(1/δ′) with B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Putting the above together, we get the upperbound for E = E( ˜fn,ˆλ) − E( f ⋆) in the statement of the claim with probability at least 1 − (2 + 2L + 2)δ′ = 1 − 2(L + 2)δ′ = 1 − δ by setting δ′ = δ/(2(L + 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 Tighter version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 (Adapted from Boucheron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' [2005], Thm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Consider the conditions and no- tations of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' If we further assume that there exists a non-decreasing function w : R+ → R+ with w(x)/√x non-increasing for x ∈ R+ such that, for any function f, � Var [| f − f ⋆|] ≤ w (E( f ) − E( f ⋆)), then the excess risk of ˜gm,ˆλ can be bounded from above as: E ≤ min θ∈(0,1) (1 + θ) � min λ∈Λ � 8 · β · Rn(Fλ) + Eapp(λ) � + ρin +2B · � 2 log((2L + 1)/δ) m + 4 · B · log 2L + 1 δ �τ∗(µ) θ + 2 3µ �� , where τ∗(n) = min{x > 0 : w(x) = x√n} for any n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For any λ ∈ Λ and some δ′ > 0, we have the following from Berstein’s inequality: E(˜gm,λ) − E( f ⋆) ≤ Ev µ(˜gm,λ) − Ev µ( f ⋆) + w(E(˜gm,λ) − E( f ⋆)) � 2 log(1/δ′) µ + 4 log(1/δ′) 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (22) with probability at least 1 − δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Let ¯λ = arg minλ∈Λ E(˜gm,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Then again applying Berstein’s inequality with δ′ > 0, we have E( f ⋆) − E(˜gm,¯λ) ≤ Ev µ( f ⋆) − Ev µ(˜gm,¯λ) + w(E(˜gm,¯λ) − E( f ⋆)) � 2 log(1/δ′) µ + 4 log(1/δ′) 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (23) with probability at least 1 − δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Combining the above two inequalities over all λ ∈ Λ, using the definition of ˜gm,ˆλ as the minimizer of Ev µ(·), and the non-decreasing nature of w(·) gives us the following with probability at least 1 − (L + 1)δ′: E(˜gm,ˆλ) − E(˜gm,¯λ) ≤ 2w(E(˜gm,ˆλ) − E( f ⋆)) � 2 log(1/δ′) µ + 8 log(1/δ′) 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (24) Given τ∗(µ) as defined in the statement of the claim, w(τ∗(µ)) = τ∗(µ)√µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Now, either E(˜gm,ˆλ) − E( f ⋆) < τ∗(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Or E(˜gm,ˆλ) − E( f ⋆) ≥ τ∗(µ), which gives us the following: w(E(˜gm,ˆλ) − E( f ⋆)) � E(˜gm,ˆλ) − E( f ⋆) (a) ≤ w(τ∗(µ)) � τ∗(µ) (b) = � τ∗(µ)√µ, (25) where (a) comes from the assumptions that w(x)/√x is non-increasing in x ∈ R+ and (b) comes from the definition of τ∗(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Using the above in (24) gives us: E(˜gm,ˆλ) − E(˜gm,¯λ) ≤ 2 � E(˜gm,ˆλ) − E( f ⋆) � 2 log(1/δ′) µ � τ∗(µ) + 8 log(1/δ′) 3µ (26) ≤ θ 2 � E(˜gm,ˆλ) − E( f ⋆) � + 8 2θ log(1/δ′)τ∗(µ) + 8 log(1/δ′) 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (27) where we utilize the fact that the arithmetic mean is greater than or equal to the geometric mean for some θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 21 Then we can get, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − (L + 1)δ′: � E(˜gm,ˆλ) − E( f ⋆) � (1 − θ/2) ≤ E(˜gm,¯λ) − E( f ⋆) + 4 log(1/δ′) �τ∗(µ) θ + 2 3µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (28) Now we have the following by definition of ¯λ and ˜gm,¯λ ∈ F¯λ: E(˜gm,¯λ) − E( f ⋆) = min λ∈Λ (E(˜gm,λ) − E( f ⋆)) (29) (c) ≤ min λ∈Λ � E( ¯fλ) − E( f ⋆) + 2 sup f ∈Fλ |E( f ) − Em( f )| + ρin � (30) (d) ≤ min λ∈Λ � Eapp(λ) + 8 · β · Rn(Fλ) + 2B · � 2 log(1/δ′) m + ρin � w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ≥ 1 − Lδ′ (31) where (d) is obtained from the application of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 on each λ ∈ Λ, and (c) is obtained as follows: E(˜gm,λ) ≤ Em(˜gm,λ) + sup f ∈Fλ |E( f ) − Em( f )| (32) ≤ Em(ˆgm,λ) + ρin + sup f ∈Fλ |E( f ) − Em( f )| (33) ≤ Em( ¯fλ) + ρin + sup f ∈Fλ |E( f ) − Em( f )| (34) ≤ E( ¯fλ) + ρin + 2 sup f ∈Fλ |E( f ) − Em( f )|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (35) Combining (28) and (31), setting δ′ = δ/(2L + 1) and noting that (1 − θ/2)−1 ≤ (1 + θ) for θ ∈ (0, 1) gives us the statement of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' B Data-driven heuristic for ρout While we have a very precise way of setting ρin given the theoretical result, the choice of ρout is somewhat more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Based on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2, we can utilize the following heuristic to set ρout: Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Assuming that ρout can be iteratively reduced during the approximate ERM for ˜fn,ˆλ, based on the terms in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 and a given ρin, we propose the following iterative scheme to set ρout with scaling parameters ν ∈ (0, 1), γ > 0: For T > 0, we iteratively reduce ρout as ρ(T+1) out ← ν · ρ(T) out if Γ(ρ(T−1) out ) − Γ(ρ(T) out) > γ · κ exit approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ERM otherwise, (36) where Γ(ρout) := I ˆλ n,m(ρin, ρout), ρ(0) out ← ρin, and κ := ρin + B � 2 log(2(L+2)/δ)(2/√n + 2/√m + 1/√µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' This heuristic leverages the fact that the empirical risk improvement I ˆλ n,m(ρin, ρout) will in- crease as ρout is reduced up until a point, and the excess risk of ˜fn,ˆλ is closely tied to this empirical risk improvement – more improvement implies better excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 tries to balance any increase in this empirical risk improvement with the other (computable) terms, denoted as κ, in the excess risk bound in (9) – we stop reducing ρout when the increase in the empirical risk 22 improvement Γ(ρ(T−1) out ) − Γ(ρ(T) out) is an order of magnitude below κ, at which point, other terms dominate the excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Heuristics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 are our answers to Q2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 just presents a way to identify when ρout is sufficiently small in terms of statistical performance while being able to gain computationally when we are able to approximate the ERM in an iterative manner and progressively decrease ρout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' C Empirical evaluation details Synthetic data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The binary classification data is generated using the make classification function [Guyon, 2003] in scikit-learn [Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We ensure that the classes are not overlapping and there is no label noise, ensuring that the Bayes optimal risk E( f ⋆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Neural network HP search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We consider two different Λs for a fully connected neural network, with (i) depth ∈ {1, 2, 3}, (ii) number of neurons in each layer ∈ {10, 100}, (iii) initial SGD learning rate ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1}, (iv) SGD batch size ∈ {8, 32, 128}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The implementation is in PyTorch [Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For one problem we consider 36 configurations (that is L = |Λ| = 36), and for another, we consider 18 configurations (that is L = |Λ| = 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' LightGBM HP search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We use the following search space for theLGBMClassifier from LightGBM [Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=', 2017] with RandomizedSearchCV from scikit-learn: (i) learning rate ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4], (ii) number of trees ∈ [100, 5000], (iii) number of leaves per tree ∈ [6, 50], (iv) minimum samples in a child node ∈ [100, 500], (v) minimum weight in a child node ∈ [10−5, 104], (vi) sub- sampling rate ∈ [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='9], (vii) maximum depth per tree ∈ [−1, 7], (viii) column sub-sampling rate per tree ∈ [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='7], (ix) α regularization ∈ [0, 100], (x) λ regularization ∈ [0, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' OpenML data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The data sets used in this experiment are listed in Table 6 with their OpenML names and IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For each data set, we utilize 3 different values of the train-validation split ratio µ/n ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' D Extended empirical evaluation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Further evaluation of Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 In Figure 2, we demonstrated the empirical utiity of Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 for a particular choice of ρin = ρout = 0 implying we leverage exact ERM in the inner level of the HPO problem (to obtain ˆgm,λ, λ ∈ Λ) and in the final training of the model on the selected HP ˆλ to obtain ˆfn,ˆλ ∈ Fˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In this subsection, we evaluate Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 for other pre-set values of ρin and ρout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We start with trying ρin ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1} and then setting ρout relative to ρin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We present the following results for the search space with 36 configuration in Figure 4: (a) Fig- ure 4a for ρout = ρin/10, (b) Figure 4b for ρout = ρin, (c) Figure 4c for ρout = 2 · ρin, (d) Figure 4d for ρout = 10ρin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We present the following results for the search space with 18 configuration in Figure 5: (a) Figure 5a for ρout = ρin/10, (b) Figure 5b for ρout = ρin, (c) Figure 5c for ρout = 2 · ρin, (d) Figure 5d for ρout = 10ρin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For the cases where ρout ≤ ρin (Figures 4a and 4b), the excess risk of ˜fn,ˆλ is usually better than the excess risk of ˜gm,ˆλ, and Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1’s “Choice” makes the right choice when there is significant difference between the performance of the two candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' For the cases where ρout > ρin (Figures 4b and 4c), there are some situations where ˜gm,ˆλ has a (significantly) better 23 Table 6: Data set name & OpenML ID of all the data sets used with 10 data sets each in the data size range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='# samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Name (OpenML ID) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1000-4999 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='kr-vs-kp (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='credit-g (31) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='sick (38) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='spambase (44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='scene (312) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='yeast-ml8 (316) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='fri-c3-1000-25 (715) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='fri-c4-1000-100 (718) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='abalone (720) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='fri-c4-1000-25 (723) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5000-9999 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='mushroom (24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='bank8FM (725) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='cpu-small (735) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='puma32H (752) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='cpu-act (761) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='delta-ailerons (803) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='kin8nm (807) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='puma8NH (816) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='delta-elevators (819) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='bank32nh (833) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='10000-49999 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='BNG(tic-tac-toe) (137) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='electricity (151) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='adult (179) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='BNG(breast-w) (251) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='mammography (310) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='webdata-wXa (350) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='pol (722) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2dplanes (727) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='ailerons (734) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='house-16H (821) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='50000-99999 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='vehicle-sensIT (357) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='KDDCup09-app (1111) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='KDDCup09-churn (1112) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='KDDCup09-upsell (1114) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='vehicleNorm (1242) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='higgs (23512) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='numerai28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='6 (23517) Run-or-walk-info (40922) APSFailure (41138) kick (41162) excess risk over ˜fn,ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' In these cases the Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 “Choice” is able to make the right choice – see for example the last row in Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 Evaluation of Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Leveraging the data-dependent selection of ρout with Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1, we evaluate the ex- cess risk incurred by approximating the ERM over Fˆλ with all n samples to ρout toler- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' We present the results for the different values of γ in Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 from the set {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01} in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The excess risks incurred and the speedups gained from using ˜fn,ˆλ inplace of ˆfn,ˆλ is visualized in Figure 6 – the solid line corresponds to ˆfn,ˆλ while the dash-dotted lines correspond to ˜fn,ˆλ for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' And the results corre- sponding to the excess risk in Figure 6a indicate that, for γ up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='005, the increase in excess-risk is quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' The speedups obtained for the different choices of γ and corresponding data- dependent ρout in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' It can be seen that we can get up to 2× speedup over exact ERM without losing much in terms of the excess risk (see for γ up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' for larger values of γ we can get up to 4× speedup if we are ready to incur some additional excess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 24 (a) ρout = ρin/10 (b) ρout = ρin (c) ρout = 2 · ρin (d) ρout = 10 · ρin Figure 4: Excess-risk of data-dependent choice between ˜fn,ˆλ and ˜gn,ˆλ based on Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 for ρin ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1} and search space with |Λ| = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Please magnify to view in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 25 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 23 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nvfn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:1e-05 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 2-3 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pat:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 2 Excess risk 2 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 v/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nvfn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 2-2 23 Choice + E(gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=') 2 E(f,) 211 212 214 21 21 214 2 212 214 21 212 214 20 222 214 +=/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nvfn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 fn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Paut:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 2~ 2~ /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 2-3 2 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 yfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pott:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pott:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Excess risk 2 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 2-3 Choice E(gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=') 214 E(f,) 21 212 214 2 212 214 2 212 214 20 212 214 210 212 214 =/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nvfn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 fn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Paut:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 2-2 2~3 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002 23 2 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pa:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pimn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pimn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Pa:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Pa:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Excess risk 2~4 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 nv/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 2~ Choice E(gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=') E(f,) 21 212 214 2 212 214 2 232 214 20 212 214 210 2 214 =/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 23 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nvfn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 2-3 fn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pat:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 2 Excess risk 2 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 v/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nvfn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pourt:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pourt:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pourt:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 22 Choice E(gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=') 22 E(f,) 2h 272 214 2 2 224 22 212 214 2 212 214 20 212 214 +=U(a) ρout = ρin/10 (b) ρout = ρin (c) ρout = 2 · ρin (d) ρout = 10 · ρin Figure 5: Excess-risk of data-dependent choice between ˜fn,ˆλ and ˜gn,ˆλ based on Heuristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 for ρin ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1} and search space with |Λ| = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Please magnify to view in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 26 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 v/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 m-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 nfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 /n- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Pa:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 risk 2 Excess nfn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nv/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 2 Choice ★ E(gn) E(f,) 214 2 214 21 21 214 214 2 212 214 =/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 v/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 m-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 nfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0002 /n- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pa:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pim:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001,Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='002 risk Excess 2 nfn=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nv/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='02 Choice E(f,) 21 214 2 214 21 21 214 214 21 212 214 =/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 n/n- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pott:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pat:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pou:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01 risk 2 Excess nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 v/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pourt:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pourt:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 Choice ★ E(gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=') E(f,) 21 214 21 212 214 21h 21 21 21 212 21 212 214 +=/n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0,Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0 n/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001,Pout:1e-05 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:1e-05 Qn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2cut:1e-05 /n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 /n- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /m- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pn:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pa:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pot:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Po:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 risk Excess 2 nvfn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 /n= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nv/n-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 /n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 v/n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pin:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Pm:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='Pout:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='001 Choice ★ E(gn) E(f,) 214 212 214 21 21 21 21 214 2 212 214 =(a) Excess-risk of ˜f vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' (b) Speedup over exact ERM Figure 6: Empirical validation of the data-dependent choice of ρout in Heuristic B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' Please mag- nify to view in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content=' 27 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 nyin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 nyln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 + E(fn) H-- E(f,),Y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0050 2-2 H--E(f),= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 H--E(ff),Y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0100 Excess risk →-E(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0005 E(f,), Y= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0500 H-- E(f,),Y= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0010 E(f),Y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1000 22 212 214 23 212 224 21 22 214 23 212 224 22 222 214 = = = = =ERM nyin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1 nyin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='2 nuln=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='3 ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='4 ny/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='5 Exact →-Approx, Y= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0050 23 exact Approx, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0001 Approx, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0100 Approx, Y= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0005 Approx, Y= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0500 Approx,y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='0010 Approx,Y= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} +page_content='1000 over 2 2 2 212 214 21 222 214 21 212 214 211 212 224 22 212 214 += += n= ny + nt = =' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfiA3u/content/2301.05131v1.pdf'} diff --git a/M9E0T4oBgHgl3EQfTABv/vector_store/index.faiss b/M9E0T4oBgHgl3EQfTABv/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..14745321c6e2e084f6941676bfe3e518c06abff1 --- /dev/null +++ b/M9E0T4oBgHgl3EQfTABv/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1b8f7602ff6eb208ebd3ba1cbbb4a3b0f1451d429a439991730063caf4a2f929 +size 1441837 diff --git a/MdFJT4oBgHgl3EQfzC0_/content/2301.11641v1.pdf b/MdFJT4oBgHgl3EQfzC0_/content/2301.11641v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aafc589fec65aa77a8870bf4136fdad27867f06d --- /dev/null +++ b/MdFJT4oBgHgl3EQfzC0_/content/2301.11641v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1156b0e9ac65f5fe015c17ca6073d6c15f1ed244912a69619df2ca515db99fcc +size 334661 diff --git a/MdFJT4oBgHgl3EQfzC0_/vector_store/index.faiss b/MdFJT4oBgHgl3EQfzC0_/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..de7b12d4c84bc3c2e62e2a6c693366dd925422e1 --- /dev/null +++ b/MdFJT4oBgHgl3EQfzC0_/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c8d90c6597338508d81814ac2b6a49c19ebe3fb28789d7a24ea0679c699b7e00 +size 2097197 diff --git a/MdFJT4oBgHgl3EQfzC0_/vector_store/index.pkl b/MdFJT4oBgHgl3EQfzC0_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1ea659d19a9cc0cd9d58b2cfc2d74797efb67e62 --- /dev/null +++ b/MdFJT4oBgHgl3EQfzC0_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8de6f1fc10283e6d8703e693fac3f278b006d0dde05fc89949fed4da7180fdaf +size 88914 diff --git a/NNAyT4oBgHgl3EQf6_pQ/content/2301.00830v1.pdf b/NNAyT4oBgHgl3EQf6_pQ/content/2301.00830v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ed9dac4a5d21a75b474596225c984aee8b21368a --- /dev/null +++ b/NNAyT4oBgHgl3EQf6_pQ/content/2301.00830v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:923d9eb70bf66d7d4f141dc874b6d676b8e86ad700f8cdf185ebc3d96e224358 +size 3762665 diff --git a/NNAyT4oBgHgl3EQf6_pQ/vector_store/index.faiss b/NNAyT4oBgHgl3EQf6_pQ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..621f6a054c89a9b0bf2185ee8693d7e8de7e5833 --- /dev/null +++ b/NNAyT4oBgHgl3EQf6_pQ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5db110ea605ba333094413602276bff193490ba32e1bc8f7283e5aa233c917b3 +size 5111853 diff --git a/NNE0T4oBgHgl3EQf0QK5/vector_store/index.faiss b/NNE0T4oBgHgl3EQf0QK5/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2aa7e2a06ddf68b8748cb0ea405a1455cffbbee0 --- /dev/null +++ b/NNE0T4oBgHgl3EQf0QK5/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8abe1401c4fd14a9657dd89b470acc2c111428fdc690fb5873032f96ef2462d8 +size 11337773 diff --git a/NNE0T4oBgHgl3EQf0QK5/vector_store/index.pkl b/NNE0T4oBgHgl3EQf0QK5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c806555fe8345afdb53c01b3da160b9e264a9039 --- /dev/null +++ b/NNE0T4oBgHgl3EQf0QK5/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b8903aa7d67c5c06e7f118c70c29fb9cfdc7e3f503c541d5408017efbc8c1be3 +size 420038 diff --git a/NdFQT4oBgHgl3EQfWTbR/vector_store/index.faiss b/NdFQT4oBgHgl3EQfWTbR/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..72526e446c21d8442f9754171eb11e29881626b8 --- /dev/null +++ b/NdFQT4oBgHgl3EQfWTbR/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:94599e647241afaaadf17f00c10708076cda8afd367d74fd11260763a1426753 +size 12124205 diff --git a/OdFLT4oBgHgl3EQfPC8X/content/tmp_files/2301.12026v1.pdf.txt b/OdFLT4oBgHgl3EQfPC8X/content/tmp_files/2301.12026v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4b5a4c5fe948531fb4e137e1c42dc1287ada6e9 --- /dev/null +++ b/OdFLT4oBgHgl3EQfPC8X/content/tmp_files/2301.12026v1.pdf.txt @@ -0,0 +1,968 @@ +arXiv:2301.12026v1 [stat.ME] 27 Jan 2023 +G-formula for causal inference via multiple imputation +Jonathan W. Bartlett1, Camila Olarte Parra1, and Rhian M. Daniel2 +1Department of Medical Statistics, London School of Hygiene & Tropical Medicine, +London, UK +2Division of Population Medicine, Cardiff University, Cardiff, UK +Abstract +G-formula is a popular approach for estimating treatment or exposure effects from longitudinal data +that are subject to time-varying confounding. G-formula estimation is typically performed by Monte- +Carlo simulation, with non-parametric bootstrapping used for inference. We show that G-formula can +be implemented by exploiting existing methods for multiple imputation (MI) for synthetic data. This +involves using an existing modified version of Rubin’s variance estimator. In practice missing data is +ubiquitous in longitudinal datasets. We show that such missing data can be readily accommodated as +part of the MI procedure, and describe how MI software can be used to implement the approach. We +explore its performance using a simulation study. +1 +Introduction +The collection of methods referred to as G-methods, developed by James Robins and co-workers, can +provide valid inference for the effects of time-varying exposures or treatments in the presence of time- +varying confounders—variables that affect treatment over time and the outcome of interest—even when +these are affected by previous values of treatment (Naimi et al., 2017). One such method is parametric +G-formula (sometimes known as G-computation). +Parametric G-formula involves postulating models +for the time-varying confounders and outcomes. +The expected outcome under specified longitudinal +treatment regimes of interest can then be estimated and contrasted. +The evaluation of G-formula estimators often involves intractable integrals. +To overcome this in +practice, G-formula implementations make use of Monte-Carlo integration (simulation) (Daniel et al., +2011; McGrath et al., 2020). Inference for the resulting estimators is performed using bootstrapping. To +reduce Monte-Carlo errors, a large number of simulations is often needed within each bootstrap sample, +making the overall procedure computationally intensive. A further common obstacle to using G-formula +is that some individuals may have missing values in the time-varying confounders, time-varying treatment +variables, or the outcome. In practice, given the already high computational complexity of parametric +G-formula, analysts faced with the additional complication of missing data may use simplistic solutions +such as last observation carried forward (Westreich et al., 2012). +Monte-Carlo simulation-based implementations of G-formula share close connections to the method +of multiple imputation (MI) for handling missing data. +Previously Westreich et al. (2015) explained +how MI could be used to implement G-formula in a point treatment setting, but stated that Rubin’s +variance combination rules could not be used here. In this paper we show that methods for adapting +MI for missing data to generate synthetic datasets can be used to implement a G-formula estimator +1 + +(Raghunathan et al., 2003). As such, we show that an accompanying simple variance combination rule +can be used to estimate the variance of the resulting estimator, obviating the need for bootstrapping. +Furthermore, we show that this MI approach can readily accommodate missing data, thereby providing +a convenient single solution to handling the missingness of both actual and counterfactual data. +This paper is organised as follows. +In Section 2 we review parametric G-formula and how it is +typically implemented. +In Section 3 we describe how G-formula can be implemented by exploiting +existing methodology for using MI to generate synthetic datasets. In Section 4 we report the results +of simulation studies investigating the performance of this approach. We conclude in Section 5 with a +discussion. +2 +Review of G-formula +We assume we have a sample of data from nobs independent individuals from some well-defined pop- +ulation. +For individual i we collect confounder measurements Lit at times t = 0, 1, . . . , T . +We also +collect measurements on the treatment Ait at each time, and a final outcome Yi. For concreteness, we +consider in the following the case of T = 2, although the developments naturally extend to T > 2. +We also suppose for now that Lit is a real-valued scalar, but natural adaptations/extensions to dis- +crete and higher-dimensional confounders apply. Let Y ¯a denote the potential outcome for a randomly +selected individual from the population when their treatment sequence has been set to ¯a. G-formula +relies on certain identification assumptions being satisfied, for the details of which we refer the reader to +Hernan and Robins (2020). +The G-formula estimator of µ = E(Y ¯a) = E(Y a0,a1,a2) for specified values of a0, a1 and a2 is then +based on the fact that under the aforementioned identifying assumptions +E(Y ¯a) = +� +l0 +� +l1 +� +l2 +E(Y |a0, a1, a2, l0, l1, l2)f(l2|a0, a1, l0, l1)f(l1|a0, l0)f(l0)dl2dl1dl0. +(1) +To implement G-formula we specify and fit models +f(Y |A0, A1, A2, L0, L1, L2; ψY ), +f(L2|A0, A1, L1, L0; ψ2), +f(L1|A0, L0; ψ1), +f(L0; ψ0) +(2) +Since the expression in (1) cannot generally be evaluated analytically, implementations of G-formula +are typically based on Monte-Carlo integration, through simulation of the longitudinal confounders and +outcome for each of nsyn individuals, under the treatment regimes of interest. That is, given maximum +likelihood estimates of the parameters in the conditional models in equation (2), denoted ˆψ0, ˆψ1, ˆψ2, ˆψY , +we sequentially simulate for each individual i = 1, . . . , nsyn as follows +˜Li0 ∼ f(L0; ˆψ0) +˜Li1 ∼ f(L1|a0, ˜Li0, ˆψ1) +˜Li2 ∼ f(L2|a0, a1, ˜Li0, ˜Li1, ˆψ2) +˜Yi ∼ f(Y |a0, a1, a2, ˜Li0, ˜Li1, ˜Li2, ˆψY ) +2 + +The Monte-Carlo G-formula estimator of E(Y a0,a1,a2) is then +1 +nsyn +nsyn +� +i=1 +˜Yi +(3) +The number of individuals to simulate for, nsyn, could be set equal to nobs, but choosing a larger value +reduces Monte-Carlo error in the estimator. For statistical inference, implementations of G-formula in +Stata and R rely on the use of non-parametric bootstrapping (Daniel et al., 2011; McGrath et al., 2020), +which as noted in Section 1, is computationally intensive. +While we have stated that a model f(L0; ψ0) is specified and used, in fact this is not needed and +is not typically used. Instead, a non-parametric model for f(L0) is used, and the simulation is per- +formed by sampling a value of L0 from its empirical distribution. Moreover, when as is often the case +interest lies in the mean E(Y ¯a) (as opposed to some other function of the distribution of Y ¯a), it suf- +fices to specify a model for E(Y |A0, A1, A2, L0, L1, L2), rather than for the full conditional distribution +f(Y |A0, A1, A2, L0, L1, L2). Our choice in the preceding to describe a version of G-formula that specifies +the latter conditional distribution (rather than mean) model is motivated by the fact this version matches +the approach taken in an MI implementation of G-formula, which we describe next. +3 +G-formula via multiple imputation +In this section we describe how a Monte-Carlo G-formula estimator can be implemented using MI meth- +ods. +In Section 3.1 we describe how the point estimator is constructed using MI. In Section 3.2 we +explain why Rubin’s standard variance estimator is biased in this instance, and describe an alterna- +tive variance estimator, which was derived in the context of using MI to generate synthetic datasets by +Raghunathan et al. (2003). In Section 3.3 we describe how standard MI software can be used to imple- +ment the approach. Lastly, in Section 3.4 we describe how the approach readily extends to accommodate +missing actual data (as opposed to missing counterfactual data). +3.1 +Point estimation +To estimate E(Y ¯a) by MI, first augment the observed dataset by adding nsyn additional rows. +Let +n = nobs + nsyn, such that n denotes the number of rows in the augmented dataset (i.e. the original +plus augmented rows). +In the augmented rows, as shown in Table 1, the baseline and time-varying +confounders (L0, . . . , LT ) and final outcome Y are set to missing, while the treatment variables are set to +their values under the regime of interest, i.e. A0 = a0, A1 = a1, . . . , AT = aT . The variable R indicates +whether the data row was in the original sample (R = 1) or not (R = 0). +Next, Bayesian MI is used to generate M imputations of the missing values in this augmented dataset, +using the chosen sequential models (equation (2)). Within imputation m (m = 1, . . . , M), calculate the +mean of Y in the augmented rows (R = 0): +ˆµm = +�n +i=1(1 − Ri)Y m +i +�n +i=1 1 − Ri +where Y m +i +denotes the imputation of Yi in imputation m. The mean outcome under the treatment regime +of interest, µ = E(Y ¯a), is then estimated as +ˆµ = 1 +M +M +� +m=1 +ˆµm +(4) +The resulting estimator is identical to the Monte-Carlo G-formula estimator described in Section +3 + +R +L0 +A0 +L1 +A1 +L2 +A2 +Y +1 +-0.3 +0 +0.5 +0 +2.2 +1 +1.3 +1 +2.3 +1 +4.2 +1 +4.6 +1 +5.5 +1 +-0.5 +1 +0.4 +0 +0.8 +1 +1.9 +1 +-0.1 +0 +1.6 +1 +4.1 +0 +7.0 +1 +-0.0 +1 +1.9 +1 +3.5 +1 +6.2 +0 +NA +1 +NA +1 +NA +1 +NA +0 +NA +1 +NA +1 +NA +1 +NA +0 +NA +1 +NA +1 +NA +1 +NA +0 +NA +1 +NA +1 +NA +1 +NA +0 +NA +1 +NA +1 +NA +1 +NA +Table 1: G-formula via MI data setup. The original dataset (top part) is augmented with additional rows +(bottom part). In the augmented part, confounders L0, L1, L2 and outcome Y are set to missing (indicated +here by NA), while the treatment variables A0, A1, A2 are set to their values under the regime of interest +(here 1, 1, 1). The variable R indicates whether the row is originally observed data (R = 1) or not (R = 0). +2, except for the fact that in MI as originally conceived by Rubin (1987), imputation m is generated +conditional on an imputation specific draw from the posterior distribution of the imputation model (here +the models in equation (2)) parameters. Although this so called ‘proper’ MI estimator has strictly larger +variance than the ‘improper’ MI estimator which generates each imputation conditional on an efficient +estimate of the parameters, the difference goes to zero as n → ∞ and M → ∞ (Wang and Robins, 1998). +3.2 +Variance estimation +The variance of an MI estimator is typically estimated using Rubin’s variance estimator (Rubin, 1987): +(1 + M −1) ˆB + ˆV , +where +ˆB = +1 +M − 1 +M +� +m=1 +(ˆµm − ˆµ)2 +denotes the between-imputation variance and +ˆV = 1 +M +M +� +m=1 +� +Var(ˆµm) +denotes the average within-imputation variance. While in many settings Rubin’s variance estimator is +asymptotically unbiased, in some it can be biased upwards or downwards relative to the true repeated +sampling variance of the MI point estimator (Robins and Wang, 2000). One such situation is where +only a subset of the records used to fit the imputation model is used to fit the analysis model, of which +the G-formula via MI estimator is one such example – the original observed dataset is used to fit the +imputation models, while only the augmented dataset rows are used to fit the analysis model (estimating +the mean of Y among those with R = 0). As such we may anticipate that Rubin’s variance estimator +will be biased for the G-formula via MI point estimator. We demonstrate this empirically in Section 4. +The G-formula via MI estimator is closely related to the use of MI to generate samples from synthetic +populations, first proposed by Rubin (1993). Here the objective is to release these synthetic samples +rather than the original data in order to protect the confidentiality of survey respondents’ data. For +4 + +synthetic MI, Raghunathan et al. (2003) suggested the following variance estimator to estimate Var(ˆµ): +ˆVsyn = (1 + M −1) ˆB − ˆV . +(5) +Raghunathan et al. (2003) derived this from both Bayesian and repeated sampling perspectives. +To build intuition for ˆVsyn, we now show it is unbiased for Var(ˆµ) in a highly simplified but instructive +setting. Thus suppose we observe data from nobs individuals on an outcome Y ∼ N(µ, σ2) and interest +lies in inference for µ. Here to estimate the mean µ we can of course trivially use the sample mean +¯Y = n−1 +obs +�nobs +i=1 Yi, which has repeated sampling variance σ2/nobs. Suppose however that we use Bayesian +MI to generate M new imputed datasets of size nsyn. +For simplicity, we assume σ2 is known. +In +this case, under the standard non-informative prior for µ, to generate imputation m we first draw +˜µ(m) ∼ N( ¯Y , +σ2 +nobs ). For i = nobs + 1, . . . , n we then simulate (impute) nsyn new Y values +Yi(m) = ˜µ(m) + ǫi(m) +where ǫi(m) ∼ N(0, σ2). +Having generated imputed/synthetic datasets for m = 1, . . . , M, the estimate of µ based on them is +then +ˆµ = 1 +M +M +� +m=1 +ˆµm += 1 +M +M +� +m=1 +1 +nsyn +n +� +i=nobs+1 +� +˜µ(m) + ǫi(m) +� += 1 +M +M +� +m=1 +˜µ(m) + +1 +nsynM +M +� +m=1 +n +� +i=nobs+1 +ǫi(m). +Letting ˜µ = {˜µ(1), . . . , ˜µ(M)}, this has variance +Var(ˆµ) = E {Var(ˆµ|˜µ)} + Var {E(ˆµ|˜µ)} += E + + +Var + + +1 +nsynM +M +� +m=1 +n +� +i=nobs+1 +ǫi(m) +������ +˜µ + + + + + + Var +� +1 +M +M +� +m=1 +˜µ(m) +� += +σ2 +nsynM + Var +� +E +� +1 +M +M +� +m=1 +˜µ(m) +����� +¯Y +�� ++ E +� +Var +� +1 +M +M +� +m=1 +˜µ(m) +����� +¯Y +�� += +σ2 +nsynM + Var( ¯Y ) + E +�σ2/nobs +M +� += +σ2 +nsynM + (1 + M −1) σ2 +nobs +. +With σ2 known, the within-imputation variance is σ2/nsyn for every imputed dataset, and so ˆV = +σ2/nsyn. Conditional on the observed data ¯Y , the between-imputation variance estimator ˆB is an unbi- +ased estimator of +Var(ˆµm| ¯Y ) = Var + + ˜µ(m) + +1 +nsyn +n +� +i=nobs+1 +ǫi(m) +������ +¯Y + + += +σ2 +nobs + σ2 +nsyn . +Thus, unlike in the missing data setting, the between-imputation variance captures variability both due to +5 + +uncertainty about µ in the observed data estimate and the additional variability due to effectively taking +new random samples of size nsyn from the population for each imputation (Reiter and Raghunathan, +2007). The expected value of ˆVsyn is then +E( ˆVsyn) = E{(1 + M −1) ˆB − ˆV } = (1 + M −1) +� σ2 +nobs + σ2 +nsyn +� +− σ2 +nsyn += +σ2 +nsynM + (1 + M −1) σ2 +nobs += Var(ˆµ), +such that ˆVsyn is unbiased for Var(ˆµ). In the Appendix we derive the variance estimator ˆVsyn for the +G-formula via MI estimator using the asymptotic results developed by Robins and Wang (2000). +As noted by Reiter (2002) and Raghunathan et al. (2003), the variance estimator ˆVsyn can be negative, +although the probability of this occurring goes to zero as M → ∞. Reiter (2002) proposes using the +within-imputation variance ˆV if ˆVsyn < 0, but reported that in simulations this did not lead to good +performance. In Section 4 we investigate the performance of a procedure where M is successively increased +until ˆVsyn > 0. According to Reiter (2002), Raghunathan and Rubin (2000) proposed inference based on +a t-distribution with degrees of freedom given by +vf = (M − 1) +� +1 − +M ˆV +(M + 1) ˆB +�2 +, +(6) +the performance of which we explore in Section 4. +3.3 +Implementation using imputation software +To implement the proposed approach, as described previously, the observed dataset of size nobs is aug- +mented by an additional nsyn rows in which all variables are set to missing except the treatment variables, +which are set to their values under the regime of interest, i.e. A0 = a0, A1 = a1, . . . , Ak = ak. MI soft- +ware, such as the mice package in R, can then be applied to the resulting dataset, with options specified +so that the time-varying confounders and outcome are imputed sequentially in time as per the models +given in equation (2). Since the missingness pattern is monotone, no iterative methods such as Markov +Chain Monte Carlo are required. Following imputation, the augmented subset is extracted from each im- +puted dataset, and the mean of Y is evaluated in each, yielding ˆµm, along with a corresponding complete +data variance estimate. The variance estimator ˆVsyn in equation (5) can then be evaluated. +Ordinarily interest focuses on the contrast of potential outcome means under two (or more) different +treatment regimes. To estimate the corresponding contrast in potential outcome means, we augment the +observed dataset twice. In the second augmentation part, the treatment variables are set according to +the second treatment regime of interest. The difference in potential outcome means can be estimated by +the difference in simulated outcomes between the two augmented parts. The variance of the resulting +estimator can be estimated by the sum of the variance estimator given in equation (5) when applied to +the two regimes of interest, since the sets of synthetic imputations for the two regimes are independent +(conditional on the parameter draws used to impute). +Implementation of the preceding steps using packages such as mice in R is relatively straightforward. +Nonetheless, to facilitate use of the approach, we provide the R package gFormulaMI. This augments the +supplied dataset as described above and imputes missing data using the mice package. The resulting +imputed datasets contain only the augmented portion of the imputations (with R = 0), which can be +used to estimate potential outcome means and contrasts of these. The point estimates and variances +from the analysis of these imputations are then passed to a function implementing the variance estimator +6 + +given in equation (5). +As noted earlier, the standard (non-Bayesian) implementation of G-formula avoids specification of a +model for f(L0), and instead simulates from the empirical distribution of L0. In the context of MI for +generation of synthetic samples, Raghunathan et al. (2003) proposed using the approximate Bayesian +bootstrap approach of Rubin and Schenker (1986). +In Section 4.1 we investigate in simulations the +performance of using this approach for Bayesian non-parametric imputation of L0. +3.4 +Missing data +Now suppose that there are some data missing. +Missing data could occur in either the longitudinal +confounders Lit, the final outcome Yi, or the time-varying treatment variables Ait. We assume that the +missing at random assumption is deemed plausible for the missing values. In this case, the application of +the results of Robins and Wang (2000) given in the Appendix still apply without modification. As such, +we can impute both the missing data in the original data (where R = 1) and missing potential outcome +data in the augmented rows (where R = 0), and continue to use the variance estimator ˆVsyn. +If the missingness pattern in the original data is monotone because of dropout, the missing values in +the original data and missing potential outcomes can be imputed simultaneously by imputing sequentially +in time, as described in the setting without missing data. +More typically, however, the pattern of +missingness in the original data will not be monotone. In this case, we propose adapting an approach +which works well when the missingness pattern is nearly monotone (Section 6.5.4 of Schafer (1997)). We +propose that first the missing values in the original data are imputed M times. The augmented rows are +then added to each of the M imputed datasets, and the missing potential outcomes in these rows are +then imputed once (in each of the M datasets) based on the sequential models (equation (2)). +The models required for G-formula given in equation (2) do not fully specify the joint distribution of +all the variables under consideration, since they do not specify models for the treatment variables. The +imputation models used to impute the missing data in the original dataset should ideally be compatible +with those used to impute the augmented rows. One way to achieve this is to specify a full joint model +for all the variables by, in addition to the models in equation (2), specifying models for the time-varying +treatment variables. That is, for t = 0, 1, . . . T , we specify models f(Ait| ¯Ai(t−1), ¯Lit), such as suitable +logistic regression models if treatment is binary. While imputation from such a joint model is possible +using Bayesian model software such as JAGS, imputation is more commonly performed using methods +such as chained equations, as implemented in the popular R package mice. As such, in Section 4.2 we +investigate performance when the models used to impute missing data are not strictly compatible with +the models specified and used by G-formula (in equation (2)). +In the setting with missing data, our R package gFormulaMI takes as input a set of M imputed +datasets. It then augments each imputed dataset with the required additional rows and imputes each +once. +4 +Simulations +In this section we report the results of simulations performed to examine the empirical performance of +the G-formula via MI approach. We first consider, in Section 4.1, the setting where there is no missing +data. Next, in Section 4.2, we consider the situation where some data are missing. +7 + +4.1 +No missing data +We simulated datasets for 500 individuals under the following models: +L0 ∼ N(0, 1) +P(A0 = 1|L0) = expit(L0) +L1 ∼ N(A0 + L0, 1) +P(A1 = 1|A0, L0, L1) = expit(A0 + L1) +L2 ∼ N(A1 + L1, 1) +P(A2 = 1|A0, A1, L0, L1, L2) = expit(A1 + L2) +Y ∼ N(A2 + L2, 1) +We report results for estimates of E(Y 1,1,1) − E(Y 0,0,0), whose true value under the data generating +mechanism is 3. The G-formula via MI approach was implemented using the mice package in R, im- +puting L0, L1, L2 and Y from normal linear models including all the preceding (in time) treatment and +confounder variables linearly. Since the missingness pattern is monotone, we specified that mice only +perform one iteration. We set nsyn = 500. To investigate how performance varied with M, we evaluated +the procedure using M = 5, 10, 25, 50, 100. If in a particular simulation ˆVsyn < 0, we added an additional +M imputations and re-calculated ˆVsyn. This was repeated until ˆVsyn > 0. +Scenario +M +Bias +Emp. SE +Est. SE +Raghu df 95% CI +Z 95% CI +Mean M +Max M +1 +5 +-0.002 +0.242 +0.236 +99.4 +87.1 +6.2 +25 +2 +10 +0.001 +0.229 +0.223 +98.4 +90.1 +10.4 +30 +3 +25 +0.000 +0.223 +0.220 +95.6 +92.9 +25.0 +50 +4 +50 +0.000 +0.217 +0.219 +95.2 +94.2 +50.0 +50 +5 +100 +0.004 +0.218 +0.219 +95.0 +94.5 +100.0 +100 +Table 2: Simulation results for G-formula via MI without any missing data. Results are shown for different +numbers of initial imputations M. Emp SE. gives the empirical standard error of the point estimates while +Est. SE gives the mean estimated standard error based on ˆVsyn. Raghu df 95% CI gives the coverage of +t-based confidence intervals based on the degrees of freedom given in equation (6) while Z 95% CI gives +coverage for confidence intervals constructed using N(0, 1) quantiles. Mean M and Max M give the mean +and maximum value of M required across the simulations in order to obtain ˆVsyn > 0. +Table 2 shows results based on 10,000 simulations per (inital) value of M. As expected since the +imputation models were correctly specified, the G-formula via MI estimator for E(Y 1,1,1) − E(Y 0,0,0) +was unbiased for all values of M. The variance estimator ˆVsyn was also essentially unbiased. Confidence +intervals calculated based on a t-distribution with degrees of freedom calculated from (6) showed overcov- +erage for M = 5, 10, 25, but achieved nominal coverage for M = 50 and M = 100. Confidence intervals +calculated based on a standard normal showed substantial undercoverage for M = 5 and M = 10, but +had close to nominal coverage for M = 50 and M = 100. Lastly, when using a smaller initial value for +M, sometimes additional imputations were required to ensure ˆVsyn > 0, as indicated by the mean and +maximum M values in Table 2. However, when an initial M = 50 (or M = 100) imputations were used, +ˆVsyn was always positive. +We additionally ran 10,000 simulations with M = 50 using the approximate Bayesian bootstrap to +impute L0. The variance estimator ˆVsyn was again unbiased. The coverage of the confidence interval +constructed using a t-distribution with degrees of freedom calculated using equation (6) was 95.3% while +the normal based confidence interval had coverage 94.1%. +8 + +4.2 +Missing data +Next we performed simulations where some data were missing. Data in each of L1, A1, L2, A2 and +Y were made missing completely at random, with the probability of each being missing set to π, with +π = {0.05, 0.1, 0.25, 0.5}. As such, the probability of an individual having complete data was (1−π)5 and +the average number of variables missing per individual was 5π. Thus π = 0.5 is a really quite extreme +scenario, with only approximately 3% of individuals having complete data. +To implement G-formula via MI we used an initial call to mice to impute the missing values M = 50 +times. The continuous variables L1, L2 and Y were imputed using normal linear models while A1 and +A2 were imputed using logistic models. Since the missingness pattern was not monotone, as per the +standard chained equations algorithm, for imputation of a given variable, all the other variables were +included as covariates. The number of iterations was left at its default value of 5, except for π = 0.5. +Here, with a very large amount of missingness, we found that 50 iterations were required to achieve +convergence. Having imputed the missing data, the additional nsyn = 500 rows were added to each +imputed dataset, and mice was applied to each of the M = 50 datasets, specifying to impute using one +iteration sequentially according to time, as used in the scenario without missing data. +Scenario +π +Bias +Emp. SE +Mean est. SE +Raghu df 95% CI +Z 95% CI +1 +0.05 +-0.001 +0.225 +0.224 +95.4 +94.2 +2 +0.10 +-0.003 +0.231 +0.231 +95.3 +94.3 +3 +0.25 +-0.008 +0.259 +0.258 +95.4 +94.3 +4 +0.50 +-0.011 +0.360 +0.361 +95.0 +94.3 +Table 3: Simulation results for G-formula via MI with missing data. π is the probability that each of L1, +A1, L2, A2 and Y are missing. +Table 3 shows the results based on 10,000 simulations per value of π. The G-formula via MI estimator +had minimal bias across all four scenarios. As we would expect, the empirical standard error increased +with increasing amounts of missing data. The variance estimator ˆVsyn was positive in all simulations +and for all values of π when using an initial value of M = 50. ˆVsyn was unbiased for the empirical SE. +Confidence intervals based on a t-distribution with degrees of freedom calculated from (6) showed slight +overcoverage, while the normal based intervals showed slight undercoverage. +5 +Discussion +G-formula via MI is an attractive approach for implementing parametric G-formula. With complete data, +inference for G-formula estimators is usually based on non-parametric bootstrapping. While the boot- +strap provides a consistent variance estimator under mild assumptions, since correct model specification +is generally required for consistency of the G-formula point estimator, it makes sense to use a variance +estimator (e.g. based on Bayesian MI) which exploits an assumption that these models are correct. +The conventional approach based on bootstrapping requires the models used in G-formula to be re- +fitted to each bootstrap sample, and typically hundreds if not thousands of bootstrap samples are used +to obtain inferences with an acceptably small amount of Monte-Carlo error. Although our simulations +were limited in scope, they suggest that reliable inferences can be obtained via MI methods using only 50 +imputations. Moreover, when data are complete such that iterative methods are not required, the models +need only be fitted once to the observed data. Although implementation is relatively straightforward +using existing multiple imputation packages, we have developed an R package gFormulaMI that interfaces +with the mice package to perform the required data manipulation steps, estimate mean outcomes under +each treatment regime of interest, and calculate the synthetic MI variance estimator. Imputation packages +9 + +such as mice are flexible in regard model specification, for example allowing the possibility for the user +to include interactions and higher order effects in models. For +In practice datasets, whether arising from experimental or observational studies, have missing data to +a lesser or greater extent. In this context the G-formula via MI approach has greater appeal, given that +it provides a coherent potential solution to handle both the missing actual and missing counterfactual +data. +While our simulation investigations suggest the G-formula via MI approach can perform well, +our conclusions regarding its empirical performance in general are necessarily limited by the fact our +simulations have only explored a relatively simple setup - with one continuous confounder and a small +number of time points. +One alternative to imputing missing data when implementing G-formula is to fit each of the models +(in equation (2)) using the subset of records for which the variables involved in each model are fully +observed. These complete case model fits yield consistent estimates of the respective conditional model +parameters provided the probability of having all the variables involved in the model is independent +of the dependent variable conditional on the covariates (White and Carlin, 2010). When the pattern +of missingness in the longitudinal dataset is complex, consisting of both intermittent missingness and +missingness due to dropout, such an assumption can sometimes be deemed more plausible than missing +at random, whose meaning becomes complex in such settings (Robins and Gill, 1997). +In this paper we have focused on G-formula where the outcome is a variable Y measured at some +final time point. G-formula can also be used when the outcome is the time to some event of interest, for +example based on discrete time logistic regression models (Westreich et al., 2012). The G-formula via +MI approach can also be used in this setting, by defining appropriate time-dependent binary indicators +of survival. +Implementations of G-formula (e.g. +the G-formula packages in Stata (Daniel et al., 2011) and R +(McGrath et al., 2020)) often fit models pooled across time points for each variable. This is achieved +by formatting the data in so-called long form. Doing so permits borrowing of information across time +points in the estimation of regression parameters, but of course relies on the validity of the assumption +that the conditional distribution of confounders given earlier variables is homogenous across time points. +Although this approach could be implemented via the MI approach we have outlined, we do not believe +it is possible using standard imputation software such as mice in R. This is because having transformed +the data into long form, it is not possible to update values from one row of the data frame from another +within the algorithm, which would be required for subsequent times points for the same subject. +While our focus in this paper has been on static treatment regimes, G-formula can be used to estimate +the effects of dynamic treatment regimes, where the exposure or treatment at a given time point is +assigned dependent on the longitudinal history observed up to that time. The G-formula via MI approach +can be extended to this case, by setting the treatment variables to missing in the augmented part of the +dataset and then specifying how they should be imputed based on the preceding (in time) variables. +This can be achieved for example in the mice package through the use of user specified deterministic (or +indeed stochastic) imputation methods. +Funding +This work was funded by a UK Medical Research Council Grant (MR/T023953/1). +Code +R code for the simulation study can be found at https://github.com/jwb133/gFormulaViaMultipleImputation. +The R package gFormulaMI is available from https://github.com/jwb133/gFormulaMI, and will in due +10 + +course be made available on CRAN. +Appendix +In this appendix we show that the variance estimator ˆVsyn derived by Raghunathan et al. (2003) is +consistent for the G-formula via MI estimator, using the results of Robins and Wang (2000). We first +describe how the G-formula via MI estimator can be embedded into the setup of Robins and Wang +(2000). To that end, consider the augmented dataset where ¯L and Y are set to missing in the augmented +part and the treatment vector ¯A is set to some level ¯a, in order to estimate µ = E(Y ¯a). Define the +variable R that takes value 1 in the original observed data and 0 in the augmented part of the data. The +full data vector is this F = ( ¯A, ¯L, Y, R) and the observed vector is thus O = ( ¯A, ¯LR, Y R, R), with ¯L and +Y missing in those with R = 0. +The missing values in ¯L and Y , in the augmented part of the dataset, are then multiply imputed. +In the first instance we assume, following Robins and Wang (2000), that the imputations are generated +conditional on the MLE of the imputation model parameter ψ. This is referred to by Rubin as ‘improper’ +imputation. After imputation, in the G-formula via MI approach we estimate µ by the mean of those +in the augmented part of the dataset, i.e. observations for which R = 0, and then average these means +across the imputed datasets. This is equivalent to simply calculating the mean of the imputed outcomes +across all imputations, using only data from those individuals with R = 0. As such, the corresponding +‘complete data’ estimator of µ is +�n +i=1(1 − Ri)Yi +�n +i=1(1 − Ri) +The complete data estimating function is therefore given by +U(F, µ) = (1 − R)(Y − µ). +Thus, under the regularity conditions detailed by Robins and Wang (2000), the (non-Bayesian improper) +G-formula via MI estimator of µ which uses M imputations is asymptotically normal. The variance of +the estimator with M = ∞ is given by equation A3 of Robins and Wang (2000) as +Σ = τ −1 +� +E +� +Uobs(ψ∗, µ∗)⊗2� ++ κΛ(ψ∗)κT + κE +� +D(ψ∗)U(ψ∗, µ∗)T � ++ E +� +D(ψ∗)U(ψ∗, µ∗)T �T +κT +� +(τ T )−1, +where A⊗2 = AAT . In the following we define the terms involved in the preceding expression, derive +their values in our setting, and show that ˆVsyn is a consistent estimator of this asymptotic variance. +Parameters with a superscript ∗ denotes the true value of the corresponding parameter. +The quantity τ is defined as +τ = −E +�∂ ¯U(ψ∗, µ∗) +∂µ +� +, +where +¯U(ψ, µ) = 1 +M +M +� +m=1 +U(F m(ψ), µ). +Here F m(ψ) denotes the mth imputed data vector for an individual, with imputations generated condi- +tional on the value ψ of the imputation model parameter. Substituting in the complete data estimating +11 + +function, we have +¯U(ψ, µ) = 1 +M +M +� +m=1 +(1 − R)(Y m(ψ) − µ) +and so +∂ ¯U(ψ∗, µ) +∂µ += ∂ +∂µ +� +1 +M +M +� +m=1 +(1 − R)(Y m(ψ∗) − µ) +� += R − 1 +Thus τ = E(1 − R) = nsyn/n. +The quantity Uobs(ψ, µ) is defined as Eψ{U(ψ, µ)|O, R}. In our case, we have +Uobs(ψ, µ) = Eψ{U(ψ, µ)|O, R} += Eψ{(1 − R)(Y − µ)|O, R} += (1 − R)Eψ{Y − µ|A = ¯a, R = 0} += (1 − R) +� +Eψ(Y | ¯A = ¯a, R = 0) − µ +� +, +and thus Uobs(ψ∗, µ∗) = 0 since Eψ∗(Y |A = ¯a, R = 0) = µ∗. +Letting ˆψ denote the observed data MLE of the imputation model parameters, D(ψ) denotes the in- +fluence function of the estimator, and under standard regularity conditions n1/2( ˆψ−ψ∗) is asymptotically +normal with mean zero and covariance matrix equal to +Λ(ψ∗) = E +� +D(ψ∗)⊗2� += I−1 +obsE +� +S⊗2 +obs(ψ∗) +� +I−1 +obs +where Sobs(ψ) denotes the observed data score and Iobs the observed information matrix. +As noted +by Robins and Wang following their Theorem 2, if, as we assume, the imputation model is correctly +specified, Iobs = E +� +S⊗2 +obs(ψ∗) +� +, in which case Λ(ψ∗) = I−1 +obs. +Observations in the augmented dataset, with R = 0, do not contribute to the estimation of the +imputation model, and so for such observations D(ψ∗) = 0. Conversely, observations in the original +data, with R = 1, do not contribute to the estimation of µ in the imputed datasets, and so for such +observations U(ψ∗, µ∗) = 0. Consequently, D(ψ∗)U(ψ∗, µ∗) = 0 for all observations, and so +E +� +D(ψ∗)U(ψ∗, µ∗)T � += 0. +Thus the asymptotic variance of the G-formula via MI estimator, with M = ∞, is given by +τ −2κI−1 +obsκT +(7) +where +κ = E{U(ψ∗, µ∗)Smis(ψ∗)T }, +Smis = ∂ +∂ψ log f(F|O, R; ψ)|ψ=ψ∗. +Equation A1 from Robins and Wang gives that the (standardised by n) between imputation variance +estimator ¯B converges as m, n → ∞ to +τ −2 � +κI−1 +obsκT + E +� +{U(ψ∗, µ∗) − Uobs(ψ∗, µ∗)}2�� += τ −2 � +κI−1 +obsκT + E +� +U(ψ∗, µ∗)2�� +, +12 + +since Uobs(ψ∗, µ∗) = 0. The (standardised) within-imputation variance �V• converges to +τ −2E +� +Uobs(ψ∗, µ∗)⊗2 + {U(ψ∗, µ∗) − Uobs(ψ∗, µ∗)}⊗2� += τ −2E +� +U(ψ∗, µ∗)2� +, +again using the fact Uobs(ψ∗, µ∗) = 0. Thus ¯B − �V• converges to +τ −2κI−1 +obsκT = Σ +as required. Lastly, since in practice we can only implement the MI estimator with finite M, we must +add an additional M −1 ¯B to account for the additional Monte-Carlo variability, resulting in the variance +estimator ˆVsyn given in equation (5). +In the main paper and appendix we have considered estimation of E(Y ¯a). However, the preceding +arguments for the consistency of ˆVsyn only depended on the complete data estimating function in so far +as it is of the form (1 − R)W (Y, µ) for some function W (Y, µ) where Eψ∗{W (Y, µ∗)| ¯A = ¯a, R = 0} = 0. +References +R. M. Daniel, B. L. De Stavola, and S. N. Cousens. gformula: Estimating causal effects in the presence +of time-varying confounding or mediation using the g-computation formula. The Stata Journal, 11(4): +479–517, 2011. +M. A. Hernan and J. M. Robins. Causal Inference: What If, chapter 13 Standardization and the para- +metric g-formula. Boca Raton: Chapman & Hall/CRC, 2020. +S. McGrath, V. Lin, Z. Zhang, L. C. Petito, R. W. Logan, M. A. Hern´an, and J. G. Young. gformula: +An r package for estimating the effects of sustained treatment strategies via the parametric g-formula. +Patterns, 1(3):100008, 2020. +A. I. Naimi, S. R. Cole, and E. H. Kennedy. An introduction to g methods. International Journal of +Epidemiology, 46(2):756–762, 2017. +T. E. Raghunathan and D. B. Rubin. Conference of the International Society for Bayesian Analysis. +2000. +T. E. Raghunathan, J. P. Reiter, and D. B. Rubin. Multiple imputation for statistical disclosure limita- +tion. Journal of Official Statistics, 19(1):1, 2003. +J. P. Reiter. Satisfying disclosure restrictions with synthetic data sets. Journal of Official Statistics, 18 +(4):531, 2002. +J. P. Reiter and T. E. Raghunathan. The multiple adaptations of multiple imputation. Journal of the +American Statistical Association, 102(480):1462–1471, 2007. +J. M. Robins and R. D. Gill. Non-response models for the analysis of non-monotone ignorable missing +data. Statistics in Medicine, 16(1):39–56, 1997. +J. M. Robins and N. Wang. Inference for imputation estimators. Biometrika, 85:113–124, 2000. +D. B. Rubin. Multiple imputation for nonresponse in surveys. New York: Wiley, 1987. +D. B. Rubin. Statistical disclosure limitation. Journal of Official Statistics, 9(2):461–468, 1993. +D. B. Rubin and N. Schenker. Multiple imputation for interval estimation from simple random samples +with ignorable nonresponse. Journal of the American Statistical Association, 81(394):366–374, 1986. +13 + +J. L. Schafer. Analysis of incomplete multivariate data. CRC press, 1997. +N. Wang and J. M. Robins. +Large-sample theory for parametric multiple imputation procedures. +Biometrika, 85:935–948, 1998. +D. Westreich, S. R. Cole, J. G. Young, F. Palella, P. C. Tien, L. Kingsley, S. J. Gange, and M. A. Hern´an. +The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident aids +or death. Statistics in Medicine, 31(18):2000–2009, 2012. +D. Westreich, J. K. Edwards, S. R. Cole, R. W. Platt, S. L. Mumford, and E. F. Schisterman. Imputation +approaches for potential outcomes in causal inference. International Journal of Epidemiology, 44(5): +1731–1737, 2015. +I. R. White and J. B. Carlin. Bias and efficiency of multiple imputation compared with complete-case +analysis for missing covariate values. Statistics in Medicine, 29(28):2920–2931, 2010. +14 + diff --git a/OdFLT4oBgHgl3EQfPC8X/content/tmp_files/load_file.txt b/OdFLT4oBgHgl3EQfPC8X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33acdde9e67022e0aca239820e05f9a9ed59bb05 --- /dev/null +++ b/OdFLT4oBgHgl3EQfPC8X/content/tmp_files/load_file.txt @@ -0,0 +1,549 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf,len=548 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='12026v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='ME] 27 Jan 2023 G-formula for causal inference via multiple imputation Jonathan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Bartlett1, Camila Olarte Parra1, and Rhian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Daniel2 1Department of Medical Statistics, London School of Hygiene & Tropical Medicine, London, UK 2Division of Population Medicine, Cardiff University, Cardiff, UK Abstract G-formula is a popular approach for estimating treatment or exposure effects from longitudinal data that are subject to time-varying confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' G-formula estimation is typically performed by Monte- Carlo simulation, with non-parametric bootstrapping used for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We show that G-formula can be implemented by exploiting existing methods for multiple imputation (MI) for synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This involves using an existing modified version of Rubin’s variance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In practice missing data is ubiquitous in longitudinal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We show that such missing data can be readily accommodated as part of the MI procedure, and describe how MI software can be used to implement the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We explore its performance using a simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 1 Introduction The collection of methods referred to as G-methods, developed by James Robins and co-workers, can provide valid inference for the effects of time-varying exposures or treatments in the presence of time- varying confounders—variables that affect treatment over time and the outcome of interest—even when these are affected by previous values of treatment (Naimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' One such method is parametric G-formula (sometimes known as G-computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Parametric G-formula involves postulating models for the time-varying confounders and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The expected outcome under specified longitudinal treatment regimes of interest can then be estimated and contrasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The evaluation of G-formula estimators often involves intractable integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To overcome this in practice, G-formula implementations make use of Monte-Carlo integration (simulation) (Daniel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' McGrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Inference for the resulting estimators is performed using bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To reduce Monte-Carlo errors, a large number of simulations is often needed within each bootstrap sample, making the overall procedure computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A further common obstacle to using G-formula is that some individuals may have missing values in the time-varying confounders, time-varying treatment variables, or the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In practice, given the already high computational complexity of parametric G-formula, analysts faced with the additional complication of missing data may use simplistic solutions such as last observation carried forward (Westreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Monte-Carlo simulation-based implementations of G-formula share close connections to the method of multiple imputation (MI) for handling missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Previously Westreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2015) explained how MI could be used to implement G-formula in a point treatment setting, but stated that Rubin’s variance combination rules could not be used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In this paper we show that methods for adapting MI for missing data to generate synthetic datasets can be used to implement a G-formula estimator 1 (Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As such, we show that an accompanying simple variance combination rule can be used to estimate the variance of the resulting estimator, obviating the need for bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Furthermore, we show that this MI approach can readily accommodate missing data, thereby providing a convenient single solution to handling the missingness of both actual and counterfactual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 2 we review parametric G-formula and how it is typically implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 3 we describe how G-formula can be implemented by exploiting existing methodology for using MI to generate synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 4 we report the results of simulation studies investigating the performance of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We conclude in Section 5 with a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 2 Review of G-formula We assume we have a sample of data from nobs independent individuals from some well-defined pop- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For individual i we collect confounder measurements Lit at times t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We also collect measurements on the treatment Ait at each time, and a final outcome Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For concreteness, we consider in the following the case of T = 2, although the developments naturally extend to T > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We also suppose for now that Lit is a real-valued scalar, but natural adaptations/extensions to dis- crete and higher-dimensional confounders apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Let Y ¯a denote the potential outcome for a randomly selected individual from the population when their treatment sequence has been set to ¯a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' G-formula relies on certain identification assumptions being satisfied, for the details of which we refer the reader to Hernan and Robins (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The G-formula estimator of µ = E(Y ¯a) = E(Y a0,a1,a2) for specified values of a0, a1 and a2 is then based on the fact that under the aforementioned identifying assumptions E(Y ¯a) = � l0 � l1 � l2 E(Y |a0, a1, a2, l0, l1, l2)f(l2|a0, a1, l0, l1)f(l1|a0, l0)f(l0)dl2dl1dl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (1) To implement G-formula we specify and fit models f(Y |A0, A1, A2, L0, L1, L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ψY ), f(L2|A0, A1, L1, L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ψ2), f(L1|A0, L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ψ1), f(L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ψ0) (2) Since the expression in (1) cannot generally be evaluated analytically, implementations of G-formula are typically based on Monte-Carlo integration, through simulation of the longitudinal confounders and outcome for each of nsyn individuals, under the treatment regimes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' That is, given maximum likelihood estimates of the parameters in the conditional models in equation (2), denoted ˆψ0, ˆψ1, ˆψ2, ˆψY , we sequentially simulate for each individual i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , nsyn as follows ˜Li0 ∼ f(L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ˆψ0) ˜Li1 ∼ f(L1|a0, ˜Li0, ˆψ1) ˜Li2 ∼ f(L2|a0, a1, ˜Li0, ˜Li1, ˆψ2) ˜Yi ∼ f(Y |a0, a1, a2, ˜Li0, ˜Li1, ˜Li2, ˆψY ) 2 The Monte-Carlo G-formula estimator of E(Y a0,a1,a2) is then 1 nsyn nsyn � i=1 ˜Yi (3) The number of individuals to simulate for, nsyn, could be set equal to nobs, but choosing a larger value reduces Monte-Carlo error in the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For statistical inference, implementations of G-formula in Stata and R rely on the use of non-parametric bootstrapping (Daniel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' McGrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2020), which as noted in Section 1, is computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' While we have stated that a model f(L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ψ0) is specified and used, in fact this is not needed and is not typically used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Instead, a non-parametric model for f(L0) is used, and the simulation is per- formed by sampling a value of L0 from its empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Moreover, when as is often the case interest lies in the mean E(Y ¯a) (as opposed to some other function of the distribution of Y ¯a), it suf- fices to specify a model for E(Y |A0, A1, A2, L0, L1, L2), rather than for the full conditional distribution f(Y |A0, A1, A2, L0, L1, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Our choice in the preceding to describe a version of G-formula that specifies the latter conditional distribution (rather than mean) model is motivated by the fact this version matches the approach taken in an MI implementation of G-formula, which we describe next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 3 G-formula via multiple imputation In this section we describe how a Monte-Carlo G-formula estimator can be implemented using MI meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 we describe how the point estimator is constructed using MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 we explain why Rubin’s standard variance estimator is biased in this instance, and describe an alterna- tive variance estimator, which was derived in the context of using MI to generate synthetic datasets by Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 we describe how standard MI software can be used to imple- ment the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Lastly, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 we describe how the approach readily extends to accommodate missing actual data (as opposed to missing counterfactual data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 Point estimation To estimate E(Y ¯a) by MI, first augment the observed dataset by adding nsyn additional rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Let n = nobs + nsyn, such that n denotes the number of rows in the augmented dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' the original plus augmented rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the augmented rows, as shown in Table 1, the baseline and time-varying confounders (L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , LT ) and final outcome Y are set to missing, while the treatment variables are set to their values under the regime of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A0 = a0, A1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , AT = aT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variable R indicates whether the data row was in the original sample (R = 1) or not (R = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Next, Bayesian MI is used to generate M imputations of the missing values in this augmented dataset, using the chosen sequential models (equation (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Within imputation m (m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , M), calculate the mean of Y in the augmented rows (R = 0): ˆµm = �n i=1(1 − Ri)Y m i �n i=1 1 − Ri where Y m i denotes the imputation of Yi in imputation m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The mean outcome under the treatment regime of interest, µ = E(Y ¯a), is then estimated as ˆµ = 1 M M � m=1 ˆµm (4) The resulting estimator is identical to the Monte-Carlo G-formula estimator described in Section 3 R L0 A0 L1 A1 L2 A2 Y 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='6 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='6 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='9 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 0 NA 1 NA 1 NA 1 NA 0 NA 1 NA 1 NA 1 NA 0 NA 1 NA 1 NA 1 NA 0 NA 1 NA 1 NA 1 NA 0 NA 1 NA 1 NA 1 NA Table 1: G-formula via MI data setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The original dataset (top part) is augmented with additional rows (bottom part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the augmented part, confounders L0, L1, L2 and outcome Y are set to missing (indicated here by NA), while the treatment variables A0, A1, A2 are set to their values under the regime of interest (here 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variable R indicates whether the row is originally observed data (R = 1) or not (R = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 2, except for the fact that in MI as originally conceived by Rubin (1987), imputation m is generated conditional on an imputation specific draw from the posterior distribution of the imputation model (here the models in equation (2)) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Although this so called ‘proper’ MI estimator has strictly larger variance than the ‘improper’ MI estimator which generates each imputation conditional on an efficient estimate of the parameters, the difference goes to zero as n → ∞ and M → ∞ (Wang and Robins, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 Variance estimation The variance of an MI estimator is typically estimated using Rubin’s variance estimator (Rubin, 1987): (1 + M −1) ˆB + ˆV , where ˆB = 1 M − 1 M � m=1 (ˆµm − ˆµ)2 denotes the between-imputation variance and ˆV = 1 M M � m=1 � Var(ˆµm) denotes the average within-imputation variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' While in many settings Rubin’s variance estimator is asymptotically unbiased, in some it can be biased upwards or downwards relative to the true repeated sampling variance of the MI point estimator (Robins and Wang, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' One such situation is where only a subset of the records used to fit the imputation model is used to fit the analysis model, of which the G-formula via MI estimator is one such example – the original observed dataset is used to fit the imputation models, while only the augmented dataset rows are used to fit the analysis model (estimating the mean of Y among those with R = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As such we may anticipate that Rubin’s variance estimator will be biased for the G-formula via MI point estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We demonstrate this empirically in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The G-formula via MI estimator is closely related to the use of MI to generate samples from synthetic populations, first proposed by Rubin (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Here the objective is to release these synthetic samples rather than the original data in order to protect the confidentiality of survey respondents’ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For 4 synthetic MI, Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2003) suggested the following variance estimator to estimate Var(ˆµ): ˆVsyn = (1 + M −1) ˆB − ˆV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (5) Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2003) derived this from both Bayesian and repeated sampling perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To build intuition for ˆVsyn, we now show it is unbiased for Var(ˆµ) in a highly simplified but instructive setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Thus suppose we observe data from nobs individuals on an outcome Y ∼ N(µ, σ2) and interest lies in inference for µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Here to estimate the mean µ we can of course trivially use the sample mean ¯Y = n−1 obs �nobs i=1 Yi, which has repeated sampling variance σ2/nobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Suppose however that we use Bayesian MI to generate M new imputed datasets of size nsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For simplicity, we assume σ2 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In this case, under the standard non-informative prior for µ, to generate imputation m we first draw ˜µ(m) ∼ N( ¯Y , σ2 nobs ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For i = nobs + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , n we then simulate (impute) nsyn new Y values Yi(m) = ˜µ(m) + ǫi(m) where ǫi(m) ∼ N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Having generated imputed/synthetic datasets for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , M, the estimate of µ based on them is then ˆµ = 1 M M � m=1 ˆµm = 1 M M � m=1 1 nsyn n � i=nobs+1 � ˜µ(m) + ǫi(m) � = 1 M M � m=1 ˜µ(m) + 1 nsynM M � m=1 n � i=nobs+1 ǫi(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Letting ˜µ = {˜µ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , ˜µ(M)}, this has variance Var(ˆµ) = E {Var(ˆµ|˜µ)} + Var {E(ˆµ|˜µ)} = E \uf8f1 \uf8f2 \uf8f3Var \uf8eb \uf8ed 1 nsynM M � m=1 n � i=nobs+1 ǫi(m) ������ ˜µ \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe + Var � 1 M M � m=1 ˜µ(m) � = σ2 nsynM + Var � E � 1 M M � m=1 ˜µ(m) ����� ¯Y �� + E � Var � 1 M M � m=1 ˜µ(m) ����� ¯Y �� = σ2 nsynM + Var( ¯Y ) + E �σ2/nobs M � = σ2 nsynM + (1 + M −1) σ2 nobs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' With σ2 known, the within-imputation variance is σ2/nsyn for every imputed dataset, and so ˆV = σ2/nsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Conditional on the observed data ¯Y , the between-imputation variance estimator ˆB is an unbi- ased estimator of Var(ˆµm| ¯Y ) = Var \uf8eb \uf8ed ˜µ(m) + 1 nsyn n � i=nobs+1 ǫi(m) ������ ¯Y \uf8f6 \uf8f8 = σ2 nobs + σ2 nsyn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Thus, unlike in the missing data setting, the between-imputation variance captures variability both due to 5 uncertainty about µ in the observed data estimate and the additional variability due to effectively taking new random samples of size nsyn from the population for each imputation (Reiter and Raghunathan, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The expected value of ˆVsyn is then E( ˆVsyn) = E{(1 + M −1) ˆB − ˆV } = (1 + M −1) � σ2 nobs + σ2 nsyn � − σ2 nsyn = σ2 nsynM + (1 + M −1) σ2 nobs = Var(ˆµ), such that ˆVsyn is unbiased for Var(ˆµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the Appendix we derive the variance estimator ˆVsyn for the G-formula via MI estimator using the asymptotic results developed by Robins and Wang (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As noted by Reiter (2002) and Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2003), the variance estimator ˆVsyn can be negative, although the probability of this occurring goes to zero as M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Reiter (2002) proposes using the within-imputation variance ˆV if ˆVsyn < 0, but reported that in simulations this did not lead to good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 4 we investigate the performance of a procedure where M is successively increased until ˆVsyn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' According to Reiter (2002), Raghunathan and Rubin (2000) proposed inference based on a t-distribution with degrees of freedom given by vf = (M − 1) � 1 − M ˆV (M + 1) ˆB �2 , (6) the performance of which we explore in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 Implementation using imputation software To implement the proposed approach, as described previously, the observed dataset of size nobs is aug- mented by an additional nsyn rows in which all variables are set to missing except the treatment variables, which are set to their values under the regime of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A0 = a0, A1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' , Ak = ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' MI soft- ware, such as the mice package in R, can then be applied to the resulting dataset, with options specified so that the time-varying confounders and outcome are imputed sequentially in time as per the models given in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Since the missingness pattern is monotone, no iterative methods such as Markov Chain Monte Carlo are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Following imputation, the augmented subset is extracted from each im- puted dataset, and the mean of Y is evaluated in each, yielding ˆµm, along with a corresponding complete data variance estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variance estimator ˆVsyn in equation (5) can then be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Ordinarily interest focuses on the contrast of potential outcome means under two (or more) different treatment regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To estimate the corresponding contrast in potential outcome means, we augment the observed dataset twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the second augmentation part, the treatment variables are set according to the second treatment regime of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The difference in potential outcome means can be estimated by the difference in simulated outcomes between the two augmented parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variance of the resulting estimator can be estimated by the sum of the variance estimator given in equation (5) when applied to the two regimes of interest, since the sets of synthetic imputations for the two regimes are independent (conditional on the parameter draws used to impute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Implementation of the preceding steps using packages such as mice in R is relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Nonetheless, to facilitate use of the approach, we provide the R package gFormulaMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This augments the supplied dataset as described above and imputes missing data using the mice package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The resulting imputed datasets contain only the augmented portion of the imputations (with R = 0), which can be used to estimate potential outcome means and contrasts of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The point estimates and variances from the analysis of these imputations are then passed to a function implementing the variance estimator 6 given in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As noted earlier, the standard (non-Bayesian) implementation of G-formula avoids specification of a model for f(L0), and instead simulates from the empirical distribution of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the context of MI for generation of synthetic samples, Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2003) proposed using the approximate Bayesian bootstrap approach of Rubin and Schenker (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 we investigate in simulations the performance of using this approach for Bayesian non-parametric imputation of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 Missing data Now suppose that there are some data missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Missing data could occur in either the longitudinal confounders Lit, the final outcome Yi, or the time-varying treatment variables Ait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We assume that the missing at random assumption is deemed plausible for the missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In this case, the application of the results of Robins and Wang (2000) given in the Appendix still apply without modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As such, we can impute both the missing data in the original data (where R = 1) and missing potential outcome data in the augmented rows (where R = 0), and continue to use the variance estimator ˆVsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' If the missingness pattern in the original data is monotone because of dropout, the missing values in the original data and missing potential outcomes can be imputed simultaneously by imputing sequentially in time, as described in the setting without missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' More typically, however, the pattern of missingness in the original data will not be monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In this case, we propose adapting an approach which works well when the missingness pattern is nearly monotone (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 of Schafer (1997)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We propose that first the missing values in the original data are imputed M times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The augmented rows are then added to each of the M imputed datasets, and the missing potential outcomes in these rows are then imputed once (in each of the M datasets) based on the sequential models (equation (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The models required for G-formula given in equation (2) do not fully specify the joint distribution of all the variables under consideration, since they do not specify models for the treatment variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The imputation models used to impute the missing data in the original dataset should ideally be compatible with those used to impute the augmented rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' One way to achieve this is to specify a full joint model for all the variables by, in addition to the models in equation (2), specifying models for the time-varying treatment variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' That is, for t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' T , we specify models f(Ait| ¯Ai(t−1), ¯Lit), such as suitable logistic regression models if treatment is binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' While imputation from such a joint model is possible using Bayesian model software such as JAGS, imputation is more commonly performed using methods such as chained equations, as implemented in the popular R package mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As such, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 we investigate performance when the models used to impute missing data are not strictly compatible with the models specified and used by G-formula (in equation (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the setting with missing data, our R package gFormulaMI takes as input a set of M imputed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' It then augments each imputed dataset with the required additional rows and imputes each once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 4 Simulations In this section we report the results of simulations performed to examine the empirical performance of the G-formula via MI approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We first consider, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1, the setting where there is no missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Next, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2, we consider the situation where some data are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 No missing data We simulated datasets for 500 individuals under the following models: L0 ∼ N(0, 1) P(A0 = 1|L0) = expit(L0) L1 ∼ N(A0 + L0, 1) P(A1 = 1|A0, L0, L1) = expit(A0 + L1) L2 ∼ N(A1 + L1, 1) P(A2 = 1|A0, A1, L0, L1, L2) = expit(A1 + L2) Y ∼ N(A2 + L2, 1) We report results for estimates of E(Y 1,1,1) − E(Y 0,0,0), whose true value under the data generating mechanism is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The G-formula via MI approach was implemented using the mice package in R, im- puting L0, L1, L2 and Y from normal linear models including all the preceding (in time) treatment and confounder variables linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Since the missingness pattern is monotone, we specified that mice only perform one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We set nsyn = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To investigate how performance varied with M, we evaluated the procedure using M = 5, 10, 25, 50, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' If in a particular simulation ˆVsyn < 0, we added an additional M imputations and re-calculated ˆVsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This was repeated until ˆVsyn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Scenario M Bias Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' SE Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' SE Raghu df 95% CI Z 95% CI Mean M Max M 1 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='236 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 25 2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='223 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 30 3 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='220 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 50 4 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='219 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 50 5 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='219 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 100 Table 2: Simulation results for G-formula via MI without any missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Results are shown for different numbers of initial imputations M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Emp SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' gives the empirical standard error of the point estimates while Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' SE gives the mean estimated standard error based on ˆVsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Raghu df 95% CI gives the coverage of t-based confidence intervals based on the degrees of freedom given in equation (6) while Z 95% CI gives coverage for confidence intervals constructed using N(0, 1) quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Mean M and Max M give the mean and maximum value of M required across the simulations in order to obtain ˆVsyn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Table 2 shows results based on 10,000 simulations per (inital) value of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As expected since the imputation models were correctly specified, the G-formula via MI estimator for E(Y 1,1,1) − E(Y 0,0,0) was unbiased for all values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variance estimator ˆVsyn was also essentially unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Confidence intervals calculated based on a t-distribution with degrees of freedom calculated from (6) showed overcov- erage for M = 5, 10, 25, but achieved nominal coverage for M = 50 and M = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Confidence intervals calculated based on a standard normal showed substantial undercoverage for M = 5 and M = 10, but had close to nominal coverage for M = 50 and M = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Lastly, when using a smaller initial value for M, sometimes additional imputations were required to ensure ˆVsyn > 0, as indicated by the mean and maximum M values in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' However, when an initial M = 50 (or M = 100) imputations were used, ˆVsyn was always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We additionally ran 10,000 simulations with M = 50 using the approximate Bayesian bootstrap to impute L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variance estimator ˆVsyn was again unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The coverage of the confidence interval constructed using a t-distribution with degrees of freedom calculated using equation (6) was 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3% while the normal based confidence interval had coverage 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 Missing data Next we performed simulations where some data were missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Data in each of L1, A1, L2, A2 and Y were made missing completely at random, with the probability of each being missing set to π, with π = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As such, the probability of an individual having complete data was (1−π)5 and the average number of variables missing per individual was 5π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Thus π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5 is a really quite extreme scenario, with only approximately 3% of individuals having complete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To implement G-formula via MI we used an initial call to mice to impute the missing values M = 50 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The continuous variables L1, L2 and Y were imputed using normal linear models while A1 and A2 were imputed using logistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Since the missingness pattern was not monotone, as per the standard chained equations algorithm, for imputation of a given variable, all the other variables were included as covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The number of iterations was left at its default value of 5, except for π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Here, with a very large amount of missingness, we found that 50 iterations were required to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Having imputed the missing data, the additional nsyn = 500 rows were added to each imputed dataset, and mice was applied to each of the M = 50 datasets, specifying to impute using one iteration sequentially according to time, as used in the scenario without missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Scenario π Bias Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' SE Mean est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' SE Raghu df 95% CI Z 95% CI 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='224 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='231 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='258 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='361 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='3 Table 3: Simulation results for G-formula via MI with missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' π is the probability that each of L1, A1, L2, A2 and Y are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Table 3 shows the results based on 10,000 simulations per value of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The G-formula via MI estimator had minimal bias across all four scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As we would expect, the empirical standard error increased with increasing amounts of missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variance estimator ˆVsyn was positive in all simulations and for all values of π when using an initial value of M = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ˆVsyn was unbiased for the empirical SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Confidence intervals based on a t-distribution with degrees of freedom calculated from (6) showed slight overcoverage, while the normal based intervals showed slight undercoverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 5 Discussion G-formula via MI is an attractive approach for implementing parametric G-formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' With complete data, inference for G-formula estimators is usually based on non-parametric bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' While the boot- strap provides a consistent variance estimator under mild assumptions, since correct model specification is generally required for consistency of the G-formula point estimator, it makes sense to use a variance estimator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' based on Bayesian MI) which exploits an assumption that these models are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The conventional approach based on bootstrapping requires the models used in G-formula to be re- fitted to each bootstrap sample, and typically hundreds if not thousands of bootstrap samples are used to obtain inferences with an acceptably small amount of Monte-Carlo error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Although our simulations were limited in scope, they suggest that reliable inferences can be obtained via MI methods using only 50 imputations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Moreover, when data are complete such that iterative methods are not required, the models need only be fitted once to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Although implementation is relatively straightforward using existing multiple imputation packages, we have developed an R package gFormulaMI that interfaces with the mice package to perform the required data manipulation steps, estimate mean outcomes under each treatment regime of interest, and calculate the synthetic MI variance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Imputation packages 9 such as mice are flexible in regard model specification, for example allowing the possibility for the user to include interactions and higher order effects in models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' For In practice datasets, whether arising from experimental or observational studies, have missing data to a lesser or greater extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In this context the G-formula via MI approach has greater appeal, given that it provides a coherent potential solution to handle both the missing actual and missing counterfactual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' While our simulation investigations suggest the G-formula via MI approach can perform well, our conclusions regarding its empirical performance in general are necessarily limited by the fact our simulations have only explored a relatively simple setup - with one continuous confounder and a small number of time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' One alternative to imputing missing data when implementing G-formula is to fit each of the models (in equation (2)) using the subset of records for which the variables involved in each model are fully observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' These complete case model fits yield consistent estimates of the respective conditional model parameters provided the probability of having all the variables involved in the model is independent of the dependent variable conditional on the covariates (White and Carlin, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' When the pattern of missingness in the longitudinal dataset is complex, consisting of both intermittent missingness and missingness due to dropout, such an assumption can sometimes be deemed more plausible than missing at random, whose meaning becomes complex in such settings (Robins and Gill, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In this paper we have focused on G-formula where the outcome is a variable Y measured at some final time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' G-formula can also be used when the outcome is the time to some event of interest, for example based on discrete time logistic regression models (Westreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The G-formula via MI approach can also be used in this setting, by defining appropriate time-dependent binary indicators of survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Implementations of G-formula (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' the G-formula packages in Stata (Daniel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2011) and R (McGrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=', 2020)) often fit models pooled across time points for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This is achieved by formatting the data in so-called long form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Doing so permits borrowing of information across time points in the estimation of regression parameters, but of course relies on the validity of the assumption that the conditional distribution of confounders given earlier variables is homogenous across time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Although this approach could be implemented via the MI approach we have outlined, we do not believe it is possible using standard imputation software such as mice in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This is because having transformed the data into long form, it is not possible to update values from one row of the data frame from another within the algorithm, which would be required for subsequent times points for the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' While our focus in this paper has been on static treatment regimes, G-formula can be used to estimate the effects of dynamic treatment regimes, where the exposure or treatment at a given time point is assigned dependent on the longitudinal history observed up to that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The G-formula via MI approach can be extended to this case, by setting the treatment variables to missing in the augmented part of the dataset and then specifying how they should be imputed based on the preceding (in time) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This can be achieved for example in the mice package through the use of user specified deterministic (or indeed stochastic) imputation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Funding This work was funded by a UK Medical Research Council Grant (MR/T023953/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Code R code for the simulation study can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='com/jwb133/gFormulaViaMultipleImputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The R package gFormulaMI is available from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='com/jwb133/gFormulaMI, and will in due 10 course be made available on CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Appendix In this appendix we show that the variance estimator ˆVsyn derived by Raghunathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' (2003) is consistent for the G-formula via MI estimator, using the results of Robins and Wang (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' We first describe how the G-formula via MI estimator can be embedded into the setup of Robins and Wang (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' To that end, consider the augmented dataset where ¯L and Y are set to missing in the augmented part and the treatment vector ¯A is set to some level ¯a, in order to estimate µ = E(Y ¯a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Define the variable R that takes value 1 in the original observed data and 0 in the augmented part of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The full data vector is this F = ( ¯A, ¯L, Y, R) and the observed vector is thus O = ( ¯A, ¯LR, Y R, R), with ¯L and Y missing in those with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The missing values in ¯L and Y , in the augmented part of the dataset, are then multiply imputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the first instance we assume, following Robins and Wang (2000), that the imputations are generated conditional on the MLE of the imputation model parameter ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This is referred to by Rubin as ‘improper’ imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' After imputation, in the G-formula via MI approach we estimate µ by the mean of those in the augmented part of the dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' observations for which R = 0, and then average these means across the imputed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' This is equivalent to simply calculating the mean of the imputed outcomes across all imputations, using only data from those individuals with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As such, the corresponding ‘complete data’ estimator of µ is �n i=1(1 − Ri)Yi �n i=1(1 − Ri) The complete data estimating function is therefore given by U(F, µ) = (1 − R)(Y − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Thus, under the regularity conditions detailed by Robins and Wang (2000), the (non-Bayesian improper) G-formula via MI estimator of µ which uses M imputations is asymptotically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The variance of the estimator with M = ∞ is given by equation A3 of Robins and Wang (2000) as Σ = τ −1 � E � Uobs(ψ∗, µ∗)⊗2� + κΛ(ψ∗)κT + κE � D(ψ∗)U(ψ∗, µ∗)T � + E � D(ψ∗)U(ψ∗, µ∗)T �T κT � (τ T )−1, where A⊗2 = AAT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the following we define the terms involved in the preceding expression, derive their values in our setting, and show that ˆVsyn is a consistent estimator of this asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Parameters with a superscript ∗ denotes the true value of the corresponding parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The quantity τ is defined as τ = −E �∂ ¯U(ψ∗, µ∗) ∂µ � , where ¯U(ψ, µ) = 1 M M � m=1 U(F m(ψ), µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Here F m(ψ) denotes the mth imputed data vector for an individual, with imputations generated condi- tional on the value ψ of the imputation model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Substituting in the complete data estimating 11 function, we have ¯U(ψ, µ) = 1 M M � m=1 (1 − R)(Y m(ψ) − µ) and so ∂ ¯U(ψ∗, µ) ∂µ = ∂ ∂µ � 1 M M � m=1 (1 − R)(Y m(ψ∗) − µ) � = R − 1 Thus τ = E(1 − R) = nsyn/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The quantity Uobs(ψ, µ) is defined as Eψ{U(ψ, µ)|O, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In our case, we have Uobs(ψ, µ) = Eψ{U(ψ, µ)|O, R} = Eψ{(1 − R)(Y − µ)|O, R} = (1 − R)Eψ{Y − µ|A = ¯a, R = 0} = (1 − R) � Eψ(Y | ¯A = ¯a, R = 0) − µ � , and thus Uobs(ψ∗, µ∗) = 0 since Eψ∗(Y |A = ¯a, R = 0) = µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Letting ˆψ denote the observed data MLE of the imputation model parameters, D(ψ) denotes the in- fluence function of the estimator, and under standard regularity conditions n1/2( ˆψ−ψ∗) is asymptotically normal with mean zero and covariance matrix equal to Λ(ψ∗) = E � D(ψ∗)⊗2� = I−1 obsE � S⊗2 obs(ψ∗) � I−1 obs where Sobs(ψ) denotes the observed data score and Iobs the observed information matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' As noted by Robins and Wang following their Theorem 2, if, as we assume, the imputation model is correctly specified, Iobs = E � S⊗2 obs(ψ∗) � , in which case Λ(ψ∗) = I−1 obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Observations in the augmented dataset, with R = 0, do not contribute to the estimation of the imputation model, and so for such observations D(ψ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Conversely, observations in the original data, with R = 1, do not contribute to the estimation of µ in the imputed datasets, and so for such observations U(ψ∗, µ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Consequently, D(ψ∗)U(ψ∗, µ∗) = 0 for all observations, and so E � D(ψ∗)U(ψ∗, µ∗)T � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Thus the asymptotic variance of the G-formula via MI estimator, with M = ∞, is given by τ −2κI−1 obsκT (7) where κ = E{U(ψ∗, µ∗)Smis(ψ∗)T }, Smis = ∂ ∂ψ log f(F|O, R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' ψ)|ψ=ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Equation A1 from Robins and Wang gives that the (standardised by n) between imputation variance estimator ¯B converges as m, n → ∞ to τ −2 � κI−1 obsκT + E � {U(ψ∗, µ∗) − Uobs(ψ∗, µ∗)}2�� = τ −2 � κI−1 obsκT + E � U(ψ∗, µ∗)2�� , 12 since Uobs(ψ∗, µ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The (standardised) within-imputation variance �V• converges to τ −2E � Uobs(ψ∗, µ∗)⊗2 + {U(ψ∗, µ∗) − Uobs(ψ∗, µ∗)}⊗2� = τ −2E � U(ψ∗, µ∗)2� , again using the fact Uobs(ψ∗, µ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Thus ¯B − �V• converges to τ −2κI−1 obsκT = Σ as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Lastly, since in practice we can only implement the MI estimator with finite M, we must add an additional M −1 ¯B to account for the additional Monte-Carlo variability, resulting in the variance estimator ˆVsyn given in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' In the main paper and appendix we have considered estimation of E(Y ¯a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' However, the preceding arguments for the consistency of ˆVsyn only depended on the complete data estimating function in so far as it is of the form (1 − R)W (Y, µ) for some function W (Y, µ) where Eψ∗{W (Y, µ∗)| ¯A = ¯a, R = 0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' References R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Daniel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' De Stavola, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Cousens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' gformula: Estimating causal effects in the presence of time-varying confounding or mediation using the g-computation formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The Stata Journal, 11(4): 479–517, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Hernan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Robins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Causal Inference: What If, chapter 13 Standardization and the para- metric g-formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Boca Raton: Chapman & Hall/CRC, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' McGrath, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Petito, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Logan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Hern´an, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' gformula: An r package for estimating the effects of sustained treatment strategies via the parametric g-formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Patterns, 1(3):100008, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Naimi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Cole, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Kennedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' An introduction to g methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' International Journal of Epidemiology, 46(2):756–762, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Raghunathan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Conference of the International Society for Bayesian Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Raghunathan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Reiter, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Multiple imputation for statistical disclosure limita- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Journal of Official Statistics, 19(1):1, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Reiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Satisfying disclosure restrictions with synthetic data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Journal of Official Statistics, 18 (4):531, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Reiter and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Raghunathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The multiple adaptations of multiple imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Journal of the American Statistical Association, 102(480):1462–1471, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Robins and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Gill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Non-response models for the analysis of non-monotone ignorable missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Statistics in Medicine, 16(1):39–56, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Robins and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Inference for imputation estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Biometrika, 85:113–124, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Multiple imputation for nonresponse in surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' New York: Wiley, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Statistical disclosure limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Journal of Official Statistics, 9(2):461–468, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Rubin and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Schenker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Multiple imputation for interval estimation from simple random samples with ignorable nonresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Journal of the American Statistical Association, 81(394):366–374, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 13 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Schafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Analysis of incomplete multivariate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' CRC press, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Robins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Large-sample theory for parametric multiple imputation procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Biometrika, 85:935–948, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Westreich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Cole, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Young, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Palella, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Tien, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Kingsley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Gange, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Hern´an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident aids or death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Statistics in Medicine, 31(18):2000–2009, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Westreich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Edwards, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Cole, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Platt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Mumford, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Schisterman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Imputation approaches for potential outcomes in causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' International Journal of Epidemiology, 44(5): 1731–1737, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' White and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Carlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Bias and efficiency of multiple imputation compared with complete-case analysis for missing covariate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' Statistics in Medicine, 29(28):2920–2931, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFLT4oBgHgl3EQfPC8X/content/2301.12026v1.pdf'} diff --git a/OtFPT4oBgHgl3EQfnDXw/content/tmp_files/2301.13128v1.pdf.txt b/OtFPT4oBgHgl3EQfnDXw/content/tmp_files/2301.13128v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..388ba2e27b60f87e5b5e63b1874fd6599d1c2ef8 --- /dev/null +++ b/OtFPT4oBgHgl3EQfnDXw/content/tmp_files/2301.13128v1.pdf.txt @@ -0,0 +1,804 @@ +Standardized CycleGAN training for unsupervised stain +adaptation in invasive carcinoma classification for breast +histopathology +N.Nerrienet, R.Peyret, M.Sockeel, S.Sockeel +January 31, 2023 +Abstract +Generalization is one of the main challenges of computational pathology. Slide preparation het- +erogeneity and the diversity of scanners lead to poor model performance when used on data from +medical centers not seen during training. +In order to achieve stain invariance in breast invasive carcinoma patch classification, we imple- +ment a stain translation strategy using cycleGANs for unsupervised image-to-image translation. We +compare three cycleGAN-based approaches to a baseline classification model obtained without any +stain invariance strategy. Two of the proposed approaches use cycleGAN’s translations at inference +or training in order to build stain-specific classification models. The last method uses them for stain +data augmentation during training. This constrains the classification model to learn stain-invariant +features. Baseline metrics are set by training and testing the baseline classification model on a refer- +ence stain. We assessed performances using three medical centers with H&E and H&E&S staining. +Every approach tested in this study improves baseline metrics without needing labels on target stains. +The stain augmentation-based approach produced the best results on every stain. Each method’s +pros and cons are studied and discussed in this paper. +However, training highly performing cycleGANs models in itself represents a challenge. In this +work, we introduce a systematical method for optimizing cycleGAN training by setting a novel stop- +ping criterion. This method has the benefit of not requiring any visual inspection of cycleGAN results +and proves superiority to methods using a predefined number of training epochs. In addition, we also +study the minimal amount of data required for cycleGAN training. +1 +Introduction +General context. Digital histopathology is changing the way pathologists work on a day-to-day basis. +Not only does it allow them to share data and insights about a diagnosis across the globe easily, but with +computational pathology, algorithms capabilities may be leveraged to ease pathologists’ work on many +tasks. Deep learning has shown promising results for various applications such as cancer classification +[1, 2, 3], detection [4, 5, 6] or segmentation [7, 8] of biomarkers required for diagnosis. +However, the integration of AI in histopathology comes with a set of challenges to overcome. +In +particular, the diversity and heterogeneity in slide preparation increases data variability across centers +or laboratories. Slide preparation is a long and tedious process consisting of multiple steps which can +each be a source of variability that negatively impacts algorithms performance [9]. Specifically, biopsy +samples are collected during surgeries and are molded into a paraffin bloc for easier slicing. In order +for the paraffin to penetrate the tissue down to molecular level, the tissue is gradually dehydrated using +series of ethanol baths with rising concentrations. Dehydration may affect tissue structure depending +on the exact protocol applied. Samples are then sliced into thin tissue pieces and placed on a glass +slide. As tissue handling is tedious, artefacts such as rips or folds often appear. Additionally, the tissue +being originally transparent, contrasting agents are to be added in order to analyse certain structures of +interest. +This process is called staining and is a crucial part of slides preparation. Hematoxylin and Eosin +(H&E) are typically used to dye cell nuclei in a specific color and tissue surrounding cells in another +one, whereas some other stains, based on Immunohistochemical (IHC) reactions, target some specific +receptors or molecular compounds of the cell membrane, making them visible. +The staining process +and the manufacturer from whom the staining agents are purchased differ from a laboratory to another, +leading to intra-stain variability. In France, pathologists are also used to adding Safran to H&E (HES) in +1 +arXiv:2301.13128v1 [eess.IV] 30 Jan 2023 + +order to dye connective tissues. This gives yet another appearance to tissue samples, which can greatly +differ depending on the balance of Hematoxylin’s and Eosin’s constitutive components. Digital pathology +adds one last source of variability. Physical slides are scanned to produce Whole Slide Image (WSI) that +can be viewed on a software. The scanner’s brand and settings, as well as the file format, are clearly +identified sources of data variability [9]. These considerations result in WSI having clearly distinguishable +appearances from one laboratory to another. +In fact, it has been observed that AI models performed poorly when applied to data coming from +medical centers not seen during training or to data stained differently than training data [9, 10]. Therefore, +before being used in a new laboratory, models need to be fine-tuned using annotated data from this +laboratory. This is a time-consuming and expensive process as it requires expert pathologists. +Hence, generalization is one of the main challenges in computational histopathology and solving it +would allow a unique model to be applied on a wide range of laboratories without fine-tuning or annotated +data [11]. +Related works. This issue is commonly addressed in literature using two global approaches: nor- +malization and augmentation. Most of the time, normalization is used to reduce dissimilarity within a +given domain [12, 13, 14]. For instance, the simplest form of stain normalization is to simply convert +RGB images to grayscale, assuming that most of the H&E signal is contained in the morphological and +structural patterns of the tissue. +On the other hand, color augmentation is often about synthesizing new images by varying the col- +ors, the contrast or the illumination. This is used in a deep learning setting, to make learnt features +independent of color variations. +Recently, due to their ability to generate highly realistic outputs, Generative Adversarial Networks +(GAN) [15] and their derivatives have been also intensively used to address the generalization challenge in +digital histopathology. Specifically, either Pix2Pix [16] or CycleGANs [17] are utilized as stain translation +devices depending on the need or availability of paired samples. The Pix2pix often displays more realistic +and consistent results but requires paired samples. However, in a digital histopathology context, it can +be problematic to obtain paired datasets, as it involves the ability to re-stain stained samples which is +a costly and tedious process. Consequently, most studies using GANs’ setup in digital histopathology +rely more heavily on Cycle-GANs. Jun-Yan Zhu et al. [17] demonstrated that cycleGANs can produce +samples in a highly realistic fashion without the need for paired training data. +In [18, 19], the authors identified three methods that can be used to address domain adaptation issues +with CycleGANs, namely Multi-Domain Supervised 1 and 2 (MDS1/MDS2) and Unsupervised Domain +Augmentation (UDA). For each CycleGAN based method, a deep learning task is used on top of the +CycleGAN’s translations. In these approaches, the source domain is either the domain on which most +of the labeled data is available or the domain on which models such as classifiers or detectors already +perform well. Conversely, the target domain is often scarce in annotated data and exhibits poor model +performance, or at least, poor generalization performance. In short, MDS1 consists of training a classifier +on the source domain and applying it on target domain samples translated to the source domain, whereas +MDS2 aims at training a classifier on source domain samples translated to the target domain. Finally, +introduced by Vasijelic et al. +[18], UDA’s goal is to use CycleGANs’ translation as a mean of data +augmentation during training. +In the literature, Shaban et al. [20] employs CycleGANs to show its superiority compared to conven- +tional methods in the case of lymph node metastasis classification. However, they solely use the MDS1 +method and lack the exploration of the other ones which were proven to lead to better performing models +by Vasiljevic et al. [18]. Moreover, little insight is given on how the final CycleGAN model was chosen +and on the data it was trained on. Several other studies use these models in the case of virtual staining for +mitosis detection [21, 22] or in the case of glomeruli segmentation [18] among other works. Interestingly, +Vasiljevic et al. [18] points out a weakness in using CycleGANs, that is, the visual quality of a CycleGAN +translation is not always related to the performance of the task on top of the generation process. This +unexpected behavior implies a substantial difficulty not only when comparing and choosing CycleGANs +for the task at hand, but also when one must decide to stop the training process or not. +This study is built upon all this prior knowledge and applies CycleGANs for breast tumor classification +generalization, while introducing a consistent method on how to evaluate and choose image-to-image +translation models for the downstream task and thus, addressing the previously mentioned concern. +Contributions. This study makes the following key contributions: +• We introduce a systematic and consistent method to train and choose a CycleGAN model. +• We study the optimal amount of WSI required to train a CycleGAN. +2 + +• We introduce a stain agnostic breast tumor classification model. +• We prove the relevance of using CycleGANs to attain stain invariance in classification tasks on a +novel large multi center dataset. +In the following section, a description of deep learning models utilized in this study is exposed, as +well as the datasets used. Section 3 describes the different experiments performed in this work, section 4 +presents the experimental results. In the final section, we will present and interpret the results obtained +from our research. +2 +Materials and Methods +2.1 +Datasets +The data used in this study comes from four different centers. Samples are collected from breast cancer +patients that underwent breast biopsy or mastectomy. Samples are colored with H&E or H&E&S stain. +Figure 1 depicts a sample from each of these centers, illustrating the high variability of observable +appearances. Patches of size 256*256 pixels are then extracted from these WSI at magnification x5. The +reference center is the one with the largest labeled dataset. This center is here referred to as reference +center or source center, interchangeably. +Figure 1: Samples from the various centers used in this study and their respective stain +Patches are given one of the following labels: invasive carcinoma (IC), which is the latest stage of +breast cancer development, and REST, which includes healthy tissue, benign lesions and non-malignant +tumors such as adenoma. +Table 1 illustrates the amount of data, slides and extracted patches, from each center. +Source center +Center 1 +Center 2 +Center 3 +Slides +train : 1690 +test : 42 +test : 119 +test : 44 +test : 309 +Patches +train : 1.7M +test : 75 888 +test : 35 209 +test : 46 808 +test : 469 217 +Table 1: Slides and patches distribution across centers +2.2 +Methods +2.2.1 +Translation model +In this work, CycleGANs [17] are used as translators between different stains prior to the breast carcinoma +classification task. +Figure 2 shows an example of CycleGAN translations with our data. In this example, a patch from +the source domain is translated to the other centers domain. A CycleGAN is trained for every pair of +centers (source center, center i) with i ∈ {1, 2, 3}, as the goal is to make translation between the reference +center and every other center. +CycleGAN architecture. CycleGANs’ generators and discriminators architecture are described in +the original paper [17]. Johnson et al. [23] proved this architecture produces high quality outputs in the +case of style transfer. Moreover, they demonstrated that instance normalization greatly improves style- +transferred samples quality as well as being more efficient for small batch size than batch normalization +[23, 24]. We thus adopt this normalization strategy. As problems such as vanishing gradients, mode +3 + +Source center +Center 1 +Center 2 +Center 3Figure 2: CycleGANs transformations examples from source to targets +collapse and failing to converge are quite standard issues with GANs [25], least-square-loss is used instead +of the negative-log-likelihood standard loss to ensure a more stable training following Xudong et al. [26]. +CycleGAN training and evaluation. Trainings are done using 1000 patches from every center +as using more patches did not show improved quality in translated samples. The number of WSI from +which we extract these 1000 patches is center-dependent and is determined by a further experiment that +will be discussed in the following section 3.2. +Besides, the difficulty of evaluating generative models is a known issue. Initially, generative models +were evaluated by mere visual inspection. However, in digital histopathology, it would require expert +knowledge to ensure that generated samples are realistic enough. In addition, it has been shown by +Vasiljevi´c et al. [18] that generated samples visual quality is not always correlated with performance for +the task on top of the generation process. Specifically, two CycleGANs from two different epochs with the +same visual generation quality can yield significantly different results in the case of a segmentation task +on the generated samples. Hence, there is a need to evaluate the quality of a cycleGAN so as to monitor +and stop its training in a releavant way. To do so, we propose to use Fr´echet Inception Distance (FID) +[27] which is a popular metric used for generative models’ evaluation. This metric uses an Inception +model to extract features from generated and real images, those feature distributions are then used to +compute the Fr´echet Distance [28]. +CycleGAN methods. +CycleGANs can be utilized to address generalization challenges in three +distinct ways, each with its own unique characteristics and applications. +First, Multi-domain Supervised 1 and 2 (MDS1/MDS2) have been introduced by Gadeymar et al. +[19]. In MDS1, the CycleGAN’s translation is used as a normalization preprocessing step to the main +classification model at inference time. Samples are translated to the source domain before being classified. +This enables the use of pre-trained source domain-specific classification models. This method is illustrated +in figure 3. +Figure 3: Multi-Domain Supervised 1 (MDS1). CycleGAN’s translations occur at inference time from +the target center to a source center for which a trained classifier is available. +In MDS2, target domain-specific classification models are built using CycleGAN’s translations as a +normalization preprocessing step at training time as illustrated by figure 4. Models built in this fashion +can then directly be applied on target domain data. +Finally, Unsupervised Domain Augmentation (UDA), illustrated in figure 5, is a method explored by +Vasiljevi´c et al. [18]. This approach makes use of CycleGANs’ translations to various domains as a mean +of data augmentation during training. This leads to the model extracting stain invariant features for +the task at hand, provided that the pool of domains is diverse enough. Contrary to the other methods +that use a single domain translation and build domain specific models, this approach aims at producing +4 + +Source center +Center1 +Center2 +Center3classifier +CycleGAN +source +target +source +TRAINING +INEERENCEFigure 4: Multi-Domain Supervised 2 (MDS2). CycleGAN’s translations occur at training time from the +source center to the target center. +models that perform on a large variety of stains and thus attain stain invariance. +Figure 5: Unsupervised Domain Augmentation (UDA). CycleGAN’s translations are used as a mean of +data augmentation during training. The resulting model can then be applied to a wide range of stains +and centers, including seen as well as unseen centers. +A major advantage of these three approaches is that they don’t require any labeling in the domain +in which samples are translated to, considerably reducing the time and money investment that labeling +requires. +In this study, each of these approaches is used to solve the generalization challenge for breast invasive +carcinoma classification following an experiment scheme described in the following section 3.3. +2.2.2 +Classification model +Classifier architecture. In order to classify patches into the two classes discussed in section 2.1, a +classification model was employed on top of the translation process as described in figures 3, 4 and 5. +The model used for this task is an EfficientNetB1 introduced by Tan et al. [29], with two outputs. In order +to train this model, we use a category cross-entropy loss [30] and an Adam optimizer [31]. Additionally, +to enhance the training data pool, various data augmentation techniques were randomly applied at each +training step: +• Flip: Flip the patch horizontally. +• Rotate: Rotate the patch by a multiple of 90 degree angle. +• Color Jitter: Shift hue, saturation and brightness of the patch. +• Noise: Add Gaussian noise to the patch. +5 + +classifier +cycleGAN +source +target +target +TRAINING +INFERENCEsource +random selection +newcenter +TRAINING +INFERENCEBaseline classification model. In order to establish a baseline performance to compare the different +methods results, we train a classification model using the architecture aforementioned. This baseline +classification model is trained on the reference center trainset (see section 2.1) for 150 epochs, and +baseline performance on each center is obtained by testing this model on every center’s test dataset. +2.2.3 +CycleGANs comparison method and notations +Throughout this study, comparing various CycleGANs in the breast classification task context will be +required. To accomplish this, we introduce a method to evaluate task-wise CycleGAN’s performance. +First, we establish the following notation: a CycleGAN responsible for translations from reference center +to center i, and vice-versa, will be mentioned as follows : CycleGAN i. Next, in order to evaluate a +CycleGAN’s performance related to the classification task, we will use MDS1 modality. To elaborate, +CycleGAN i is utilized to translate samples from a dataset to the reference center. Then, the baseline +classification model is tested on these translated samples to obtain metrics that will be used for comparison +purposes. We further refer to this method as ”MDS1 comparison”. +MDS1 modality is preferred for this task as baseline classification model can be used as an evaluator +whereas labeled training data is too scarce to train ”baseline models” for other centers in our settings. One +can argue that using MDS1 only is insufficient and indeed previous work showed that translation direction +is an important factor [18, 17]. CycleGANs can generate higher quality transformations depending on +translation’s direction and can even hide information in the translated samples [18, 32, 33]. The latter +effect can have an unexpected impact on deep learning models. However, this is often the case when one +domain has more information than the other, which implies that a direction will suppress information +and the other will restore what has been lost [32, 33]. In our case, this effect is greatly mitigated as +the domains, H&E stains, hold the same information content. Therefore, we will solely rely on MDS1 +comparison throughout the study. +6 + +3 +Experiments +3.1 +CycleGANs training stopping criterion +A common challenge when working with GANs is to know when the model has converged and when we +should stop the training procedure. Usually, mere visual inspection of generated samples’ quality is key +to determine when the training should end. However, this procedure comes with two major downsides. +First, it implies saving the model at each training epoch in order to be able to retrieve the epoch weights +that gave the most satisfying results, which is memory consuming and requires a manual intervention. +Secondly, as stated in the section 2.2.1, visual inspection is not sufficient as image quality does not +systematically reflects performance for the task on top of the generation process. +In this experiment, we evaluate the effectiveness of using the FID metric as an early stopping criterion +for training CycleGAN models. We compare a model trained using the FID criterion to models trained +for 25, 50, 75 and 100 epochs. This comparison is done for CycleGANs i (see section 2.2.3) where i takes +the following values {1, 2, 3)}. +The FID for CycleGAN i training is computed as the Fr´echet distance between 1000 samples randomly +selected in the source center, and 1000 samples translated from center i to the the source center. Latter +samples plays the role of the ”generated images”. +The resulting CycleGAN models are then evaluated on center i test dataset using MDS1 comparison +described in section 2.2.3. +For each center, we repeat this experiment three times to compute error +margins and ensure results repeatability. +3.2 +Slides Requirements +When it comes to digital histopathology, an under-discussed but nonetheless decisive matter is the number +of slides required for training a CycleGAN that produces satisfying enough samples. This point is crucial +for the method convenience considering the difficulty of collecting a sufficient amount of WSIs to train +deep learning algorithms. +Therefore, we aim to answer this question. To that end, we build various datasets each containing +1000 patches extracted from a various number N of slides, train CycleGANs on each dataset and assess +its performance. Specifically, patch datasets are extracted from N = {2, 5, 10, 25, 50, 75} slides randomly +picked from reference and target center training sets. For each of these configurations, three datasets are +built in total to ensure that observed results are not dependent on the chosen set of slides. +Resulting CycleGANs are again compared through MDS1 comparison. Finally, to ensure the results +are independent of the choice of training centers, the experiment is conducted for target centers 1, 2 and +3. +3.3 +Methods comparison +Approaches described in 2.2.1 are tested and compared on the task of invasive carcinoma classification. +For comparison matter, a baseline performance is obtain on each center using the baseline classification +model (see section 2.2.2). +In MDS1 approach, patches translated from center i to the reference center are fed to the baseline +classification model to test generalization on center i; in our experiments i takes alternatively the values +1, 2, 3. +Regarding UDA approach, the classifier is trained on both data from the reference center and data +translated from the source center to centers i, j. The resulting classifier generalization ability is tested on +center k as well as on its training centers i, j; in our experiments (i, j, k) are alternatively set to (1, 2, 3), +(3, 1, 2), (3, 2, 1). +7 + +4 +Results +4.1 +CycleGANs training stopping criterion +The performance of the models trained with and without the FID stopping criterion are illustrated in +the following figure 6. These metrics are computed using MDS1 modality. +Figure 6: Performance of CycleGAN models trained with and without the FID stopping criterion. Models +are tested and compared with MDS1 comparison : by translating a center i test set to the reference center’s +style and then evaluating the baseline classification model performance. +We observe that the models trained with the FID stopping criterion consistently outperform models +trained for any fixed number of epochs in terms of MDS1 performance and this, independently from the +choice of the target center. For instance, for the CycleGAN i, the FID stopped model achieve a recall of +0.81±0.06, and precision of 0.84±0.02. Despite having similar or even more visually appealing generated +samples, the models trained for 100 epochs perform significantly worse, with a recall of 0.49 ± 0.06 and +precision of 0.65 ± 0.02. Models’ performance when trained for 25, 50 and 75 epochs are even worse with +(0.62 ± 0.12, 0.65 ± 0.08), (0.71 ± 0.12, 0.61 ± 0.08) and (0.55 ± 0.12, 0.59 ± 0.08) for recall and precision +respectively. Likewise, similar results are observed for CycleGANs 2 and CycleGANs 3 whose results are +detailed in the Table 2 below. +Centers +Model +Recall +Precision +FID stopped +0.81 ± 0.06 +0.84 ± 0.02 +Fixed epochs (25) +0.55 ± 0.12 +0.70 ± 0.08 +Source to center 1 +Fixed epochs (50) +0.66 ± 0.12 +0.58 ± 0.08 +Fixed epochs (75) +0.49 ± 0.12 +0.65 ± 0.08 +Fixed epochs (100) +0.74 ± 0.12 +0.69 ± 0.08 +FID stopped +0.79 ± 0.06 +0.84 ± 0.02 +Fixed epochs (25) +0.62 ± 0.12 +0.65 ± 0.08 +Source to center 2 +Fixed epochs (50) +0.71 ± 0.12 +0.61 ± 0.08 +Fixed epochs (75) +0.55 ± 0.12 +0.59 ± 0.08 +Fixed epochs (100) +0.72 ± 0.12 +0.54 ± 0.08 +FID stopped +0.86 ± 0.06 +0.92 ± 0.02 +Fixed epochs (25) +0.73 ± 0.12 +0.80 ± 0.08 +Source to center 3 +Fixed epochs (50) +0.67 ± 0.12 +0.79 ± 0.08 +Fixed epochs (75) +0.70 ± 0.12 +0.80 ± 0.08 +Fixed epochs (100) +0.65 ± 0.12 +0.72 ± 0.08 +Table 2: Performance of CycleGAN models trained with and without the FID stopping criterion. Utilizing +FID criterion led to better performance in every case. +Given the superiority of this method for training and selecting CycleGAN models, we will use the +FID criterion throughout the experiments in this work. +8 + +Source to Center 1 +Source to Center 2 +Source to Center 3 +0.B +1 +0.6 +工 +I +T +0.4 +0.2 +0.D +Recall +Precisian +Recall +Precisian +Recall +Precisian +FID stopped +Fixed epochs (25) +Fixed epochs (50) +Fixed epochs (75) +Fixed epochs (100)4.2 +Slides Requirements +The result of the experiment described in 3.2 regarding slides requirement are displayed in the following +Figure 7. +Figure 7: Performance of CycleGAN models regarding centers and slide number +For CycleGAN 1 and CycleGAN 3, metrics are stable from using 2 slides to 75 slides. +However, +for CycleGAN 2, using less than 10 slides negatively impacts performance. Overall, the results indicate +that the amount of slides from which training patches are extracted impacts CycleGANs performance +in a center-dependent manner. We posit that this center dependance might be related to the amount +of intra-stain variability that can be found in the center’s slides. This will be explored and discussed in +detail in the following section. +4.3 +Methods comparison +In this section, we compare the different methods presented in section 2.2.1 following the experiment +scheme described in section 3.3. Results are described in table 3. +Source center +Center 1 +Center 2 +Center 3 +Baseline +precision: 93% +precision: 48% +precision: 63% +precision: 75% +recall: 80% +recall: 39% +recall: 86% +recall: 91% +MDS1 +precision: 60% +precision: 86% +precision: 94% +recall: 78% +recall: 94% +recall: 94% +MDS2 +precision: 90% +precision: 46% +precision: 94% +recall: 80% +recall: 86% +recall: 94% +UDA +precision: 92% +precision: 89% +precision: 92% +precision: 97% +recall: 84% +recall: 89% +recall: 95% +recall: 93% +Table 3: Quantitative results of the different methods. UDA model is trained on samples from both +reference center and augmented samples from center (1, 2) and its generalization ability is tested on +center 3 +Baseline. The baseline classification model achieved 93% precision and 80% recall on the reference +center. On centers 1, 2 and 3, precision and recall are 48%, 39%; 63%, 86% and 75%, 91% respectively. +These results show that the performance of the baseline classification model on centers other than the +one on which it was trained is significantly hampered. From these results, it can be argued that the +generalization task is harder for some centers, as illustrated by the model’s abysmal performance on +center 2. +MDS1. On centers 2 and 3, the performance of MDS1 was significantly better than the baseline, +with precision and recall being 86%, 94%; 94%, 94% respectively. The performance of MDS1 on center 1 +was lower overall, with 60% precision and 78% recall. However, the baseline performance on this center +was already significantly lower compared to the other centers. +MDS2. In this case, metrics are also substantially better than the baseline on most of centers. Center +3 results are equal to those of MDS1, but on center 1, the model has 90% precision and 80% recall which +is significantly better than previous method. Results on center 2 contrast with that of other centers as +precision is only 46%. +9 + +0.95 +0.9 +0.90 +0.85 +0.8 +recision +0.80 +Source center Center 1 +Source center Center2 +0.7 +Source center Center3 +0.70 +0.6 +0.65 +0.60 +0.5 +2 +5 +10 +25 +50 +75 +2 +5 +10 +25 +50 +75 +Slides Number +Slides NumberUDA. UDA method shows consistent performance across centers. For center 1, 2 and 3, precision +and recall are 96%, 70%, 86%, 95% and 97%, 93% respectively. Compared to the baseline, generalization +performance is significantly higher and is often even higher than baseline performance on the reference +center. This method outperform the others in term of performance and generalization ability. Moreover, +UDA performance is not impacted by the choice of training and unknwown centers, as illustrated by the +following figure. +Augmentation centers +Test center +Source center +Center 1 +Center 2 +Center 3 +(1,2) +3 +precision: 92% +precision: 89% +precision: 92% +precision: 97% +recall: 84% +recall: 89% +recall: 95% +recall: 93% +(3,1) +2 +precision: 93% +precision: 90% +precision: 93% +precision: 98% +recall : 80% +recall: 85% +recall: 88% +recall: 93% +(3,2) +1 +precision: 91% +precision: 90% +precision: 92% +precision: 98% +recall : 84% +recall: 85% +recall: 95% +recall: 93% +Table 4: UDA method results depending on the choice of training and unknown centers. UDA model is +trained on samples from both reference center and augmented samples from center i, j and its general- +ization ability is tested on unknown center k, where (i, j, k) are alternately set to (1, 2, 3), (3, 1, 2) and +(3, 2, 1) +10 + +5 +Discussion. +CycleGANs training stopping criterion. Overall, utilizing FID stopping criterion to end CycleGANs +training consistently led performance improvement compared to stopping at a arbitrarily fixed epoch. +Furthermore, FID stopped models had significantly lower error margin, indicating that the method is +consistent and produces models that perform similarly. Such discrepancy in fixed training epochs is yet +another proof of CycleGANs training instability and highlights the importance of utilizing more robust +and automatic stopping criteria such as the one used in this work. Additionally, using the FID criterion +resulted in significant time savings as training often stopped in the 25-40 epochs range. In our settings, +this reduced training time by approximately 10 hours compared to the model trained for 100 epochs. +On the other hand, it should be noted that this method may not always produce the optimal CycleGAN +model for the task at hand. This would require an evaluation of models at each epoch during training, +which is a tedious and costly process. However, it is useful to produce performing enough models for +our task at hand in an automated fashion. Consequently, this makes it quite easy to include in a more +systematic and production-oriented workflow when working with new and exotic stain variations. +Slides requirements. Section 4.2 indicates that the amount of slides impacts results for one out of +three centers. This clearly shows that this impact is center dependent. We hypothesize that this center +dependence is related to the amount of intra-stain variability that can be found in a particular center’s +slides. Hence, centers showing low intra-stain variability may require only 2 slides for the cycleGAN to +capture this diversity, whereas in the case of high intra-stain variability, it may not be sufficient. +To validate this hypothesis, patches from x slides of each center are converted to the Hue-Saturation- +Value color (HSV) channel, Hue’s mean and its standard deviation is computed for every patch. Here, x +takes the values {2, 5, 10, 25, 50, 75)}. The results are displayed in the following figure 8. +Figure 8: Hue standard deviation by center and slides number. +The results indeed show a clear increase in hue’s standard deviation when increasing the number of +slides for center 2, from 0.076 with 2 slides to a stable 0.178 when hitting the 25 slides mark, whereas +it stays low and stable in the case of center 1 and 3. The evidence suggests that some centers exhibit +higher intra-stain variability, and the amount of slides required to train good stain translators depends +on this prior. Besides, it also suggests that building more refined tools capable of assessing this stain +variability in a straight-forward and scientific manner could be an interesting and useful work to better +assess cycleGANs training data requirements. +Methods comparison. Overall, these results indicate that every method outperform baseline per- +formance. Nevertheless, it seems that the performance gains are center dependent and are also impacted +by translation direction. This indicate that some stains translations are more challenging for the Cy- +cleGAN to learn. The center-wise difficulty is highlighted by the overall lower performance on center 1, +while the translation direction effect is made clear by comparing MDS1 and MDS2 results for center 2. +This translation’s direction impact on performance has been first observed by Gadermayr et al.. [19], +whose advice is to translate from the ”hard-to-segment” to the ”easy-to-segment” stain. However, in +their work, the task on top of the generation process was glomeruli segmentation and translation were +made between a Periodic-Acid-Sciff (PAS) stain and IHC stains, in particular CD31. In this case, ”hard” +and ”easy” to segment domains are quite easy to determine, as PAS will make the glomeruli structure +obvious while CD31 will focus on targeting lymphocytes. Hence, one would focus on MDS1 for success- +ful segmentation, and indeed, the authors found it to be the ”preferred method”. In the present work +11 + +Center 1 +Center 2 +Center 3 +0.200 +0.200 +0.200 +0.175 +0.175 +0.175 +0.150 +0.150 +0.150 +0.125 +0.125 +0.125 +Std +Std +Std +an +0.100 +Hue +0.100 +Hue +0.100 +0.075 +0.075 +0.075 +0.050 +0.050 +0.050 +0.025 +0.025 +0.025 +0.000 +0.000 +0.000 +2 +10 +25 +50 +75 +2 +5 +10 +25 +50 +75 +2 +5 +10 +25 +50 +75 +Number of slides +Number of slides +Number of slideshowever, every stain is H&E and typically highlight the same structures of interest and thus, ”hard” +and ”easy” to classify stains don’t hold the same meaning and cannot be predicted beforehand. This +translation direction effect might actually be explained by the high intra-center stain diversity observed +in center 2’s samples and discussed in section 4.2. To elaborate, a plausible explanation might be that +the cycleGAN focused on an ”easy-to-translate” variation of center 2 stain, which would guarantee a low +cycle-loss, but is not representative of the entire real distribution. Plus, this hypothesis also explains +why this problem doesn’t occur on the target to source translation, as source stain doesn’t show such +intra-center stain diversity. This behaviour can be seen as a clear case of mode collapse for CycleGANs +and deserves a further investigation. +Next, on the methods themselves, each has it’s specificity and use-case. MDS1 doesn’t require any +classifier re-training. However, this method requires labeled data from the target center. On the other +hand, MDS2 transfers the labelling knowledge on a source center to the target center as labels are +preserved with the translation. Thus, it is possible to use a reference center’s dataset to train a classifier +on another domain and thus, no labeling on this domain is needed. These two methods are stain specific +as obtained classifiers can only be used with one unique stain at the end of the process. This is the +main limitation of these methods as they can suffer from center-wise difficulty and high intra-center stain +diversity as explained previously. +Finally, UDA method comes with two merits. First, this method yields the best and most consistent +results overall. Furthermore, the model obtained with this method can be used on several stains and +even on unseen ones, which is a great advantage from a practical point of view. We posit that the greater +the number of stains available for training, the greater the potential for generalization. Undoubtedly, +UDA method is the most convenient and preferred method for high performance and stain agnosticism +purposes. +6 +Conclusion +The hereby study is built around three experiments that each bring an answer to major issues in digital +pathology. Firstly, it introduces a systematic and consistent method to train and choose a CycleGAN +model using FID metric, thus answering a quite often discussed concern regarding CycleGAN usage in a +medical context. The study also explores a relatively unconsidered but nonetheless decisive topic regard- +ing digital pathology, that is, the CycleGANs’ training WSI requirement. Basically, as the intra-center +(i.e intra-scanner) stain diversity increases, so does the WSI requirement for training the CycleGAN. +Finally and most importantly, CycleGANs appear to be a solution to the generalization challenge in dig- +ital histopathology. The study shows improved breast carcinoma classification performance when using +all three CycleGAN’s methods on every stains we had at our disposal compared to the baseline. Each +method has its own use-case and merits and can answer any stain related domain adaptation problem. +To sum up, we deliver a CycleGAN based solution to the generalization challenges in digital histopathol- +ogy. While the results have been obtained on the specific task of breast invasive carcinoma classification, +the method can surely be applied to a wide range of tasks. +Future work may focus on tweaking generator and discriminator architecture and hyper-parameters, as +this topic was left aside during this study although it may improve performance compared to the actual +work. Another interesting area of research would be to investigate the benefits of using this method +with the most recent generative models capable of image-to-image tasks compared to the conventional +CycleGAN architecture. +12 + +References +[1] S. Rajpal, V. Kumar, M. Agarwal, and N. Kumar, “Deep learning based model for breast cancer +subtype classification,” 2021. +[2] E. Rashed and M. S. A. E. Seoud, “Deep learning approach for breast cancer diagnosis,” in Proceed- +ings of the 2019 8th International Conference on Software and Information Engineering, ACM, apr +2019. +[3] M. Z. Alom, T. Aspiras, T. M. Taha, V. K. Asari, T. Bowen, D. Billiter, and S. Arkell, “Advanced +deep convolutional neural network approaches for digital pathology image analysis: a comprehensive +evaluation with different use cases,” 2019. +[4] T. Yao, Y. Lu, J. Long, A. Jha, Z. Zhu, Z. Asad, H. Yang, A. B. Fogo, and Y. Huo, “Glo-in-one: +Holistic glomerular detection, segmentation, and lesion characterization with large-scale web image +mining,” 2022. +[5] A. Gupta, A. Reddy, C. Jawahar, and P. Vinod, “Classification of histopathology images using +convnets to detect lupus nephritis,” 2021. +[6] S. H. Mirjahanmardi, S. Mitha, S. Razavi, S. Done, and A. Khademi, “Mitosis detection for breast +cancer pathology images using uv-net,” 2021. +[7] M. A. Loodaricheh, N. Karimi, and S. Samavi, “Nuclei segmentation in histopathology images using +deep learning with local and global views,” 2021. +[8] S. Rezaei, A. Emami, H. Zarrabi, S. Rafiei, K. Najarian, N. Karimi, S. Samavi, and S. M. R. +Soroushmehr, “Gland segmentation in histopathology images using deep networks and handcrafted +features,” 2019. +[9] P. Leo, G. Lee, N. Shih, R. Elliott, M. D. Feldman, and A. Madabhushi, “Evaluating stability of +histomorphometric features across scanner and staining variations: prostate cancer diagnosis from +whole slide images,” Journal of Medical Imaging, vol. 3, 2016. +[10] D. Tellez, G. Litjens, P. B´andi, W. Bulten, J.-M. Bokhorst, F. Ciompi, and J. van der Laak, “Quanti- +fying the effects of data augmentation and stain color normalization in convolutional neural networks +for computational pathology,” Medical Image Analysis, vol. 58, p. 101544, dec 2019. +[11] N. Tajbakhsh, L. Jeyaseelan, Q. Li, J. Chiang, Z. Wu, and X. Ding, “Embracing imperfect datasets: +A review of deep learning solutions for medical image segmentation,” Medical Image Analysis, vol. 63, +p. 101693, 04 2020. +[12] F. Ciompi, O. Geessink, B. E. Bejnordi, G. S. de Souza, A. Baidoshvili, G. Litjens, B. van Ginneken, +I. Nagtegaal, and J. van der Laak, “The importance of stain normalization in colorectal tissue +classification with convolutional networks,” 2017. +[13] S. Krishna, S. S. S, S. Krishnamoorthy, and A. Bhavsar, “Stain normalized breast histopathology +image recognition using convolutional neural networks for cancer detection,” 2022. +[14] M. Macenko, M. Niethammer, J. S. Marron, D. Borland, J. T. Woosley, X. Guan, C. Schmitt, and +N. E. Thomas, “A method for normalizing histology slides for quantitative analysis,” pp. 1107–1110, +2009. +[15] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and +Y. Bengio, “Generative adversarial networks,” 2014. +[16] P. Isola, J.-Y. Zhu, T. Zhou, and A. Efros, “Image-to-image translation with conditional adversarial +networks,” pp. 5967–5976, 07 2017. +[17] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, “Unpaired image-to-image translation using cycle- +consistent adversarial networks,” 2017. +[18] J. Vasiljevi´c, F. Feuerhake, C. Wemmert, and T. Lampert, “Towards histopathological stain invari- +ance by unsupervised domain augmentation using generative adversarial networks,” 2020. +13 + +[19] G. M, G. L, A. V, B. P, K. BM, and Merhof, “Generative adversarial networks for facilitating stain- +independent supervised and unsupervised segmentation: A study on kidney histology,” IEEE Trans +Med Imaging, 2019. +[20] M. T. Shaban, C. Baur, N. Navab, and S. Albarqouni, “Staingan: Stain style transfer for digital +histological images,” 2018. +[21] M. Aubreville, N. Stathonikos, C. A. Bertram, R. Klopleisch, N. ter Hoeve, F. Ciompi, F. Wilm, +C. Marzahl, T. A. Donovan, A. Maier, J. Breen, N. Ravikumar, Y. Chung, J. Park, R. Nateghi, +F. Pourakpour, R. H. J. Fick, S. B. Hadj, M. Jahanifar, N. Rajpoot, J. Dexl, T. Wittenberg, +S. Kondo, M. W. Lafarge, V. H. Koelzer, J. Liang, Y. Wang, X. Long, J. Liu, S. Razavi, A. Khademi, +S. Yang, X. Wang, M. Veta, and K. Breininger, “Mitosis domain generalization in histopathology +images – the midog challenge,” 2022. +[22] C. Mercan, G. Reijnen-Mooij, D. T. Martin, J. Lotz, N. Weiss, M. van Gerven, and F. Ciompi, +“Virtual staining for mitosis detection in breast histopathology,” 2020. +[23] J. Johnson, A. Alahi, and L. Fei-Fei, “Perceptual losses for real-time style transfer and super- +resolution,” 2016. +[24] D. Ulyanov, A. Vedaldi, and V. Lempitsky, “Instance normalization: The missing ingredient for fast +stylization,” 2016. +[25] M. Wiatrak, S. V. Albrecht, and A. Nystrom, “Stabilizing generative adversarial networks: A sur- +vey,” 2019. +[26] X. Mao, Q. Li, H. Xie, R. Y. K. Lau, Z. Wang, and S. P. Smolley, “Least squares generative +adversarial networks,” 2016. +[27] Y. Yu, W. Zhang, and Y. Deng, “Frechet inception distance (fid) for evaluating gans,” 09 2021. +[28] T. Eiter and H. Mannila, “Computing discrete fr´echet distance,” 1994. +[29] M. Tan and Q. V. Le, “Efficientnet: Rethinking model scaling for convolutional neural networks,” +2019. +[30] Z. Zhang and M. R. Sabuncu, “Generalized cross entropy loss for training deep neural networks with +noisy labels,” 2018. +[31] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” 2014. +[32] D. Bashkirova, B. Usman, and K. Saenko, “Adversarial self-defense for cycle-consistent gans,” in +Advances in Neural Information Processing Systems (H. Wallach, H. Larochelle, A. Beygelzimer, +F. d'Alch´e-Buc, E. Fox, and R. Garnett, eds.), vol. 32, Curran Associates, Inc., 2019. +[33] C. Chu, A. Zhmoginov, and M. Sandler, “Cyclegan, a master of steganography,” 2017. +14 + diff --git a/OtFPT4oBgHgl3EQfnDXw/content/tmp_files/load_file.txt b/OtFPT4oBgHgl3EQfnDXw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbd173377cece12911805e120b334470edf9aee6 --- /dev/null +++ b/OtFPT4oBgHgl3EQfnDXw/content/tmp_files/load_file.txt @@ -0,0 +1,777 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf,len=776 +page_content='Standardized CycleGAN training for unsupervised stain adaptation in invasive carcinoma classification for breast histopathology N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Nerrienet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Peyret, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Sockeel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Sockeel January 31, 2023 Abstract Generalization is one of the main challenges of computational pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Slide preparation het- erogeneity and the diversity of scanners lead to poor model performance when used on data from medical centers not seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In order to achieve stain invariance in breast invasive carcinoma patch classification, we imple- ment a stain translation strategy using cycleGANs for unsupervised image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We compare three cycleGAN-based approaches to a baseline classification model obtained without any stain invariance strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Two of the proposed approaches use cycleGAN’s translations at inference or training in order to build stain-specific classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The last method uses them for stain data augmentation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This constrains the classification model to learn stain-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Baseline metrics are set by training and testing the baseline classification model on a refer- ence stain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We assessed performances using three medical centers with H&E and H&E&S staining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Every approach tested in this study improves baseline metrics without needing labels on target stains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The stain augmentation-based approach produced the best results on every stain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Each method’s pros and cons are studied and discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, training highly performing cycleGANs models in itself represents a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In this work, we introduce a systematical method for optimizing cycleGAN training by setting a novel stop- ping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This method has the benefit of not requiring any visual inspection of cycleGAN results and proves superiority to methods using a predefined number of training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In addition, we also study the minimal amount of data required for cycleGAN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 1 Introduction General context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Digital histopathology is changing the way pathologists work on a day-to-day basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Not only does it allow them to share data and insights about a diagnosis across the globe easily, but with computational pathology, algorithms capabilities may be leveraged to ease pathologists’ work on many tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Deep learning has shown promising results for various applications such as cancer classification [1, 2, 3], detection [4, 5, 6] or segmentation [7, 8] of biomarkers required for diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, the integration of AI in histopathology comes with a set of challenges to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In particular, the diversity and heterogeneity in slide preparation increases data variability across centers or laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Slide preparation is a long and tedious process consisting of multiple steps which can each be a source of variability that negatively impacts algorithms performance [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Specifically, biopsy samples are collected during surgeries and are molded into a paraffin bloc for easier slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In order for the paraffin to penetrate the tissue down to molecular level, the tissue is gradually dehydrated using series of ethanol baths with rising concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Dehydration may affect tissue structure depending on the exact protocol applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Samples are then sliced into thin tissue pieces and placed on a glass slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' As tissue handling is tedious, artefacts such as rips or folds often appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Additionally, the tissue being originally transparent, contrasting agents are to be added in order to analyse certain structures of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This process is called staining and is a crucial part of slides preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Hematoxylin and Eosin (H&E) are typically used to dye cell nuclei in a specific color and tissue surrounding cells in another one, whereas some other stains, based on Immunohistochemical (IHC) reactions, target some specific receptors or molecular compounds of the cell membrane, making them visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The staining process and the manufacturer from whom the staining agents are purchased differ from a laboratory to another, leading to intra-stain variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In France, pathologists are also used to adding Safran to H&E (HES) in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='13128v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='IV] 30 Jan 2023 order to dye connective tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This gives yet another appearance to tissue samples, which can greatly differ depending on the balance of Hematoxylin’s and Eosin’s constitutive components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Digital pathology adds one last source of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Physical slides are scanned to produce Whole Slide Image (WSI) that can be viewed on a software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The scanner’s brand and settings, as well as the file format, are clearly identified sources of data variability [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' These considerations result in WSI having clearly distinguishable appearances from one laboratory to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In fact, it has been observed that AI models performed poorly when applied to data coming from medical centers not seen during training or to data stained differently than training data [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Therefore, before being used in a new laboratory, models need to be fine-tuned using annotated data from this laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This is a time-consuming and expensive process as it requires expert pathologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Hence, generalization is one of the main challenges in computational histopathology and solving it would allow a unique model to be applied on a wide range of laboratories without fine-tuning or annotated data [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This issue is commonly addressed in literature using two global approaches: nor- malization and augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Most of the time, normalization is used to reduce dissimilarity within a given domain [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For instance, the simplest form of stain normalization is to simply convert RGB images to grayscale, assuming that most of the H&E signal is contained in the morphological and structural patterns of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' On the other hand, color augmentation is often about synthesizing new images by varying the col- ors, the contrast or the illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This is used in a deep learning setting, to make learnt features independent of color variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Recently, due to their ability to generate highly realistic outputs, Generative Adversarial Networks (GAN) [15] and their derivatives have been also intensively used to address the generalization challenge in digital histopathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Specifically, either Pix2Pix [16] or CycleGANs [17] are utilized as stain translation devices depending on the need or availability of paired samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The Pix2pix often displays more realistic and consistent results but requires paired samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, in a digital histopathology context, it can be problematic to obtain paired datasets, as it involves the ability to re-stain stained samples which is a costly and tedious process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Consequently, most studies using GANs’ setup in digital histopathology rely more heavily on Cycle-GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Jun-Yan Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [17] demonstrated that cycleGANs can produce samples in a highly realistic fashion without the need for paired training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In [18, 19], the authors identified three methods that can be used to address domain adaptation issues with CycleGANs, namely Multi-Domain Supervised 1 and 2 (MDS1/MDS2) and Unsupervised Domain Augmentation (UDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For each CycleGAN based method, a deep learning task is used on top of the CycleGAN’s translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In these approaches, the source domain is either the domain on which most of the labeled data is available or the domain on which models such as classifiers or detectors already perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Conversely, the target domain is often scarce in annotated data and exhibits poor model performance, or at least, poor generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In short, MDS1 consists of training a classifier on the source domain and applying it on target domain samples translated to the source domain, whereas MDS2 aims at training a classifier on source domain samples translated to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Finally, introduced by Vasijelic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [18], UDA’s goal is to use CycleGANs’ translation as a mean of data augmentation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In the literature, Shaban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [20] employs CycleGANs to show its superiority compared to conven- tional methods in the case of lymph node metastasis classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, they solely use the MDS1 method and lack the exploration of the other ones which were proven to lead to better performing models by Vasiljevic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Moreover, little insight is given on how the final CycleGAN model was chosen and on the data it was trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Several other studies use these models in the case of virtual staining for mitosis detection [21, 22] or in the case of glomeruli segmentation [18] among other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Interestingly, Vasiljevic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [18] points out a weakness in using CycleGANs, that is, the visual quality of a CycleGAN translation is not always related to the performance of the task on top of the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This unexpected behavior implies a substantial difficulty not only when comparing and choosing CycleGANs for the task at hand, but also when one must decide to stop the training process or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This study is built upon all this prior knowledge and applies CycleGANs for breast tumor classification generalization, while introducing a consistent method on how to evaluate and choose image-to-image translation models for the downstream task and thus, addressing the previously mentioned concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This study makes the following key contributions: We introduce a systematic and consistent method to train and choose a CycleGAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We study the optimal amount of WSI required to train a CycleGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 2 We introduce a stain agnostic breast tumor classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We prove the relevance of using CycleGANs to attain stain invariance in classification tasks on a novel large multi center dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In the following section, a description of deep learning models utilized in this study is exposed, as well as the datasets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Section 3 describes the different experiments performed in this work, section 4 presents the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In the final section, we will present and interpret the results obtained from our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 2 Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1 Datasets The data used in this study comes from four different centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Samples are collected from breast cancer patients that underwent breast biopsy or mastectomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Samples are colored with H&E or H&E&S stain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 1 depicts a sample from each of these centers, illustrating the high variability of observable appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Patches of size 256*256 pixels are then extracted from these WSI at magnification x5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The reference center is the one with the largest labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This center is here referred to as reference center or source center, interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 1: Samples from the various centers used in this study and their respective stain Patches are given one of the following labels: invasive carcinoma (IC), which is the latest stage of breast cancer development, and REST, which includes healthy tissue, benign lesions and non-malignant tumors such as adenoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Table 1 illustrates the amount of data, slides and extracted patches, from each center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Source center Center 1 Center 2 Center 3 Slides train : 1690 test : 42 test : 119 test : 44 test : 309 Patches train : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='7M test : 75 888 test : 35 209 test : 46 808 test : 469 217 Table 1: Slides and patches distribution across centers 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1 Translation model In this work, CycleGANs [17] are used as translators between different stains prior to the breast carcinoma classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 2 shows an example of CycleGAN translations with our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In this example, a patch from the source domain is translated to the other centers domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A CycleGAN is trained for every pair of centers (source center, center i) with i ∈ {1, 2, 3}, as the goal is to make translation between the reference center and every other center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGANs’ generators and discriminators architecture are described in the original paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [23] proved this architecture produces high quality outputs in the case of style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Moreover, they demonstrated that instance normalization greatly improves style- transferred samples quality as well as being more efficient for small batch size than batch normalization [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We thus adopt this normalization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' As problems such as vanishing gradients, mode 3 Source center Center 1 Center 2 Center 3Figure 2: CycleGANs transformations examples from source to targets collapse and failing to converge are quite standard issues with GANs [25], least-square-loss is used instead of the negative-log-likelihood standard loss to ensure a more stable training following Xudong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGAN training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Trainings are done using 1000 patches from every center as using more patches did not show improved quality in translated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The number of WSI from which we extract these 1000 patches is center-dependent and is determined by a further experiment that will be discussed in the following section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Besides, the difficulty of evaluating generative models is a known issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Initially, generative models were evaluated by mere visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, in digital histopathology, it would require expert knowledge to ensure that generated samples are realistic enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In addition, it has been shown by Vasiljevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [18] that generated samples visual quality is not always correlated with performance for the task on top of the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Specifically, two CycleGANs from two different epochs with the same visual generation quality can yield significantly different results in the case of a segmentation task on the generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Hence, there is a need to evaluate the quality of a cycleGAN so as to monitor and stop its training in a releavant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To do so, we propose to use Fr´echet Inception Distance (FID) [27] which is a popular metric used for generative models’ evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This metric uses an Inception model to extract features from generated and real images, those feature distributions are then used to compute the Fr´echet Distance [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGAN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGANs can be utilized to address generalization challenges in three distinct ways, each with its own unique characteristics and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' First, Multi-domain Supervised 1 and 2 (MDS1/MDS2) have been introduced by Gadeymar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In MDS1, the CycleGAN’s translation is used as a normalization preprocessing step to the main classification model at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Samples are translated to the source domain before being classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This enables the use of pre-trained source domain-specific classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This method is illustrated in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 3: Multi-Domain Supervised 1 (MDS1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGAN’s translations occur at inference time from the target center to a source center for which a trained classifier is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In MDS2, target domain-specific classification models are built using CycleGAN’s translations as a normalization preprocessing step at training time as illustrated by figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Models built in this fashion can then directly be applied on target domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Finally, Unsupervised Domain Augmentation (UDA), illustrated in figure 5, is a method explored by Vasiljevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This approach makes use of CycleGANs’ translations to various domains as a mean of data augmentation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This leads to the model extracting stain invariant features for the task at hand, provided that the pool of domains is diverse enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Contrary to the other methods that use a single domain translation and build domain specific models, this approach aims at producing 4 Source center Center1 Center2 Center3classifier CycleGAN source target source TRAINING INEERENCEFigure 4: Multi-Domain Supervised 2 (MDS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGAN’s translations occur at training time from the source center to the target center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' models that perform on a large variety of stains and thus attain stain invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 5: Unsupervised Domain Augmentation (UDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGAN’s translations are used as a mean of data augmentation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The resulting model can then be applied to a wide range of stains and centers, including seen as well as unseen centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A major advantage of these three approaches is that they don’t require any labeling in the domain in which samples are translated to, considerably reducing the time and money investment that labeling requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In this study, each of these approaches is used to solve the generalization challenge for breast invasive carcinoma classification following an experiment scheme described in the following section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 Classification model Classifier architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In order to classify patches into the two classes discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1, a classification model was employed on top of the translation process as described in figures 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The model used for this task is an EfficientNetB1 introduced by Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [29], with two outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In order to train this model, we use a category cross-entropy loss [30] and an Adam optimizer [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Additionally, to enhance the training data pool, various data augmentation techniques were randomly applied at each training step: Flip: Flip the patch horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Rotate: Rotate the patch by a multiple of 90 degree angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Color Jitter: Shift hue, saturation and brightness of the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Noise: Add Gaussian noise to the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 5 classifier cycleGAN source target target TRAINING INFERENCEsource random selection newcenter TRAINING INFERENCEBaseline classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In order to establish a baseline performance to compare the different methods results, we train a classification model using the architecture aforementioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This baseline classification model is trained on the reference center trainset (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1) for 150 epochs, and baseline performance on each center is obtained by testing this model on every center’s test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3 CycleGANs comparison method and notations Throughout this study, comparing various CycleGANs in the breast classification task context will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To accomplish this, we introduce a method to evaluate task-wise CycleGAN’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' First, we establish the following notation: a CycleGAN responsible for translations from reference center to center i, and vice-versa, will be mentioned as follows : CycleGAN i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Next, in order to evaluate a CycleGAN’s performance related to the classification task, we will use MDS1 modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To elaborate, CycleGAN i is utilized to translate samples from a dataset to the reference center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Then, the baseline classification model is tested on these translated samples to obtain metrics that will be used for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We further refer to this method as ”MDS1 comparison”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' MDS1 modality is preferred for this task as baseline classification model can be used as an evaluator whereas labeled training data is too scarce to train ”baseline models” for other centers in our settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' One can argue that using MDS1 only is insufficient and indeed previous work showed that translation direction is an important factor [18, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGANs can generate higher quality transformations depending on translation’s direction and can even hide information in the translated samples [18, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The latter effect can have an unexpected impact on deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, this is often the case when one domain has more information than the other, which implies that a direction will suppress information and the other will restore what has been lost [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In our case, this effect is greatly mitigated as the domains, H&E stains, hold the same information content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Therefore, we will solely rely on MDS1 comparison throughout the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 6 3 Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1 CycleGANs training stopping criterion A common challenge when working with GANs is to know when the model has converged and when we should stop the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Usually, mere visual inspection of generated samples’ quality is key to determine when the training should end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, this procedure comes with two major downsides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' First, it implies saving the model at each training epoch in order to be able to retrieve the epoch weights that gave the most satisfying results, which is memory consuming and requires a manual intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Secondly, as stated in the section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1, visual inspection is not sufficient as image quality does not systematically reflects performance for the task on top of the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In this experiment, we evaluate the effectiveness of using the FID metric as an early stopping criterion for training CycleGAN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We compare a model trained using the FID criterion to models trained for 25, 50, 75 and 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This comparison is done for CycleGANs i (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3) where i takes the following values {1, 2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The FID for CycleGAN i training is computed as the Fr´echet distance between 1000 samples randomly selected in the source center, and 1000 samples translated from center i to the the source center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Latter samples plays the role of the ”generated images”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The resulting CycleGAN models are then evaluated on center i test dataset using MDS1 comparison described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For each center, we repeat this experiment three times to compute error margins and ensure results repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 Slides Requirements When it comes to digital histopathology, an under-discussed but nonetheless decisive matter is the number of slides required for training a CycleGAN that produces satisfying enough samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This point is crucial for the method convenience considering the difficulty of collecting a sufficient amount of WSIs to train deep learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Therefore, we aim to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To that end, we build various datasets each containing 1000 patches extracted from a various number N of slides, train CycleGANs on each dataset and assess its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Specifically, patch datasets are extracted from N = {2, 5, 10, 25, 50, 75} slides randomly picked from reference and target center training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For each of these configurations, three datasets are built in total to ensure that observed results are not dependent on the chosen set of slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Resulting CycleGANs are again compared through MDS1 comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Finally, to ensure the results are independent of the choice of training centers, the experiment is conducted for target centers 1, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3 Methods comparison Approaches described in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1 are tested and compared on the task of invasive carcinoma classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For comparison matter, a baseline performance is obtain on each center using the baseline classification model (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In MDS1 approach, patches translated from center i to the reference center are fed to the baseline classification model to test generalization on center i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' in our experiments i takes alternatively the values 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Regarding UDA approach, the classifier is trained on both data from the reference center and data translated from the source center to centers i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The resulting classifier generalization ability is tested on center k as well as on its training centers i, j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' in our experiments (i, j, k) are alternatively set to (1, 2, 3), (3, 1, 2), (3, 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 7 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1 CycleGANs training stopping criterion The performance of the models trained with and without the FID stopping criterion are illustrated in the following figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' These metrics are computed using MDS1 modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 6: Performance of CycleGAN models trained with and without the FID stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Models are tested and compared with MDS1 comparison : by translating a center i test set to the reference center’s style and then evaluating the baseline classification model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We observe that the models trained with the FID stopping criterion consistently outperform models trained for any fixed number of epochs in terms of MDS1 performance and this, independently from the choice of the target center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For instance, for the CycleGAN i, the FID stopped model achieve a recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='06, and precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Despite having similar or even more visually appealing generated samples, the models trained for 100 epochs perform significantly worse, with a recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='06 and precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Models’ performance when trained for 25, 50 and 75 epochs are even worse with (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08) and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08) for recall and precision respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Likewise, similar results are observed for CycleGANs 2 and CycleGANs 3 whose results are detailed in the Table 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Centers Model Recall Precision FID stopped 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='02 Fixed epochs (25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Source to center 1 Fixed epochs (50) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Fixed epochs (75) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Fixed epochs (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 FID stopped 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='02 Fixed epochs (25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Source to center 2 Fixed epochs (50) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Fixed epochs (75) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Fixed epochs (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 FID stopped 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='02 Fixed epochs (25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Source to center 3 Fixed epochs (50) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Fixed epochs (75) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Fixed epochs (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='08 Table 2: Performance of CycleGAN models trained with and without the FID stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Utilizing FID criterion led to better performance in every case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Given the superiority of this method for training and selecting CycleGAN models, we will use the FID criterion throughout the experiments in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 8 Source to Center 1 Source to Center 2 Source to Center 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='B 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='6 工 I T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='D Recall Precisian Recall Precisian Recall Precisian FID stopped Fixed epochs (25) Fixed epochs (50) Fixed epochs (75) Fixed epochs (100)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 Slides Requirements The result of the experiment described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 regarding slides requirement are displayed in the following Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 7: Performance of CycleGAN models regarding centers and slide number For CycleGAN 1 and CycleGAN 3, metrics are stable from using 2 slides to 75 slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, for CycleGAN 2, using less than 10 slides negatively impacts performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Overall, the results indicate that the amount of slides from which training patches are extracted impacts CycleGANs performance in a center-dependent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We posit that this center dependance might be related to the amount of intra-stain variability that can be found in the center’s slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This will be explored and discussed in detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3 Methods comparison In this section, we compare the different methods presented in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1 following the experiment scheme described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Results are described in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Source center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Center 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Center 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Center 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 93% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 48% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 63% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 75% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 39% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 86% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 91% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='MDS1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 86% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 94% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 78% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 94% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 94% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='MDS2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 46% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 94% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 86% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 94% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='UDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 92% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 89% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 92% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='precision: 97% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 84% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 89% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 95% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='recall: 93% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='Table 3: Quantitative results of the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' UDA model is trained on samples from both reference center and augmented samples from center (1, 2) and its generalization ability is tested on center 3 Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The baseline classification model achieved 93% precision and 80% recall on the reference center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' On centers 1, 2 and 3, precision and recall are 48%, 39%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 63%, 86% and 75%, 91% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' These results show that the performance of the baseline classification model on centers other than the one on which it was trained is significantly hampered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' From these results, it can be argued that the generalization task is harder for some centers, as illustrated by the model’s abysmal performance on center 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' MDS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' On centers 2 and 3, the performance of MDS1 was significantly better than the baseline, with precision and recall being 86%, 94%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 94%, 94% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The performance of MDS1 on center 1 was lower overall, with 60% precision and 78% recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, the baseline performance on this center was already significantly lower compared to the other centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' MDS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In this case, metrics are also substantially better than the baseline on most of centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Center 3 results are equal to those of MDS1, but on center 1, the model has 90% precision and 80% recall which is significantly better than previous method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Results on center 2 contrast with that of other centers as precision is only 46%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='8 recision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='80 Source center Center 1 Source center Center2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='7 Source center Center3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='5 2 5 10 25 50 75 2 5 10 25 50 75 Slides Number Slides NumberUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' UDA method shows consistent performance across centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' For center 1, 2 and 3, precision and recall are 96%, 70%, 86%, 95% and 97%, 93% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Compared to the baseline, generalization performance is significantly higher and is often even higher than baseline performance on the reference center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This method outperform the others in term of performance and generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Moreover, UDA performance is not impacted by the choice of training and unknwown centers, as illustrated by the following figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Augmentation centers Test center Source center Center 1 Center 2 Center 3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2) 3 precision: 92% precision: 89% precision: 92% precision: 97% recall: 84% recall: 89% recall: 95% recall: 93% (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='1) 2 precision: 93% precision: 90% precision: 93% precision: 98% recall : 80% recall: 85% recall: 88% recall: 93% (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2) 1 precision: 91% precision: 90% precision: 92% precision: 98% recall : 84% recall: 85% recall: 95% recall: 93% Table 4: UDA method results depending on the choice of training and unknown centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' UDA model is trained on samples from both reference center and augmented samples from center i, j and its general- ization ability is tested on unknown center k, where (i, j, k) are alternately set to (1, 2, 3), (3, 1, 2) and (3, 2, 1) 10 5 Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' CycleGANs training stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Overall, utilizing FID stopping criterion to end CycleGANs training consistently led performance improvement compared to stopping at a arbitrarily fixed epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Furthermore, FID stopped models had significantly lower error margin, indicating that the method is consistent and produces models that perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Such discrepancy in fixed training epochs is yet another proof of CycleGANs training instability and highlights the importance of utilizing more robust and automatic stopping criteria such as the one used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Additionally, using the FID criterion resulted in significant time savings as training often stopped in the 25-40 epochs range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In our settings, this reduced training time by approximately 10 hours compared to the model trained for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' On the other hand, it should be noted that this method may not always produce the optimal CycleGAN model for the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This would require an evaluation of models at each epoch during training, which is a tedious and costly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, it is useful to produce performing enough models for our task at hand in an automated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Consequently, this makes it quite easy to include in a more systematic and production-oriented workflow when working with new and exotic stain variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Slides requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2 indicates that the amount of slides impacts results for one out of three centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This clearly shows that this impact is center dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We hypothesize that this center dependence is related to the amount of intra-stain variability that can be found in a particular center’s slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Hence, centers showing low intra-stain variability may require only 2 slides for the cycleGAN to capture this diversity, whereas in the case of high intra-stain variability, it may not be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To validate this hypothesis, patches from x slides of each center are converted to the Hue-Saturation- Value color (HSV) channel, Hue’s mean and its standard deviation is computed for every patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Here, x takes the values {2, 5, 10, 25, 50, 75)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The results are displayed in the following figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Figure 8: Hue standard deviation by center and slides number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The results indeed show a clear increase in hue’s standard deviation when increasing the number of slides for center 2, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='076 with 2 slides to a stable 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='178 when hitting the 25 slides mark, whereas it stays low and stable in the case of center 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The evidence suggests that some centers exhibit higher intra-stain variability, and the amount of slides required to train good stain translators depends on this prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Besides, it also suggests that building more refined tools capable of assessing this stain variability in a straight-forward and scientific manner could be an interesting and useful work to better assess cycleGANs training data requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Methods comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Overall, these results indicate that every method outperform baseline per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Nevertheless, it seems that the performance gains are center dependent and are also impacted by translation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This indicate that some stains translations are more challenging for the Cy- cleGAN to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The center-wise difficulty is highlighted by the overall lower performance on center 1, while the translation direction effect is made clear by comparing MDS1 and MDS2 results for center 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This translation’s direction impact on performance has been first observed by Gadermayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='. [19], whose advice is to translate from the ”hard-to-segment” to the ”easy-to-segment” stain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, in their work, the task on top of the generation process was glomeruli segmentation and translation were made between a Periodic-Acid-Sciff (PAS) stain and IHC stains, in particular CD31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In this case, ”hard” and ”easy” to segment domains are quite easy to determine, as PAS will make the glomeruli structure obvious while CD31 will focus on targeting lymphocytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Hence, one would focus on MDS1 for success- ful segmentation, and indeed, the authors found it to be the ”preferred method”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' In the present work 11 Center 1 Center 2 Center 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='125 Std Std Std an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='100 Hue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='100 Hue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='000 2 10 25 50 75 2 5 10 25 50 75 2 5 10 25 50 75 Number of slides Number of slides Number of slideshowever, every stain is H&E and typically highlight the same structures of interest and thus, ”hard” and ”easy” to classify stains don’t hold the same meaning and cannot be predicted beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This translation direction effect might actually be explained by the high intra-center stain diversity observed in center 2’s samples and discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To elaborate, a plausible explanation might be that the cycleGAN focused on an ”easy-to-translate” variation of center 2 stain, which would guarantee a low cycle-loss, but is not representative of the entire real distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Plus, this hypothesis also explains why this problem doesn’t occur on the target to source translation, as source stain doesn’t show such intra-center stain diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This behaviour can be seen as a clear case of mode collapse for CycleGANs and deserves a further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Next, on the methods themselves, each has it’s specificity and use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' MDS1 doesn’t require any classifier re-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' However, this method requires labeled data from the target center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' On the other hand, MDS2 transfers the labelling knowledge on a source center to the target center as labels are preserved with the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Thus, it is possible to use a reference center’s dataset to train a classifier on another domain and thus, no labeling on this domain is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' These two methods are stain specific as obtained classifiers can only be used with one unique stain at the end of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' This is the main limitation of these methods as they can suffer from center-wise difficulty and high intra-center stain diversity as explained previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Finally, UDA method comes with two merits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' First, this method yields the best and most consistent results overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Furthermore, the model obtained with this method can be used on several stains and even on unseen ones, which is a great advantage from a practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' We posit that the greater the number of stains available for training, the greater the potential for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Undoubtedly, UDA method is the most convenient and preferred method for high performance and stain agnosticism purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 6 Conclusion The hereby study is built around three experiments that each bring an answer to major issues in digital pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Firstly, it introduces a systematic and consistent method to train and choose a CycleGAN model using FID metric, thus answering a quite often discussed concern regarding CycleGAN usage in a medical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The study also explores a relatively unconsidered but nonetheless decisive topic regard- ing digital pathology, that is, the CycleGANs’ training WSI requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Basically, as the intra-center (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='e intra-scanner) stain diversity increases, so does the WSI requirement for training the CycleGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Finally and most importantly, CycleGANs appear to be a solution to the generalization challenge in dig- ital histopathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' The study shows improved breast carcinoma classification performance when using all three CycleGAN’s methods on every stains we had at our disposal compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Each method has its own use-case and merits and can answer any stain related domain adaptation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' To sum up, we deliver a CycleGAN based solution to the generalization challenges in digital histopathol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' While the results have been obtained on the specific task of breast invasive carcinoma classification, the method can surely be applied to a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Future work may focus on tweaking generator and discriminator architecture and hyper-parameters, as this topic was left aside during this study although it may improve performance compared to the actual work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Another interesting area of research would be to investigate the benefits of using this method with the most recent generative models capable of image-to-image tasks compared to the conventional CycleGAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 12 References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Rajpal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Kumar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Agarwal, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Kumar, “Deep learning based model for breast cancer subtype classification,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Rashed and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Seoud, “Deep learning approach for breast cancer diagnosis,” in Proceed- ings of the 2019 8th International Conference on Software and Information Engineering, ACM, apr 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Alom, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Aspiras, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Taha, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Asari, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bowen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Billiter, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Arkell, “Advanced deep convolutional neural network approaches for digital pathology image analysis: a comprehensive evaluation with different use cases,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Long, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Jha, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Asad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Yang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Fogo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Huo, “Glo-in-one: Holistic glomerular detection, segmentation, and lesion characterization with large-scale web image mining,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Reddy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Jawahar, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Vinod, “Classification of histopathology images using convnets to detect lupus nephritis,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Mirjahanmardi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Mitha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Razavi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Done, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Khademi, “Mitosis detection for breast cancer pathology images using uv-net,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Loodaricheh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Karimi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Samavi, “Nuclei segmentation in histopathology images using deep learning with local and global views,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Rezaei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Emami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zarrabi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Rafiei, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Najarian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Karimi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Samavi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Soroushmehr, “Gland segmentation in histopathology images using deep networks and handcrafted features,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Leo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Shih, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Elliott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Feldman, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Madabhushi, “Evaluating stability of histomorphometric features across scanner and staining variations: prostate cancer diagnosis from whole slide images,” Journal of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 3, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Tellez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Litjens, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' B´andi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bulten, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bokhorst, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ciompi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' van der Laak, “Quanti- fying the effects of data augmentation and stain color normalization in convolutional neural networks for computational pathology,” Medical Image Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 58, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 101544, dec 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Tajbakhsh, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Jeyaseelan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Chiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ding, “Embracing imperfect datasets: A review of deep learning solutions for medical image segmentation,” Medical Image Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 63, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 101693, 04 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ciompi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Geessink, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bejnordi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' de Souza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Baidoshvili, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Litjens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' van Ginneken, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Nagtegaal, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' van der Laak, “The importance of stain normalization in colorectal tissue classification with convolutional networks,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Krishna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' S, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Krishnamoorthy, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bhavsar, “Stain normalized breast histopathology image recognition using convolutional neural networks for cancer detection,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Macenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Niethammer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Marron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Borland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Woosley, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Guan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Schmitt, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Thomas, “A method for normalizing histology slides for quantitative analysis,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 1107–1110, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [15] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Pouget-Abadie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Mirza, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Warde-Farley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ozair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Courville, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bengio, “Generative adversarial networks,” 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Isola, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhou, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Efros, “Image-to-image translation with conditional adversarial networks,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 5967–5976, 07 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Park, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Isola, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Efros, “Unpaired image-to-image translation using cycle- consistent adversarial networks,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Vasiljevi´c, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Feuerhake, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wemmert, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lampert, “Towards histopathological stain invari- ance by unsupervised domain augmentation using generative adversarial networks,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 13 [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' M, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' L, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' V, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' P, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' BM, and Merhof, “Generative adversarial networks for facilitating stain- independent supervised and unsupervised segmentation: A study on kidney histology,” IEEE Trans Med Imaging, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Shaban, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Baur, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Navab, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Albarqouni, “Staingan: Stain style transfer for digital histological images,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Aubreville, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Stathonikos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bertram, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Klopleisch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' ter Hoeve, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ciompi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wilm, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Marzahl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Donovan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Maier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Breen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ravikumar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Chung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Park, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Nateghi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Pourakpour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Fick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Hadj, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Jahanifar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Rajpoot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Dexl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wittenberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Kondo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lafarge, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Koelzer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Long, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Razavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Khademi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Veta, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Breininger, “Mitosis domain generalization in histopathology images – the midog challenge,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Mercan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Reijnen-Mooij, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lotz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Weiss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' van Gerven, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ciompi, “Virtual staining for mitosis detection in breast histopathology,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Johnson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Alahi, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Fei-Fei, “Perceptual losses for real-time style transfer and super- resolution,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ulyanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Vedaldi, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lempitsky, “Instance normalization: The missing ingredient for fast stylization,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wiatrak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Albrecht, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Nystrom, “Stabilizing generative adversarial networks: A sur- vey,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [26] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Mao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Xie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Lau, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Smolley, “Least squares generative adversarial networks,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Yu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Deng, “Frechet inception distance (fid) for evaluating gans,” 09 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Eiter and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Mannila, “Computing discrete fr´echet distance,” 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Tan and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Le, “Efficientnet: Rethinking model scaling for convolutional neural networks,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [30] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Sabuncu, “Generalized cross entropy loss for training deep neural networks with noisy labels,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Ba, “Adam: A method for stochastic optimization,” 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Bashkirova, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Usman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Saenko, “Adversarial self-defense for cycle-consistent gans,” in Advances in Neural Information Processing Systems (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Larochelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Beygelzimer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=" d'Alch´e-Buc, E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Fox, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Garnett, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' ), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 32, Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Chu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Zhmoginov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' Sandler, “Cyclegan, a master of steganography,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFPT4oBgHgl3EQfnDXw/content/2301.13128v1.pdf'} diff --git a/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf b/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..48636f37827499f9e1b1b653bb4310a3c2a76c6a --- /dev/null +++ b/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1931186e0b7a1460337267bb47585e2d434367fcf6a9265ac4c66098dec8e7f5 +size 760401 diff --git a/PtFJT4oBgHgl3EQf2C2D/vector_store/index.faiss b/PtFJT4oBgHgl3EQf2C2D/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..47d4f42fe03659fa419dfec0446d9c9b0773962c --- /dev/null +++ b/PtFJT4oBgHgl3EQf2C2D/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7719da0e837dd83d8f18f7f459b6cf0dce96cd1a05f2646c9b2b7fc8436081dc +size 1245229 diff --git a/PtFJT4oBgHgl3EQf2C2D/vector_store/index.pkl b/PtFJT4oBgHgl3EQf2C2D/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..35938577c4c46d3b25b78a5fc7713f11117061de --- /dev/null +++ b/PtFJT4oBgHgl3EQf2C2D/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d863603f9089ef18faa5940836033b22c082c1c4fd47e99c8f39289f4238482e +size 43526 diff --git a/QtAyT4oBgHgl3EQfU_dt/content/2301.00135v1.pdf b/QtAyT4oBgHgl3EQfU_dt/content/2301.00135v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3b892a9819adcf136b27bc59974d2bff51fd8c61 --- /dev/null +++ b/QtAyT4oBgHgl3EQfU_dt/content/2301.00135v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:870f05c6cbb29dd2b6e654b50951b7ea5b403dcf75712e23a0d800ef1c9ee247 +size 2119813 diff --git a/QtAyT4oBgHgl3EQfU_dt/vector_store/index.pkl b/QtAyT4oBgHgl3EQfU_dt/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..89accb19b0e75776d5324e1420598d7a72974678 --- /dev/null +++ b/QtAyT4oBgHgl3EQfU_dt/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7fe911b31ee76e75608c0d744cbdd896dce540c910ecf21a3612c069ca64462f +size 107231 diff --git a/SNE3T4oBgHgl3EQfzAs2/vector_store/index.faiss b/SNE3T4oBgHgl3EQfzAs2/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..47b2687487bf343e53a1569194cd9fc11f1e86d0 --- /dev/null +++ b/SNE3T4oBgHgl3EQfzAs2/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:697d6c32d616fa23928694f1e4cf5e9f94a5f037de69dda5d58ee3974c1c107a +size 6750253 diff --git a/SNE3T4oBgHgl3EQfzAs2/vector_store/index.pkl b/SNE3T4oBgHgl3EQfzAs2/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..84cf757b8eec20a32cfcf5184144c4cad319c833 --- /dev/null +++ b/SNE3T4oBgHgl3EQfzAs2/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:978690f9466931cc33c942007a8218d5c46a09a1fc473dd3bacf02f81247986b +size 244084 diff --git a/U9AzT4oBgHgl3EQfJvv5/vector_store/index.pkl b/U9AzT4oBgHgl3EQfJvv5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..58ceabd1fb832fe4617368890dc7258719a80a9b --- /dev/null +++ b/U9AzT4oBgHgl3EQfJvv5/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7bd4a009a8a79cad257045cc023bd318856a2e5747c871a5959061f4e43d46fe +size 201705 diff --git a/V9E4T4oBgHgl3EQfng2k/content/tmp_files/2301.05177v1.pdf.txt b/V9E4T4oBgHgl3EQfng2k/content/tmp_files/2301.05177v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea46f51fdec823ab21252736ac320564ad1b1668 --- /dev/null +++ b/V9E4T4oBgHgl3EQfng2k/content/tmp_files/2301.05177v1.pdf.txt @@ -0,0 +1,1500 @@ +Searching for Heavy Neutral Leptons at A Future +Muon Collider +Tsz Hong Kwok,a Lingfeng Li,b Tao Liua,c and Ariel Rockc +aDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, +Kowloon, Hong Kong S.A.R., PRC +bDepartment of Physics, Brown University, Providence, RI, 02912, USA +cInstitute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water +Bay, Kowloon, Hong Kong S.A.R., PRC +E-mail: thkwokae@connect.ust.hk, lingfeng_li@brown.edu, taoliu@ust.hk, +iasarock@ust.hk +Abstract: As the planning stages for a high energy muon collider enter a more concrete +era, an important question arises as to what new physics could be uncovered. A TeV-scale +muon collider is also a vector boson fusion (VBF) factory with a very clean background, and +as such it is a promising environment to look for new physics that couples to the electroweak +(EW) sector. In this paper, we explore the ability of a future TeV-scale muon collider to +search for Majorana and Dirac Heavy Neutral Leptons (HNLs) produced via EW bosons. +Employing a model-independent, conservative approach, we present an estimation of the +production and decay rate of HNLs over a mass range between 200 GeV and 9.5 TeV in two +benchmark collider proposals with √s = 3, 10 TeV, as well as an estimation of the dominant +Standard Model (SM) background. We find that exclusion limits for the mixing between +the HNLs and SM neutrinos can be as low as O(10−6). Additionally, we demonstrate that a +TeV-scale muon collider allows for the ability to discriminate between Majorana and Dirac +type HNLs for a large range of mixing values. +Keywords: Muon Collider, BSM Physics, Heavy Neutral Leptons +arXiv:2301.05177v1 [hep-ph] 12 Jan 2023 + +Contents +1 +Introduction +1 +2 +Model +3 +3 +Simulation Framework +4 +3.1 +Signal Event Generation +4 +3.2 +Background Event Generation +6 +3.3 +Detector Simulation +8 +4 +Analysis +8 +5 +Results +14 +5.1 +Exclusion Limits on |Vℓ|2 +14 +5.2 +Distinguishing Majorana versus Dirac Heavy Neutral Leptons +16 +6 +Conclusion and Future Directions +17 +Contents +1 +Introduction +It is now well-known that active neutrinos oscillate and that at least two generations have +very small but nonzero mass [1–7]. As the Standard Model (SM) predicts that neutrinos +have exactly zero mass, this is evidence of Beyond the Standard Model (BSM) physics. +The question then arises as to what could be the origin of neutrino masses and mixings. It +might be the case that new physics at an energy scale much higher than that of the SM may +give rise to neutrino masses and mixings. If that new physics is renormalizable and allowed +to violate lepton number, the lowest order effective operator at the scale of the SM is the +d = 5 dimension Weinberg operator [8]. There are many possible ultraviolet (UV) theories +that may result in this effective operator, but one such class of theories is called the Type I +Seesaw mechanism [9–12], in which the smallness of masses of the SM neutrinos is enforced +by the presence of right-handed Heavy Neutral Leptons (HNLs) at a higher scale. +To date, there have been numerous limits set on the coupling |Vℓ|2 between HNLs and +the SM. Prompt trilepton searches of HNL masses less than 60 GeV have been studied in +hadron colliders, with sensitivity down to O(10−5) [13, 14]. Displaced searches in hadron +colliders have been shown to have sensitivity reaching almost O(10−7) for HNL masses +below 20 GeV [15, 16]. Limits from LHCb also indicate sensitivity down to O(10−4) for +masses below 50 GeV [17]. Other future experiments promise even stricter limits, such as +– 1 – + +SHiP, with limits down to O(10−9) for masses below 5 GeV [18], or other potential hadron +colliders [19, 20], with limits at O(10−7) for similar mass ranges. In addition, future lepton +colliders show promise for higher mass HNLs, possibly down to O(10−5 − 10−6) for masses +between 200-2000 GeV [21]. +A future high-energy muon collider could serve as both an intensity and energy frontier, +providing various opportunities for both direct and indirect searches of new physics. As +compared to an electron-positron collider, a muon collider would have higher energy and +luminosity, and reduced background [22–27]. As synchrotron radiation scales as m−4, a +muon collider would have an energy loss rate reduced by a factor of (me/mµ)4 ≈ (207)−4 +as compared to an electron-positron collider. This, in combination with the property that +muon colliders would have negligible beamstrahlung [28, 29], makes TeV-scale beam energies +more feasible. Currently, estimates put the expected luminosity of a TeV-scale muon collider +at order 1 ab−1 [30, 31]. +As compared to a proton collider, colliding muons would allow for a much more efficient +probe of high energy processes, as a large portion of the beam energy would be carried by +the muon itself, whereas a hadronic collider only admits high energy collisions at the tail of +the proton’s parton distribution function (PDF) — which is highly suppressed. In fact, the +efficiency gap is so pronounced that for some 2 → 2 processes, a muon collider with a given +√sµ has the same reach as a proton collider of √sp ∼ 5−20×√sµ [30, 32, 33]. Furthermore, +a muon collider is a perfect laboratory for studying new electroweak (EW) physics, such as +in searches for HNLs. For a TeV-scale muon collider, roughly 5% of the beam’s energy is +carried by EW bosons [34], whereas a proton beam carries less than 1% [35]. +It is important to note, however, the various technical challenges facing the develop- +ment of a muon collider. Unlike the beams of electron-positron or proton-proton colliders, +muons are unstable, with a lifetime of around 2 µs. As such, it remains a formidable task +to produce and store low emmittance muon beams. In recent years there have been rapid +developments in overcoming these difficulties [36–38]. In the United States, the US Muon +Accelerator Program (MAP) is investigating a potential muon source wherein a proton +beam would collide with a high-Z fixed target to produce secondary muons as pion decay +products [39–42]. However, the kinematics of this process admit final state muons with high +emittance, and therefore significant cooling is needed. A possible cooling technology has +been demonstrated by the Muon Ionization Cooling Experiment (MICE) at the Rutherford +Appleton Laboratory in the United Kingdom [43–45]. An interesting approach to combine +these two steps has been put forward in the Low Emmittance Muon Accelerator (LEMMA) +proposal, which would produce muon pairs at threshold via a 45 GeV positron beam in- +teracting with a fixed-target electron [46]. These muons would have a long lifetime and +small emittance. However, LEMMA is unable to achieve a high enough luminosity to be +feasible [31]. +Nevertheless, it is clear that the new physics reach of a future muon collider is sub- +stantial, and the advantages over conventional lepton colliders have resulted in a multitude +of phenomenological studies, such as searches for axion-like particles [47–49], explorations +of Higgs physics [50–56], precision EW studies [57–61], flavor studies [62–66], and searches +for dark matter [67–71], among others [30, 33, 72, 73]. Nevertheless, unlike for electron- +– 2 – + +positron colliders [21], the phenomenology of HNLs in a muon collider has not been studied +in the literature. In this paper we present an analysis of the sensitivity and reach of a future +muon collider in looking for HNLs. +This paper is structured as follows. In section 2, we introduce the the model we use +that describes the interactions between the HNLs and the SM, and we describe the signal +production channel of interest in the work. In section 3, we describe the framework we use +to simulate the production and detection of signal and background events. In section 4, we +detail the reconstruction and analysis. Section 5 presents the predicted exclusion bounds on +the coupling strengths of the HNLs as well as an estimation of the sensitivity in discerning +between Majorana and Dirac HNLs. We conclude and consider future directions in section 6. +2 +Model +From a collider physics perspective, an unmodified Type I Seesaw mechanism is excluded +in the region of HNL mass and mixing parameter-space under consideration in this work. +More specifically, for the TeV scale mass range of HNLs we focus on, the mixing VNℓ must +be bounded below 10−6 in order to be compatible with current neutrino mass upper limits. +Even for a future muon collider, such a mixing would be far too small to be probed. Thus, +many collider studies focus on scenarios in which the HNLs couple to the SM via a new +mediator with sizable couplings to the SM [20, 74–80]; in this case, the HNL production +would rate become independent of the mixing parameters. +This approach, however, is +highly model-dependent. Another, more general, possible theory is a modification of the +Type I Seesaw, the Inverse Seesaw mechanism [81–83]. Via the introduction of small lepton +number violating scale, the Inverse Seesaw is able to accommodate the SM neutrino data +for any values of mixing and HNL mass [84, 85]. +As the purpose of this work is to investigate the collider reach in observing electroweak +production of HNLs in a model-independent fashion, we nevertheless remain agnostic as +to the UV theory. +Therefore, we employ an effective, phenomenological model — the +Universal FeynRules Output (UFO) [86, 87] models HeavyN, for Majorana [88–90] and +Dirac [19] HNLs. In these implementations, the HNLs, Ni for i = 1, . . . , 3, are sterile in the +SM but mix with the active neutrinos, namely, +νℓL = +3 +� +m=1 +UℓmνmL + +3 +� +m′=1 +Vℓm′Nc +m′L , +(2.1) +where m, and m′ index mass eigenstates, and ℓ = e, µ, τ index gauge eigenstates. The +matrix Uℓm is the usual Maki–Nakagawa–Sakata-Pontecorvo (MNSP) mixing matrix for SM +neutrinos [91–93], and Vℓm′ is a matrix parameterizing the mixing between the HNLs and +the SM neutrinos. +After electroweak symmetry breaking, the effective Lagrangian in the mass eigenstate +basis that describes the interactions between the HNLs and the SM electroweak bosons is +given by +– 3 – + +W +Ni +ℓ +Z +Ni +νℓ +Figure 1. New interaction vertices between the HNLs and SM electroweak bosons in the simplified +effective Lagrangian of equation 2.2. +−Lint,EW = g +√ +2W µ+ +τ +� +ℓ=e +� +3 +� +m=1 +U ∗ +ℓm¯νmγµPLℓ + +3 +� +m=1 +V ∗ +ℓm ¯Nc +mγµPLℓ +� +(2.2) ++ +g +2 cos θW +Zµ +τ +� +ℓ=e +� +3 +� +m=1 +U ∗ +ℓm¯νmγµPLνℓ + +3 +� +m=1 +V ∗ +ℓm ¯Nc +mγµPLνℓ +� ++ h.c. +This interaction Lagrangian introduces two new classes of vertices, given in figure 1. +There are several production mechanisms for HNLs in a high-energy muon collider, +the dominant of which depends on the mass of the HNL and its mixing with different +generations. In the region of parameter space under consideration in this work, namely +√s, mNi ≫ mW , the dominant production process is the production of an HNL and SM +neutrino pair. This Niνℓ pair occurs via two production channels: either a t-channel ex- +change of a W boson, or annihilation via a Z boson. These channels (along with a decay +N → qqℓ) are shown in figure 2 on the left- and right-hand sides, respectively. The W +exchange process benefits from a large W boson electroweak (EW) parton distribution +function (PDF) [33, 34, 94]. Conversely, the masses of the HNLs under consideration are +much larger than mZ, and as such annihilation processes via Z decay are far off shell and +highly suppressed. +At mN ≫ mW , we also have that the Goldstone Equivalence Theorem applies, and +thus it then follows [19, 90] that +BR(N → W +ℓ−) ≈ 2BR(N → Zνℓ) ≈ 2BR(N → hνℓ). +(2.3) +Note that this relation holds for both Majorana and Dirac HNLs, despite the absolute +widths of the Majorana HNLs being twice as large as for the Dirac case. As the HNLs +preferentially decay via charged-current interactions, and that the W boson decays largely +hadronically [7], we pick our decay channel of investigation to be N → qqℓ. An additional +benefit to this channel is that all decay products are visible. +3 +Simulation Framework +3.1 +Signal Event Generation +In this work we use the Monte-Carlo event generator Whizard 3 [95, 96] to generate +events at the parton level. The advantage of using Whizard 3 for this study is that it is +– 4 – + +W +N +W +µ +µ +ν +q +q +ℓ +Z +N +W +µ +µ +ν +q +q +ℓ +Figure 2. Feynman diagrams for the process µ+µ− → N(qqℓ±)ν. For the collider energies above +mZ, such as the scenarios under consideration in this paper, the dominant production channel is +given by the left-hand diagram of t-channel W exchange. +possible to include the structure function for initial state radiation (ISR) for muon beams. +Additionally, Whizard 3 allows for the use of the equivalent photon approximation (EPA) +for the inclusion of photon-induced background events due to the collinear splitting of +µ → µγ. Furthermore, the implementations of ISR and EPA in Whizard 3 allow for the +insertion of pT recoil of the hard scattering processes into the event record. The following +perturbative parton shower and hadronization steps are done using Pythia 8 [97]. +For the generation of signal events, we use the HeavyN UFO files SM_HeavyN_NLO +and SM_HeavyN_Dirac_NLO, for Majorana and Dirac HNLs, respectively. We focus on +the case in which only one HNL mixes with the SM. Furthermore, we assume that |V1e| = +|V1µ| and |V1τ| = 0. As we only consider one HNL, we use the notation |V1e| = |V1µ| ≡ |Vℓ| +with no ambiguity. We simulate the signal process by first generating the production of +HNLs, µ+µ− → Nν, the Feynman diagrams of which are shown in figure 2. +We then decay the HNL via N → qqℓ±, where q = u, d, c, s and ℓ± = e±, µ±. As we +decay N using the narrow-width approximation (NWA), we choose |Vℓ| = 0.002, as this en- +forces ΓN ≪ mN.1 We are free to make this choice as the quantity σ (µ+µ− → N(qqℓ±)ν) / |Vℓ|2 +is independent of |Vℓ| for a given mN. Furthermore, for the values of √s and mN considered +in this work, this choice of |Vℓ|2 is still large enough to ensure prompt decays of the HNLs, +as even in the most boosted, lowest width scenario under consideration (i.e √s = 10 TeV +and mN = 200 GeV), the decay length in the lab frame is of order 10−4 µm. This is sig- +nificantly smaller than the anticipated spatial resolution of detectors under consideration +in [31]. Note, however, that in non-trivial model extensions where the HNL production is +governed by another operator or mediator, the HNL production rate may be independent +of Vℓ. In such a case it is then possible to have Vℓ ≲ 10−6 while still producing an appre- +ciable collider signature. In this case, it might be such that the decay length of HNL is +macroscopic and the HNL becomes a long-lived particle [77, 98]. However, this approach +deviates from the focus of this work, which is conservative and model-independent. +In this work, we consider two of the muon collider benchmarks given in [31]: a scenario +in which √s = 3 TeV and L = 1 ab−1, and a scenario in which √s = 10 TeV and L = +10 ab−1. For analysis in a potential 3 TeV (10 TeV) muon collider, we consider benchmark +1This choice of |Vℓ| is actually much smaller than required for the validity of the NWA, but, as noted +in [19], our choice is advantageous in that larger mixings that still satisfy the NWA might allow for increased +virtuality of the HNL, leading to large variations in the kinematic distributions of the HNL’s decay products +across events. +– 5 – + +200 +1000 +10000 +mN (GeV) +1 +10 +50 +σ/|Vℓ|2 (pb) +σ(pb) for µ+µ− → N(qqℓ±)ν, √s = 10 TeV +Majorana +Dirac +200 +1000 +3000 +mN (GeV) +1 +10 +50 +σ/|Vℓ|2 (pb) +σ(pb) for µ+µ− → N(qqℓ±)ν, √s = 3 TeV +Majorana +Dirac +Figure 3. +The total cross section divided by |Vℓ|2 = |Ve|2 = |Vµ|2 for the process µ+µ− → +N(qqℓ±)ν as a function of mN, shown at two benchmark future muon colliders with √s = 3, 10 TeV. +scenarios with masses mN between 200 GeV and 2900 GeV (200 GeV and 9500 GeV). +When generating signal events, we set all final state lepton and quark masses to zero. +However, since we have ISR and ISR recoiled enabled, we set mµ = 0.105 GeV for the +beams, to be used as the mass for the ISR structure functions. At each mass benchmark, +ΓN is recalculated and used to compute BR(N → qqℓ±). The production cross section for +each benchmark is then reweighted by BR(N → qqℓ±) to give the total cross section for +the process σ(µ+µ− → N(qqℓ±)ν). The resulting total process cross section divided by +|Vℓ|2 as a function of mN is shown in figure 3. Note that the cross section for mN ≈ mW +is moderately enhanced by the branching of N → W ±ℓ∓ being favored over other decay +channels at those masses, as noted in [90]. +3.2 +Background Event Generation +As our final state topology of interest is two jets and a lepton, we consider SM background +processes with two quarks and between one and three charged leptons in the final state. +Specifically, we consider the following channels classified into two types according to the +simulation methods used: +ME +(a) µ+µ− → qqℓν, +(b) µ+µ− → qqℓℓ, +(c) µ+µ− → qqℓℓνν, +(d) µ+µ− → qqℓℓℓν, +EPA +(e) γγ → qqℓν, +(f) γµ± → qqℓ. +The first four processes listed are computed using the full matrix element (ME). They +include generator-level cuts such that the Monte-Carlo integration is convergent. In order +to prevent underestimation of the SM background, the generator-level cuts are taken to be +softer than those used in preselection. +– 6 – + +Process +Generator Level Cuts +Method +µ+µ− → qqℓν +Mqq,ℓℓ > 10 GeV, pT,ℓ > 4 GeV, |ηℓ| < 8, qℓ > 4 GeV +ME + ISR +µ+µ− → qqℓℓ +µ+µ− → qqℓℓνν +Mqq,ℓℓ > 40 GeV, pT,ℓ > 4 GeV, |ηℓ| < 8, qℓ > 4 GeV +ME + ISR +µ+µ− → qqℓℓℓν +γγ → qqℓν +Mqq > 10 GeV, qγ < 4 GeV +EPA +γµ± → qqℓ +Mqq > 10 GeV, qγ < 4 GeV, 3◦ < θℓ < 177◦ +EPA + ISR +Table 1. +Summary of the cuts and simulation methods used to generate the SM background in +Whizard 3. qℓ refers to the momentum-transfer between incoming and outgoing charged leptons, +and qγ is the upper momentum-transfer for the EPA structure function. For the process γµ± → +qqℓ, EPA is applied to one muon beam to produce the partonic photon, and ISR is applied to +the other beam. +For all processes, pT recoil on the hard scattering states due to emitted ISR +(EPA) photons is applied to the generated events by setting “?Isr(Epa)_Handler=True” and +“?Isr(Epa)_Handler_Mode=Recoil.” +Furthermore, in order to preserve gauge-invariance, the simulations of the processes +µ+µ− → qqℓℓℓν and µ+µ− → qqℓℓ include the processes γγ → qqℓν and γµ± → qqℓ as +subdiagrams. However, the cuts chosen on the full processes exclude the regions of low +momentum transfer, qℓ, where the contribution of partonic photon scattering dominates. +Therefore, we include the last two processes, γγ → qqℓν and γµ± → qqℓ, where γ refers +to partonic photons split collinearly from the muon beam. These are simulated using the +Equivalent Photon Approximation (EPA) [99], based on the Weizsäcker-Williams approxi- +mation [100, 101]. We choose the lower cutoff qℓ for the full ME processes and the upper +cutoff qγ for the EPA structure function to be complementary to avoid double counting +of events. We also include the effect of initial state radiation (ISR) on the beam(s) that +do not split into EPA photons. The specific cuts used for the SM background processes +are detailed in table 1. The cross sections and estimated event yields for each background +channel are given in table 2. +It is also worth commenting on the major contributors to each of the background +channels. As expected, given that √s is TeV scale, every background channel is dominated +by contributions from vector boson fusion or scattering (VBF or VBS). More specifically, +we find that the dominant processes in the channel µ+µ− → qqℓν are from WZ and Wγ +to W VBF, where the W subsequently decays to a quark pair. +The process µ+µ− → +qqℓℓ is dominated by γγ to qq VBF via the exchange of a t-channel light quark. +For +µ+µ− → qqℓℓνν, the major contribution is given by Wγ/WZ to Wγ/WZ VBS, where +the neutral boson decays to a qq pair, and the W decays leptonically to ℓν. The channel +µ+µ− → qqℓℓℓν is dominated by γγ to WW VBS, where one W decays hadronically and +the other leptonically. Additionally, the EPA process γγ → qqℓν is also dominated by the +same γγ to WW VBS, which is consistent with it being considered a subdiagram of the +µ+µ− → qqℓℓℓν channel. Likewise, the EPA process γµ± → qqℓ is dominated by the same +γγ to qq VBS process that µ+µ− → qqℓℓ is. +– 7 – + +3.3 +Detector Simulation +After generating parton-level events with Whizard 3, and using Pythia 8 to shower and +hadronize, we use Delphes 3 [102] to simulate the detector response. We use the default +muon collider detector card that is included in the Delphes 3 distribution. This card is a +hybrid of the FCC-hh [103] and CLIC [104] cards. For this detector simulation, it requires +isolated leptons to have |η| < 2.5. +It also includes a hypothetical forward muon spectrometer sensitive to muons with +2.5 < |η| < 8.0 at 90% efficiency for muons with 0.5 GeV < pT < 1.0 GeV and at 95% +efficiency for muons with 2.5 < |η| < 8.0 and pT > 1.0 GeV. This ability to detect forward +muons is possibly advantageous in distinguishing signal from background in our analysis, +as events that pass preselection with muons in the forward region are almost always due to +the SM background. However, as the future of such a detector is uncertain, we therefore +do not include its contribution in our analysis. +4 +Analysis +Following the detector simulation, we preselect events to keep those with signal-like topolo- +gies. Namely, in each event we require: +• At least one isolated ℓ = e, µ candidate. Our definition of an isolated lepton is +the same as the definition used by the built-in Delphes muon collider card. Firstly, it +requires that the ℓ candidate have pT > 0.5 GeV. Secondly, inside a cone of ∆R = 0.1 +around the ℓ candidate, the ratio of the scalar sum of pT of all particles in that cone +to the pT of the ℓ candidate must be no more than 0.2. In case of having more than +one ℓ candidate, the one with the highest pT will be used. +• A W boson jet system (WJ) candidate. We reconstruct the jet system in two +possible ways, using the Valencia algorithm [105]. For most of the mN parameter +space, W bosons from the decays of the HNLs are highly boosted. Therefore, we +choose to reconstruct the WJ as one jet with a moderate radius parameter R = 1.2 +and both remaining parameters β and γ set to 1.0, denoted as Jfat. +The second option is to reconstruct two narrower jets in a similar way, but with +R = 0.2, denoted as j1 and j2. If it is possible to reconstruct both classes of jet(s) in +the event, we compare the invariant masses of WJ = Jfat with that of WJ = j1 + j2. +The jet reconstruction which gives an invariant mass closest to that of the W boson +is then accepted. If only one class of jet(s) is able to be reconstructed, that class will +be used as WJ. +• Kinematic cuts on ℓ and WJ. We require ℓ and WJ to have pT > 100 GeV, and +that the reconstructed jet system is on-shell, namely |mWJ − mW | < 5ΓW . +In figure 4, we show the reconstruction efficiency for Majorana and Dirac HNL signal +channels. It is expected that the reconstruction efficiency is lessened for HNL masses well +below √s, as such HNLs are mostly produced by the muon beam with a virtual W boson +– 8 – + +200 +1500 +10000 +mN (GeV) +1 +10 +70 +Reconstruction Efficiency (%) +Efficiency for µ+µ− → N(qqℓ±)ν, √s = 10 TeV +Majorana +Dirac +200 +500 +1000 +3000 +mN (GeV) +1 +10 +70 +Reconstruction Efficiency (%) +Efficiency for µ+µ− → N(qqℓ±)ν, √s = 3 TeV +Majorana +Dirac +Figure 4. The preselection and reconstruction efficiency of the process µ+µ− → N(qqℓ±)ν as a +function of mN, shown at two benchmark future muon colliders with √s = 3, 10 TeV. As expected, +the ability to detect and reconstruct events for HNLs with masses well below the collider energy is +poor, as such HNLs are too boosted to have detectable decay products. +with small pz, as shown in the left panel of figure 2. The overall pz of the produced HNL +becomes comparable to the beam energy, with the expected pT of its decay products limited +by its small mass. Consequently the ℓ and q¯q′ final states are less likely to reach the central +region with low |η|. The reconstruction efficiencies and event yields for the background +channels are given in table 2. Note that the reconstruction efficiency of the µ+µ− → qqℓℓℓν +background channel is higher than that of the µ+µ− → qqℓν channel. This is due to the fact +that the major contributing process in the µ+µ− → qqℓℓℓν background is µ+µ− → qqℓµµν, +where the two outgoing muons are from the beam muons. The outgoing muons are therefore +in the forward region, and will either be missed by the detector or detected with very small +pT . Such an event will still pass our reconstruction by identifying the remaining lepton +(i.e. the one with highest pT ). Similarly, majority of the outgoing leptons in µ+µ− → qqℓν +channel are also beam muons. Therefore, this event will not pass preselection if such low pT +forward muon is missed by the detector. Alternatively, one could also introduce a different +reconstruction method which could strictly require that only one lepton is detected in the +event in order to veto some of the µ+µ− → qqℓℓℓν background. +After preselection, we then reconstruct the HNL by combining WJ and the isolated +lepton. +A plot of the distribution of the invariant mass mN is show in figure 5 for a +benchmark scenario of mN = 1 TeV and √s = 3 TeV. A peak in the signal events is visible +at the expected value of mN = 1 TeV. Note that there is also a tail in the reconstructed +mass distribution below the expected value. +This is due to missing W-jet components +that have either been missed by the detector reconstruction or not clustered by the jet +algorithm. In both cases, the invariant mass reconstructed drops below the physical value. +In figure 6, we show the mN distribution for the individual channels that comprise the total +SM background in both a √s = 3 TeV and √s = 10 TeV collider. On the left-hand side of +figure 6, for √s = 3 TeV, we see that the dominant background channel is µ+µ− → qqℓν. +For the right-hand side, with √s = 10 TeV, the dominant background for mN < 6 TeV is +– 9 – + +0 +500 +1000 +1500 +2000 +2500 +3000 +Reconstructed mN (GeV) +10−2 +10−1 +100 +101 +102 +103 +104 +Events/50 GeV +√s =3 TeV; mN =1 TeV +Majorana +Dirac +Background +Figure 5. A histogram of reconstructed mN after preselection, for a benchmark scenario of mN = +1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider. Blue corresponds to Majorana signal events, +red for Dirac, and yellow for the combined SM background. Signal events are normalized to |Vℓ|2 = +5×10−6. A sharp peak is visible at mN = 1 TeV with a tail to the left due missing W-jet components. +due to the µ+µ− → qqℓℓℓν channel, with the µ+µ− → qqℓν becoming dominant for larger +mN. +To further separate signal-like and background-like events after preselection, we employ +a Boosted Decision Tree (BDT) analysis using the Python implementation of XGBoost [106]. +We train BDT to distinguish between three classes of event — signal events due to the decay +of a Majorana HNL, signal events due to the decay of a Dirac HNL, and signal events due +to the SM background. The BDT training utilizes the following input features: +• Lepton Information: +– The lepton’s transverse momentum, pT,ℓ; pseudorapidity, ηℓ; and its energy, Eℓ. +– The charge of the lepton and its flavor. +• W Jet System Information: +– The W jet system’s transverse momentum, pT,WJ; pseudorapidity, ηWJ; and +mass, mWJ. +– The energies of the two narrow sub-jets Ej1 and Ej2, in the case that WJ is able +to be reconstructed as two jets. +• Reconstructed HNL Information: +– The HNL’s transverse and z-components of momenta, pT,N and pz,N. +• Geometric Information: +– 10 – + +Collider COM Energy +√s = 3 TeV +√s = 10 TeV +Integrated Luminosity +L = 1 ab−1 +L = 10 ab−1 +Process +σ (pb) +Nevents +Eff. (%) +σ (pb) +Nevents +Eff. (%) +µ+µ− → qqℓν +6.025 +263400 +4.373 +9.534 +932800 +0.9784 +µ+µ− → qqℓℓ +2.842 +12160 +0.4278 +3.784 +32090 +0.0846 +µ+µ− → qqℓℓνν +0.02255 +3201 +14.20 +0.07968 +85100 +10.68 +µ+µ− → qqℓℓℓν +0.3133 +90090 +28.76 +3.207 +14950000 +47.63 +γγ → qqℓν +0.1589 +5068 +3.190 +0.4274 +113600 +2.658 +γµ± → qqℓ +3.811 +11390 +0.2986 +0.5823 +21360 +0.3668 +Table 2. Cross sections, expected event yields and reconstruction efficiencies after the preselection +for the SM background processes detailed in section 4. Although the expected signal has only one +final state charged lepton, we allow events with more than one charged lepton past preselection, as +the Boosted Decision Tree analysis afterwards does an effective job of removing such events. +0 +2000 +4000 +6000 +8000 +10000 +12000 +Reconstructed mN (GeV) +101 +102 +103 +104 +105 +106 +Events/200 GeV +√s =10 TeV; L =10 ab−1 +µ+µ− → qqℓν +µ+µ− → qqℓℓ +µ+µ− → qqℓℓνν +µ+µ− → qqℓℓℓν +γµ± → qqℓ +γγ → qqℓν +0 +1000 +2000 +3000 +Reconstructed mN (GeV) +10−1 +100 +101 +102 +103 +104 +Events/50 GeV +√s =3 TeV; L =1 ab−1 +Figure 6. +The mN distribution for the individual channels that comprise the total SM background +in both a √s = 3 TeV (left) and √s = 10 TeV (right) muon collider. For √s = 3 TeV, we see that the +dominant background channel is µ+µ− → qqℓν, whereas for √s = 10 TeV the dominant background +is due to the µ+µ− → qqℓℓℓν channel for mN < 6 TeV, with the µ+µ− → qqℓν channel becoming +dominant for larger mN. +– The angular distance and azimuthal angle difference between the lepton and W +jet system, ∆R(ℓ, WJ) and |φℓ − φWJ|. +We show the distributions for the BDT features in figures 7 and 8. +Generally, for +most variables, the kinematic distributions for Majorana and Dirac HNLs are similar, but +the distributions of the energy and pseudorapidity of the lepton ηℓ (figure 7 (b)) and Eℓ +(figure 7 (c)), and of the pseudorapidity of the jet system ηWJ (figure 7 (e)) differ for the two +types of HNL. The flatness of the distribution in Eℓ for Majorana HNLs is due to the lack of +forward-backward asymmetry in their decay. More broadly, This effect is well known [107– +111] and has been noted in the context of hadron [112] and lepton [21, 113] colliders. +Additionally, since the pT distributions for Majorana and Dirac HNLs are functionally +– 11 – + +identical, it then follows that pseudorapidity distributions vary between the two types of +HNLs in an analogous fashion to the Eℓ distributions. +0 +250 +500 +750 +1000 +1250 +1500 +pT,ℓ (GeV) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/25 GeV +√s =3 TeV; mN =1 TeV +Majorana +Dirac +Background +(a) +−3 +−2 +−1 +0 +1 +2 +3 +ηℓ +10−2 +10−1 +100 +101 +102 +103 +104 +Events/0.1 +(b) +0 +250 +500 +750 +1000 +1250 +1500 +Eℓ (GeV) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/25 GeV +(c) +0 +250 +500 +750 +1000 +1250 +1500 +pT,WJ (GeV) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/25 GeV +(d) +−3 +−2 +−1 +0 +1 +2 +3 +ηWJ +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/0.1 +(e) +70 +75 +80 +85 +90 +mWJ (GeV) +10−2 +10−1 +100 +101 +102 +103 +104 +Events/0.35 GeV +(f) +Figure 7. +Distributions of the features used in the BDT analysis, for a benchmark scenario of +mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider. Blue corresponds to Majorana signal +events, red for Dirac, and yellow for the combined SM background. For all signal events, cross +sections are normalized to their 95% exclusion limits. +As we run the BDT analysis as a three-class categorization task, we use the quantity +– 12 – + +0 +250 +500 +750 +1000 +1250 +1500 +pT,N (GeV) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/25 GeV +√s =3 TeV; mN =1 TeV +Majorana +Dirac +Background +(a) +−1500 −1000 +−500 +0 +500 +1000 +1500 +pz,N (GeV) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/ 50 GeV +(b) +0 +1 +2 +3 +4 +5 +6 +∆R(ℓ, WJ) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/0.1 +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +|φℓ − φWJ| +10−2 +100 +102 +104 +Events/0.05 +(d) +0 +250 +500 +750 +1000 +1250 +1500 +EJ1 (GeV) +10−3 +10−2 +10−1 +100 +101 +102 +103 +104 +Events/25 GeV +(e) +0 +250 +500 +750 +1000 +1250 +1500 +EJ2 (GeV) +10−2 +100 +102 +104 +Events/25 GeV +(f) +Figure 8. +Continuation of figure 7. Distributions of the features used in the BDT analysis, for a +benchmark scenario of mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider. Blue corresponds +to Majorana signal events, red for Dirac, and yellow for the combined SM background. For all +signal events, cross sections are normalized to their 95% exclusion limits. +1 − PB as the BDT response score, where PB is the probability the algorithm classifies +an event as background. The BDT response for a benchmark scenario of mN = 1 TeV +in a √s = 3 TeV collider is shown in figure 9. The dashed vertical line in figure 9 is the +– 13 – + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BDT Score +10−4 +10−3 +10−2 +10−1 +100 +Normalized Events +√s =3 TeV; mN =1 TeV +Signal +Background +Figure 9. The normalized BDT response, 1 − PB, for a benchmark scenario of mN = 1 TeV in +a √s = 3 TeV, L = 1 ab−1 muon collider. The events in yellow correspond to the combined SM +background, and the purple events correspond to the combined the Majorana and Dirac signals. The +dashed vertical line is the cut on the BDT score that we use to distinguish signal from background. +The optimization of the BDT cut here is operated on a grid four times finer than the binning shown +in the plot. +cutoff on the BDT score that we use to separate signal from background. In the analysis +of section 5.1, only the 1 − PB cut is applied. For use in the analysis of section 5.2, on +events that pass the background cut we also apply a cutoff on the quantity PM − PD to +separate Majorana-like and Dirac-like signal events, where PM (PD) is the probability the +BDT algorithm classifies the signal as Majorana (Dirac). We optimize both the 1−PB and +PM − PD cuts simultaneously. +In addition to cutting on the BDT score, we further apply a cut on the mass of the +reconstructed HNL, namely within [0.9 × mN, 1.05 × mN]. This window is asymmetric, +as the reconstructed distribution has a low mass tail. Figure 10 shows the invariant mass +distribution mN of the HNL signals and SM background after the application of the BDT +analysis. +5 +Results +5.1 +Exclusion Limits on |Vℓ|2 +We now present the results of the BDT analysis in order to construct 95% exclusion limits +on the HNL mixing parameter |Vℓ|2. At this level of analysis, we assume no systematic +uncertainty in the number of recorded events. Our goal is to determine exclusion limits at +95% confidence, and as such we require Z ≥ 1.96. Therefore, for each benchmark simulation +we run the BDT analysis and extract the lower limit on |Vℓ|2 such that the signal significance +– 14 – + +600 +800 +1000 +1200 +1400 +Reconstructed mN (GeV) +10−3 +10−2 +10−1 +100 +101 +Events/25 GeV +√s =3 TeV; mN =1 TeV +Majorana +Dirac +Background +Figure 10. A histogram of reconstructed mN after applying BDT, for a benchmark scenario of +mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider. The dashed vertical lines are the cuts +on the mN mass used to further separate signal from background. Blue corresponds to Majorana +signal events, red for Dirac, and yellow for the combined SM background. For all signal events, +cross sections are normalized to their 95% exclusion limits. +is above threshold. This analysis was done using a three-class BDT. We applied a two-class +analysis to cross check and found exclusion curves with comparable sensitivies. +The results of this process are shown in figure 11, as well as comparisons to similar +searches in other (proposed) experiments. For masses above mN > 1 TeV, the exclusion +limits are stronger than any other limits shown so far, with reach down to |Vℓ|2 ∼ O(10−6) +or even lower. Note that in this mass region, it is often the case that the sensitivities for the +Majorana signals are slightly better than those of the Dirac signals. The reason is evident +from figure 7 (c, e), and figure 8 (e, f), which show that the corresponding distributions +of a Majorana HNL differ from the SM background distributions more significantly than +those of a Dirac HNL. +In the region of lighter HNL masses, namely mN < 1 TeV, our limits are generally less +strict than those for other experiments, due to the poor reconstruction efficiency for low +mass HNLs. Nevertheless, the uncertainty in the signal yield for these benchmarks is high, +and as such our limits can be considered conservative in this region. +Furthermore, in this low mN region, the HNL signal would be dominated by the VBF +channel, wherein two vector bosons from the muon beams give rise to N + ℓ or N + ν +final states. The reconstruction efficiency, in this case, could be significantly larger than +that of the t-channel ones, as the pz of the HNL produced would be moderate, leaving +more decay products in the central region. In addition, in the scenario where |Vµ| ≪ |Ve| +or |Vτ|, the t-channel signal rate would be greatly suppressed, whereas the VBF channel +signal rate could still be relevant. Nevertheless, the biggest problem for the VBF channel +is that its production rate is at least two orders of magnitude smaller than that of the t- +– 15 – + +101 +102 +103 +104 +mN (GeV) +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +|Vℓ|2 +FCC-hh +LHC prompt +CLIC3000 +ILC1000 +ILC500 +mlightest = 0 GeV +Thermal initial conditions +Vanishing initial conditions +µC 3 TeV Majorana 1 ab−1 +µC 3 TeV Dirac 1 ab−1 +µC 10 TeV Majorana 10 ab−1 +µC 10 TeV Dirac 10 ab−1 +Figure 11. +95% exclusion limits for |Vℓ|2 = |Ve|2 = |Vµ|2 as a function of the HNL mass +mN. The red dashed (solid) line corresponds to a Majorana (Dirac) HNL at a muon collider with +√s = 3 TeV, L = 1 ab−1. The teal dashed (solid) line corresponds to a Majorana (Dirac) HNL +at a muon collider with √s = 10 TeV, L = 10 ab−1. The solid black and purple lines correspond +to limits from considerations of viable leptogenesis scenarios [114]. The grey area is the region +excluded by a global scan [115]. The red line shows the limits from prompt trilepton searches at the +LHC [116]. The green line shows the limits from a future FCC-hh [117]. Lastly, the grey, orange +and blue lines are exclusion limits in future e+e− linear colliders [21]. +channel process, since the process of higher order. Consequently, for this work’s benchmark +of |Vℓ| = |Ve| = |Vµ|, the estimated exclusion limit of |Vℓ| is at most comparable to that of +the t-channel when mN ≃ 200 GeV, and drops much faster as mN increases. We will leave +a through study of VBF channels to future work. +5.2 +Distinguishing Majorana versus Dirac Heavy Neutral Leptons +As noted in section 4 and figure 7, there are signatures in the final state kinematic distri- +butions that differ between Majorana and Dirac type HNLs. It is therefore an interesting +question to ask what would be the discrimination potential to distinguish Majorana versus +Dirac HNLs in a muon collider. We consider the quantity r = nM/(nM + nD), where nM +– 16 – + +(nD) is the number of detected Majorana (Dirac) HNLs. Note that this definition differs +from the usual Rll seen in the literature, which is based on the ratio of lepton number +violating (LNV) decays to lepton number conserving (LNC) decays (see, e.g. [118]).2 In +order to quantify our analysis, we consider the two-dimensional likelihood in the plane of +Majorana and Dirac HNL signal yields in a √s = 3 TeV and a √s = 10 TeV collider, shown +in figure 12 on the left and right-hand sides respectively. Both the horizontal and vertical +axes are normalized by the same quantity, namely the 95% exclusion limits |Vℓ|2 as given in +figure 11 for either Majorana or Dirac type — whichever value is more conservative. Each +point on the the dotted line represents a possible value of r, and the x and y intercepts of +that line correspond to the fully Majorana (r = 1) and Dirac (r = 0) cases respectively. +For a given r value, we can construct the corresponding 1σ likelihood contour based on the +Majorana and Dirac regions defined in section 4. In this analysis, a signal event is classified +as Majorana or Dirac if it appropriately satisfies the PM − PD cut correctly. If not, it is +counted as additional background. +We plot three such contours at r = 0, 0.5 and 1.0. The centers of these ellipses lie on the +diagonal dotted line, and the boundaries of each correspond to fluctuations in signal yields +with significance 1σ. Note that the ellipses are tilted, and showing a negative correlation +between Majorana and Dirac signal yields. This is due to the fact that the classification +is imperfect, as some Majorana or Dirac decays could be misclassified as each other while +keeping total HNL signal rate the same. The contours for the √s = 10 TeV benchmark +are tilted more strongly (i.e more highly correlated). +This indicates that a downwards +fluctuation of Majorana HNL decay is more likely to be due to misclassification as a Dirac +HNL than misclassification as background, as compared to the case in the √s = 3 TeV +benchmark. +Lastly, the uncertainties shown in figure 12 are marginal, as we are considering discrim- +ination potential at the 95% exclusion limits. We have found that repeating this analysis +for larger |Vℓ|2 results in much tighter contours since the larger statistics improve the pre- +cision of both signal rates. For example, in figure 13 we repeat the above analysis with +|Vℓ|2 = 5 × |Vℓ|2 +95%. Note that the values for here |Vℓ|2 are still small — of order O(10−5). +We see that we therefore have a nontrivial discrimination potential for the HNL’s type as +long as its signal yield is above the exclusion limit. +6 +Conclusion and Future Directions +In this paper we investigate the potential for searching for HNLs in a future high-energy +muon collider. We consider an effective theory benchmark that couples a single HNL to +the first two generations of active neutrinos with equal mixing. We consider HNLs of both +Majorana and Dirac types, with masses ranging from 200 GeV to 9.5 TeV and determine +the reach in the mixing parameter |Vℓ|2 for each mass, in √s = 3 TeV, L = 1 ab−1 and +√s = 10 TeV, L = 10 ab−1 muon colliders. We find that for HNL masses greater than +2The two definitions agree for r = Rll = 0, 1, but our r = 1/2 corresponds to Rll = 1/3. Alternatively, r +could also be defined as +2nLNV +nLNC+nLNV , where nLNC and nLNV are numbers of detected LNC and LNV decays, +respectively. +– 17 – + +0.25 +0.5 +0.75 +1 +1.25 +1.5 +r +0.25 +0.5 +0.75 +1 +1.25 +1.5 +(1 − r) +√s =3 TeV; mN =1 TeV +1σ Uncertainty +r=0.0 +r=0.5 +r=1.0 +0.25 +0.5 +0.75 +1 +1.25 +1.5 +1.75 +r +0.25 +0.5 +0.75 +1 +1.25 +1.5 +1.75 +(1 − r) +√s =10 TeV; mN =3 TeV +1σ Uncertainty +r=0.0 +r=0.5 +r=1.0 +Figure 12. The two-dimensional likelihood in the plane of Majorana and Dirac HNL signal yields +in a √s = 3 TeV (left) and a √s = 10 TeV collider (right). The 3 ellipses on each panel represent the +1σ contours centered at r = 0, 0.5 and 1.0. Note that the ellipses are tilted, and showing a negative +correlation between the measured Majorana and Dirac signal yields. All contours are evaluated at +their 95% exclusion limits where |Vℓ|2 ∼ O(10−6), respectively. +0.25 +0.5 +0.75 +1 +1.25 +1.5 +r +0.25 +0.5 +0.75 +1 +1.25 +1.5 +(1 − r) +√s =3 TeV; mN =1 TeV +1σ Uncertainty +r=0.0 +r=0.5 +r=1.0 +0.25 +0.5 +0.75 +1 +1.25 +1.5 +1.75 +r +0.25 +0.5 +0.75 +1 +1.25 +1.5 +1.75 +(1 − r) +√s =10 TeV; mN =3 TeV +1σ Uncertainty +r=0.0 +r=0.5 +r=1.0 +Figure 13. The two-dimensional likelihood in the plane of Majorana and Dirac HNL signal yields +in a √s = 3 TeV (left) and a √s = 10 TeV collider (right). The 3 ellipses on each panel represent +the 1σ contours centered at r = 0, 0.5 and 1.0. Note that the ellipses are tilted, and showing a +negative correlation between Majorana and Dirac signal yields. The |Vl|2 used here are five times +larger than their counterparts in figure 12, leading to much sharper contours. +1 TeV the limits in |Vℓ|2 are the strictest collider limits yet. We also show that a future +muon collider has strong discrimination potential to distinguish between Majorana and +Dirac type HNLs. +In the future, there are a few aspects of this work that could be broadened. Firstly, we +– 18 – + +focus our analysis on an effective theory in which the HNL signal is dominated by single +production. +The purpose of this work was to be conservative and model-independent, +but different UV completions could have distinct signatures accessible at a muon collider. +Secondly, a more realistic analysis would include the production and decay of multiple +HNLs. +Similarly, this work assumes a benchmark point of enhanced symmetry for the +mixing of the HNLs, namely |V1e| = |V1µ| ̸= 0 and |V1τ| = 0. Consequently, we do not +consider the decay of an HNL to a tau in this work. For other benchmarks, the dominant +processes in the production of HNLs might differ. Lastly, while we include a fast detector +simulation in this work, we do not incorporate the effects of detector-based systematic +uncertainties. Likewise, we do not include systemic uncertainties due the beams nor do we +incorporate beam spectra. As a spectrum profile for muon beams is not yet available in +Whizard, we also leave such refinements for a future work. +Note Added: At the same time as this work was shared on the arXiv, another preprint +on the same topic was also uploaded [119]. While the two works’ aims are similar, they +differ in their methods and analyses. +We are also aware of a work by Peiran Li, Zhen +Liu and Kun-Feng Lyu on the same subject that will also appear on the arXiv soon. The +preliminary results of that work can be found in the Snowmass report of BSM physics [120]. +Acknowledgments +LL would like to thank Samuel Homiller, Zhen Liu, and Kun-Feng Lyu for valuable dis- +cussions. AR would like to thank Adam Lister and Harry Hausner for useful comments. +THK, TL and AR are supported partly by the Area of Excellence (AoE) under the Grant +No. AoE/P-404/18-3 and supported partly by the General Research Fund (GRF) under +Grant No. 16304321. Both of the AoE and GRF grants are issued by the Research Grants +Council of Hong Kong S.A.R. LL is supported by the DOE grant DE-SC-0010010. +– 19 – + +References +[1] R. Davis, Jr., D. S. Harmer, and K. C. Hoffman, Search for neutrinos from the sun, Phys. +Rev. Lett. 20 (1968) 1205–1209. +[2] KamLAND Collaboration, K. Eguchi et al., First results from KamLAND: Evidence for +reactor anti-neutrino disappearance, Phys. Rev. Lett. 90 (2003) 021802, [hep-ex/0212021]. +[3] Daya Bay Collaboration, F. P. An et al., Observation of electron-antineutrino +disappearance at Daya Bay, Phys. Rev. Lett. 108 (2012) 171803, [arXiv:1203.1669]. +[4] RENO Collaboration, J. K. Ahn et al., Observation of Reactor Electron Antineutrino +Disappearance in the RENO Experiment, Phys. Rev. Lett. 108 (2012) 191802, +[arXiv:1204.0626]. +[5] Double Chooz Collaboration, Y. Abe et al., Improved measurements of the neutrino +mixing angle θ13 with the Double Chooz detector, JHEP 10 (2014) 086, [arXiv:1406.7763]. +[Erratum: JHEP 02, 074 (2015)]. +[6] SNO Collaboration, Q. R. Ahmad et al., Direct evidence for neutrino flavor transformation +from neutral current interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett. 89 +(2002) 011301, [nucl-ex/0204008]. +[7] Particle Data Group Collaboration, R. L. Workman et al., Review of Particle Physics, +PTEP 2022 (2022) 083C01. +[8] S. Weinberg, Baryon and Lepton Nonconserving Processes, Phys. Rev. Lett. 43 (1979) +1566–1570. +[9] M. Gell-Mann, P. Ramond, and R. Slansky, Complex Spinors and Unified Theories, Conf. +Proc. C 790927 (1979) 315–321, [arXiv:1306.4669]. +[10] P. Minkowski, µ → eγ at a Rate of One Out of 109 Muon Decays?, Phys. Lett. B 67 (1977) +421–428. +[11] T. Yanagida, Horizontal gauge symmetry and masses of neutrinos, Conf. Proc. C 7902131 +(1979) 95–99. +[12] R. N. Mohapatra and G. Senjanovic, Neutrino Mass and Spontaneous Parity +Nonconservation, Phys. Rev. Lett. 44 (1980) 912. +[13] CMS Collaboration, A. M. Sirunyan et al., Search for heavy neutral leptons in events with +three charged leptons in proton-proton collisions at √s = 13 TeV, Phys. Rev. Lett. 120 +(2018), no. 22 221801, [arXiv:1802.02965]. +[14] ATLAS Collaboration, G. Aad et al., Search for heavy neutral leptons in decays of W +bosons produced in 13 TeV pp collisions using prompt and displaced signatures with the +ATLAS detector, JHEP 10 (2019) 265, [arXiv:1905.09787]. +[15] CMS Collaboration, A. Tumasyan et al., Search for long-lived heavy neutral leptons with +displaced vertices in proton-proton collisions at √s =13 TeV, JHEP 07 (2022) 081, +[arXiv:2201.05578]. +[16] ATLAS Collaboration, Search for heavy neutral leptons in decays of W bosons using a +dilepton displaced vertex in √s = 13 TeV pp collisions with the ATLAS detector, +arXiv:2204.11988. +[17] LHCb Collaboration, R. Aaij et al., Search for heavy neutral leptons in W + → µ+µ±jet +decays, Eur. Phys. J. C 81 (2021), no. 3 248, [arXiv:2011.05263]. +– 20 – + +[18] SHiP Collaboration, C. Ahdida et al., Sensitivity of the SHiP experiment to Heavy Neutral +Leptons, JHEP 04 (2019) 077, [arXiv:1811.00930]. +[19] S. Pascoli, R. Ruiz, and C. Weiland, Heavy neutrinos with dynamic jet vetoes: multilepton +searches at √s = 14 , 27, and 100 TeV, JHEP 06 (2019) 049, [arXiv:1812.08750]. +[20] G. Cottin, J. C. Helo, M. Hirsch, A. Titov, and Z. S. Wang, Heavy neutral leptons in +effective field theory and the high-luminosity LHC, JHEP 09 (2021) 039, +[arXiv:2105.13851]. +[21] K. Mękała, J. Reuter, and A. F. Żarnecki, Heavy neutrinos at future linear e+e− colliders, +JHEP 06 (2022) 010, [arXiv:2202.06703]. +[22] A. N. Skrinsky and V. V. Parkhomchuk, Cooling Methods for Beams of Charged Particles. +(In Russian), Sov. J. Part. Nucl. 12 (1981) 223–247. +[23] D. Neuffer, Principles and Applications of Muon Cooling, Conf. Proc. C 830811 (1983) +481–484. +[24] D. Neuffer, Multi-TeV muon colliders, AIP Conf. Proc. 156 (1987) 201–208. +[25] V. D. Barger, M. S. Berger, J. F. Gunion, and T. Han, s channel Higgs boson production at +a muon muon collider, Phys. Rev. Lett. 75 (1995) 1462–1465, [hep-ph/9504330]. +[26] V. D. Barger, M. S. Berger, J. F. Gunion, and T. Han, Higgs Boson physics in the s +channel at µ+µ− colliders, Phys. Rept. 286 (1997) 1–51, [hep-ph/9602415]. +[27] C. M. Ankenbrandt et al., Status of muon collider research and development and future +plans, Phys. Rev. ST Accel. Beams 2 (1999) 081001, [physics/9901022]. +[28] P. Chen, Beam-beam interaction in muon colliders, Nucl. Phys. B Proc. Suppl. 51 (1996) +179–185. +[29] R. Palmer et al., Muon collider design, Nucl. Phys. B Proc. Suppl. 51 (1996) 61–84, +[acc-phys/9604001]. +[30] C. Aime et al., Muon Collider Physics Summary, arXiv:2203.07256. +[31] K. M. Black et al., Muon Collider Forum Report, arXiv:2209.01318. +[32] J. P. Delahaye, M. Diemoz, K. Long, B. Mansoulié, N. Pastrone, L. Rivkin, D. Schulte, +A. Skrinsky, and A. Wulzer, Muon Colliders, arXiv:1901.06150. +[33] A. Costantini, F. De Lillo, F. Maltoni, L. Mantani, O. Mattelaer, R. Ruiz, and X. Zhao, +Vector boson fusion at multi-TeV muon colliders, JHEP 09 (2020) 080, +[arXiv:2005.10289]. +[34] T. Han, Y. Ma, and K. Xie, High energy leptonic collisions and electroweak parton +distribution functions, Phys. Rev. D 103 (2021), no. 3 L031301, [arXiv:2007.14300]. +[35] NNPDF Collaboration, V. Bertone, S. Carrazza, N. P. Hartland, and J. Rojo, Illuminating +the photon content of the proton within a global PDF analysis, SciPost Phys. 5 (2018), no. 1 +008, [arXiv:1712.07053]. +[36] R. B. Palmer, Muon Colliders, Rev. Accel. Sci. Tech. 7 (2014) 137–159. +[37] Muon Collider Collaboration, D. Stratakis et al., A Muon Collider Facility for Physics +Discovery, arXiv:2203.08033. +[38] D. Schulte et al., Chapter 5: Bright muon beams and muon colliders, CERN Yellow Rep. +Monogr. 1 (2022) 145–183. +– 21 – + +[39] J.-P. Delahaye et al., Enabling Intensity and Energy Frontier Science with a Muon +Accelerator Facility in the U.S.: A White Paper Submitted to the 2013 U.S. Community +Summer Study of the Division of Particles and Fields of the American Physical Society, in +Community Summer Study 2013: Snowmass on the Mississippi, 8, 2013. arXiv:1308.0494. +[40] J.-P. Delahaye et al., A Staged Muon Accelerator Facility For Neutrino and Collider +Physics, in 5th International Particle Accelerator Conference, p. WEZA02, 6, 2014. +arXiv:1502.01647. +[41] R. D. Ryne et al., Design Concepts for Muon-Based Accelerators, in 6th International +Particle Accelerator Conference, p. WEPWA057, 2015. +[42] K. Long, D. Lucchesi, M. Palmer, N. Pastrone, D. Schulte, and V. Shiltsev, Muon colliders +to expand frontiers of particle physics, Nature Phys. 17 (2021), no. 3 289–292, +[arXiv:2007.15684]. +[43] MICE Collaboration, T. A. Mohayai, First Demonstration of Ionization Cooling in MICE, +in 9th International Particle Accelerator Conference, 6, 2018. arXiv:1806.01807. +[44] MICE Collaboration, V. Blackmore, Recent results from the study of emittance evolution +in MICE, in 9th International Particle Accelerator Conference, 6, 2018. arXiv:1806.04409. +[45] MICE Collaboration, M. Bogomilov et al., Demonstration of cooling by the Muon +Ionization Cooling Experiment, Nature 578 (2020), no. 7793 53–59, [arXiv:1907.08562]. +[46] M. Antonelli, M. Boscolo, R. Di Nardo, and P. Raimondi, Novel proposal for a low +emittance muon beam using positron beam on target, Nucl. Instrum. Meth. A 807 (2016) +101–107, [arXiv:1509.04454]. +[47] Y. Bao, J. Fan, and L. Li, Electroweak ALP searches at a muon collider, JHEP 08 (2022) +276, [arXiv:2203.04328]. +[48] T. Han, T. Li, and X. Wang, Axion-Like Particles at High Energy Muon Colliders – A +White paper for Snowmass 2021, in 2022 Snowmass Summer Study, 3, 2022. +arXiv:2203.05484. +[49] S. C. İnan and A. V. Kisselev, Probe of axion-like particles in vector boson scattering at a +muon collider, arXiv:2207.03325. +[50] M. Chiesa, F. Maltoni, L. Mantani, B. Mele, F. Piccinini, and X. Zhao, Measuring the +quartic Higgs self-coupling at a multi-TeV muon collider, JHEP 09 (2020) 098, +[arXiv:2003.13628]. +[51] T. Han, D. Liu, I. Low, and X. Wang, Electroweak couplings of the Higgs boson at a +multi-TeV muon collider, Phys. Rev. D 103 (2021), no. 1 013002, [arXiv:2008.12204]. +[52] M. Chiesa, B. Mele, and F. Piccinini, Multi Higgs production via photon fusion at future +multi-TeV muon colliders, arXiv:2109.10109. +[53] M. Cepeda, S. Gori, V. M. Outschoorn, and J. Shelton, Exotic Higgs Decays, +arXiv:2111.12751. +[54] J. Chen, T. Li, C.-T. Lu, Y. Wu, and C.-Y. Yao, Measurement of Higgs boson self-couplings +through 2→3 vector bosons scattering in future muon colliders, Phys. Rev. D 105 (2022), +no. 5 053009, [arXiv:2112.12507]. +[55] M. Forslund and P. Meade, High precision higgs from high energy muon colliders, JHEP 08 +(2022) 185, [arXiv:2203.09425]. +– 22 – + +[56] Muon Collider Physics and Detector Working Group Collaboration, +L. Buonincontri, P. Andreetto, N. Bartosik, M. Casarsa, A. Gianelle, D. Lucchesi, and +L. Sestini, Higgs boson couplings at muon collider, PoS EPS-HEP2021 (2022) 619. +[57] L. Di Luzio, R. Gröber, and G. Panico, Probing new electroweak states via precision +measurements at the LHC and future colliders, JHEP 01 (2019) 011, [arXiv:1810.10993]. +[58] D. Buttazzo, R. Franceschini, and A. Wulzer, Two Paths Towards Precision at a Very High +Energy Lepton Collider, JHEP 05 (2021) 219, [arXiv:2012.11555]. +[59] S. Spor and M. Köksal, Investigation of anomalous triple gauge couplings in µγ collision at +multi-TeV muon colliders, arXiv:2201.00787. +[60] S. Chen, A. Glioti, R. Rattazzi, L. Ricci, and A. Wulzer, Learning from radiation at a very +high energy lepton collider, JHEP 05 (2022) 180, [arXiv:2202.10509]. +[61] M. Chen and D. Liu, Top Yukawa Coupling at the Muon Collider, arXiv:2212.11067. +[62] G.-y. Huang, S. Jana, F. S. Queiroz, and W. Rodejohann, Probing the RK(*) anomaly at a +muon collider, Phys. Rev. D 105 (2022), no. 1 015013, [arXiv:2103.01617]. +[63] P. Asadi, R. Capdevilla, C. Cesarotti, and S. Homiller, Searching for leptoquarks at future +muon colliders, JHEP 10 (2021) 182, [arXiv:2104.05720]. +[64] G. Haghighat and M. Mohammadi Najafabadi, Search for lepton-flavor-violating ALPs at a +future muon collider and utilization of polarization-induced effects, Nucl. Phys. B 980 +(2022) 115827, [arXiv:2106.00505]. +[65] P. Bandyopadhyay, A. Karan, R. Mandal, and S. Parashar, Distinguishing signatures of +scalar leptoquarks at hadron and muon colliders, Eur. Phys. J. C 82 (2022), no. 10 916, +[arXiv:2108.06506]. +[66] A. Das, T. Nomura, and T. Shimomura, Multi muon/anti-muon signals via productions of +gauge and scalar bosons in a U(1)Lµ−Lτ model at muonic colliders, arXiv:2212.11674. +[67] T. Han, Z. Liu, L.-T. Wang, and X. Wang, WIMPs at High Energy Muon Colliders, Phys. +Rev. D 103 (2021), no. 7 075004, [arXiv:2009.11287]. +[68] R. Capdevilla, F. Meloni, R. Simoniello, and J. Zurita, Hunting wino and higgsino dark +matter at the muon collider with disappearing tracks, JHEP 06 (2021) 133, +[arXiv:2102.11292]. +[69] A. D. Medina, N. I. Mileo, A. Szynkman, and S. A. Tanco, Elusive muonic WIMP, Phys. +Rev. D 106 (2022), no. 7 075018, [arXiv:2112.09103]. +[70] K. Black, T. Bose, Y. Chen, S. Dasu, H. Jia, D. Pinna, V. Sharma, +N. Venkatasubramanian, and C. Vuosalo, Prospects for Heavy WIMP Dark Matter Searches +at Muon Colliders, in 2022 Snowmass Summer Study, 5, 2022. arXiv:2205.10404. +[71] J. Liu, Z.-L. Han, Y. Jin, and H. Li, Unraveling the Scotogenic model at muon collider, +JHEP 12 (2022) 057, [arXiv:2207.07382]. +[72] H. Al Ali et al., The muon Smasher’s guide, Rept. Prog. Phys. 85 (2022), no. 8 084201, +[arXiv:2103.14043]. +[73] S. Homiller, Q. Lu, and M. Reece, Complementary signals of lepton flavor violation at a +high-energy muon collider, JHEP 07 (2022) 036, [arXiv:2203.08825]. +[74] S. Blanchet, Z. Chacko, S. S. Granor, and R. N. Mohapatra, Probing Resonant Leptogenesis +at the LHC, Phys. Rev. D 82 (2010) 076008, [arXiv:0904.2174]. +– 23 – + +[75] P. Fileviez Perez, T. Han, and T. Li, Testability of Type I Seesaw at the CERN LHC: +Revealing the Existence of the B-L Symmetry, Phys. Rev. D 80 (2009) 073015, +[arXiv:0907.4186]. +[76] G. Anamiati, M. Hirsch, and E. Nardi, Quasi-Dirac neutrinos at the LHC, JHEP 10 (2016) +010, [arXiv:1607.05641]. +[77] F. F. Deppisch, P. S. Bhupal Dev, and A. Pilaftsis, Neutrinos and Collider Physics, New J. +Phys. 17 (2015), no. 7 075019, [arXiv:1502.06541]. +[78] C.-W. Chiang, G. Cottin, A. Das, and S. Mandal, Displaced heavy neutrinos from Z’ decays +at the LHC, JHEP 12 (2019) 070, [arXiv:1908.09838]. +[79] W. Liu, K.-P. Xie, and Z. Yi, Testing leptogenesis at the LHC and future muon colliders: A +Z’ scenario, Phys. Rev. D 105 (2022), no. 9 095034, [arXiv:2109.15087]. +[80] G. Chauhan and P. S. B. Dev, Resonant Leptogenesis, Collider Signals and Neutrinoless +Double Beta Decay from Flavor and CP Symmetries, arXiv:2112.09710. +[81] R. N. Mohapatra and J. W. F. Valle, Neutrino Mass and Baryon Number Nonconservation +in Superstring Models, Phys. Rev. D 34 (1986) 1642. +[82] R. N. Mohapatra, Mechanism for Understanding Small Neutrino Mass in Superstring +Theories, Phys. Rev. Lett. 56 (1986) 561–563. +[83] S. Nandi and U. Sarkar, A Solution to the Neutrino Mass Problem in Superstring E6 +Theory, Phys. Rev. Lett. 56 (1986) 564. +[84] M. C. Gonzalez-Garcia and J. W. F. Valle, Fast Decaying Neutrinos and Observable Flavor +Violation in a New Class of Majoron Models, Phys. Lett. B 216 (1989) 360–366. +[85] J. Bernabeu, A. Santamaria, J. Vidal, A. Mendez, and J. W. F. Valle, Lepton Flavor +Nonconservation at High-Energies in a Superstring Inspired Standard Model, Phys. Lett. B +187 (1987) 303–308. +[86] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, and T. Reiter, UFO - The +Universal FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201–1214, +[arXiv:1108.2040]. +[87] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, FeynRules 2.0 - A +complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) +2250–2300, [arXiv:1310.1921]. +[88] D. Alva, T. Han, and R. Ruiz, Heavy Majorana neutrinos from Wγ fusion at hadron +colliders, JHEP 02 (2015) 072, [arXiv:1411.7305]. +[89] C. Degrande, O. Mattelaer, R. Ruiz, and J. Turner, Fully-Automated Precision Predictions +for Heavy Neutrino Production Mechanisms at Hadron Colliders, Phys. Rev. D 94 (2016), +no. 5 053002, [arXiv:1602.06957]. +[90] A. Atre, T. Han, S. Pascoli, and B. Zhang, The Search for Heavy Majorana Neutrinos, +JHEP 05 (2009) 030, [arXiv:0901.3589]. +[91] Z. Maki, M. Nakagawa, and S. Sakata, Remarks on the unified model of elementary +particles, Prog. Theor. Phys. 28 (1962) 870–880. +[92] B. Pontecorvo, Mesonium and anti-mesonium, Sov. Phys. JETP 6 (1957) 429. +[93] B. Pontecorvo, Inverse beta processes and nonconservation of lepton charge, Zh. Eksp. Teor. +Fiz. 34 (1957) 247. +– 24 – + +[94] T. Han, Y. Ma, and K. Xie, Quark and gluon contents of a lepton at high energies, JHEP +02 (2022) 154, [arXiv:2103.09844]. +[95] W. Kilian, T. Ohl, and J. Reuter, WHIZARD: Simulating Multi-Particle Processes at LHC +and ILC, Eur. Phys. J. C 71 (2011) 1742, [arXiv:0708.4233]. +[96] M. Moretti, T. Ohl, and J. Reuter, O’Mega: An Optimizing matrix element generator, +hep-ph/0102195. +[97] C. Bierlich et al., A comprehensive guide to the physics and usage of PYTHIA 8.3, +arXiv:2203.11601. +[98] F. F. Deppisch, S. Kulkarni, and W. Liu, Heavy neutrino production via Z +′ at the lifetime +frontier, Phys. Rev. D 100 (Aug, 2019) 035005. +[99] V. M. Budnev, I. F. Ginzburg, G. V. Meledin, and V. G. Serbo, The Two photon particle +production mechanism. Physical problems. Applications. Equivalent photon approximation, +Phys. Rept. 15 (1975) 181–281. +[100] C. F. von Weizsacker, Radiation emitted in collisions of very fast electrons, Z. Phys. 88 +(1934) 612–625. +[101] E. J. Williams, Nature of the high-energy particles of penetrating radiation and status of +ionization and radiation formulae, Phys. Rev. 45 (1934) 729–730. +[102] DELPHES 3 Collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, +V. Lemaître, A. Mertens, and M. Selvaggi, DELPHES 3, A modular framework for fast +simulation of a generic collider experiment, JHEP 02 (2014) 057, [arXiv:1307.6346]. +[103] M. Selvaggi, A Delphes parameterisation of the FCC-hh detector, tech. rep., CERN, +Geneva, 2020. +[104] P. G. Roloff, U. Schnoor, M. A. Weber, and E. Leogrande, A Delphes card for the CLIC +detector, tech. rep., CERN, Geneva, 2018. +[105] M. Boronat, J. Fuster, I. Garcia, E. Ros, and M. Vos, A robust jet reconstruction algorithm +for high-energy lepton colliders, Phys. Lett. B 750 (2015) 95–99, [arXiv:1404.4294]. +[106] T. Chen and C. Guestrin, XGBoost: A Scalable Tree Boosting System, arXiv:1603.02754. +[107] S. T. Petcov, Possible Signature for Production of Majorana Particles in e+e− and p¯p +Collisions, Phys. Lett. B 139 (1984) 421–426. +[108] A. Baha Balantekin and B. Kayser, On the Properties of Neutrinos, Ann. Rev. Nucl. Part. +Sci. 68 (2018) 313–338, [arXiv:1805.00922]. +[109] A. B. Balantekin, A. de Gouvêa, and B. Kayser, Addressing the Majorana vs. Dirac +Question with Neutrino Decays, Phys. Lett. B 789 (2019) 488–495, [arXiv:1808.10518]. +[110] A. de Gouvêa, P. J. Fox, B. J. Kayser, and K. J. Kelly, Three-body decays of heavy Dirac +and Majorana fermions, Phys. Rev. D 104 (2021), no. 1 015038, [arXiv:2104.05719]. +[111] A. de Gouvêa, P. J. Fox, B. J. Kayser, and K. J. Kelly, Characterizing heavy neutral +fermions via their decays, Phys. Rev. D 105 (2022), no. 1 015019, [arXiv:2109.10358]. +[112] C. Arbelaéz, C. Dib, I. Schmidt, and J. C. Vasquez, Probing the Dirac or Majorana nature +of the Heavy Neutrinos in pure leptonic decays at the LHC, Phys. Rev. D 97 (2018), no. 5 +055011, [arXiv:1712.08704]. +– 25 – + +[113] P. Hernández, J. Jones-Pérez, and O. Suarez-Navarro, Majorana vs Pseudo-Dirac Neutrinos +at the ILC, Eur. Phys. J. C 79 (2019), no. 3 220, [arXiv:1810.07210]. +[114] M. Drewes, Y. Georis, and J. Klarić, Mapping the Viable Parameter Space for Testable +Leptogenesis, Phys. Rev. Lett. 128 (2022), no. 5 051801, [arXiv:2106.16226]. +[115] M. Chrzaszcz, M. Drewes, T. E. Gonzalo, J. Harz, S. Krishnamurthy, and C. Weniger, A +frequentist analysis of three right-handed neutrinos with GAMBIT, Eur. Phys. J. C 80 +(2020), no. 6 569, [arXiv:1908.02302]. +[116] E. Izaguirre and B. Shuve, Multilepton and Lepton Jet Probes of Sub-Weak-Scale +Right-Handed Neutrinos, Phys. Rev. D 91 (2015), no. 9 093010, [arXiv:1504.02470]. +[117] S. Antusch, E. Cazzato, and O. Fischer, Sterile neutrino searches at future e−e+, pp, and +e−p colliders, Int. J. Mod. Phys. A 32 (2017), no. 14 1750078, [arXiv:1612.02728]. +[118] M. Drewes, Distinguishing Dirac and Majorana Heavy Neutrinos at Lepton Colliders, in +41st International Conference on High Energy Physics, 10, 2022. arXiv:2210.17110. +[119] K. Mękała, J. Reuter, and A. F. Żarnecki, Optimal search reach for heavy neutral leptons at +a muon collider, arXiv:2301.02602. +[120] T. Bose et al., Report of the Topical Group on Physics Beyond the Standard Model at +Energy Frontier for Snowmass 2021, arXiv:2209.13128. +– 26 – + diff --git a/V9E4T4oBgHgl3EQfng2k/content/tmp_files/load_file.txt b/V9E4T4oBgHgl3EQfng2k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbf3a37db8be2f761fb3e53b1a48707e6226aa0f --- /dev/null +++ b/V9E4T4oBgHgl3EQfng2k/content/tmp_files/load_file.txt @@ -0,0 +1,1357 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf,len=1356 +page_content='Searching for Heavy Neutral Leptons at A Future Muon Collider Tsz Hong Kwok,a Lingfeng Li,b Tao Liua,c and Ariel Rockc aDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', PRC bDepartment of Physics, Brown University, Providence, RI, 02912, USA cInstitute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', PRC E-mail: thkwokae@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='hk, lingfeng_li@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='edu, taoliu@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='hk, iasarock@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='hk Abstract: As the planning stages for a high energy muon collider enter a more concrete era, an important question arises as to what new physics could be uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A TeV-scale muon collider is also a vector boson fusion (VBF) factory with a very clean background, and as such it is a promising environment to look for new physics that couples to the electroweak (EW) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In this paper, we explore the ability of a future TeV-scale muon collider to search for Majorana and Dirac Heavy Neutral Leptons (HNLs) produced via EW bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Employing a model-independent, conservative approach, we present an estimation of the production and decay rate of HNLs over a mass range between 200 GeV and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 TeV in two benchmark collider proposals with √s = 3, 10 TeV, as well as an estimation of the dominant Standard Model (SM) background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We find that exclusion limits for the mixing between the HNLs and SM neutrinos can be as low as O(10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Additionally, we demonstrate that a TeV-scale muon collider allows for the ability to discriminate between Majorana and Dirac type HNLs for a large range of mixing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Keywords: Muon Collider, BSM Physics, Heavy Neutral Leptons arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05177v1 [hep-ph] 12 Jan 2023 Contents 1 Introduction 1 2 Model 3 3 Simulation Framework 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 Signal Event Generation 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 Background Event Generation 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3 Detector Simulation 8 4 Analysis 8 5 Results 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 Exclusion Limits on |Vℓ|2 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 Distinguishing Majorana versus Dirac Heavy Neutral Leptons 16 6 Conclusion and Future Directions 17 Contents 1 Introduction It is now well-known that active neutrinos oscillate and that at least two generations have very small but nonzero mass [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As the Standard Model (SM) predicts that neutrinos have exactly zero mass, this is evidence of Beyond the Standard Model (BSM) physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The question then arises as to what could be the origin of neutrino masses and mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' It might be the case that new physics at an energy scale much higher than that of the SM may give rise to neutrino masses and mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' If that new physics is renormalizable and allowed to violate lepton number, the lowest order effective operator at the scale of the SM is the d = 5 dimension Weinberg operator [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' There are many possible ultraviolet (UV) theories that may result in this effective operator, but one such class of theories is called the Type I Seesaw mechanism [9–12], in which the smallness of masses of the SM neutrinos is enforced by the presence of right-handed Heavy Neutral Leptons (HNLs) at a higher scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' To date, there have been numerous limits set on the coupling |Vℓ|2 between HNLs and the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Prompt trilepton searches of HNL masses less than 60 GeV have been studied in hadron colliders, with sensitivity down to O(10−5) [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Displaced searches in hadron colliders have been shown to have sensitivity reaching almost O(10−7) for HNL masses below 20 GeV [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Limits from LHCb also indicate sensitivity down to O(10−4) for masses below 50 GeV [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Other future experiments promise even stricter limits, such as – 1 – SHiP, with limits down to O(10−9) for masses below 5 GeV [18], or other potential hadron colliders [19, 20], with limits at O(10−7) for similar mass ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In addition, future lepton colliders show promise for higher mass HNLs, possibly down to O(10−5 − 10−6) for masses between 200-2000 GeV [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A future high-energy muon collider could serve as both an intensity and energy frontier, providing various opportunities for both direct and indirect searches of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As compared to an electron-positron collider, a muon collider would have higher energy and luminosity, and reduced background [22–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As synchrotron radiation scales as m−4, a muon collider would have an energy loss rate reduced by a factor of (me/mµ)4 ≈ (207)−4 as compared to an electron-positron collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This, in combination with the property that muon colliders would have negligible beamstrahlung [28, 29], makes TeV-scale beam energies more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Currently, estimates put the expected luminosity of a TeV-scale muon collider at order 1 ab−1 [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As compared to a proton collider, colliding muons would allow for a much more efficient probe of high energy processes, as a large portion of the beam energy would be carried by the muon itself, whereas a hadronic collider only admits high energy collisions at the tail of the proton’s parton distribution function (PDF) — which is highly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In fact, the efficiency gap is so pronounced that for some 2 → 2 processes, a muon collider with a given √sµ has the same reach as a proton collider of √sp ∼ 5−20×√sµ [30, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Furthermore, a muon collider is a perfect laboratory for studying new electroweak (EW) physics, such as in searches for HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For a TeV-scale muon collider, roughly 5% of the beam’s energy is carried by EW bosons [34], whereas a proton beam carries less than 1% [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' It is important to note, however, the various technical challenges facing the develop- ment of a muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Unlike the beams of electron-positron or proton-proton colliders, muons are unstable, with a lifetime of around 2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As such, it remains a formidable task to produce and store low emmittance muon beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In recent years there have been rapid developments in overcoming these difficulties [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In the United States, the US Muon Accelerator Program (MAP) is investigating a potential muon source wherein a proton beam would collide with a high-Z fixed target to produce secondary muons as pion decay products [39–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' However, the kinematics of this process admit final state muons with high emittance, and therefore significant cooling is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A possible cooling technology has been demonstrated by the Muon Ionization Cooling Experiment (MICE) at the Rutherford Appleton Laboratory in the United Kingdom [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' An interesting approach to combine these two steps has been put forward in the Low Emmittance Muon Accelerator (LEMMA) proposal, which would produce muon pairs at threshold via a 45 GeV positron beam in- teracting with a fixed-target electron [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' These muons would have a long lifetime and small emittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' However, LEMMA is unable to achieve a high enough luminosity to be feasible [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nevertheless, it is clear that the new physics reach of a future muon collider is sub- stantial, and the advantages over conventional lepton colliders have resulted in a multitude of phenomenological studies, such as searches for axion-like particles [47–49], explorations of Higgs physics [50–56], precision EW studies [57–61], flavor studies [62–66], and searches for dark matter [67–71], among others [30, 33, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nevertheless, unlike for electron- – 2 – positron colliders [21], the phenomenology of HNLs in a muon collider has not been studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In this paper we present an analysis of the sensitivity and reach of a future muon collider in looking for HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In section 2, we introduce the the model we use that describes the interactions between the HNLs and the SM, and we describe the signal production channel of interest in the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In section 3, we describe the framework we use to simulate the production and detection of signal and background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In section 4, we detail the reconstruction and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Section 5 presents the predicted exclusion bounds on the coupling strengths of the HNLs as well as an estimation of the sensitivity in discerning between Majorana and Dirac HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We conclude and consider future directions in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 2 Model From a collider physics perspective, an unmodified Type I Seesaw mechanism is excluded in the region of HNL mass and mixing parameter-space under consideration in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' More specifically, for the TeV scale mass range of HNLs we focus on, the mixing VNℓ must be bounded below 10−6 in order to be compatible with current neutrino mass upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Even for a future muon collider, such a mixing would be far too small to be probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Thus, many collider studies focus on scenarios in which the HNLs couple to the SM via a new mediator with sizable couplings to the SM [20, 74–80];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' in this case, the HNL production would rate become independent of the mixing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This approach, however, is highly model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Another, more general, possible theory is a modification of the Type I Seesaw, the Inverse Seesaw mechanism [81–83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Via the introduction of small lepton number violating scale, the Inverse Seesaw is able to accommodate the SM neutrino data for any values of mixing and HNL mass [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As the purpose of this work is to investigate the collider reach in observing electroweak production of HNLs in a model-independent fashion, we nevertheless remain agnostic as to the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Therefore, we employ an effective, phenomenological model — the Universal FeynRules Output (UFO) [86, 87] models HeavyN, for Majorana [88–90] and Dirac [19] HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In these implementations, the HNLs, Ni for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' , 3, are sterile in the SM but mix with the active neutrinos, namely, νℓL = 3 � m=1 UℓmνmL + 3 � m′=1 Vℓm′Nc m′L , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1) where m, and m′ index mass eigenstates, and ℓ = e, µ, τ index gauge eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The matrix Uℓm is the usual Maki–Nakagawa–Sakata-Pontecorvo (MNSP) mixing matrix for SM neutrinos [91–93], and Vℓm′ is a matrix parameterizing the mixing between the HNLs and the SM neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' After electroweak symmetry breaking, the effective Lagrangian in the mass eigenstate basis that describes the interactions between the HNLs and the SM electroweak bosons is given by – 3 – W Ni ℓ Z Ni νℓ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' New interaction vertices between the HNLs and SM electroweak bosons in the simplified effective Lagrangian of equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' −Lint,EW = g √ 2W µ+ τ � ℓ=e � 3 � m=1 U ∗ ℓm¯νmγµPLℓ + 3 � m=1 V ∗ ℓm ¯Nc mγµPLℓ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2) + g 2 cos θW Zµ τ � ℓ=e � 3 � m=1 U ∗ ℓm¯νmγµPLνℓ + 3 � m=1 V ∗ ℓm ¯Nc mγµPLνℓ � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This interaction Lagrangian introduces two new classes of vertices, given in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' There are several production mechanisms for HNLs in a high-energy muon collider, the dominant of which depends on the mass of the HNL and its mixing with different generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In the region of parameter space under consideration in this work, namely √s, mNi ≫ mW , the dominant production process is the production of an HNL and SM neutrino pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This Niνℓ pair occurs via two production channels: either a t-channel ex- change of a W boson, or annihilation via a Z boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' These channels (along with a decay N → qqℓ) are shown in figure 2 on the left- and right-hand sides, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The W exchange process benefits from a large W boson electroweak (EW) parton distribution function (PDF) [33, 34, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Conversely, the masses of the HNLs under consideration are much larger than mZ, and as such annihilation processes via Z decay are far off shell and highly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' At mN ≫ mW , we also have that the Goldstone Equivalence Theorem applies, and thus it then follows [19, 90] that BR(N → W +ℓ−) ≈ 2BR(N → Zνℓ) ≈ 2BR(N → hνℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3) Note that this relation holds for both Majorana and Dirac HNLs, despite the absolute widths of the Majorana HNLs being twice as large as for the Dirac case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As the HNLs preferentially decay via charged-current interactions, and that the W boson decays largely hadronically [7], we pick our decay channel of investigation to be N → qqℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' An additional benefit to this channel is that all decay products are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 3 Simulation Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 Signal Event Generation In this work we use the Monte-Carlo event generator Whizard 3 [95, 96] to generate events at the parton level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The advantage of using Whizard 3 for this study is that it is – 4 – W N W µ µ ν q q ℓ Z N W µ µ ν q q ℓ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Feynman diagrams for the process µ+µ− → N(qqℓ±)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For the collider energies above mZ, such as the scenarios under consideration in this paper, the dominant production channel is given by the left-hand diagram of t-channel W exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' possible to include the structure function for initial state radiation (ISR) for muon beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Additionally, Whizard 3 allows for the use of the equivalent photon approximation (EPA) for the inclusion of photon-induced background events due to the collinear splitting of µ → µγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Furthermore, the implementations of ISR and EPA in Whizard 3 allow for the insertion of pT recoil of the hard scattering processes into the event record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The following perturbative parton shower and hadronization steps are done using Pythia 8 [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For the generation of signal events, we use the HeavyN UFO files SM_HeavyN_NLO and SM_HeavyN_Dirac_NLO, for Majorana and Dirac HNLs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We focus on the case in which only one HNL mixes with the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Furthermore, we assume that |V1e| = |V1µ| and |V1τ| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As we only consider one HNL, we use the notation |V1e| = |V1µ| ≡ |Vℓ| with no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We simulate the signal process by first generating the production of HNLs, µ+µ− → Nν, the Feynman diagrams of which are shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We then decay the HNL via N → qqℓ±, where q = u, d, c, s and ℓ± = e±, µ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As we decay N using the narrow-width approximation (NWA), we choose |Vℓ| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='002, as this en- forces ΓN ≪ mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 We are free to make this choice as the quantity σ (µ+µ− → N(qqℓ±)ν) / |Vℓ|2 is independent of |Vℓ| for a given mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Furthermore, for the values of √s and mN considered in this work, this choice of |Vℓ|2 is still large enough to ensure prompt decays of the HNLs, as even in the most boosted, lowest width scenario under consideration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='e √s = 10 TeV and mN = 200 GeV), the decay length in the lab frame is of order 10−4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This is sig- nificantly smaller than the anticipated spatial resolution of detectors under consideration in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note, however, that in non-trivial model extensions where the HNL production is governed by another operator or mediator, the HNL production rate may be independent of Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In such a case it is then possible to have Vℓ ≲ 10−6 while still producing an appre- ciable collider signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In this case, it might be such that the decay length of HNL is macroscopic and the HNL becomes a long-lived particle [77, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' However, this approach deviates from the focus of this work, which is conservative and model-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In this work, we consider two of the muon collider benchmarks given in [31]: a scenario in which √s = 3 TeV and L = 1 ab−1, and a scenario in which √s = 10 TeV and L = 10 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For analysis in a potential 3 TeV (10 TeV) muon collider, we consider benchmark 1This choice of |Vℓ| is actually much smaller than required for the validity of the NWA, but, as noted in [19], our choice is advantageous in that larger mixings that still satisfy the NWA might allow for increased virtuality of the HNL, leading to large variations in the kinematic distributions of the HNL’s decay products across events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 5 – 200 1000 10000 mN (GeV) 1 10 50 σ/|Vℓ|2 (pb) σ(pb) for µ+µ− → N(qqℓ±)ν, √s = 10 TeV Majorana Dirac 200 1000 3000 mN (GeV) 1 10 50 σ/|Vℓ|2 (pb) σ(pb) for µ+µ− → N(qqℓ±)ν, √s = 3 TeV Majorana Dirac Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The total cross section divided by |Vℓ|2 = |Ve|2 = |Vµ|2 for the process µ+µ− → N(qqℓ±)ν as a function of mN, shown at two benchmark future muon colliders with √s = 3, 10 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' scenarios with masses mN between 200 GeV and 2900 GeV (200 GeV and 9500 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' When generating signal events, we set all final state lepton and quark masses to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' However, since we have ISR and ISR recoiled enabled, we set mµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='105 GeV for the beams, to be used as the mass for the ISR structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' At each mass benchmark, ΓN is recalculated and used to compute BR(N → qqℓ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The production cross section for each benchmark is then reweighted by BR(N → qqℓ±) to give the total cross section for the process σ(µ+µ− → N(qqℓ±)ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The resulting total process cross section divided by |Vℓ|2 as a function of mN is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that the cross section for mN ≈ mW is moderately enhanced by the branching of N → W ±ℓ∓ being favored over other decay channels at those masses, as noted in [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 Background Event Generation As our final state topology of interest is two jets and a lepton, we consider SM background processes with two quarks and between one and three charged leptons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Specifically, we consider the following channels classified into two types according to the simulation methods used: ME (a) µ+µ− → qqℓν, (b) µ+µ− → qqℓℓ, (c) µ+µ− → qqℓℓνν, (d) µ+µ− → qqℓℓℓν, EPA (e) γγ → qqℓν, (f) γµ± → qqℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The first four processes listed are computed using the full matrix element (ME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' They include generator-level cuts such that the Monte-Carlo integration is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In order to prevent underestimation of the SM background, the generator-level cuts are taken to be softer than those used in preselection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 6 – Process Generator Level Cuts Method µ+µ− → qqℓν Mqq,ℓℓ > 10 GeV, pT,ℓ > 4 GeV, |ηℓ| < 8, qℓ > 4 GeV ME + ISR µ+µ− → qqℓℓ µ+µ− → qqℓℓνν Mqq,ℓℓ > 40 GeV, pT,ℓ > 4 GeV, |ηℓ| < 8, qℓ > 4 GeV ME + ISR µ+µ− → qqℓℓℓν γγ → qqℓν Mqq > 10 GeV, qγ < 4 GeV EPA γµ± → qqℓ Mqq > 10 GeV, qγ < 4 GeV, 3◦ < θℓ < 177◦ EPA + ISR Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Summary of the cuts and simulation methods used to generate the SM background in Whizard 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' qℓ refers to the momentum-transfer between incoming and outgoing charged leptons, and qγ is the upper momentum-transfer for the EPA structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For the process γµ± → qqℓ, EPA is applied to one muon beam to produce the partonic photon, and ISR is applied to the other beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For all processes, pT recoil on the hard scattering states due to emitted ISR (EPA) photons is applied to the generated events by setting “?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='Isr(Epa)_Handler=True” and “?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='Isr(Epa)_Handler_Mode=Recoil.” Furthermore, in order to preserve gauge-invariance, the simulations of the processes µ+µ− → qqℓℓℓν and µ+µ− → qqℓℓ include the processes γγ → qqℓν and γµ± → qqℓ as subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' However, the cuts chosen on the full processes exclude the regions of low momentum transfer, qℓ, where the contribution of partonic photon scattering dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Therefore, we include the last two processes, γγ → qqℓν and γµ± → qqℓ, where γ refers to partonic photons split collinearly from the muon beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' These are simulated using the Equivalent Photon Approximation (EPA) [99], based on the Weizsäcker-Williams approxi- mation [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We choose the lower cutoff qℓ for the full ME processes and the upper cutoff qγ for the EPA structure function to be complementary to avoid double counting of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We also include the effect of initial state radiation (ISR) on the beam(s) that do not split into EPA photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The specific cuts used for the SM background processes are detailed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The cross sections and estimated event yields for each background channel are given in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' It is also worth commenting on the major contributors to each of the background channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As expected, given that √s is TeV scale, every background channel is dominated by contributions from vector boson fusion or scattering (VBF or VBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' More specifically, we find that the dominant processes in the channel µ+µ− → qqℓν are from WZ and Wγ to W VBF, where the W subsequently decays to a quark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The process µ+µ− → qqℓℓ is dominated by γγ to qq VBF via the exchange of a t-channel light quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For µ+µ− → qqℓℓνν, the major contribution is given by Wγ/WZ to Wγ/WZ VBS, where the neutral boson decays to a qq pair, and the W decays leptonically to ℓν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The channel µ+µ− → qqℓℓℓν is dominated by γγ to WW VBS, where one W decays hadronically and the other leptonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Additionally, the EPA process γγ → qqℓν is also dominated by the same γγ to WW VBS, which is consistent with it being considered a subdiagram of the µ+µ− → qqℓℓℓν channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Likewise, the EPA process γµ± → qqℓ is dominated by the same γγ to qq VBS process that µ+µ− → qqℓℓ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 7 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3 Detector Simulation After generating parton-level events with Whizard 3, and using Pythia 8 to shower and hadronize, we use Delphes 3 [102] to simulate the detector response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We use the default muon collider detector card that is included in the Delphes 3 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This card is a hybrid of the FCC-hh [103] and CLIC [104] cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For this detector simulation, it requires isolated leptons to have |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' It also includes a hypothetical forward muon spectrometer sensitive to muons with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 < |η| < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 at 90% efficiency for muons with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 GeV < pT < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 GeV and at 95% efficiency for muons with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 < |η| < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 and pT > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This ability to detect forward muons is possibly advantageous in distinguishing signal from background in our analysis, as events that pass preselection with muons in the forward region are almost always due to the SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' However, as the future of such a detector is uncertain, we therefore do not include its contribution in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 4 Analysis Following the detector simulation, we preselect events to keep those with signal-like topolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Namely, in each event we require: At least one isolated ℓ = e, µ candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Our definition of an isolated lepton is the same as the definition used by the built-in Delphes muon collider card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Firstly, it requires that the ℓ candidate have pT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Secondly, inside a cone of ∆R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 around the ℓ candidate, the ratio of the scalar sum of pT of all particles in that cone to the pT of the ℓ candidate must be no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In case of having more than one ℓ candidate, the one with the highest pT will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A W boson jet system (WJ) candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We reconstruct the jet system in two possible ways, using the Valencia algorithm [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For most of the mN parameter space, W bosons from the decays of the HNLs are highly boosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Therefore, we choose to reconstruct the WJ as one jet with a moderate radius parameter R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 and both remaining parameters β and γ set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0, denoted as Jfat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The second option is to reconstruct two narrower jets in a similar way, but with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2, denoted as j1 and j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' If it is possible to reconstruct both classes of jet(s) in the event, we compare the invariant masses of WJ = Jfat with that of WJ = j1 + j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The jet reconstruction which gives an invariant mass closest to that of the W boson is then accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' If only one class of jet(s) is able to be reconstructed, that class will be used as WJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kinematic cuts on ℓ and WJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We require ℓ and WJ to have pT > 100 GeV, and that the reconstructed jet system is on-shell, namely |mWJ − mW | < 5ΓW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In figure 4, we show the reconstruction efficiency for Majorana and Dirac HNL signal channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' It is expected that the reconstruction efficiency is lessened for HNL masses well below √s, as such HNLs are mostly produced by the muon beam with a virtual W boson – 8 – 200 1500 10000 mN (GeV) 1 10 70 Reconstruction Efficiency (%) Efficiency for µ+µ− → N(qqℓ±)ν, √s = 10 TeV Majorana Dirac 200 500 1000 3000 mN (GeV) 1 10 70 Reconstruction Efficiency (%) Efficiency for µ+µ− → N(qqℓ±)ν, √s = 3 TeV Majorana Dirac Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The preselection and reconstruction efficiency of the process µ+µ− → N(qqℓ±)ν as a function of mN, shown at two benchmark future muon colliders with √s = 3, 10 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As expected, the ability to detect and reconstruct events for HNLs with masses well below the collider energy is poor, as such HNLs are too boosted to have detectable decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' with small pz, as shown in the left panel of figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The overall pz of the produced HNL becomes comparable to the beam energy, with the expected pT of its decay products limited by its small mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Consequently the ℓ and q¯q′ final states are less likely to reach the central region with low |η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The reconstruction efficiencies and event yields for the background channels are given in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that the reconstruction efficiency of the µ+µ− → qqℓℓℓν background channel is higher than that of the µ+µ− → qqℓν channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This is due to the fact that the major contributing process in the µ+µ− → qqℓℓℓν background is µ+µ− → qqℓµµν, where the two outgoing muons are from the beam muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The outgoing muons are therefore in the forward region, and will either be missed by the detector or detected with very small pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Such an event will still pass our reconstruction by identifying the remaining lepton (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' the one with highest pT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Similarly, majority of the outgoing leptons in µ+µ− → qqℓν channel are also beam muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Therefore, this event will not pass preselection if such low pT forward muon is missed by the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Alternatively, one could also introduce a different reconstruction method which could strictly require that only one lepton is detected in the event in order to veto some of the µ+µ− → qqℓℓℓν background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' After preselection, we then reconstruct the HNL by combining WJ and the isolated lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A plot of the distribution of the invariant mass mN is show in figure 5 for a benchmark scenario of mN = 1 TeV and √s = 3 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A peak in the signal events is visible at the expected value of mN = 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that there is also a tail in the reconstructed mass distribution below the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This is due to missing W-jet components that have either been missed by the detector reconstruction or not clustered by the jet algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In both cases, the invariant mass reconstructed drops below the physical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In figure 6, we show the mN distribution for the individual channels that comprise the total SM background in both a √s = 3 TeV and √s = 10 TeV collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' On the left-hand side of figure 6, for √s = 3 TeV, we see that the dominant background channel is µ+µ− → qqℓν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For the right-hand side, with √s = 10 TeV, the dominant background for mN < 6 TeV is – 9 – 0 500 1000 1500 2000 2500 3000 Reconstructed mN (GeV) 10−2 10−1 100 101 102 103 104 Events/50 GeV √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV Majorana Dirac Background Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A histogram of reconstructed mN after preselection, for a benchmark scenario of mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Blue corresponds to Majorana signal events, red for Dirac, and yellow for the combined SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Signal events are normalized to |Vℓ|2 = 5×10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A sharp peak is visible at mN = 1 TeV with a tail to the left due missing W-jet components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' due to the µ+µ− → qqℓℓℓν channel, with the µ+µ− → qqℓν becoming dominant for larger mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' To further separate signal-like and background-like events after preselection, we employ a Boosted Decision Tree (BDT) analysis using the Python implementation of XGBoost [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We train BDT to distinguish between three classes of event — signal events due to the decay of a Majorana HNL, signal events due to the decay of a Dirac HNL, and signal events due to the SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The BDT training utilizes the following input features: Lepton Information: – The lepton’s transverse momentum, pT,ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' pseudorapidity, ηℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' and its energy, Eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – The charge of the lepton and its flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' W Jet System Information: – The W jet system’s transverse momentum, pT,WJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' pseudorapidity, ηWJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' and mass, mWJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – The energies of the two narrow sub-jets Ej1 and Ej2, in the case that WJ is able to be reconstructed as two jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reconstructed HNL Information: – The HNL’s transverse and z-components of momenta, pT,N and pz,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Geometric Information: – 10 – Collider COM Energy √s = 3 TeV √s = 10 TeV Integrated Luminosity L = 1 ab−1 L = 10 ab−1 Process σ (pb) Nevents Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' (%) σ (pb) Nevents Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' (%) µ+µ− → qqℓν 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='025 263400 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='373 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='534 932800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='9784 µ+µ− → qqℓℓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='842 12160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4278 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='784 32090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0846 µ+µ− → qqℓℓνν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02255 3201 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='07968 85100 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='68 µ+µ− → qqℓℓℓν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3133 90090 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='207 14950000 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='63 γγ → qqℓν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1589 5068 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4274 113600 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='658 γµ± → qqℓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='811 11390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5823 21360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3668 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Cross sections, expected event yields and reconstruction efficiencies after the preselection for the SM background processes detailed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Although the expected signal has only one final state charged lepton, we allow events with more than one charged lepton past preselection, as the Boosted Decision Tree analysis afterwards does an effective job of removing such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 0 2000 4000 6000 8000 10000 12000 Reconstructed mN (GeV) 101 102 103 104 105 106 Events/200 GeV √s =10 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' L =10 ab−1 µ+µ− → qqℓν µ+µ− → qqℓℓ µ+µ− → qqℓℓνν µ+µ− → qqℓℓℓν γµ± → qqℓ γγ → qqℓν 0 1000 2000 3000 Reconstructed mN (GeV) 10−1 100 101 102 103 104 Events/50 GeV √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' L =1 ab−1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The mN distribution for the individual channels that comprise the total SM background in both a √s = 3 TeV (left) and √s = 10 TeV (right) muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For √s = 3 TeV, we see that the dominant background channel is µ+µ− → qqℓν, whereas for √s = 10 TeV the dominant background is due to the µ+µ− → qqℓℓℓν channel for mN < 6 TeV, with the µ+µ− → qqℓν channel becoming dominant for larger mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – The angular distance and azimuthal angle difference between the lepton and W jet system, ∆R(ℓ, WJ) and |φℓ − φWJ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We show the distributions for the BDT features in figures 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Generally, for most variables, the kinematic distributions for Majorana and Dirac HNLs are similar, but the distributions of the energy and pseudorapidity of the lepton ηℓ (figure 7 (b)) and Eℓ (figure 7 (c)), and of the pseudorapidity of the jet system ηWJ (figure 7 (e)) differ for the two types of HNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The flatness of the distribution in Eℓ for Majorana HNLs is due to the lack of forward-backward asymmetry in their decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' More broadly, This effect is well known [107– 111] and has been noted in the context of hadron [112] and lepton [21, 113] colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Additionally, since the pT distributions for Majorana and Dirac HNLs are functionally – 11 – identical, it then follows that pseudorapidity distributions vary between the two types of HNLs in an analogous fashion to the Eℓ distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 0 250 500 750 1000 1250 1500 pT,ℓ (GeV) 10−3 10−2 10−1 100 101 102 103 104 Events/25 GeV √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV Majorana Dirac Background (a) −3 −2 −1 0 1 2 3 ηℓ 10−2 10−1 100 101 102 103 104 Events/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 (b) 0 250 500 750 1000 1250 1500 Eℓ (GeV) 10−3 10−2 10−1 100 101 102 103 104 Events/25 GeV (c) 0 250 500 750 1000 1250 1500 pT,WJ (GeV) 10−3 10−2 10−1 100 101 102 103 104 Events/25 GeV (d) −3 −2 −1 0 1 2 3 ηWJ 10−3 10−2 10−1 100 101 102 103 104 Events/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 (e) 70 75 80 85 90 mWJ (GeV) 10−2 10−1 100 101 102 103 104 Events/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='35 GeV (f) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Distributions of the features used in the BDT analysis, for a benchmark scenario of mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Blue corresponds to Majorana signal events, red for Dirac, and yellow for the combined SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For all signal events, cross sections are normalized to their 95% exclusion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As we run the BDT analysis as a three-class categorization task, we use the quantity – 12 – 0 250 500 750 1000 1250 1500 pT,N (GeV) 10−3 10−2 10−1 100 101 102 103 104 Events/25 GeV √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV Majorana Dirac Background (a) −1500 −1000 −500 0 500 1000 1500 pz,N (GeV) 10−3 10−2 10−1 100 101 102 103 104 Events/ 50 GeV (b) 0 1 2 3 4 5 6 ∆R(ℓ, WJ) 10−3 10−2 10−1 100 101 102 103 104 Events/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 |φℓ − φWJ| 10−2 100 102 104 Events/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05 (d) 0 250 500 750 1000 1250 1500 EJ1 (GeV) 10−3 10−2 10−1 100 101 102 103 104 Events/25 GeV (e) 0 250 500 750 1000 1250 1500 EJ2 (GeV) 10−2 100 102 104 Events/25 GeV (f) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Continuation of figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Distributions of the features used in the BDT analysis, for a benchmark scenario of mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Blue corresponds to Majorana signal events, red for Dirac, and yellow for the combined SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For all signal events, cross sections are normalized to their 95% exclusion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 − PB as the BDT response score, where PB is the probability the algorithm classifies an event as background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The BDT response for a benchmark scenario of mN = 1 TeV in a √s = 3 TeV collider is shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The dashed vertical line in figure 9 is the – 13 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 BDT Score 10−4 10−3 10−2 10−1 100 Normalized Events √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV Signal Background Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The normalized BDT response, 1 − PB, for a benchmark scenario of mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The events in yellow correspond to the combined SM background, and the purple events correspond to the combined the Majorana and Dirac signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The dashed vertical line is the cut on the BDT score that we use to distinguish signal from background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The optimization of the BDT cut here is operated on a grid four times finer than the binning shown in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' cutoff on the BDT score that we use to separate signal from background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In the analysis of section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1, only the 1 − PB cut is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For use in the analysis of section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2, on events that pass the background cut we also apply a cutoff on the quantity PM − PD to separate Majorana-like and Dirac-like signal events, where PM (PD) is the probability the BDT algorithm classifies the signal as Majorana (Dirac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We optimize both the 1−PB and PM − PD cuts simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In addition to cutting on the BDT score, we further apply a cut on the mass of the reconstructed HNL, namely within [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='9 × mN, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05 × mN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This window is asymmetric, as the reconstructed distribution has a low mass tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Figure 10 shows the invariant mass distribution mN of the HNL signals and SM background after the application of the BDT analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5 Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1 Exclusion Limits on |Vℓ|2 We now present the results of the BDT analysis in order to construct 95% exclusion limits on the HNL mixing parameter |Vℓ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' At this level of analysis, we assume no systematic uncertainty in the number of recorded events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Our goal is to determine exclusion limits at 95% confidence, and as such we require Z ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Therefore, for each benchmark simulation we run the BDT analysis and extract the lower limit on |Vℓ|2 such that the signal significance – 14 – 600 800 1000 1200 1400 Reconstructed mN (GeV) 10−3 10−2 10−1 100 101 Events/25 GeV √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV Majorana Dirac Background Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A histogram of reconstructed mN after applying BDT, for a benchmark scenario of mN = 1 TeV in a √s = 3 TeV, L = 1 ab−1 muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The dashed vertical lines are the cuts on the mN mass used to further separate signal from background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Blue corresponds to Majorana signal events, red for Dirac, and yellow for the combined SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For all signal events, cross sections are normalized to their 95% exclusion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' is above threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This analysis was done using a three-class BDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We applied a two-class analysis to cross check and found exclusion curves with comparable sensitivies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The results of this process are shown in figure 11, as well as comparisons to similar searches in other (proposed) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For masses above mN > 1 TeV, the exclusion limits are stronger than any other limits shown so far, with reach down to |Vℓ|2 ∼ O(10−6) or even lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that in this mass region, it is often the case that the sensitivities for the Majorana signals are slightly better than those of the Dirac signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The reason is evident from figure 7 (c, e), and figure 8 (e, f), which show that the corresponding distributions of a Majorana HNL differ from the SM background distributions more significantly than those of a Dirac HNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In the region of lighter HNL masses, namely mN < 1 TeV, our limits are generally less strict than those for other experiments, due to the poor reconstruction efficiency for low mass HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nevertheless, the uncertainty in the signal yield for these benchmarks is high, and as such our limits can be considered conservative in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Furthermore, in this low mN region, the HNL signal would be dominated by the VBF channel, wherein two vector bosons from the muon beams give rise to N + ℓ or N + ν final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The reconstruction efficiency, in this case, could be significantly larger than that of the t-channel ones, as the pz of the HNL produced would be moderate, leaving more decay products in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In addition, in the scenario where |Vµ| ≪ |Ve| or |Vτ|, the t-channel signal rate would be greatly suppressed, whereas the VBF channel signal rate could still be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nevertheless, the biggest problem for the VBF channel is that its production rate is at least two orders of magnitude smaller than that of the t- – 15 – 101 102 103 104 mN (GeV) 10−7 10−6 10−5 10−4 10−3 10−2 10−1 |Vℓ|2 FCC-hh LHC prompt CLIC3000 ILC1000 ILC500 mlightest = 0 GeV Thermal initial conditions Vanishing initial conditions µC 3 TeV Majorana 1 ab−1 µC 3 TeV Dirac 1 ab−1 µC 10 TeV Majorana 10 ab−1 µC 10 TeV Dirac 10 ab−1 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 95% exclusion limits for |Vℓ|2 = |Ve|2 = |Vµ|2 as a function of the HNL mass mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The red dashed (solid) line corresponds to a Majorana (Dirac) HNL at a muon collider with √s = 3 TeV, L = 1 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The teal dashed (solid) line corresponds to a Majorana (Dirac) HNL at a muon collider with √s = 10 TeV, L = 10 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The solid black and purple lines correspond to limits from considerations of viable leptogenesis scenarios [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The grey area is the region excluded by a global scan [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The red line shows the limits from prompt trilepton searches at the LHC [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The green line shows the limits from a future FCC-hh [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lastly, the grey, orange and blue lines are exclusion limits in future e+e− linear colliders [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' channel process, since the process of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Consequently, for this work’s benchmark of |Vℓ| = |Ve| = |Vµ|, the estimated exclusion limit of |Vℓ| is at most comparable to that of the t-channel when mN ≃ 200 GeV, and drops much faster as mN increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We will leave a through study of VBF channels to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 Distinguishing Majorana versus Dirac Heavy Neutral Leptons As noted in section 4 and figure 7, there are signatures in the final state kinematic distri- butions that differ between Majorana and Dirac type HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' It is therefore an interesting question to ask what would be the discrimination potential to distinguish Majorana versus Dirac HNLs in a muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We consider the quantity r = nM/(nM + nD), where nM – 16 – (nD) is the number of detected Majorana (Dirac) HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that this definition differs from the usual Rll seen in the literature, which is based on the ratio of lepton number violating (LNV) decays to lepton number conserving (LNC) decays (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [118]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2 In order to quantify our analysis, we consider the two-dimensional likelihood in the plane of Majorana and Dirac HNL signal yields in a √s = 3 TeV and a √s = 10 TeV collider, shown in figure 12 on the left and right-hand sides respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Both the horizontal and vertical axes are normalized by the same quantity, namely the 95% exclusion limits |Vℓ|2 as given in figure 11 for either Majorana or Dirac type — whichever value is more conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Each point on the the dotted line represents a possible value of r, and the x and y intercepts of that line correspond to the fully Majorana (r = 1) and Dirac (r = 0) cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For a given r value, we can construct the corresponding 1σ likelihood contour based on the Majorana and Dirac regions defined in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In this analysis, a signal event is classified as Majorana or Dirac if it appropriately satisfies the PM − PD cut correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' If not, it is counted as additional background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We plot three such contours at r = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The centers of these ellipses lie on the diagonal dotted line, and the boundaries of each correspond to fluctuations in signal yields with significance 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that the ellipses are tilted, and showing a negative correlation between Majorana and Dirac signal yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This is due to the fact that the classification is imperfect, as some Majorana or Dirac decays could be misclassified as each other while keeping total HNL signal rate the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The contours for the √s = 10 TeV benchmark are tilted more strongly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='e more highly correlated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' This indicates that a downwards fluctuation of Majorana HNL decay is more likely to be due to misclassification as a Dirac HNL than misclassification as background, as compared to the case in the √s = 3 TeV benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lastly, the uncertainties shown in figure 12 are marginal, as we are considering discrim- ination potential at the 95% exclusion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We have found that repeating this analysis for larger |Vℓ|2 results in much tighter contours since the larger statistics improve the pre- cision of both signal rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For example, in figure 13 we repeat the above analysis with |Vℓ|2 = 5 × |Vℓ|2 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that the values for here |Vℓ|2 are still small — of order O(10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We see that we therefore have a nontrivial discrimination potential for the HNL’s type as long as its signal yield is above the exclusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 6 Conclusion and Future Directions In this paper we investigate the potential for searching for HNLs in a future high-energy muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We consider an effective theory benchmark that couples a single HNL to the first two generations of active neutrinos with equal mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We consider HNLs of both Majorana and Dirac types, with masses ranging from 200 GeV to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 TeV and determine the reach in the mixing parameter |Vℓ|2 for each mass, in √s = 3 TeV, L = 1 ab−1 and √s = 10 TeV, L = 10 ab−1 muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We find that for HNL masses greater than 2The two definitions agree for r = Rll = 0, 1, but our r = 1/2 corresponds to Rll = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Alternatively, r could also be defined as 2nLNV nLNC+nLNV , where nLNC and nLNV are numbers of detected LNC and LNV decays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 17 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 (1 − r) √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV 1σ Uncertainty r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 (1 − r) √s =10 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =3 TeV 1σ Uncertainty r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The two-dimensional likelihood in the plane of Majorana and Dirac HNL signal yields in a √s = 3 TeV (left) and a √s = 10 TeV collider (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The 3 ellipses on each panel represent the 1σ contours centered at r = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that the ellipses are tilted, and showing a negative correlation between the measured Majorana and Dirac signal yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' All contours are evaluated at their 95% exclusion limits where |Vℓ|2 ∼ O(10−6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 (1 − r) √s =3 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =1 TeV 1σ Uncertainty r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='75 (1 − r) √s =10 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' mN =3 TeV 1σ Uncertainty r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The two-dimensional likelihood in the plane of Majorana and Dirac HNL signal yields in a √s = 3 TeV (left) and a √s = 10 TeV collider (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The 3 ellipses on each panel represent the 1σ contours centered at r = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note that the ellipses are tilted, and showing a negative correlation between Majorana and Dirac signal yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The |Vl|2 used here are five times larger than their counterparts in figure 12, leading to much sharper contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 TeV the limits in |Vℓ|2 are the strictest collider limits yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We also show that a future muon collider has strong discrimination potential to distinguish between Majorana and Dirac type HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' In the future, there are a few aspects of this work that could be broadened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Firstly, we – 18 – focus our analysis on an effective theory in which the HNL signal is dominated by single production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The purpose of this work was to be conservative and model-independent, but different UV completions could have distinct signatures accessible at a muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Secondly, a more realistic analysis would include the production and decay of multiple HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Similarly, this work assumes a benchmark point of enhanced symmetry for the mixing of the HNLs, namely |V1e| = |V1µ| ̸= 0 and |V1τ| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Consequently, we do not consider the decay of an HNL to a tau in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' For other benchmarks, the dominant processes in the production of HNLs might differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lastly, while we include a fast detector simulation in this work, we do not incorporate the effects of detector-based systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Likewise, we do not include systemic uncertainties due the beams nor do we incorporate beam spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' As a spectrum profile for muon beams is not yet available in Whizard, we also leave such refinements for a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Note Added: At the same time as this work was shared on the arXiv, another preprint on the same topic was also uploaded [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' While the two works’ aims are similar, they differ in their methods and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' We are also aware of a work by Peiran Li, Zhen Liu and Kun-Feng Lyu on the same subject that will also appear on the arXiv soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' The preliminary results of that work can be found in the Snowmass report of BSM physics [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Acknowledgments LL would like to thank Samuel Homiller, Zhen Liu, and Kun-Feng Lyu for valuable dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' AR would like to thank Adam Lister and Harry Hausner for useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' THK, TL and AR are supported partly by the Area of Excellence (AoE) under the Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' AoE/P-404/18-3 and supported partly by the General Research Fund (GRF) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 16304321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Both of the AoE and GRF grants are issued by the Research Grants Council of Hong Kong S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' LL is supported by the DOE grant DE-SC-0010010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 19 – References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Davis, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Harmer, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Hoffman, Search for neutrinos from the sun, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 20 (1968) 1205–1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [2] KamLAND Collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Eguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', First results from KamLAND: Evidence for reactor anti-neutrino disappearance, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 90 (2003) 021802, [hep-ex/0212021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [3] Daya Bay Collaboration, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Observation of electron-antineutrino disappearance at Daya Bay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 108 (2012) 171803, [arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1669].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [4] RENO Collaboration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 108 (2012) 191802, [arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0626].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [5] Double Chooz Collaboration, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Improved measurements of the neutrino mixing angle θ13 with the Double Chooz detector, JHEP 10 (2014) 086, [arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='7763].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [Erratum: JHEP 02, 074 (2015)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [6] SNO Collaboration, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 89 (2002) 011301, [nucl-ex/0204008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [7] Particle Data Group Collaboration, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Workman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Review of Particle Physics, PTEP 2022 (2022) 083C01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Weinberg, Baryon and Lepton Nonconserving Processes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 43 (1979) 1566–1570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gell-Mann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ramond, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Slansky, Complex Spinors and Unified Theories, Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 790927 (1979) 315–321, [arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4669].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Minkowski, µ → eγ at a Rate of One Out of 109 Muon Decays?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 67 (1977) 421–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Yanagida, Horizontal gauge symmetry and masses of neutrinos, Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 7902131 (1979) 95–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mohapatra and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Senjanovic, Neutrino Mass and Spontaneous Parity Nonconservation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 44 (1980) 912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [13] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Search for heavy neutral leptons in events with three charged leptons in proton-proton collisions at √s = 13 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 120 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 22 221801, [arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02965].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [14] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Search for heavy neutral leptons in decays of W bosons produced in 13 TeV pp collisions using prompt and displaced signatures with the ATLAS detector, JHEP 10 (2019) 265, [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='09787].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [15] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Tumasyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Search for long-lived heavy neutral leptons with displaced vertices in proton-proton collisions at √s =13 TeV, JHEP 07 (2022) 081, [arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05578].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [16] ATLAS Collaboration, Search for heavy neutral leptons in decays of W bosons using a dilepton displaced vertex in √s = 13 TeV pp collisions with the ATLAS detector, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [17] LHCb Collaboration, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Search for heavy neutral leptons in W + → µ+µ±jet decays, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 81 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 3 248, [arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 20 – [18] SHiP Collaboration, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ahdida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Sensitivity of the SHiP experiment to Heavy Neutral Leptons, JHEP 04 (2019) 077, [arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='00930].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pascoli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ruiz, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Weiland, Heavy neutrinos with dynamic jet vetoes: multilepton searches at √s = 14 , 27, and 100 TeV, JHEP 06 (2019) 049, [arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='08750].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Cottin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Helo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Hirsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Titov, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wang, Heavy neutral leptons in effective field theory and the high-luminosity LHC, JHEP 09 (2021) 039, [arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='13851].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mękała, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reuter, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Żarnecki, Heavy neutrinos at future linear e+e− colliders, JHEP 06 (2022) 010, [arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='06703].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Skrinsky and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Parkhomchuk, Cooling Methods for Beams of Charged Particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' (In Russian), Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 12 (1981) 223–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Neuffer, Principles and Applications of Muon Cooling, Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 830811 (1983) 481–484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Neuffer, Multi-TeV muon colliders, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 156 (1987) 201–208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Barger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Berger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gunion, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, s channel Higgs boson production at a muon muon collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 75 (1995) 1462–1465, [hep-ph/9504330].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [26] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Barger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Berger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gunion, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, Higgs Boson physics in the s channel at µ+µ− colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 286 (1997) 1–51, [hep-ph/9602415].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ankenbrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Status of muon collider research and development and future plans, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' ST Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Beams 2 (1999) 081001, [physics/9901022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chen, Beam-beam interaction in muon colliders, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 51 (1996) 179–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Palmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Muon collider design, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 51 (1996) 61–84, [acc-phys/9604001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [30] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Aime et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Muon Collider Physics Summary, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='07256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Muon Collider Forum Report, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='01318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Delahaye, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Diemoz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Long, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mansoulié, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pastrone, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rivkin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Schulte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Skrinsky, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wulzer, Muon Colliders, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='06150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Costantini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' De Lillo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Maltoni, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mantani, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mattelaer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ruiz, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Zhao, Vector boson fusion at multi-TeV muon colliders, JHEP 09 (2020) 080, [arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10289].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [34] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ma, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Xie, High energy leptonic collisions and electroweak parton distribution functions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 103 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 3 L031301, [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='14300].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [35] NNPDF Collaboration, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bertone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Carrazza, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Hartland, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rojo, Illuminating the photon content of the proton within a global PDF analysis, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 008, [arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='07053].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Palmer, Muon Colliders, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 7 (2014) 137–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [37] Muon Collider Collaboration, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Stratakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', A Muon Collider Facility for Physics Discovery, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='08033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [38] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Schulte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Chapter 5: Bright muon beams and muon colliders, CERN Yellow Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Monogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 (2022) 145–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 21 – [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Delahaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Enabling Intensity and Energy Frontier Science with a Muon Accelerator Facility in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=': A White Paper Submitted to the 2013 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Community Summer Study of the Division of Particles and Fields of the American Physical Society, in Community Summer Study 2013: Snowmass on the Mississippi, 8, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Delahaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', A Staged Muon Accelerator Facility For Neutrino and Collider Physics, in 5th International Particle Accelerator Conference, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' WEZA02, 6, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='01647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ryne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Design Concepts for Muon-Based Accelerators, in 6th International Particle Accelerator Conference, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' WEPWA057, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [42] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Long, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lucchesi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Palmer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pastrone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Schulte, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Shiltsev, Muon colliders to expand frontiers of particle physics, Nature Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 17 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 3 289–292, [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='15684].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [43] MICE Collaboration, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mohayai, First Demonstration of Ionization Cooling in MICE, in 9th International Particle Accelerator Conference, 6, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='01807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [44] MICE Collaboration, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Blackmore, Recent results from the study of emittance evolution in MICE, in 9th International Particle Accelerator Conference, 6, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='04409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [45] MICE Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bogomilov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Demonstration of cooling by the Muon Ionization Cooling Experiment, Nature 578 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 7793 53–59, [arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='08562].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Antonelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Boscolo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Di Nardo, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Raimondi, Novel proposal for a low emittance muon beam using positron beam on target, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A 807 (2016) 101–107, [arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='04454].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [47] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fan, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Li, Electroweak ALP searches at a muon collider, JHEP 08 (2022) 276, [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='04328].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [48] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wang, Axion-Like Particles at High Energy Muon Colliders – A White paper for Snowmass 2021, in 2022 Snowmass Summer Study, 3, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [49] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' İnan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kisselev, Probe of axion-like particles in vector boson scattering at a muon collider, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='03325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chiesa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Maltoni, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mantani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mele, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Piccinini, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Zhao, Measuring the quartic Higgs self-coupling at a multi-TeV muon collider, JHEP 09 (2020) 098, [arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='13628].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [51] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Liu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Low, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wang, Electroweak couplings of the Higgs boson at a multi-TeV muon collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 103 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 013002, [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='12204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chiesa, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mele, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Piccinini, Multi Higgs production via photon fusion at future multi-TeV muon colliders, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Cepeda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gori, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Outschoorn, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Shelton, Exotic Higgs Decays, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='12751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Yao, Measurement of Higgs boson self-couplings through 2→3 vector bosons scattering in future muon colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 105 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5 053009, [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='12507].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Forslund and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Meade, High precision higgs from high energy muon colliders, JHEP 08 (2022) 185, [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='09425].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 22 – [56] Muon Collider Physics and Detector Working Group Collaboration, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Buonincontri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Andreetto, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bartosik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Casarsa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gianelle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lucchesi, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sestini, Higgs boson couplings at muon collider, PoS EPS-HEP2021 (2022) 619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [57] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Di Luzio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gröber, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Panico, Probing new electroweak states via precision measurements at the LHC and future colliders, JHEP 01 (2019) 011, [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [58] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Buttazzo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Franceschini, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wulzer, Two Paths Towards Precision at a Very High Energy Lepton Collider, JHEP 05 (2021) 219, [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11555].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [59] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Spor and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Köksal, Investigation of anomalous triple gauge couplings in µγ collision at multi-TeV muon colliders, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='00787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Glioti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rattazzi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ricci, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wulzer, Learning from radiation at a very high energy lepton collider, JHEP 05 (2022) 180, [arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10509].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [61] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Liu, Top Yukawa Coupling at the Muon Collider, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [62] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Jana, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Queiroz, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rodejohann, Probing the RK(*) anomaly at a muon collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 105 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 015013, [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='01617].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [63] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Asadi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Capdevilla, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Cesarotti, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Homiller, Searching for leptoquarks at future muon colliders, JHEP 10 (2021) 182, [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05720].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [64] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Haghighat and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mohammadi Najafabadi, Search for lepton-flavor-violating ALPs at a future muon collider and utilization of polarization-induced effects, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 980 (2022) 115827, [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='00505].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [65] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bandyopadhyay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Karan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mandal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Parashar, Distinguishing signatures of scalar leptoquarks at hadron and muon colliders, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 82 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 10 916, [arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='06506].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Das, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nomura, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Shimomura, Multi muon/anti-muon signals via productions of gauge and scalar bosons in a U(1)Lµ−Lτ model at muonic colliders, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [67] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Wang, WIMPs at High Energy Muon Colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 103 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 7 075004, [arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11287].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [68] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Capdevilla, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Meloni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Simoniello, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Zurita, Hunting wino and higgsino dark matter at the muon collider with disappearing tracks, JHEP 06 (2021) 133, [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11292].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [69] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Medina, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mileo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Szynkman, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Tanco, Elusive muonic WIMP, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 106 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 7 075018, [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='09103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [70] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Black, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bose, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Dasu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Jia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pinna, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sharma, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Venkatasubramanian, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Vuosalo, Prospects for Heavy WIMP Dark Matter Searches at Muon Colliders, in 2022 Snowmass Summer Study, 5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [71] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Jin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Li, Unraveling the Scotogenic model at muon collider, JHEP 12 (2022) 057, [arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='07382].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [72] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Al Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', The muon Smasher’s guide, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 85 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 8 084201, [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='14043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [73] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Homiller, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reece, Complementary signals of lepton flavor violation at a high-energy muon collider, JHEP 07 (2022) 036, [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='08825].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [74] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Blanchet, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chacko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Granor, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mohapatra, Probing Resonant Leptogenesis at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 82 (2010) 076008, [arXiv:0904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 23 – [75] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fileviez Perez, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Li, Testability of Type I Seesaw at the CERN LHC: Revealing the Existence of the B-L Symmetry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 80 (2009) 073015, [arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [76] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Anamiati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Hirsch, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nardi, Quasi-Dirac neutrinos at the LHC, JHEP 10 (2016) 010, [arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05641].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [77] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Deppisch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bhupal Dev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pilaftsis, Neutrinos and Collider Physics, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 17 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 7 075019, [arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='06541].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [78] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chiang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Cottin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Das, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mandal, Displaced heavy neutrinos from Z’ decays at the LHC, JHEP 12 (2019) 070, [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='09838].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [79] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Xie, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Yi, Testing leptogenesis at the LHC and future muon colliders: A Z’ scenario, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 105 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 9 095034, [arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='15087].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [80] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chauhan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Dev, Resonant Leptogenesis, Collider Signals and Neutrinoless Double Beta Decay from Flavor and CP Symmetries, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='09710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [81] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mohapatra and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Valle, Neutrino Mass and Baryon Number Nonconservation in Superstring Models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 34 (1986) 1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [82] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mohapatra, Mechanism for Understanding Small Neutrino Mass in Superstring Theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 56 (1986) 561–563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [83] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nandi and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sarkar, A Solution to the Neutrino Mass Problem in Superstring E6 Theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 56 (1986) 564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [84] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gonzalez-Garcia and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Valle, Fast Decaying Neutrinos and Observable Flavor Violation in a New Class of Majoron Models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 216 (1989) 360–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [85] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bernabeu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Santamaria, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Vidal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mendez, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Valle, Lepton Flavor Nonconservation at High-Energies in a Superstring Inspired Standard Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 187 (1987) 303–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [86] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Degrande, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Duhr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fuks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Grellscheid, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mattelaer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reiter, UFO - The Universal FeynRules Output, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 183 (2012) 1201–1214, [arXiv:1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='2040].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [87] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Alloul, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Christensen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Degrande, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Duhr, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fuks, FeynRules 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='0 - A complete toolbox for tree-level phenomenology, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 185 (2014) 2250–2300, [arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='1921].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [88] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Alva, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ruiz, Heavy Majorana neutrinos from Wγ fusion at hadron colliders, JHEP 02 (2015) 072, [arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='7305].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [89] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Degrande, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mattelaer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ruiz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Turner, Fully-Automated Precision Predictions for Heavy Neutrino Production Mechanisms at Hadron Colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 94 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5 053002, [arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='06957].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [90] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Atre, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pascoli, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Zhang, The Search for Heavy Majorana Neutrinos, JHEP 05 (2009) 030, [arXiv:0901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3589].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [91] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Maki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nakagawa, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sakata, Remarks on the unified model of elementary particles, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 28 (1962) 870–880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [92] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pontecorvo, Mesonium and anti-mesonium, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' JETP 6 (1957) 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [93] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Pontecorvo, Inverse beta processes and nonconservation of lepton charge, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 34 (1957) 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 24 – [94] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ma, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Xie, Quark and gluon contents of a lepton at high energies, JHEP 02 (2022) 154, [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='09844].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [95] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kilian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ohl, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reuter, WHIZARD: Simulating Multi-Particle Processes at LHC and ILC, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 71 (2011) 1742, [arXiv:0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [96] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Moretti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ohl, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reuter, O’Mega: An Optimizing matrix element generator, hep-ph/0102195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [97] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bierlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', A comprehensive guide to the physics and usage of PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='3, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='11601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [98] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Deppisch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kulkarni, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Liu, Heavy neutrino production via Z ′ at the lifetime frontier, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 100 (Aug, 2019) 035005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [99] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Budnev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ginzburg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Meledin, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Serbo, The Two photon particle production mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Physical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Equivalent photon approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 15 (1975) 181–281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [100] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' von Weizsacker, Radiation emitted in collisions of very fast electrons, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 88 (1934) 612–625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [101] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Williams, Nature of the high-energy particles of penetrating radiation and status of ionization and radiation formulae, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 45 (1934) 729–730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [102] DELPHES 3 Collaboration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' de Favereau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Delaere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Demin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Giammanco, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lemaître, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mertens, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Selvaggi, DELPHES 3, A modular framework for fast simulation of a generic collider experiment, JHEP 02 (2014) 057, [arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='6346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [103] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Selvaggi, A Delphes parameterisation of the FCC-hh detector, tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', CERN, Geneva, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [104] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Roloff, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Schnoor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Weber, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Leogrande, A Delphes card for the CLIC detector, tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', CERN, Geneva, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [105] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Boronat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fuster, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Garcia, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Ros, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Vos, A robust jet reconstruction algorithm for high-energy lepton colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 750 (2015) 95–99, [arXiv:1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='4294].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [106] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Guestrin, XGBoost: A Scalable Tree Boosting System, arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [107] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Petcov, Possible Signature for Production of Majorana Particles in e+e− and p¯p Collisions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 139 (1984) 421–426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [108] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Baha Balantekin and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kayser, On the Properties of Neutrinos, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 68 (2018) 313–338, [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='00922].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [109] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' de Gouvêa, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kayser, Addressing the Majorana vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Dirac Question with Neutrino Decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' B 789 (2019) 488–495, [arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10518].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [110] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' de Gouvêa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fox, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kayser, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kelly, Three-body decays of heavy Dirac and Majorana fermions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 104 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 015038, [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='05719].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [111] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' de Gouvêa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fox, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kayser, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Kelly, Characterizing heavy neutral fermions via their decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 105 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 1 015019, [arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='10358].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [112] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Arbelaéz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Dib, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Schmidt, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Vasquez, Probing the Dirac or Majorana nature of the Heavy Neutrinos in pure leptonic decays at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 97 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5 055011, [arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='08704].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 25 – [113] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Hernández, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Jones-Pérez, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Suarez-Navarro, Majorana vs Pseudo-Dirac Neutrinos at the ILC, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 79 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 3 220, [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='07210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [114] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Drewes, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Georis, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Klarić, Mapping the Viable Parameter Space for Testable Leptogenesis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 128 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 5 051801, [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='16226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [115] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Chrzaszcz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Drewes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Gonzalo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Harz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Krishnamurthy, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Weniger, A frequentist analysis of three right-handed neutrinos with GAMBIT, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' C 80 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 6 569, [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [116] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Izaguirre and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Shuve, Multilepton and Lepton Jet Probes of Sub-Weak-Scale Right-Handed Neutrinos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' D 91 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 9 093010, [arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02470].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [117] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Antusch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Cazzato, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Fischer, Sterile neutrino searches at future e−e+, pp, and e−p colliders, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' A 32 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' 14 1750078, [arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02728].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [118] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Drewes, Distinguishing Dirac and Majorana Heavy Neutrinos at Lepton Colliders, in 41st International Conference on High Energy Physics, 10, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='17110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [119] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Mękała, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Reuter, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Żarnecki, Optimal search reach for heavy neutral leptons at a muon collider, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='02602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' [120] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=', Report of the Topical Group on Physics Beyond the Standard Model at Energy Frontier for Snowmass 2021, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content='13128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} +page_content=' – 26 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf'} diff --git a/VdE1T4oBgHgl3EQfIwNu/content/2301.02943v1.pdf b/VdE1T4oBgHgl3EQfIwNu/content/2301.02943v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3d4518d2083fd32b55f4ec5dc662351b9076f6e8 --- /dev/null +++ b/VdE1T4oBgHgl3EQfIwNu/content/2301.02943v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d9a4551d0e3f94928418b9dd75706cda74ed416cbdfdd1d19a9c9970cf21e7a7 +size 1794629 diff --git a/VdE1T4oBgHgl3EQfIwNu/vector_store/index.pkl b/VdE1T4oBgHgl3EQfIwNu/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1ffbe2371f8f37c789f8de14fd9b38e78a03db86 --- /dev/null +++ b/VdE1T4oBgHgl3EQfIwNu/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:28297e46ac3d0ce74f6d62c0feba4d0ca2bcd4a21cfd3f4dc673744e1569fe55 +size 313384 diff --git a/W9FQT4oBgHgl3EQfcjYS/vector_store/index.faiss b/W9FQT4oBgHgl3EQfcjYS/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ba9baca8f1fc108f1bd340e00d05b8eeda2eb062 --- /dev/null +++ b/W9FQT4oBgHgl3EQfcjYS/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9ff427a01ee391dfa92b0864078ac6974b291b891087c231f0afc6be8a72d248 +size 4063277 diff --git a/YtA0T4oBgHgl3EQfFv9O/content/tmp_files/2301.02036v1.pdf.txt b/YtA0T4oBgHgl3EQfFv9O/content/tmp_files/2301.02036v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f574ec2cf91db22f2383d285896b155316468262 --- /dev/null +++ b/YtA0T4oBgHgl3EQfFv9O/content/tmp_files/2301.02036v1.pdf.txt @@ -0,0 +1,391 @@ +arXiv:2301.02036v1 [math.DG] 5 Jan 2023 +COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS +LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE +Abstract. We study the singularities of commuting vectors fields of a real submanifold of a +K¨ahler manifold Z. +1. Introduction +Let (Z, ω) be a connected K¨ahler manifold with an holomorphic action of a complex reductive +group U C, where U C is the complexification of a compact connected Lie group U with Lie algebra +u. We also assume ω is U-invariant and that there is a U-equivariant momentum map µ : Z → u∗. +By definition, for any ξ ∈ u and z ∈ Z, dµξ = iξZω, where µξ(z) := µ(z)(ξ) and ξZ denotes the +fundamental vector field induced on Z by the action of U, i.e., +ξZ(z) := d +dt +���� +t=0 +exp(tξ)z +(see, for example, [9] for more details on the momentum map). Since U is compact we may +identify u ∼= u∗ by an Ad(U)-invariant scalar product on u. Hence, we consider a momentum +map as a u-valued map, i.e., µ : Z → u. +Recently, the momentum map has been generalized to the following settings [7, 8]; we say +that a subgroup G of U C is compatible if G is closed and the map K ×p → G, (k, β) �→ kexp(β) +is a diffeomorphism, where K := G∩U and p := g∩iu; g is the Lie algebra of G. The Lie algebra +uC of U C is the direct sum u⊕iu. It follows that G is compatible with the Cartan decomposition +U C = Uexp(iu), K is a maximal compact subgroup of G with Lie algebra k and that g = k ⊕ p. +The inclusion ip ֒→ u induces by restriction, a K-equivariant map µip : Z → (ip)∗. Using a +Ad(U)-invariant scalar product on iu requiring multiplication by i to be an isometry between +u and iu, µip can be viewed as the orthogonal projection of iµ(z) onto p given as µp : Z → p. +2010 Mathematics Subject Classification. 53D20; 14L24. +Key words and phrases. Momentum map, Reductive Lie group. +The first author was partially supported by PRIN 2017 “Real and Complex Manifolds: Topology, Geometry +and holomorphic dynamics ” and GNSAGA INdAM. +The first author was supported by the Project MIUR “Geometric Properties of Real and Complex Manifolds” +and by GNSAGA of INdAM. The second author was supported by the PRIN 2007 MIUR of INdAM. +1 + +2 +LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE +Let µβ +p (z) := ⟨µp(z), β⟩ = ⟨iµ(z), β⟩ = ⟨µ(z), −iβ⟩ = µ−iβ(z) for any β ∈ p and z ∈ Z. Then +grad µβ +p = βZ, where the gradient (grad) is computed with respect to the Riemannian metric +induced by the K¨ahler structure. The map µp is called the gradient map associated with µ. In +this paper, a G-invariant compact connected locally closed real submanifold X of Z is fixed and +the restriction of µp to X is denoted by µp. Then µp : X −→ p is a K-equivariant map such that +gradµβ +p = βX, where the gradient is computed with respect to the induced Riemannian metric +on X denoted by (·, ·). +Let T be a torus of U. This means that T is a connected compact Abelian subgroup of U. +By a Theorem of Koszul, [6], the connected component of ZT := {x ∈ Z : T · x = x} are closed +and K¨ahler submanifolds of Z. Let t be the Lie algebra of T. It is well-known that the set +� +β ∈ t : exp(Rβ) = T +� +, +contains a dense subset [1]. Hence, +ZT = ZT C = {p ∈ Z : βZ(p) = 0} , +for some β ∈ t. In this paper, we investigate the fixed point set of the action of an Abelian +compatible subgroup of U C acting on real submanifold of Z. +Let a ⊂ p be an Abelian subalgebra and A = exp(a). Then the A-gradient map on X is given +by µa = πa ◦ µp, where πa : p −→ a denotes the orthogonal projection of p onto a. If β ∈ a, let +Xβ := {z ∈ X : βX(z) = 0}. By the linearization Theorem [8, 10], any connected component of +Xβ is an embedded submanifold. Our main results are the following. +Theorem 1.1. The set +� +β ∈ a : Xβ = XA� +is dense. +Let β ∈ a. The flow +R × X −→ X, +(t, x) �→ exp(tβ)x, +is the gradient flow of the function µβ +a . By the linearization Theorem, the limit +lim +t→+∞ exp(tβ)x exists. +Let α1, . . . , αn ∈ a be a basis of a an let x ∈ X. +Set x1 = limt→+∞ exp(tα1)x and xi = +limt→+∞ exp(tαi)xi−1, for i = 2, . . . , n. +Theorem 1.2. There exists δ > 0 such that for any 0 < ǫ2, . . . , ǫn < δ we have +lim +t→+∞ exp(t(α1 + ǫ2α2 + · · · + ǫnαn))x = xn, + +COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS +3 +for any x ∈ X. +1.1. Gradient map. Let U be a compact connected Lie group and U C its complexification. Let +(Z, ω) be a K¨ahler manifold on which U C acts holomorphically on Z. Assume that U preserves +ω and that there is a U-equivariant momentum map µ : Z → u. Let ⟨·, ·⟩ denote an Ad(U)- +invariant scalar product on u. Up to a locally constant map, µ is determined by dµξ = iξZω, +where µξ(x) := ⟨µ(z), ξ⟩, ξ ∈ u and iξZ denotes the contraction of ω with the vector field ξZ +which is given by ξ and the U-action on Z. +Let G ⊂ U C be a compatible subgroup of U C. Then p ⊂ iu. Let ⟨·, ·⟩ also denote the Ad(U)- +invariant scalar product on iu requiring the multiplication by i to be an isometry of u onto iu. If +z ∈ Z, then the orthogonal projection of iµ(z) onto p defines a K-equivariant map µp : Z −→ p. +In other words, we define µp requiring that for any β ∈ p, we have +µβ +p := ⟨µp(z), β⟩ = ⟨iµ(z), β⟩ = −⟨µ(z), iβ⟩ = µ−iβ. +The map µp : Z −→ p is called G-gradient map. βZ is the gradient of µβ +p . Indeed, since U C acts +holomorphically on Z, +gradµβ +p = gradµ−iβ = J(−iβZ) = βZ, +where the gradient is computed with respect to the Riemannian structure denoted by (·, ·) given +by the K¨ahler form ω on Z, i.e., (v, w) = ω(v, Jw). For the rest of this paper, fix a G-invariant +compact connected locally closed real submanifold X of Z and denote the restriction of µp to X +by µp. Then µp : X −→ p is a K-equivariant map such that gradµβ +p = βX, where the gradient is +computed with respect to the induced Riemannian metric on X that we also denote by (·, ·). For +any subspace m of g and x ∈ X, let m · x := {ξX(x) : ξ ∈ m} and mx := {α ∈ m : αX(x) = 0}. +Let a ⊂ p be an Abelian subalgebra. Then the A = exp(a)-gradient map on X is given by +µa = πa ◦ µp, where πa : p −→ a denotes the orthogonal projection of p onto a. +Lemma 1.3. Let x ∈ X. Then the stabilizer of x denoted by Ax, is compatible. +Proof. If a ∈ Ax, then a = exp(β) for a β ∈ a. Let f(t) = ⟨µa(exp(tβ)x), β⟩. Then f(1) = +⟨µa(exp(β)x), β⟩ = ⟨µa(ax), β⟩ = ⟨µa(x), β⟩ = f(0) and f ′(t) =∥ βX(exp(tβ)x) ∥2≥ 0. This +implies βX(x) = 0 and so β ∈ ax. Therefore, Ax = exp(ax). +□ +We recall the Slice Theorem, see [8] for details. + +4 +LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE +Theorem 1.4. [Slice Theorem [8, Thm. 3.1], [10]] If x ∈ X and µp(x) = 0, there are a Gx- +invariant decomposition TxX = g · x ⊕ W, open Gx-invariant subsets S ⊂ W, Ω ⊂ X and a +G-equivariant diffeomorphism Ψ : G ×Gx S → Ω, such that 0 ∈ S, x ∈ Ω and Ψ([e, 0]) = x. +Here G ×Gx S denotes the associated bundle with principal bundle G → G/Gx. +Corollary 1.4.1. If x ∈ X and µp(x) = β, there are a Gβ-invariant decomposition TxX = +gβ · x ⊕ W, open Gβ-invariant subsets S ⊂ W, Ω ⊂ X and a Gβ-equivariant diffeomorphism +Ψ : Gβ ×Gx S → Ω, such that 0 ∈ S, x ∈ Ω and Ψ([e, 0]) = x. +This follows from applying the previous theorem to the action of Gβ on X. Indeed, it is +well known that Gβ = Kβ exp(pβ) is compatible [4] and the orthogonal projection of iµ onto pβ +is the Gβ-gradient map µpβ. The group Gβ is also compatible with the Cartan decomposition +of (U C)β = (U C)iβ = (U iβ)C and iβ is fixed by the U iβ-action on uiβ. +This implies that +� +µuiβ : Z −→ uiβ given by � +µuiβ(z) = πuiβ ◦ µ + iβ, where πuiβ is the orthogonal projection of +u onto uiβ, is the U iβ-shifted momentum map. The associated Gβ-gradient map is given by +� +µpβ := µpβ − β. Hence, if G is commutative, then we have a Slice Theorem for G at every point +of X, see [8, p.169] and [10] for more details. +If β ∈ p, then βX is a vector field on X, i.e. +a section of the bundle TX. +For x ∈ X, +the differential is a map TxX → TβX(x)(TX). +If βX(x) = 0, there is a canonical splitting +TβX(x)(TX) = TxX ⊕ TxX. Accordingly the differential of βX, regarded as a section of TX, +splits into a horizontal and a vertical part. The horizontal part is the identity map. We denote +the vertical part by dβX(x). The linear map dβX(x) ∈ End(TxX) is indeed the so-called intrinsic +differential of βX, regarded as a section in the tangent bundle TX, at the vanishing point x. +Let {ϕt = exp(tβ)} be the flow of βX. There is a corresponding flow on TX. Since ϕt(x) = x, +the flow on TX preserves TxX and there it is given by dϕt(x) ∈ Gl(TxX). Thus we get a linear +R-action on TxX with infinitesimal generator dβX(x). +Corollary 1.4.2. If β ∈ p and x ∈ X is a critical point of µβ +p , then there are open invariant +neighborhoods S ⊂ TxX and Ω ⊂ X and an R-equivariant diffeomorphism Ψ : S → Ω, such that +0 ∈ S, x ∈ Ω, Ψ(0) = x. Here t ∈ R acts as dϕt(x) on S and as ϕt on Ω.) +Proof. Since exp : p −→ G is a diffeomorphism onto the image, the subgroup H := exp(Rβ) +is closed and so it is compatible. Hence, it is enough to apply the previous corollary to the +H-action on X and the value at x of the corresponding gradient map. +□ + +COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS +5 +Let β ∈ p and let Xβ = {x ∈ X : βX(x) = 0}. By Corollary 1.4.2, Xβ is a smooth, possibly +disconnected, submanifold of X. +Proposition 1.5. Xβ is Gβ-invariant. +Proof. Let g ∈ Gβ. Since βX(gx) = (dg)x(βX(x)), it follows that gx ∈ Xβ. +□ +Let x ∈ Xβ. Let D2µβ +p (x) denote the Hessian, which is a symmetric operator on TxX such +that +(D2µβ +p(x)(v, v) = d2 +dt2 (µβ +p ◦ γ)(0) +where γ is a smooth curve, γ(0) = x and ˙γ(0) = v. Using the inner scalar product (·, ·)(x) +one obtains an associate symmetric endomorphism L(µβ +p )(x) of the Euclidian vector space +(TxX, (·, ·)(x)). Denote by V− (respectively V+) the sum of the eigenspaces of L(µβ +p )(x) cor- +responding to negative (resp. positive) eigenvalues. Denote by V0 the kernel. Since L(µβ +p )(x) is +a symmetric endomorphism, we get an orthogonal decomposition +TxX = V− ⊕ V0 ⊕ V+. +(1) +Let α : G → X be the orbit map: α(g) := gx. The differential dαe is the map ξ �→ ξX(x). The +following result is well-known. A proof is given in [2]. +Proposition 1.6. If β ∈ p and x ∈ Xβ then +L(µβ +p )(x) = dβX(x). +Moreover dαe(rβ±) ⊂ V± and dαe(gβ) ⊂ V0. If G acts transitively on X, then these are equalities. +Corollary 1.6.1. For every β ∈ p, µβ +p is a Morse-Bott function. +Proof. By Corollary 1.4.2, we have TxXβ = V0 for x ∈ Xβ. +Hence, the first statement of +Proposition 1.6 shows that the Hessian is non-degenerate in the normal directions. +□ +Let c1 > · · · > cr be the critical values of µβ +p . The corresponding level sets of µβ +p , Ci := +(µβ +p )−1(ci) are submanifolds which are union of components of Crit(µβ +p ). The function µβ +p defines +a gradient flow generated by its gradient which is given by βX. By Theorem 1.4, it follows that +for any x ∈ X the limit: +ϕ∞(x) := +lim +t→+∞ exp(tβ)x, + +6 +LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE +exists. Let us denote by W β +i the unstable manifold of the critical component Ci for the gradient +flow of µβ +p : +W β +i := {x ∈ X : ϕ∞(x) ∈ Ci}. +(2) +Applying Theorem 1.4, we have the following well-known decomposition of X into unstable +manifolds with respect to µβ +p . +Theorem 1.7. In the above assumption, we have +X = +r� +i=1 +W β +i , +(3) +and for any i the map: +(ϕ∞)|Wi : W β +i → Ci, +is a smooth fibration with fibres diffeomorphic to Rli where li is the index (of negativity) of the +critical submanifold Ci +2. Proof of the main results +Suppose X ⊂ Z is G-invariant compact connected real submanifold of Z with the gradient +map µp : X → p. In this section we study the fixed point set with respect to vector fields induced +by α, β ∈ p, using the G-action, satisfying [α, β] = 0. +Lemma 2.1. Let β, α ∈ p be such that [β, α] = 0. There exists δ > 0 such that for any ǫ ∈ (0, δ) +Xβ+ǫα = Xβ ∩ Xα. +Proof. Let ǫ > 0 and let A = exp(a), where a = span(α, β). Since the exponential map is a +diffeomorphism restricted on p, it follows that A is a closed and compatible subgroup of G. +Let XA denote the fixed point set of A, i.e., XA = {z ∈ X : A · x = x}. By Lemma 1.3, +XA = Xβ ∩ Xα. +By Corollary 1.4.2, both Xβ ∩ Xα and Xα+ǫβ are compact submanifolds +satisfying Xβ ∩ Xα ⊆ Xα+ǫβ. Since Xα+ǫβ is A-invariant, and so there exists A-gradient map +[8], any connected component of Xα+ǫβ contains a connected component of Xα ∩ Xβ. +Let x ∈ Xα∩Xβ. Let C be the connected component of x and let C′ be the connected compo- +nent of Xα+ǫβ containing C. Since x is fixed by A, by the linearization theorem, Corollary 1.4.2, +there exists A-invariant open subsets Ω ⊂ X and S ⊂ TxX and a A-equivariant diffeomorphism +ϕ : S → Ω such that 0 ∈ S, x ∈ Ω, ϕ(0) = x, dϕ0 = idTxX. Thus we may assume that Ω = Rn, + +COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS +7 +α, β are symmetric matrices of order n satisfying [α, β] = 0. Moreover, TxXα+ǫβ = Ker (α + ǫβ) +and TxXα ∩ TxXβ = Ker α ∩ Ker β. +The matrices α and β are simultaneously diagonalizable. Let {e1, . . . , en} be a basis of Rn +such that αei = aiei and βei = biei for i = 1, . . . , n. Let J = {1 ≤ i ≤ n : aibi ̸= 0}. Pick +δ = min{|ai| +|bi| : i ∈ J}. Now, (α + ǫβ)ei = 0 if and only if ai + ǫbi = 0. If ai ̸= 0, then bi ̸= 0 and +vice-versa. Therefore, for any ǫ < δ, we get (α + ǫβ)ei = 0, if and only if ai = bi = 0. Therefore, +Ker (α + ǫβ) = Ker α ∩ Ker β. Since C ⊂ C′ and TxC = TxC′, keeping in mind that both C and +C′ are compact, it follows that C = C′. Since Xα+ǫβ has finitely many connected components, +it follows that there exists δ > 0 such that for any 0 < ǫ < δ, we have +Xα ∩ Xβ = Xα+ǫβ, +concluding the proof. +□ +Theorem 2.2. Let a ⊂ p be an Abelian subalgebra and let A = exp(a). Then the set +� +α ∈ a : XA = Xα� +is dense. +Proof. Let α1, . . . , αn be a basis of a. Then +XA = Xα1 ∩ · · · ∩ Xαn. +By the above Lemma, there exists δ > 0 such that for any ǫ2, . . . , ǫn < δ, we have +(4) +XA = Xα1+ǫ2α2+···+ǫnαn +Let α ∈ a different form 0. It is well known that there exists α2, . . . αn ∈ a such that α, α2, . . . , αn +is a basis of a. By (4), for any neighborhood U of α, there exists β ∈ U such that +XA = Xβ, +concluding the proof. +□ +The following Lemma is proved in [3], see also [5, pag. 1036]. For sake of completeness we +give the proof. +Lemma 2.3. Let x ∈ X and β, α ∈ p be such that [β, α] = 0. Set y := limt→∞ exp(tβ)x and +z := limt→∞ exp(tα)y. Let δ be as in Lemma 2.1. Then for 0 < ǫ < δ, +lim +t→∞ exp(t(β + ǫα)) = z. + +8 +LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE +Proof. Fix x ∈ X. Then y ∈ Xβ and z ∈ Xα ∩Xβ. Let a = span(α, β) and A = exp(a). Since the +exponential map is a diffeomorphism restricted on p, it follows that A is a closed and compatible +subgroup of G. Then z is fixed by A and by the linearization theorem, Corollary 1.4.2, there +exists A-invariant open subsets Ω ⊂ X and S ⊂ TzX and a A-equivariant diffeomorphism +ϕ : S → Ω such that 0 ∈ S, z ∈ Ω, ϕ(0) = z, dϕ0 = idTzX. Since z = limt→+∞ exp(tα)y, +there is t0 such that exp(t0α)y ∈ Ω. Since Ω is A-invariant, we get y ∈ Ω and also, x ∈ Ω. +Thus we can study all the limits in the linearization S. Hence, keeping in mind Proposition +1.6, we may assume that Ω = Rn, α, β are symmetric matrices of order n satisfying [α, β] = +0. The matrices α and β are simultaneously diagonalizable. Decompose V = � +λ∈Spec(β) Vλ. +This means β|Vλ = λkIdVλk. Since limt�→+∞ exp(tβ)x = y, it follows that x = v0 + v1, where +v0 ∈ V0 and v1 is the sum of some eigenvetors corresponding to negative eigenvalues. Therefore +limt�→+∞ exp(tβ)x = v0 + limt�→+∞ exp(tβ)v1 and limt�→+∞ exp(tβ)v1 = 0. This implies v0 = y. +Let +δ = min +� −λ +2|µ| : λ ∈ Spec(β) ∩ (0, −∞), µ ∈ Spec(α)\{0} +� +. +If λ ∈ Spec(β) ∩ (0, −∞), then +Vλ = W0 ∩ Vλ +� +µ∈Spec(α)\{0} +(Vλ ∩ Wµ), +where W0 = Ker α and α|Wµ = µIdWµ. Let ǫ < δ and let v ∈ Vλ. Then v = w0+� +µ∈Spec(α)\{0} wµ +and so +(α + ǫβ)v = λw0 + +� +µ∈Spec(α)\{0} +(λ + ǫµ)wµ +with λ + ǫµ < 0 for every µ ∈ Spec(α)\{0}. Therefore limt�→+∞ exp(t(β + ǫα))v = 0. This holds +for any λ ∈ Spec(β) ∩ (−∞, 0). Now, keeping in mind that x = y + v1, where v1 is the sum of +eigenvectors of β associated to negative eigenvalues, we have +lim +t�→+∞ exp(t(β + ǫα))x = +lim +t�→+∞ exp(t(β + ǫα))(y + v1) += +lim +t�→+∞ exp(tǫα))y + lim +t�→+∞ exp(t(β + ǫα))v1 += +lim +t�→+∞ exp(tα)y += z. +□ +As a consequence of the above Lemma we get the following result. + +COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS +9 +Theorem 2.4. Let α1, . . . , αn be a basis of a. Let x ∈ X. Set x1 := limt→∞ exp(tα1)x and xi = +limt→∞ exp(tαi)xi−1 for i = 2, . . . , n. Then there exists δ > 0 such that for 0 < ǫ2, . . . , ǫn < δ, +we have +lim +t→∞ exp(t(α1 + ǫ2α2 + · · · + ǫnαn))x = xn, +for any x ∈ X. +Proof. By Theorem 2.2, there exists δ > 0 such that for any 0 < ǫ2, . . . , ǫn < δ, we have +XA = Xα1+ǫ2α2+···+ǫαn. +Let A = exp(a). Let z ∈ XA. By Corollary 1.4.2, there exists A-invariant open subsets Ω ⊂ X +and S ⊂ TzX and a A-equivariant diffeomorphism ϕ : S → Ω such that 0 ∈ S, z ∈ Ω, ϕ(0) = z, +dϕ0 = idTzX. Let x ∈ X. +Set x1 := limt→∞ exp(tα1)x and xi = limt→∞ exp(tα)xi−1 for +i = 2, . . . , n. If xn ∈ Ω, we may choce δ > 0 such that for any 0 < ǫ2, . . . , ǫn < δ, we have +lim +t�→ exp(t(α1 + ǫ2α2 + · · · + ǫαn))x = xn. +By compactness of XA there exist open subsets Ω1, . . . , Ωk satisfying the above property and +such that +XA ⊆ Ω1 ∪ · · · ∪ Ωk. +Let x ∈ X. Set x1 := limt→∞ exp(tα1)x and xi = limt→∞ exp(tαi)xi−1 for i = 2, . . . , n. If +xn ∈ Ωj, for some j = 1, . . . , k, then there exits δj > 0 such that any 0 < ǫ2, . . . , ǫn < δj we have +lim +t�→+∞ exp(t(α1 + ǫ2α2 + · · · + ǫαn))x = xn. +Let δ = min{δ1, . . . , δk}. Then for any 0 < ǫ2, . . . , ǫn < δ we have +lim +t�→+∞ exp(t(α1 + ǫ2α2 + · · · + ǫnαn))x = xn, +for any x ∈ X, concluding the proof. +□ +References +[1] Adams, J.F., Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+182 pp. +[2] Biliotti L., Ghigi A. and Heinzner P., Polar orbitopes, Comm. Ann. Geom. 21 (3), (2013), 1–28. +[3] Biliotti L., Windare, O.J., Stability, analytic stability for real reductive Lie groups, to appear on J. Geom. +Anal. . +[4] Borel A., Ji L., Compactifications of symmetric and locally symmetric spaces. Mathematics: Theory & Ap- +plications. Birkh¨auser Boston Inc., Boston, MA., (2006). + +10 +LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE +[5] Bruasse A., Teleman, Harder-Narasimhan and optimal destabilizing vectors in complex geometry, Ann. Inst. +Fourier (Grenoble) 55 (3), (2005), 1017—1053. +[6] Duistermaat, J.J., Kolk, J. A. C., Lie groups, Universitext. Springer-Verlag, Berlin, 2000. +[7] +Heinzner P., Schwarz G. W. Cartan decomposition of the moment map, Math. Ann. 337, (2007), 197-232. +[8] +Heinzner P.,Schwarz G. W. and St¨otzel H. Stratifications with respect to actions of real reductive groups. +Compos. Math., 144(1), (2008), 163–185. +[9] Kirwan F., Cohomology of quotients in symplectic and algebraic Geometry, Math. Notes 31, Princeton, (1984). +[10] Sjamaar, R. Convexity properties of the momentum mapping re-examinated. Adv. Math. 138 (1), (1998), +46-91. +Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a di Parma (Italy) +Email address: leonardo.biliotti@unipr.it +Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a di Parma (Italy) +Email address: oluwagbengajoshua.windare@unipr.it + diff --git a/YtA0T4oBgHgl3EQfFv9O/content/tmp_files/load_file.txt b/YtA0T4oBgHgl3EQfFv9O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bec358a315f373bf55d740cabfb6104675ecd4c --- /dev/null +++ b/YtA0T4oBgHgl3EQfFv9O/content/tmp_files/load_file.txt @@ -0,0 +1,410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf,len=409 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='02036v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='DG] 5 Jan 2023 COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' We study the singularities of commuting vectors fields of a real submanifold of a K¨ahler manifold Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Introduction Let (Z, ω) be a connected K¨ahler manifold with an holomorphic action of a complex reductive group U C, where U C is the complexification of a compact connected Lie group U with Lie algebra u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' We also assume ω is U-invariant and that there is a U-equivariant momentum map µ : Z → u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By definition, for any ξ ∈ u and z ∈ Z, dµξ = iξZω, where µξ(z) := µ(z)(ξ) and ξZ denotes the fundamental vector field induced on Z by the action of U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', ξZ(z) := d dt ���� t=0 exp(tξ)z (see, for example, [9] for more details on the momentum map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since U is compact we may identify u ∼= u∗ by an Ad(U)-invariant scalar product on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Hence, we consider a momentum map as a u-valued map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', µ : Z → u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Recently, the momentum map has been generalized to the following settings [7, 8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' we say that a subgroup G of U C is compatible if G is closed and the map K ×p → G, (k, β) �→ kexp(β) is a diffeomorphism, where K := G∩U and p := g∩iu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' g is the Lie algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The Lie algebra uC of U C is the direct sum u⊕iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' It follows that G is compatible with the Cartan decomposition U C = Uexp(iu), K is a maximal compact subgroup of G with Lie algebra k and that g = k ⊕ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The inclusion ip ֒→ u induces by restriction, a K-equivariant map µip : Z → (ip)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Using a Ad(U)-invariant scalar product on iu requiring multiplication by i to be an isometry between u and iu, µip can be viewed as the orthogonal projection of iµ(z) onto p given as µp : Z → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 53D20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 14L24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Momentum map, Reductive Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The first author was partially supported by PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics ” and GNSAGA INdAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The first author was supported by the Project MIUR “Geometric Properties of Real and Complex Manifolds” and by GNSAGA of INdAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The second author was supported by the PRIN 2007 MIUR of INdAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 1 2 LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE Let µβ p (z) := ⟨µp(z), β⟩ = ⟨iµ(z), β⟩ = ⟨µ(z), −iβ⟩ = µ−iβ(z) for any β ∈ p and z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then grad µβ p = βZ, where the gradient (grad) is computed with respect to the Riemannian metric induced by the K¨ahler structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The map µp is called the gradient map associated with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' In this paper, a G-invariant compact connected locally closed real submanifold X of Z is fixed and the restriction of µp to X is denoted by µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then µp : X −→ p is a K-equivariant map such that gradµβ p = βX, where the gradient is computed with respect to the induced Riemannian metric on X denoted by (·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let T be a torus of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This means that T is a connected compact Abelian subgroup of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By a Theorem of Koszul, [6], the connected component of ZT := {x ∈ Z : T · x = x} are closed and K¨ahler submanifolds of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let t be the Lie algebra of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' It is well-known that the set � β ∈ t : exp(Rβ) = T � , contains a dense subset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Hence, ZT = ZT C = {p ∈ Z : βZ(p) = 0} , for some β ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' In this paper, we investigate the fixed point set of the action of an Abelian compatible subgroup of U C acting on real submanifold of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let a ⊂ p be an Abelian subalgebra and A = exp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then the A-gradient map on X is given by µa = πa ◦ µp, where πa : p −→ a denotes the orthogonal projection of p onto a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If β ∈ a, let Xβ := {z ∈ X : βX(z) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By the linearization Theorem [8, 10], any connected component of Xβ is an embedded submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Our main results are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The set � β ∈ a : Xβ = XA� is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let β ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The flow R × X −→ X, (t, x) �→ exp(tβ)x, is the gradient flow of the function µβ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By the linearization Theorem, the limit lim t→+∞ exp(tβ)x exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , αn ∈ a be a basis of a an let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Set x1 = limt→+∞ exp(tα1)x and xi = limt→+∞ exp(tαi)xi−1, for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' There exists δ > 0 such that for any 0 < ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δ we have lim t→+∞ exp(t(α1 + ǫ2α2 + · · · + ǫnαn))x = xn, COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS 3 for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Gradient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let U be a compact connected Lie group and U C its complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let (Z, ω) be a K¨ahler manifold on which U C acts holomorphically on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Assume that U preserves ω and that there is a U-equivariant momentum map µ : Z → u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let ⟨·, ·⟩ denote an Ad(U)- invariant scalar product on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Up to a locally constant map, µ is determined by dµξ = iξZω, where µξ(x) := ⟨µ(z), ξ⟩, ξ ∈ u and iξZ denotes the contraction of ω with the vector field ξZ which is given by ξ and the U-action on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let G ⊂ U C be a compatible subgroup of U C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then p ⊂ iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let ⟨·, ·⟩ also denote the Ad(U)- invariant scalar product on iu requiring the multiplication by i to be an isometry of u onto iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If z ∈ Z, then the orthogonal projection of iµ(z) onto p defines a K-equivariant map µp : Z −→ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' In other words, we define µp requiring that for any β ∈ p, we have µβ p := ⟨µp(z), β⟩ = ⟨iµ(z), β⟩ = −⟨µ(z), iβ⟩ = µ−iβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The map µp : Z −→ p is called G-gradient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' βZ is the gradient of µβ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Indeed, since U C acts holomorphically on Z, gradµβ p = gradµ−iβ = J(−iβZ) = βZ, where the gradient is computed with respect to the Riemannian structure denoted by (·, ·) given by the K¨ahler form ω on Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', (v, w) = ω(v, Jw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' For the rest of this paper, fix a G-invariant compact connected locally closed real submanifold X of Z and denote the restriction of µp to X by µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then µp : X −→ p is a K-equivariant map such that gradµβ p = βX, where the gradient is computed with respect to the induced Riemannian metric on X that we also denote by (·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' For any subspace m of g and x ∈ X, let m · x := {ξX(x) : ξ ∈ m} and mx := {α ∈ m : αX(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let a ⊂ p be an Abelian subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then the A = exp(a)-gradient map on X is given by µa = πa ◦ µp, where πa : p −→ a denotes the orthogonal projection of p onto a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then the stabilizer of x denoted by Ax, is compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If a ∈ Ax, then a = exp(β) for a β ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let f(t) = ⟨µa(exp(tβ)x), β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then f(1) = ⟨µa(exp(β)x), β⟩ = ⟨µa(ax), β⟩ = ⟨µa(x), β⟩ = f(0) and f ′(t) =∥ βX(exp(tβ)x) ∥2≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This implies βX(x) = 0 and so β ∈ ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Therefore, Ax = exp(ax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ We recall the Slice Theorem, see [8] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 4 LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [Slice Theorem [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1], [10]] If x ∈ X and µp(x) = 0, there are a Gx- invariant decomposition TxX = g · x ⊕ W, open Gx-invariant subsets S ⊂ W, Ω ⊂ X and a G-equivariant diffeomorphism Ψ : G ×Gx S → Ω, such that 0 ∈ S, x ∈ Ω and Ψ([e, 0]) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Here G ×Gx S denotes the associated bundle with principal bundle G → G/Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If x ∈ X and µp(x) = β, there are a Gβ-invariant decomposition TxX = gβ · x ⊕ W, open Gβ-invariant subsets S ⊂ W, Ω ⊂ X and a Gβ-equivariant diffeomorphism Ψ : Gβ ×Gx S → Ω, such that 0 ∈ S, x ∈ Ω and Ψ([e, 0]) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This follows from applying the previous theorem to the action of Gβ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Indeed, it is well known that Gβ = Kβ exp(pβ) is compatible [4] and the orthogonal projection of iµ onto pβ is the Gβ-gradient map µpβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The group Gβ is also compatible with the Cartan decomposition of (U C)β = (U C)iβ = (U iβ)C and iβ is fixed by the U iβ-action on uiβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This implies that � µuiβ : Z −→ uiβ given by � µuiβ(z) = πuiβ ◦ µ + iβ, where πuiβ is the orthogonal projection of u onto uiβ, is the U iβ-shifted momentum map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The associated Gβ-gradient map is given by � µpβ := µpβ − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Hence, if G is commutative, then we have a Slice Theorem for G at every point of X, see [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='169] and [10] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If β ∈ p, then βX is a vector field on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' a section of the bundle TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' For x ∈ X, the differential is a map TxX → TβX(x)(TX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If βX(x) = 0, there is a canonical splitting TβX(x)(TX) = TxX ⊕ TxX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Accordingly the differential of βX, regarded as a section of TX, splits into a horizontal and a vertical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The horizontal part is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' We denote the vertical part by dβX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The linear map dβX(x) ∈ End(TxX) is indeed the so-called intrinsic differential of βX, regarded as a section in the tangent bundle TX, at the vanishing point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let {ϕt = exp(tβ)} be the flow of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' There is a corresponding flow on TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since ϕt(x) = x, the flow on TX preserves TxX and there it is given by dϕt(x) ∈ Gl(TxX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Thus we get a linear R-action on TxX with infinitesimal generator dβX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If β ∈ p and x ∈ X is a critical point of µβ p , then there are open invariant neighborhoods S ⊂ TxX and Ω ⊂ X and an R-equivariant diffeomorphism Ψ : S → Ω, such that 0 ∈ S, x ∈ Ω, Ψ(0) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Here t ∈ R acts as dϕt(x) on S and as ϕt on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since exp : p −→ G is a diffeomorphism onto the image, the subgroup H := exp(Rβ) is closed and so it is compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Hence, it is enough to apply the previous corollary to the H-action on X and the value at x of the corresponding gradient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS 5 Let β ∈ p and let Xβ = {x ∈ X : βX(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, Xβ is a smooth, possibly disconnected, submanifold of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Xβ is Gβ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let g ∈ Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since βX(gx) = (dg)x(βX(x)), it follows that gx ∈ Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ Let x ∈ Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let D2µβ p (x) denote the Hessian, which is a symmetric operator on TxX such that (D2µβ p(x)(v, v) = d2 dt2 (µβ p ◦ γ)(0) where γ is a smooth curve, γ(0) = x and ˙γ(0) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Using the inner scalar product (·, ·)(x) one obtains an associate symmetric endomorphism L(µβ p )(x) of the Euclidian vector space (TxX, (·, ·)(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Denote by V− (respectively V+) the sum of the eigenspaces of L(µβ p )(x) cor- responding to negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' positive) eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Denote by V0 the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since L(µβ p )(x) is a symmetric endomorphism, we get an orthogonal decomposition TxX = V− ⊕ V0 ⊕ V+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' (1) Let α : G → X be the orbit map: α(g) := gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The differential dαe is the map ξ �→ ξX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The following result is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' A proof is given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If β ∈ p and x ∈ Xβ then L(µβ p )(x) = dβX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Moreover dαe(rβ±) ⊂ V± and dαe(gβ) ⊂ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If G acts transitively on X, then these are equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' For every β ∈ p, µβ p is a Morse-Bott function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, we have TxXβ = V0 for x ∈ Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Hence, the first statement of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='6 shows that the Hessian is non-degenerate in the normal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ Let c1 > · · · > cr be the critical values of µβ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The corresponding level sets of µβ p , Ci := (µβ p )−1(ci) are submanifolds which are union of components of Crit(µβ p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The function µβ p defines a gradient flow generated by its gradient which is given by βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4, it follows that for any x ∈ X the limit: ϕ∞(x) := lim t→+∞ exp(tβ)x, 6 LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let us denote by W β i the unstable manifold of the critical component Ci for the gradient flow of µβ p : W β i := {x ∈ X : ϕ∞(x) ∈ Ci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' (2) Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4, we have the following well-known decomposition of X into unstable manifolds with respect to µβ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' In the above assumption, we have X = r� i=1 W β i , (3) and for any i the map: (ϕ∞)|Wi : W β i → Ci, is a smooth fibration with fibres diffeomorphic to Rli where li is the index (of negativity) of the critical submanifold Ci 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof of the main results Suppose X ⊂ Z is G-invariant compact connected real submanifold of Z with the gradient map µp : X → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' In this section we study the fixed point set with respect to vector fields induced by α, β ∈ p, using the G-action, satisfying [α, β] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let β, α ∈ p be such that [β, α] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' There exists δ > 0 such that for any ǫ ∈ (0, δ) Xβ+ǫα = Xβ ∩ Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let ǫ > 0 and let A = exp(a), where a = span(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since the exponential map is a diffeomorphism restricted on p, it follows that A is a closed and compatible subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let XA denote the fixed point set of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', XA = {z ∈ X : A · x = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='3, XA = Xβ ∩ Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, both Xβ ∩ Xα and Xα+ǫβ are compact submanifolds satisfying Xβ ∩ Xα ⊆ Xα+ǫβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since Xα+ǫβ is A-invariant, and so there exists A-gradient map [8], any connected component of Xα+ǫβ contains a connected component of Xα ∩ Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let x ∈ Xα∩Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let C be the connected component of x and let C′ be the connected compo- nent of Xα+ǫβ containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since x is fixed by A, by the linearization theorem, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, there exists A-invariant open subsets Ω ⊂ X and S ⊂ TxX and a A-equivariant diffeomorphism ϕ : S → Ω such that 0 ∈ S, x ∈ Ω, ϕ(0) = x, dϕ0 = idTxX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Thus we may assume that Ω = Rn, COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS 7 α, β are symmetric matrices of order n satisfying [α, β] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Moreover, TxXα+ǫβ = Ker (α + ǫβ) and TxXα ∩ TxXβ = Ker α ∩ Ker β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The matrices α and β are simultaneously diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , en} be a basis of Rn such that αei = aiei and βei = biei for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let J = {1 ≤ i ≤ n : aibi ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Pick δ = min{|ai| |bi| : i ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Now, (α + ǫβ)ei = 0 if and only if ai + ǫbi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If ai ̸= 0, then bi ̸= 0 and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Therefore, for any ǫ < δ, we get (α + ǫβ)ei = 0, if and only if ai = bi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Therefore, Ker (α + ǫβ) = Ker α ∩ Ker β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since C ⊂ C′ and TxC = TxC′, keeping in mind that both C and C′ are compact, it follows that C = C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since Xα+ǫβ has finitely many connected components, it follows that there exists δ > 0 such that for any 0 < ǫ < δ, we have Xα ∩ Xβ = Xα+ǫβ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let a ⊂ p be an Abelian subalgebra and let A = exp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then the set � α ∈ a : XA = Xα� is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , αn be a basis of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then XA = Xα1 ∩ · · · ∩ Xαn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By the above Lemma, there exists δ > 0 such that for any ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δ, we have (4) XA = Xα1+ǫ2α2+···+ǫnαn Let α ∈ a different form 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' It is well known that there exists α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' αn ∈ a such that α, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , αn is a basis of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By (4), for any neighborhood U of α, there exists β ∈ U such that XA = Xβ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ The following Lemma is proved in [3], see also [5, pag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 1036].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' For sake of completeness we give the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let x ∈ X and β, α ∈ p be such that [β, α] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Set y := limt→∞ exp(tβ)x and z := limt→∞ exp(tα)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let δ be as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then for 0 < ǫ < δ, lim t→∞ exp(t(β + ǫα)) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 8 LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Fix x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then y ∈ Xβ and z ∈ Xα ∩Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let a = span(α, β) and A = exp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since the exponential map is a diffeomorphism restricted on p, it follows that A is a closed and compatible subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then z is fixed by A and by the linearization theorem, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, there exists A-invariant open subsets Ω ⊂ X and S ⊂ TzX and a A-equivariant diffeomorphism ϕ : S → Ω such that 0 ∈ S, z ∈ Ω, ϕ(0) = z, dϕ0 = idTzX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since z = limt→+∞ exp(tα)y, there is t0 such that exp(t0α)y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since Ω is A-invariant, we get y ∈ Ω and also, x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Thus we can study all the limits in the linearization S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Hence, keeping in mind Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='6, we may assume that Ω = Rn, α, β are symmetric matrices of order n satisfying [α, β] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' The matrices α and β are simultaneously diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Decompose V = � λ∈Spec(β) Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This means β|Vλ = λkIdVλk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Since limt�→+∞ exp(tβ)x = y, it follows that x = v0 + v1, where v0 ∈ V0 and v1 is the sum of some eigenvetors corresponding to negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Therefore limt�→+∞ exp(tβ)x = v0 + limt�→+∞ exp(tβ)v1 and limt�→+∞ exp(tβ)v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This implies v0 = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let δ = min � −λ 2|µ| : λ ∈ Spec(β) ∩ (0, −∞), µ ∈ Spec(α)\\{0} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If λ ∈ Spec(β) ∩ (0, −∞), then Vλ = W0 ∩ Vλ � µ∈Spec(α)\\{0} (Vλ ∩ Wµ), where W0 = Ker α and α|Wµ = µIdWµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let ǫ < δ and let v ∈ Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then v = w0+� µ∈Spec(α)\\{0} wµ and so (α + ǫβ)v = λw0 + � µ∈Spec(α)\\{0} (λ + ǫµ)wµ with λ + ǫµ < 0 for every µ ∈ Spec(α)\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Therefore limt�→+∞ exp(t(β + ǫα))v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' This holds for any λ ∈ Spec(β) ∩ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Now, keeping in mind that x = y + v1, where v1 is the sum of eigenvectors of β associated to negative eigenvalues, we have lim t�→+∞ exp(t(β + ǫα))x = lim t�→+∞ exp(t(β + ǫα))(y + v1) = lim t�→+∞ exp(tǫα))y + lim t�→+∞ exp(t(β + ǫα))v1 = lim t�→+∞ exp(tα)y = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ As a consequence of the above Lemma we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' COMMON SINGULARITIES OF COMMUTING VECTOR FIELDS 9 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , αn be a basis of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Set x1 := limt→∞ exp(tα1)x and xi = limt→∞ exp(tαi)xi−1 for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then there exists δ > 0 such that for 0 < ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δ, we have lim t→∞ exp(t(α1 + ǫ2α2 + · · · + ǫnαn))x = xn, for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, there exists δ > 0 such that for any 0 < ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δ, we have XA = Xα1+ǫ2α2+···+ǫαn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let A = exp(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let z ∈ XA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='2, there exists A-invariant open subsets Ω ⊂ X and S ⊂ TzX and a A-equivariant diffeomorphism ϕ : S → Ω such that 0 ∈ S, z ∈ Ω, ϕ(0) = z, dϕ0 = idTzX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Set x1 := limt→∞ exp(tα1)x and xi = limt→∞ exp(tα)xi−1 for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If xn ∈ Ω, we may choce δ > 0 such that for any 0 < ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δ, we have lim t�→ exp(t(α1 + ǫ2α2 + · · · + ǫαn))x = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' By compactness of XA there exist open subsets Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , Ωk satisfying the above property and such that XA ⊆ Ω1 ∪ · · · ∪ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Set x1 := limt→∞ exp(tα1)x and xi = limt→∞ exp(tαi)xi−1 for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' If xn ∈ Ωj, for some j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , k, then there exits δj > 0 such that any 0 < ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δj we have lim t�→+∞ exp(t(α1 + ǫ2α2 + · · · + ǫαn))x = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Let δ = min{δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , δk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Then for any 0 < ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' , ǫn < δ we have lim t�→+∞ exp(t(α1 + ǫ2α2 + · · · + ǫnαn))x = xn, for any x ∈ X, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' □ References [1] Adams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Lectures on Lie groups, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Benjamin, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', New York-Amsterdam 1969 xii+182 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [2] Biliotti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Ghigi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' and Heinzner P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Polar orbitopes, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 21 (3), (2013), 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [3] Biliotti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Windare, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Stability, analytic stability for real reductive Lie groups, to appear on J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [4] Borel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Ji L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Compactifications of symmetric and locally symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Mathematics: Theory & Ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Birkh¨auser Boston Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Boston, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 10 LEONARDO BILIOTTI AND OLUWAGBENGA JOSHUA WINDARE [5] Bruasse A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Teleman, Harder-Narasimhan and optimal destabilizing vectors in complex geometry, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Fourier (Grenoble) 55 (3), (2005), 1017—1053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [6] Duistermaat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Kolk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Lie groups, Universitext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [7] Heinzner P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Schwarz G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Cartan decomposition of the moment map, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 337, (2007), 197-232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [8] Heinzner P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=',Schwarz G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' and St¨otzel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Stratifications with respect to actions of real reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', 144(1), (2008), 163–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [9] Kirwan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=', Cohomology of quotients in symplectic and algebraic Geometry, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Notes 31, Princeton, (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' [10] Sjamaar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Convexity properties of the momentum mapping re-examinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' 138 (1), (1998), 46-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content=' Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a di Parma (Italy) Email address: leonardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='biliotti@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='it Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a di Parma (Italy) Email address: oluwagbengajoshua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='windare@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf'} diff --git a/_NE1T4oBgHgl3EQfUwPr/vector_store/index.faiss b/_NE1T4oBgHgl3EQfUwPr/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..cff680b0e6510b134ae3d62d0114d06854adc362 --- /dev/null +++ b/_NE1T4oBgHgl3EQfUwPr/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b82257ee469b048b88aa13a631eb7e4f97dccfdd3c6a3f2ff6eadeeb6f2ea9d2 +size 1835053 diff --git a/_dAzT4oBgHgl3EQfhPzZ/vector_store/index.pkl b/_dAzT4oBgHgl3EQfhPzZ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..785a69c25062db88cfc6ee8e5f6bc2a7d643ff78 --- /dev/null +++ b/_dAzT4oBgHgl3EQfhPzZ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bcf33dd4402f83aee477b1fc19df9caf0a631628834c04f8737e52d8b5496f7a +size 46126 diff --git a/atE2T4oBgHgl3EQfZwex/content/tmp_files/2301.03868v1.pdf.txt b/atE2T4oBgHgl3EQfZwex/content/tmp_files/2301.03868v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a13de9258db8dfba608999721aef32ba93ccbead --- /dev/null +++ b/atE2T4oBgHgl3EQfZwex/content/tmp_files/2301.03868v1.pdf.txt @@ -0,0 +1,321 @@ +arXiv:2301.03868v1 [cond-mat.stat-mech] 10 Jan 2023 +THERMAL ENTANGLEMENT OF A GEOMETRICALLY FRUSTRATED SPIN-1 +HEISENBERG DIAMOND CLUSTER +Azadeh Ghannadan,1 Katar´ına Karl’ov´a,1 and Jozef Streˇcka1 +1Department of Theoretical Physics and Astrophysics, Faculty of Science, +P. J. ˇSaf´arik University, Park Angelinum 9, 04001 Koˇsice, Slovakia +I. +INTRODUCTION +Magnetic properties of a spin-1 Heisenberg diamond +cluster were recently studied in Ref. +[1]. +It has +been shown that the tetranuclear nickel complex [Ni4(µ- +CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine) +[2], which provides an experimental realization of the +spin-1 Heisenberg diamond cluster, displays in a low- +temperature magnetic curve one-half and three-quarter +plateaus. In the present work the bipartite entanglement +between two spin pairs either on a diagonal or a side of +the spin-1 Heisenberg diamond cluster N12 and N13 will +be quantified through the measure called the negativity +[3]. The separability criterion of a state expresses that +a state is entangled if and only if there is at least one +negative eigenvalue of the partially transposed density +matrix of the state, otherwise it is separable [4]. Hence, +the negativity is defined as the sum of absolute value of +negative eigenvalues of the partially transposed density +matrix [5]. +II. +MODEL AND METHODS +The spin-1 Heisenberg diamond cluster is defined +through the Hamiltonian: +ˆH = J1 ( ˆS1 · ˆS2)+ J2( ˆS1 + ˆS2)·( ˆS3 + ˆS4)− h +4 +� +i=1 +ˆSz +i , (1) +where ˆSi are the spin-1 operators, J1 and J2 are the +coupling constants along the diagonal and side of the +diamond cluster and h is the magnetic field. The Hamil- +tonian (1) was fully diagonalized in Ref. [1] and hence, +the negativity can be calculated from a full set of eigen- +values and eigenvectors of the Hamiltonian (1). The cal- +culation details will be presented together with further +results in our future work [3]. +To calculate the nega- +tivity one should first obtain the density operator and +the corresponding density matrix of the given bipartite +state. Then the density matrix is partially transposed +with respect to one of its subsystems. Finally negativity +is calculated as the sum of the absolute value of negative +eigenvalues of the partially transposed density matrix ac- +cording to the following formula [5] +N = +� +λi<0 +|λi| = +� +i +|λi| − λi +2 +(2) +III. +RESULTS AND DISCUSSIONS +In this short paper we will discuss the thermal entan- +glement of the spin-1 Heisenberg diamond cluster. Ther- +mal dependencies of the negativity N12 between a spin +pair on the diagonal, and the negativity N13 between a +spin pair on the side are depicted in Fig. 1 for two in- +teraction ratios J2/J1 = 0.5 and J2/J1 = 1.5. In both +figures the curves starting from the same point in a zero- +temperature limit T =0 imply the same ground state, note +that the upper (lower) curve refers to the higher (lower) +magnetic field. The curves with a single asymptotic value +refer to a phase transition from one ground state to the +other one. If the interaction ratio is J2/J1 = 0.5, the +system goes through four different ground states upon +increasing of the magnetic field, whereby N12=0 holds +in one ground state and N13=0 in three of them. The +results of nonzero negativities are exhibited in Fig. 1(a) +and (c). On the other hand the system goes through five +ground states for J2/J1 = 1.5, whereby N12=0 holds in +two ground states and N13=0 in one ground state. The +results of nonzero negativities are shown in Fig. 1(b) and +(d). +It is obvious from Fig. 1 that negativity mostly de- +creases upon increasing of temperature until it terminates +at threshold temperature. However, one may also a more +striking re-entrance of the negativity when it is initially +zero at low enough temperatures, then it shows a small +temperature-induced rise until it repeatedly tends to zero +at higher temperatures. A competition of high magnetic +fields with slight thermal effects may thus cause a clas- +sical ground state to gain quantum properties. Another +prominent feature which is observed in some thermal de- +pendencies of negativity, is presence of the nonanalytical +point known as a kink. The kinks occur when one or some +negative eigenvalues of the partially transposed density +matrix become zero at a given temperature. The kinks +are also manifested in the corresponding magnetic-field +dependencies of the negativity [3]. +Now, let us shed light on thermal dependencies of +the negativity of the tetranuclear nickel complex [Ni4(µ- +CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine) +to be further abbreviated as the Ni4 complex, which af- +fords a geometrically frustrated compound modeled by +the Hamiltonian (1). +High-field data measured at low +enough temperature T =1.3 showed presence of one-half +and three-quarter plateaus in the magnetization curve of +the Ni4 compound. Moreover, the values of the coupling +constant between the nickel ions on the diagonal and side + +0.0 +0.3 +0.6 +0.9 +1.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + h / J +1 + 0 + 0.1 + 0.35 + 0.6 + 1.95 + 2.05 + 2.95 + 3.05 + + +N +12 +k +B +T / J +1 +J +2 + / J +1 += 0.5 +(a) +0.0 +0.2 +0.4 +0.6 +0.00 +0.04 +0.08 +0.12 + h / J +1 + 1.6 + 1.8 + 2.9 + 3.3 + 4.4 + 4.5 + 6.0 + 6.1 + + +N +12 +k +B +T / J +1 +J +2 + / J +1 += 1.5 +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.00 +0.05 +0.10 +0.15 + h / J +1 + 0.0 + 0.1 + 0.45 + 0.5 + 0.55 + + +N +13 +k +B +T / J +1 +J +2 + / J +1 += 0.5 +(c) +0.0 +0.4 +0.8 +1.2 +1.6 +2.0 +0.0 +0.1 +0.2 +0.3 + + 0 + 1.4 + 1.8 + 2.9 + 3.2 + 4.4 + 4.6 + 5.8 + 6.2 + + +N +13 +k +B +T / J +1 +J +2 + / J +1 += 1.5 +(d) +h / J +1 +FIG. 1. Temperature variations of the negativities N12 and N13 for the diagonal and side of the spin-1 Heisenberg diamond +cluster for two different values of the interaction ratio: J2/J1=0.5 (a,c) and J2/J1=1.5 (b,d). +of Ni4 complex are known J1/kB=41.4 K and J2/kB=9.2 +K respectively. The coupling constant ratio J2/J1 =0.222 +accordingly falls into a parameter region, where the sys- +tem goes through three different ground states upon in- +creasing of the magnetic field and entanglement exists +just between the spin pairs on the diagonal. Fig. 2 dis- +plays temperature dependencies of negativity N12 of the +Ni4 complex, which is fully entangled N12=1 from B=0 +T up to B = 40T due to formation of two singlet bonds +within the diagonal spin pair. If the magnetic field ranges +from 40 T to 65 T there exists another ground state, +in which one singlet bond breaks and N12=0.5. All de- +scending curves of negativity terminate in the threshold +temperature about 55 K. Beyond 65 T all four spins are +directed to the magnetic field within the classical fully +saturated ground state with N12=0. It is clear from Fig. +2 that the interesting re-entrance of the negativity occur +for the magnetic fields slightly higher than 65 T. +IV. +CONCLUSIONS +In summery we have investigated in detail temperature +dependencies of the negativity of the spin-1 Heisenberg +diamond cluster for two different spin pairs on the diag- +onal and the side of the diamond cluster, which typically +monotonically decrease upon rising temperature until +0 +10 +20 +30 +40 +50 +60 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + B [T] + 0 + 35 + 40 + 50 + 65 + 70 + + +N +12 +T [K] +FIG. 2. Temperature variations of the negativity N12 of the +Ni4 complex at a few different magnetic fields. +they vanish at the threshold temperature. The striking +re-entrance of the negativity was observed at high +enough magnetic fields when relatively weak thermal +entanglement can be induced above the classical fully +polarized ground state by thermal fluctuations. Besides, +the threshold temperature of the negativity of the Ni4 +complex was conjectured about 55 K. + +ACKNOWLEDGMENTS +This work was financially supported by the grant Nos. +APVV-20-0150 and VVGS-PF-2022-2101. +[1] K. Karl’ov´a et al. Magnetochemistry. 6, 59 (2020). +[2] A. Escuer et al. J. Chem. Soc., Datrlton Trans., 20, 3473 +(1998). +[3] A. Ghannadan, K. Karl’ov´a, J. Seˇcka, Magnetochemistry. +8, 156 (2022). +[4] A. Peres, Phys. Rev. Lett. 77, 1413 (1996). +[5] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 +(2002). + diff --git a/atE2T4oBgHgl3EQfZwex/content/tmp_files/load_file.txt b/atE2T4oBgHgl3EQfZwex/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fbc7c77cf390a0cf1bfa2ce5016d34070ff1573 --- /dev/null +++ b/atE2T4oBgHgl3EQfZwex/content/tmp_files/load_file.txt @@ -0,0 +1,175 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf,len=174 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='03868v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='stat-mech] 10 Jan 2023 THERMAL ENTANGLEMENT OF A GEOMETRICALLY FRUSTRATED SPIN-1 HEISENBERG DIAMOND CLUSTER Azadeh Ghannadan,1 Katar´ına Karl’ov´a,1 and Jozef Streˇcka1 1Department of Theoretical Physics and Astrophysics, Faculty of Science, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' ˇSaf´arik University, Park Angelinum 9, 04001 Koˇsice, Slovakia I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' INTRODUCTION Magnetic properties of a spin-1 Heisenberg diamond cluster were recently studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' It has been shown that the tetranuclear nickel complex [Ni4(µ- CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine) [2], which provides an experimental realization of the spin-1 Heisenberg diamond cluster, displays in a low- temperature magnetic curve one-half and three-quarter plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' In the present work the bipartite entanglement between two spin pairs either on a diagonal or a side of the spin-1 Heisenberg diamond cluster N12 and N13 will be quantified through the measure called the negativity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The separability criterion of a state expresses that a state is entangled if and only if there is at least one negative eigenvalue of the partially transposed density matrix of the state, otherwise it is separable [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Hence, the negativity is defined as the sum of absolute value of negative eigenvalues of the partially transposed density matrix [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' MODEL AND METHODS The spin-1 Heisenberg diamond cluster is defined through the Hamiltonian: ˆH = J1 ( ˆS1 · ˆS2)+ J2( ˆS1 + ˆS2)·( ˆS3 + ˆS4)− h 4 � i=1 ˆSz i , (1) where ˆSi are the spin-1 operators, J1 and J2 are the coupling constants along the diagonal and side of the diamond cluster and h is the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The Hamil- tonian (1) was fully diagonalized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [1] and hence, the negativity can be calculated from a full set of eigen- values and eigenvectors of the Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The cal- culation details will be presented together with further results in our future work [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' To calculate the nega- tivity one should first obtain the density operator and the corresponding density matrix of the given bipartite state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Then the density matrix is partially transposed with respect to one of its subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Finally negativity is calculated as the sum of the absolute value of negative eigenvalues of the partially transposed density matrix ac- cording to the following formula [5] N = � λi<0 |λi| = � i |λi| − λi 2 (2) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS In this short paper we will discuss the thermal entan- glement of the spin-1 Heisenberg diamond cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Ther- mal dependencies of the negativity N12 between a spin pair on the diagonal, and the negativity N13 between a spin pair on the side are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 1 for two in- teraction ratios J2/J1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 and J2/J1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' In both figures the curves starting from the same point in a zero- temperature limit T =0 imply the same ground state, note that the upper (lower) curve refers to the higher (lower) magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The curves with a single asymptotic value refer to a phase transition from one ground state to the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' If the interaction ratio is J2/J1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5, the system goes through four different ground states upon increasing of the magnetic field, whereby N12=0 holds in one ground state and N13=0 in three of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The results of nonzero negativities are exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 1(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' On the other hand the system goes through five ground states for J2/J1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5, whereby N12=0 holds in two ground states and N13=0 in one ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The results of nonzero negativities are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 1(b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' It is obvious from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 1 that negativity mostly de- creases upon increasing of temperature until it terminates at threshold temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' However, one may also a more striking re-entrance of the negativity when it is initially zero at low enough temperatures, then it shows a small temperature-induced rise until it repeatedly tends to zero at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' A competition of high magnetic fields with slight thermal effects may thus cause a clas- sical ground state to gain quantum properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Another prominent feature which is observed in some thermal de- pendencies of negativity, is presence of the nonanalytical point known as a kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The kinks occur when one or some negative eigenvalues of the partially transposed density matrix become zero at a given temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The kinks are also manifested in the corresponding magnetic-field dependencies of the negativity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Now, let us shed light on thermal dependencies of the negativity of the tetranuclear nickel complex [Ni4(µ- CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine) to be further abbreviated as the Ni4 complex, which af- fords a geometrically frustrated compound modeled by the Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' High-field data measured at low enough temperature T =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='3 showed presence of one-half and three-quarter plateaus in the magnetization curve of the Ni4 compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Moreover, the values of the coupling constant between the nickel ions on the diagonal and side 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 h / J 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='05 N 12 k B T / J 1 J 2 / J 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='12 h / J 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='1 N 12 k B T / J 1 J 2 / J 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='15 h / J 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='55 N 13 k B T / J 1 J 2 / J 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='3 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 N 13 k B T / J 1 J 2 / J 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 (d) h / J 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Temperature variations of the negativities N12 and N13 for the diagonal and side of the spin-1 Heisenberg diamond cluster for two different values of the interaction ratio: J2/J1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 (a,c) and J2/J1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5 (b,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' of Ni4 complex are known J1/kB=41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 K and J2/kB=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 K respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The coupling constant ratio J2/J1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='222 accordingly falls into a parameter region, where the sys- tem goes through three different ground states upon in- creasing of the magnetic field and entanglement exists just between the spin pairs on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 2 dis- plays temperature dependencies of negativity N12 of the Ni4 complex, which is fully entangled N12=1 from B=0 T up to B = 40T due to formation of two singlet bonds within the diagonal spin pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' If the magnetic field ranges from 40 T to 65 T there exists another ground state, in which one singlet bond breaks and N12=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' All de- scending curves of negativity terminate in the threshold temperature about 55 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Beyond 65 T all four spins are directed to the magnetic field within the classical fully saturated ground state with N12=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 2 that the interesting re-entrance of the negativity occur for the magnetic fields slightly higher than 65 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' CONCLUSIONS In summery we have investigated in detail temperature dependencies of the negativity of the spin-1 Heisenberg diamond cluster for two different spin pairs on the diag- onal and the side of the diamond cluster, which typically monotonically decrease upon rising temperature until 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content='0 B [T] 0 35 40 50 65 70 N 12 T [K] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Temperature variations of the negativity N12 of the Ni4 complex at a few different magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' they vanish at the threshold temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' The striking re-entrance of the negativity was observed at high enough magnetic fields when relatively weak thermal entanglement can be induced above the classical fully polarized ground state by thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Besides, the threshold temperature of the negativity of the Ni4 complex was conjectured about 55 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was financially supported by the grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' APVV-20-0150 and VVGS-PF-2022-2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Karl’ov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Magnetochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 6, 59 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Escuer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=', Datrlton Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=', 20, 3473 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Ghannadan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Karl’ov´a, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Seˇcka, Magnetochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 8, 156 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Peres, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' 77, 1413 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Vidal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Werner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} +page_content=' A 65, 032314 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE2T4oBgHgl3EQfZwex/content/2301.03868v1.pdf'} diff --git a/b9E1T4oBgHgl3EQfdgQp/vector_store/index.pkl b/b9E1T4oBgHgl3EQfdgQp/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ed834fa6babb30c3513e4b80b1d30c14ab051936 --- /dev/null +++ b/b9E1T4oBgHgl3EQfdgQp/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:11a3c56e9e76e218426ae617ff3649f4aadb0fa5161dba0852e8851449e08d63 +size 98109 diff --git a/bdE_T4oBgHgl3EQfzRz6/vector_store/index.faiss b/bdE_T4oBgHgl3EQfzRz6/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..650fde50a4d2b8b4d39107c0e740af91fdb2b3cc --- /dev/null +++ b/bdE_T4oBgHgl3EQfzRz6/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:17e6bd84dadaaf7b5f54eabd611e58e25804c0f9b409b4dfb0d628d57cf3bc94 +size 3866669 diff --git a/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf b/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..52e0347a8f69b92174d17b4d2ec43d20c0ffb0fb --- /dev/null +++ b/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:48df4c4ba0f58a65bc54ed43ca1d700f16c2fc4d5f04d6a17d7b01699a191528 +size 3714727 diff --git a/cNE5T4oBgHgl3EQfEg7V/content/tmp_files/2301.05415v1.pdf.txt b/cNE5T4oBgHgl3EQfEg7V/content/tmp_files/2301.05415v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..66ed58301eeea989c79fe59314b29004e1705ab4 --- /dev/null +++ b/cNE5T4oBgHgl3EQfEg7V/content/tmp_files/2301.05415v1.pdf.txt @@ -0,0 +1,2268 @@ +Transactions on Robotics, VOL. XX, NO. XX, XXXX +1 +Guaranteed Encapsulation of Targets with Unknown +Motion by a Minimalist Robotic Swarm +Himani Sinhmar, Hadas Kress-Gazit +Abstract—We present a decentralized control algorithm for +a robotic swarm given the task of encapsulating static and +moving targets in a bounded unknown environment. We consider +minimalist robots without memory, explicit communication, or lo- +calization information. The state-of-the-art approaches generally +assume that the robots in the swarm are able to detect the relative +position of neighboring robots and targets in order to provide +convergence guarantees. In this work, we propose a novel control +law for the guaranteed encapsulation of static and moving targets +while avoiding all collisions, when the robots do not know the +exact relative location of any robot or target in the environment. +We make use of the Lyapunov stability theory to prove the +convergence of our control algorithm and provide bounds on +the ratio between the target and robot speeds. Furthermore, our +proposed approach is able to provide stochastic guarantees under +the bounds that we determine on task parameters for scenarios +where a target moves faster than a robot. Finally, we present +an analysis of how the emergent behavior changes with different +parameters of the task and noisy sensor readings. +Index Terms—Collision avoidance, Decentralized control, Min- +imalist robot swarm, Lyapunov stability, Target tracking +I. INTRODUCTION +T +YPICAL approaches to swarm robotics propose simple +local behaviors for large numbers of simple robots such +that they collectively accomplish a complex task; many ap- +proaches study the properties of the emergent behavior [1]–[3]. +In this work, we consider a swarm consisting of homogeneous +robots which are minimalist; they have no memory, cannot +broadcast or receive location information from their neighbors +and are unable to plan ahead. Minimalistic robotic swarms [4], +[5] have a number of applications, ranging from nanomedicine +to underwater monitoring and surveillance [6], [7], where +robots might not be able to efficiently communicate with a +central controller or with each other, and might not have +the ability to self localize. For example, in an underwater +mission, communication may be limited to acoustic signals, +which are sensitive to interference and lead to errors in the +relative positioning of nearby entities. +In this paper, we focus on the problem of encapsulating +multiple targets, which are moving in unknown motion +patterns, by a minimalist robotic swarm. A robot in the +swarm has no knowledge of the exact relative location of +nearby robots, targets, or the boundary of the environment. +This work extends our previous work on encapsulating static +targets [5] by addressing moving targets. We develop an +orbiting behavior for robots to encapsulate the targets, in +The +authors +are +with +the +Sibley +School +of +Mechanical +and +Aerospace +Engineering, +Cornell +University, +Ithaca, +NY, +14853 +USA. +{hs962,hadaskg}@cornell.edu. This work is supported by NSF +EFMA-1935252. +addition to the searching for targets and avoiding collisions +within the swarm, as in [5]. We compare the efficiency of +our previous algorithm with the one introduced in this paper. +Furthermore, we also show the behavior of our algorithm +when applied to non-circular robots. +Related Work: There has been extensive work on developing +various techniques to localize and track a moving target +while ensuring collision avoidance [8]–[13]. In [14], authors +introduced a motion planning strategy for a single robot based +on velocity pursuit to intercept a target moving with unknown +maneuvers. For target tracking using a multi-robot system, +most approaches use artificial potential fields to design a +controller consisting of a virtual attraction force to move +towards a target and a repulsion force to avoid collision with +obstacles [15]. Another widely used approach to guarantee +collision avoidance with dynamic obstacles is using a limit +cycle method [16]. The authors of [17] introduced a hybrid +approach where they instead used the limit cycle method to +encircle a moving target using a swarm of holonomic robots, +and artificial potential fields for collision avoidance. Since the +use of the limit cycle method, either for surrounding a target or +avoiding collision with obstacles requires the exact knowledge +of the neighbor’s relative position information, we cannot use +it for our minimalist robotic swarm. +Pursuit-evasion games [18]–[20] provide guarantees for +catching a faster-moving evader by constructing an encircling +formation of pursuers composed of a series of Apollonius +circles around a target and slowly closing the escape paths +of the evader. In this approach, an evader is captured if a +pursuer meets the evader at the same point at the same time. +Most of the pursuit-evasion methods in the literature assume +knowledge of the target’s motion model. In this work, we do +not assume such knowledge. +Existing research [21] in “hunting” of dynamic targets +generally makes use of communication within the team and +formation-keeping control strategies, while approaching the +target, to ensure that all of the escaping routes of the targets +are occupied by the robots. Work in [22] developed a leader- +follower strategy based on the behavior of wolves to hunt +a randomly moving target with unexpected behaviors. The +authors of [23] proposed a limit cycle based algorithm using +a neural oscillator to surround a target moving with unknown +but constant velocity. The authors of [24] utilized rule-based +mechanisms using only relative positions of neighbors and no +direct communication within the swarm for surrounding an +escaping target by introducing a circulating behavior in the +swarm. +Recent research in colloidal swarms has shown the capture +arXiv:2301.05415v1 [cs.RO] 13 Jan 2023 + +Transactions on Robotics, VOL. XX, NO. XX, XXXX +2 +of multiple randomly moving targets using self-organization +control schemes. In [25], the authors designed a stochastic +centralized controller for an intelligent colloidal micro-robotic +swarm to capture multiple Brownian targets in a maze. In +[26], the authors show via simulations, the feedback-controlled +reconfigurability of colloidal particles that act as a swarm +capable of capturing and transporting microscopic Brownian +cargo. +To implement a distributed approach of searching and +encircling targets in an inexpensive and efficient way, in +[27] the authors developed a new dual-rotating proximity +sensor to obtain relative position information of neighbors for +tracking multiple targets with a minimalist swarm. Authors +of [28] proposed a scheme to estimate the global quantities +required by the controller in a decentralized way using only +local information exchange between robots for the guaranteed +encirclement of a 2D or 3D target. +While the above approaches successfully solve the target +encirclement while avoiding collisions, most of them rely +on the assumption that robots have knowledge of the exact +relative location of both their neighbors and the target. +Furthermore, it is a common assumption that the average +speed of the agents in the swarm is greater than that of the +moving target to guarantee encapsulation [29]. In contrast, +in this work, we provide guarantees on the encapsulation of +dynamic targets without the requirement of accurate (relative) +location information and without direct communication within +the swarm. +Contributions: This paper’s contributions are: (i) a discrete- +time decentralized control law for a minimalist robotic swarm +that guarantees the encapsulation of dynamic targets, for +different target motion models, without accurate detection +of the relative location of either the targets or neighboring +robots, given certain bounds are met (ii) sensor-placement +dependent bounds on the ratio between the target and robot +speeds to guarantee encapsulation, (iii) proof of stochastic +convergence of our control law for scenarios when a target is +moving faster than a robot, and (iv) simulations and analysis +of emergent behavior of the swarm in the presence of sensor +noise and different task parameters. +II. DEFINITIONS +In this section, we provide definitions from [5] that we use +throughout. +Environment: +We +consider +a +2D +convex +bounded +environment E +⊆ +R2. The environment has a fixed +global frame I. +Robot: We model a robot, R = (cr, γr, rr, p, Z), as a disk +of radius rr centered at cr ∈ E with heading γr ∈ S. The +shape of a robot does not affect the analysis presented in +the paper since the robot can always be circumscribed by a +circle of radius rr. Each robot is reactive, memoryless, has no +knowledge of the relative locations of other robots or targets, +and cannot communicate with its neighbors. The kinematics +of a robot is given by Eq. (1), which is a typical model +for a differential drive robot. At each time step, the robot +is controlled in a turn-then-move scheme with control inputs +θr ∈ S and dr ∈ R+. The maximum step-size of a robot is +dmax +r +. +γr,T = γr,T −1 + θr +cr,T = cr,T −1 + dr[cosγr,T +sinγr,T ] +(1) +A robot has p isotropic sensors arranged on its boundary +such that φk ∀k ∈ {1 · · · p} is the angle between the kth +sensor and the robot’s heading direction. Z is the set of +measurements from all sensors on a robot. +Signal Sources: We consider three types of signal-emitting +sources present in the environment that a robot can detect: +sg from a point source at the center of a target, sr from +a point source at the center of a robot, and se from a line +source present on the entire environment boundary. For clarity +in notation, we hereby denote the signal set {sg, sr, se} by +{g, r, e}. +The strength of any signal s ∈ {g, r, e} located at a distance +d from a signal source is given by the function Bs(d). The +influence distance of a source is limited to βs, such that +Bs(d) = 0 ∀d ≥ βs. Let N k +s be the set of all the sources +of type s in the sensing range of the kth sensor and dk +j be the +distance of this sensor from a source j ∈ N k +s . Then the sensor +reading zk +s = � +j∈N k +s Bs(dk +j ) is the sum of signal strengths +from all sources in N k +s . This summation becomes an integral +over the boundary segment for a line source present inside the +influence region βe. +The tuple (zk +g, zk +r , zk +e ) corresponds to the measurements of +the kth sensor. Let Zg = {z1 +g · · · zp +g}, Zr = {z1 +r · · · zp +r} and +Ze = {z1 +e · · · zp +e}, then the measurement set is Z = Zg ∪ +Zr ∪ Ze. We define rsafe +s +∀s ∈ {g, r, e} as the user-specified +minimum safety distance that a robot must maintain from a +source at all times. +III. PROBLEM FORMULATION +We model a target g = (cg, rg) as a disk of radius rg +centered at cg ∈ E. G is the set of all targets contained in +E. The kinematics of a target is given in Eq. (2). At any time +step T, dg ∈ R+ is the distance moved by the target, and +γg,T ∈ S is the target heading. +γg,T = γg,T −1 + θg +cg,T = cg,T −1 + dg[cosγg,T +sinγg,T ] +(2) +The maximum distance that a target can move is dmax +g +. +Target Motion Models: In this paper, we design controllers +and analyze the swarm behavior for different types of target +motion models. A target can exhibit one of the following +motions: +1) Target moves randomly such that at any time step T, +γg,T ∈ [0 2π), dg ∈ [0 dmax +g +] and cg,T ∈ E. +2) Target moves randomly as in motion model 1 until a +robot is in its escape domain = (cg, rescape +g +) of radius + +Transactions on Robotics, VOL. XX, NO. XX, XXXX +3 +rescape +g +centered at cg, in which case the target chooses a +heading direction to escape from all the robots that sat- +isfies ∥cg,T − cr,T ∥ ≤ rescape +g +, and moves the maximum +step-size dg. +3) Target follows an unknown motion pattern until a robot +satisfies ∥cg,T − cr,T ∥ ≤ rescape +g +, in which case it +chooses a heading direction to escape nearby robots. +Target Encapsulation: For each target g ∈ G, we define an +encapsulation ring Ag,T = (cg,T , rsafe +g +, rencap +g +) of inner radius +rsafe +g +and outer radius rencap +g +centered at cg,T . A robot R is +considered to be in Ag,T if, rsafe +g +< ∥cr,T − cg,T ∥ ≤ rencap +g +. +A target is encapsulated if the total number of robots present +in the encapsulation ring is ng, which is a user-specified +input as shown in Fig. 1. +Fig. 1: A target is encapsulated if ng robots are present +simultaneously in the encapsulation ring while maintaining at +least a distance of rsafe +r +from each other. +Problem statement: Consider a bounded environment E ⊆ +R2 with m dynamic targets where the initial distribution of +the robots and targets is arbitrary. Given the total number of +sensors p on a robot, the user-provided safe distance rsafe +s +∀s ∈ {g, r, e}, the encapsulation ring Ag, and the number +of robots ng needed to encapsulate each target g such that +the total number of robots n ≥ � +g∈G ng, our objective is to +find a real-time decentralized control law for encapsulating all +targets while ensuring safety distances are always maintained. +We make the following assumptions about the environment +and the system: +Assumption 1. The sensors are arranged on a robot such +that when a robot’s center is rsafe +s +away from a source s, +at least one sensor is in the influence region of the source. +For ease of exposition, we consider circular robots with a +symmetric placement of sensors to explain our algorithm, and +show in simulations how asymmetric sensor placements and +non-circular robots affect swarm behavior. +Assumption 2. The distance between any two moving targets +is greater than (2βg + 2rr). That is, a robot can sense at most +one target at a time. +Assumption 3. We constrain a target to maintain a minimum +distance of (rencap +g ++ rsafe +e ++ dmax +r +) from the environment +boundary. This ensures that robots will be able to encapsulate +the target without colliding with the environment boundary. +Assumption 4. We place no restriction on the target’s knowl- +edge of the environment; it may be able to perfectly sense +the relative location of any robot present in its user-specified +escape domain, rescape +g +thereby knowing the optimal escape +route. However, if a target is encapsulated, we assume it emits +a single burst of a shut-off signal and stops emitting any signal +subsequently. The influence distance of this signal is limited +to Ag, and we assume that thereafter both the robots within +the encapsulation ring and the target stop moving, i.e. dr = 0 +and dg = 0, respectively. +Assumption 5. The signal strength Bs strictly decreases with +the radial distance, d from a source and the inverse of the +signal function Bs(d) exists and is known to the robots. +IV. APPROACH +Our strategy for designing a local control law is based on +geometry and the relative kinematics of the interaction of a +robot with its neighboring robots and a dynamic target. We +extend our previous work [5] where we only considered static +targets; a robot’s behavior there was to either move randomly +in the bounded environment when it does not sense any target, +or to move towards a target if sensing one while ensuring +safety. Here, we introduce an additional robot behavior of +orbital encirclement of a target, inspired by [16]. As we show +in Section V, this behavior ensures the encapsulation of an +escaping target. In Section IV-A we describe virtual sources as +defined in [5] and use them to under-approximate the relative +distance between a source and the robot’s center as a function +of the sensor placement. In Section IV-B we find the bounds +on control parameters (dr and θr) for a robot to ensure that +it maintains rsafe +s +distance from a source s ∈ {g, r, e}. In +Section IV-C we introduce the concept of orbital encirclement +of a moving target; we provide a summary of the overall +reactive control law for a robot in the swarm in Section IV-D. +A. Virtual Source +Since we assume a robot is equipped with isotropic sensors, +a sensor measurement corresponds to the aggregated signal +strength from all the nearby sources. Hence, the same mea- +surement could correspond to a single source nearby or a +cluster of sources further away. Therefore, for each sensor +reading, zk +s ∀s ∈ {g, r, e}, we define a virtual source on +a circle centered at the sensor k as shown in Fig. 2. It is +shown in [5] that the closest possible location of the virtual +source with respect to the robot’s center is given by Eq. (3). +Furthermore, the range of possible directions of the location of +the virtual source with respect to the robot’s center is restricted +to [φk − π/p, φk + π/p] for symmetric sensor placement. +ds = rrcos(π/p) + +� +(dks)2 − r2rsin2(π/p) +(3) +For asymmetric sensor placement, we replace π/p with half of +the maximum angle that the kth sensor makes with either of +its adjacent sensors. Similarly, for robots that are not circular + +safe +7 +Encapsulation +ring, Ag,T +Encapsulating +9 +robots, ng = 6 +e +Ds +TargetTransactions on Robotics, VOL. XX, NO. XX, XXXX +4 +Fig. 2: Virtual source for the kth sensor [5]. +in shape, we replace rr by the distance between the kth sensor +and the robot’s center in the above equation. We can see in +(3) that as p → ∞, ds → dk +s +rr. That is, the error in locating +the source is dependent on the total sensors on a robot. +B. Collision Avoidance +We use the technique introduced in [5] for collision avoid- +ance with nearby robots and the environment boundary. At +each time step, the robot estimates the relative distance be- +tween its center and the nearby sources using Eq. (3) for the +sensor with the maximum sensor reading zk +s ∀s ∈ {g, r, e}. If +this distance is less than or equal to (rsafe +s ++dmax +s +), the collision +avoidance behavior is triggered for this robot to ensure safety. +We have shown in [5] that to avoid collisions with static +obstacles (such as environment boundary), the robot’s heading +direction θr must be chosen from the angular range given by +Eq. (4). +Θavo +e += [φk + π/p + π/2, φk − π/p + 3π/2] +(4) +Whereas to avoid the neighboring moving robots, the distance +dr that a robot moves at time step T in a given heading +direction γr,T must be chosen such that at T + 1 it maintains +at least a distance of rsafe +r +from the closest neighboring robot. +As shown in Fig. 3, let k and l be the indices of the sensors +closest to the intended heading direction γr at time T and dk +r +and dl +r are their radii of virtual sources respectively such that +dk +r > dl +r. Then, we can compute the bounds on the step-size +dr that the robot can take in the heading direction γr,T using +Eq. (5). +Fig. 3: Computing dr such that collision is avoided with nearby +moving robots [5]. +0 ≤ dr ≤ rrcos(φl − θr) ++ +� +(dlr − rsafe +r +− dmax +r +)2 − r2rsin2(φl − θr) +(5) +To ensure that two robots never deadlock, the bounds on +the maximum step size a robot can take, and the influence +region of a robot’s source, are given by Eq. (6) and Eq. (7), +respectively. The proof is detailed in Lemma V.3 of [5]. +dmax +r +< rsafe +r ++ rrcos(π/p) +2 +− +� +(rsafe +r +)2 + r2r − 2rrrsafe +r +cos(π/p) +2 +(6) +� +(rsafe +r +)2 + r2r − 2rrrsafe +r +cos(π/p) + 2dmax +r +< βr +< rsafe +r ++ rrcos(π/p) +(7) +C. Encirclement of a Target +In [5], our approach to encapsulate a static target, was for a +robot to either move towards the target or move away from an +obstacle between itself and the target in the direction of the +sensor receiving the minimum reading from nearby moving +robots. However, in order to surround a dynamic target, the +behavior of a robot should be such that the swarm is able to +disperse around the target in order to block off its escaping +paths. Since we consider minimalist robots that can neither +communicate with their neighbors nor know their exact relative +position, we can not make use of formation control strategies, +such as [17], [23]. +Consider a scenario where all the robots in the swarm start +on one side of a target. Then, for a swarm to disperse around a +target, it is necessary that an individual robot be able to catch +up with the escaping target, and once the robot reaches the +encapsulation ring, it should be able to encircle the target so +that the target is prevented from escaping. +To ensure encapsulation, we define primary and secondary +orbits around each target, as shown in Fig. 4. For each +orbit, we define a tie-breaking orbital rotation which can +be either clockwise (denoted by a value of -1) or counter- +clockwise (denoted by a value of 1). The primary orbit, +Fig. 4: Primary (purple ring) and secondary (cyan rings) orbits +around a target, and the lower bound on the target’s escape +domain (red circle) . A robot moves either clockwise or +counter-clockwise in an orbit depending on its neighbors. The +solid arrows denote the tie-breaking rotation for an orbit. +Or0 = (cg,T , Orinner +0 +, Orouter +0 +, −1) is an annular ring centered +at cg,T with an inner radius of Orinner +0 +≥ rsafe +g ++ dmax +g +, an + +virtual +source +actual +k +sources +R +S +robot +kth sensorheading at T+1 +closest virtual source at T+1 +Ith sensor +Cr,T+1 +(p +kth sensor +p +SOrbit +imax +rotations +Secondary +Primary +orbits Ori>0 +Target +orbit OroTransactions on Robotics, VOL. XX, NO. XX, XXXX +5 +outer radius Orouter +0 += rencap +g +and a clockwise orbital rotation +(chosen arbitrarily). Let w be the width of a secondary orbit, +then an ith secondary orbit is given by, Ori = (cg,T , Orouter +0 ++ +(i − 1)w, Orouter +0 ++ (i)w, (−1)i−1), ∀i > 0. We consider a +robot to be in ith orbit if, Orinner +i +< ∥cg,T − cr,T ∥ ≤ Orouter +i +. +Each robot in the swarm computes its current orbit using +its estimate of ∥cg,T − cr,T ∥. At time-step T, let Ori be the +current orbit as estimated by a robot, then its control consists +of one of the following behaviors: +1) if i > 0, the robot moves towards the target in a +heading direction chosen from the line of sight angular +range as estimated from the virtual source (ΘLOS +g +) while +maintaining a safe distance from nearby robots. +2) else if i > 0 and the robot cannot move a non-zero +distance towards the target, it moves tangentially in its +current orbit while maintaining a safe distance from +nearby robots. The direction of the tangent is chosen +such that it maximizes the possible step-size dr. In case +of symmetry, the robot moves in the orbital rotation of +the ith orbit. +3) else if i > 0 and the robot can neither move in a direc- +tion from ΘLOS +g +nor tangential to the orbit, it chooses a +direction of motion that maximizes the possible step-size +dr. +4) else if i = 0, the robot moves tangentially in its current +orbit while maintaining a safe distance from the target. +5) else if the relative distance between the target and a +robot is less than or equal to Orinner +0 +, it moves away +from the target. +6) else the robot performs a simple random walk while +avoiding nearby moving robots. +In general, a robot moves toward the target until it reaches +the primary orbit. If other robots are present between itself +and the target, the robot moves tangentially in its current orbit +until it can move toward the target. All the robots that place +themselves in the primary orbit constantly move tangentially +and eventually close off the target’s escape routes. The width +of a secondary orbit, w, must be less than βr, so that a robot’s +neighbors in adjacent orbits lie within its sensing range. This +ensures that a robot doesn’t move towards a target when it +senses other robots in the front and instead moves tangentially +in its current orbit. +Now, using the sensor readings and their corresponding +virtual sources, we find the set of directions that a robot +needs to choose from to move towards a target, away from +a target, or tangentially in an orbit. Let k be the index of the +sensor such that zk +g > zl +g, ∀l ̸= k. Here we have ignored +the unlikely scenario where two sensors receive the same +maximum intensity from a target. Then the angular range, +ΘLOS +g +, for the possible location of the target with respect to +the robot’s center is given by Eq. (8). +ΘLOS +g += [φk − π/p, φk + π/p] +(8) +The angular range, Θavo +g +(Eq. +(9)), to move away from the +target can be derived in a similar fashion to Eq. (4). +Θavo +g += [φk + π/p + π/2, φk − π/p + 3π/2] +(9) +The angular range to move tangentially in an orbit in a +clockwise or counter-clockwise direction is given by Eq. +(10) and Eq. +(11), respectively, where we define Θtan +g += +Θtan,+1 +g +∪ Θtan,−1 +g +. +Θtan,−1 +g += [φk − π/p + π/2, +φk + π/p + π/2] +(10) +Θtan,+1 +g += [φk − π/p + 3π/2, +φk + π/p + 3π/2] +(11) +Fig. 5 shows the different angular range sets for a target-robot +interaction. It is worth mentioning that for noiseless sensors, +Fig. 5: The angular range set for a target-robot interaction. +The robot is equipped with 5 sensors placed asymmetrically. +if zk−1 +g +> zk+1 +g +then ΘLOS +g += [φk − π/p, φk]. This results +in a more accurate estimation of the location of a target and +reduces the angular resolution error by half. The estimation of +Θtan +g +and Θavo +g +also changes accordingly. +As shown in our previous work [5], a heading direction +in the angular ranges ΘLOS +g +and Θavo +g +is guaranteed to make +a robot move towards the target and away from the target, +respectively. In contrast, a robot might end up moving towards +or away from the target when it moves tangentially in an orbit. +Since secondary orbits are at least at a distance of Orouter +0 +from a target, a robot moving tangentially in these orbits will +always maintain a safe distance from the target. However, if +a robot is moving tangentially in the primary orbit, we need +to make sure that it maintains at least a distance of Orinner +0 +from the target after moving dr units in the intended heading +direction γr,T such that θr ∈ Θtan +g +. +In Fig. 6, we can see that at T + 1, the closest possible +location of the target is at S∈ I. If the heading direction θr /∈ +ΘLOS +g +, the closest possible location of the target with respect to +the robot’s center at T +1 would be along one of the extremes +of the angular range ΘLOS +g +. To ensure safety, ∥cr,T +1 − S∥ ≥ +Fig. 6: The distance dr that the robot can move in the intended +heading is computed using the geometry of ∆Scr,T cr,T +1 +Orinner +0 +. Using the cosine rule of triangle for △Scr,T cr,T +1, + +g +IICg,T - Cr,Tll +g +SO70 +g +kth sensor +TargetSO70 +g +Heading of robot +at T+1 with a ++1 +step-size of dr +Actual +Heading of +target + dreq +r +then +16 +dr = dreq +r +17 else if currentOrbit = Or0 then +18 +θr = arg max +θ∈Θtan +g +min +� +DistAvoRob(Zr, Br, θ, rsafe +r +), +DistAvoTar (Zg, Bg, ΘLOS +g +, θ) +� +19 +dr = DistAvoRob(Zr, Br, θr, rsafe +r +) +20 else +21 +θr = arg max +θ∈ΘLOS +g +DistAvoRob(Zr, Br, θ, rsafe +r +) +22 +dr = DistAvoRob(Zr, Br, θr, rsafe +r +) +23 +if dr = 0 then +24 +θr = arg max +θ∈Θtan +g +DistAvoRob(Zr, Br, θ, rsafe +r +) +25 +dr = DistAvoRob(Zr, Br, θr, rsafe +r +) +26 +if dr = 0 then +27 +k = argmin(Zr) +28 +θr = φk +29 +dr = DistAvoRob(Zr, Br, θr, rsafe +r +) +Algorithm 1 encodes the local reactive control law for +a robot in the swarm that is tasked with searching and +encapsulating targets while avoiding collisions. +The algorithm describes the computation that happened +at each time step T. Each robot in the swarm has: Z–the +tuple of sensor measurements, Bs–the function describing the +signal source strength as a function of radial distance from +s ∈ {g, r, e}, the maximum step-size of a robot dmax +r +and +a target dmax +g +, the user-specified safety constraints for each +source rsafe +s +, and the set orbits defined by an inner and +outer radius of each orbit. +The control synthesis proceeds as follows: First, the robot +estimates its distance, DistToEnvBound (Section IV-A), +from the environment boundary. If the robot is too close to the +boundary, it computes the allowed set of heading directions, +Θavo +e +. The direction of motion, θr is then chosen such that +the robot moves away from the boundary with a maximum +possible step size, dr while avoiding nearby robots (lines 1-2). +The function DistAvoRob computes this maximum possible +value of dr from Eq. (5), as described in Section IV-B. +Once the robot is at a safe distance from the boundary, +it then estimates the relative distance DistToTar (Sec- +tion IV-A ) from a target. If no target is sensed the robot +performs a random walk while maintaining a safe distance +from nearby robots (lines 4-10). If, on the other hand, the +robot is inside the influence region of a target, it computes +its current orbit, currentOrbit based on the estimated +DistToTar and the input orbits. If the robot estimates +that the relative distance between itself and the target is less +than Orinner +0 +, it computes the set of allowed heading direction, +Θavo +g +, and chooses a direction of motion from this set while +maximizing the step size to avoid nearby robots (lines 11- +16). The distance to move away from a target is capped at +dreq +r += +��DistToTar − Orinner +0 +�� (lines 15-16) to ensure that +the robot doesn’t move outside the primary orbit +If the robot is in the primary orbit (line 17), it moves +tangentially to the orbit Or0 (heading direction chosen from +the computed set Θtan +g +) with a step size dr such that it +maintains a safe distance from the nearby robots and the +target (lines 18-19). The function DistAvoTar computes the +maximum possible value of dr from Eq. (12), as described in +Section IV-C. +When a robot is in a secondary orbit, Ori>0 it chooses +a heading direction from the set ΘLOS +g +to move towards the +target while avoiding nearby robots (line 21-22). In case the +robot cannot find a direction of motion to move a non-zero +distance toward the target (line 23), it either moves a non-zero +distance tangentially in its current orbit (lines 24-25) or moves +a safe distance in a heading direction based on the reading +from the sensor receiving the minimum signal strength zk +r , +i.e. the direction where the virtual source corresponding to +other robots is the farthest (lines 26-29). +A robot’s local control, as summarized in Algorithm 1, is +agnostic to the motion type of the targets. This, together with +our convergence guarantees in the following section, implies +that our algorithm guarantees the encapsulation of multiple +targets moving in the bounded environment with different +types of motion models, as described in Section III. +V. CONVERGENCE GUARANTEES +We use the Lyapunov stability theory to provide guarantees +on the emergent behavior of the swarm. In this section, for + +Transactions on Robotics, VOL. XX, NO. XX, XXXX +7 +clarity, we consider circular robots with noiseless sensors. In +practice, the desired behavior emerges for non-circular robots +as well, which we demonstrate in simulations in Section VI. +Lemma V.1. From [30]: A disc robot with a non-zero radius +performing a random walk in a bounded 2D environment will +always eventually explore the entire area. +Lemma V.2. For any arbitrary initial condition such that a +robot is at least rsafe +g +away from a target, a necessary condition +to ensure a collision-free target’s motion is that the escape +radius of the target, rescape +g +≥ rsafe +g ++ dmax +g +and the inner +radius of the primary orbit, Orinner +0 +≥ rsafe +g ++ dmax +g +Proof. The lower bound on rescape +g +ensures that a target gets +enough margin to escape an approaching robot. As described +in Section IV-C, a robot’s behavior is such that it moves +away from the target if the robot crosses the inner ring, +Orinner +0 +, of the primary orbit. Hence the above lower bound on +Orinner +0 +ensures that collision avoidance behavior for a robot +is triggered before the distance between a target and a robot +becomes rsafe +g +. +Lemma V.3. For any arbitrary initial condition such that a +robot is at least rsafe +g +away from a target the following are the +necessary conditions to ensure a target’s encapsulation: +1) the outer radius of the encapsulation ring Ag satisfies +rencap +g +≥ dmax +r ++ rr ++ +� +(Orinner +0 +)2 + r2r − 2rrOrinner +0 +cos(π/p) +(13) +2) the number of robots ng specified for encapsulation +satisfies, +ng ≤ n0 = +2π +cos−1 +� +1 − (βr+rr)2 +2(rencap +g +)2 +� +(14) +Proof. Each robot’s estimate of the relative distance from +a target depends on the sensor with the maximum reading, +max(Zg). Given that a virtual source is always either closer +or at the radial location of an actual source, it is possible +that even if a robot is present in the primary orbit, the robot +estimates itself to be present at a relative distance of less +than Orinner +0 +with respect to the target. This will trigger the +collision avoidance behavior for the robot and it will move +away from the target. To successfully encapsulate a target g, +it is required that the outer radius of the encapsulation ring, +rencap +g +incorporate the robot with the worst possible estimate +of the target’s location. This will ensure that the robot remains +in the primary orbit even after being over-cautious in moving +away from the target. +At each time step, a robot chooses its control parame- +ters such that it maintains at least a distance of Orinner +0 +from a target, that is, ∥cg,T − cr,T ∥ +≥ +Orinner +0 +. Since +Orinner +0 +is defined between a robot’s center and the tar- +get, we set ds = Orinner +0 +in Eq. +(3) to obtain dk +g += +� +(Orinner +0 +)2 + r2r − 2rrOrinner +0 +cos(π/p). A robot will start +to move away from the target when max(Zg) ≥ Bg(dk +g). At +this point, the upper bound on ∥cg,T − cr,T ∥ is given by Eq. +(15). We can see that as p → ∞, ∥cg,T − cr,T ∥ → Orinner +0 +and for a finite p, collision avoidance behavior is triggered +before the robot is at a distance of Orinner +0 +from the target. +∥cg,T − cr,T ∥ ≤ rr+ +� +(Orinner +0 +)2 + r2r − 2rrOrinner +0 +cos(π/p) +(15) +For asymmetric sensor placement, we replace π/p with half +of the maximum angle between two adjacent sensors on the +robot. +To incorporate the robot with the worst estimate of a +target’s location, we set a lower bound on the outer radius +of the encapsulation ring using Eq. +(15) as rencap +g +≥ +rr+ +� +(Orinner +0 +)2 + r2r − 2rrOrinner +0 +cos(π/p). Since the robot +may chatter in the encapsulation ring due to constant attraction +and repulsion from the target and nearby robots, we add dmax +r +to the lower bound on rencap +g +(condition 1). This will ensure +that a robot remains in the encapsulation ring when there are +robots nearby. +Furthermore, the maximum number of robots that can be +specified for target encapsulation (condition 2) is bounded by +the total number of robots that can be physically placed in +the encapsulation ring such that the encapsulating robots are +outside each other’s influence region to ensure no chattering +that can be caused by repulsion from each other. When ng > +n0 a dynamic equilibrium exists around a target such that there +are always almost n0 robots present in the primary orbit [5]. +As described in Section III, we consider three types of mo- +tion patterns that a target can exhibit. For each of these target +motion patterns, we provide guarantees for liveness (eventual +encapsulation) based on the Lyapunov stability theory and +stochastic analysis (Lemmas V.4 - V.6). +Lemma V.4. Consider a target g ∈ G moving randomly in the +bounded environment until it senses any robot in its escape +domain (as described by motion model 2 in Section III). If, +1) the maximum step size of the target, dmax +g +≤ λdmax +r +where +λ = min +�π +2 , +α +sin(π − α) +�sinϕ +ϕ cosϕ +α = cos−1 +� +1 − (βr + rr)2 +2(rescape +g +)2 +� +ϕ = max(φk − φk+1) +2 +, +k = {1 · · · p} += π/p, +for symmetric sensor placement +the target g will be encapsulated eventually. +Proof. Consider +a +target +g +∈ +G. +Let +ug += +[dgcos(γg,T ++ +θg) +dgsin(γg,T ++ +θg)] +and +ur += +[drcos(γr,T + θr) +drsin(γr,T + θr)] be the control input +of a target and robot respectively at time T, and η be the +total robots currently present in the escape domain of the +target, that satisfy ∥cg,T − cr,T ∥ ≤ rescape +g +. As outlined in +Section IV-C, the motion strategy of a robot can be broken +down as follows: +Case I: Robot is in an orbit Ori≥0 such that η = 0 +We use the definition of stochastic stability in the sense of +Lyapunov [31] to show that a robot eventually reaches the + +Transactions on Robotics, VOL. XX, NO. XX, XXXX +8 +primary orbit Or0. Let V = ∥cg,T − cr,T ∥2 be the candidate +Lyapunov function defined on the the domain Dr ⊆ R2 such +that ∥cg,T − cr,T ∥ ≥ rencap +g +. Using Eq. (2) and Eq. (1) we +have, ∆V = ∥(cg,T + ug) − (cr,T + ur)∥2 − ∥cg,T − cr,T ∥2. +For +ease +of +exposition, +we +will +drop +the +subscript +T +in +the +following +analysis. +On +simplifying, +∆V += +∥ug − ur∥2 + 2(cg − cr) · (ug − ur). Fig. 7 +Fig. 7: Relative kinematics of a robot-target interaction. +depicts the relative kinematics model between the target and +a robot where ω is the angle that the LOS vector, (cg − cr) +makes with x-axis. Let ˆl = [cosω +sinω] be the vector along +(cg − cr) and ˆt = [−sinω +cosω] be the vector tangential to +it. Then, +∆V = d2 +g + d2 +r − 2ug · ur + 2 ∥cg − cr∥ˆl · (ug − ur) +(16) +(a) Robot is in a secondary orbit: For this case, a robot would +move towards the target, that is θr ∈ ΘLOS +g +given in Eq. (8). If +η = 0, that is, there are no robots in the target’s escape domain, +the target moves randomly. Hence, θg ∈ [0 2π). That is, both +θg and θr are stochastic. Moreover, the control inputs ug and +ur are independent random vectors and their corresponding +expected values are given by, +E[ug] = dgE[cos(γg + θg) sin(γg + θg)] += dg +� 2π +� +0 +cos(γg + θg) 1 +2π dθg +2π +� +0 +sin(γg + θg) 1 +2π dθg +� += 0 +(17) +E[ur] = drE[cos(γr + θr) +sin(γr + θr)] += dr +� φk+ π +p +� +φk− π +p +cos(γr + θr) +1 +2π/pdθr +φk+ π +p +� +φk− π +p +sin(γr + θr) +1 +2π/pdθr +� += dr +sinϕ +ϕ [cos(γr + φk) +sin(γr + φk)] += dr +sinϕ +ϕ ˆuk +r +(18) +where ϕ = π/p for symmetric sensor placement and ˆuk +r is +the unit vector in the direction of kth sensor. Intuitively this +means that on an average the robot moves in the direction of +the kth sensor (receiving maximum intensity from the target) +with a step-size reduced by the factor sinϕ/ϕ. Furthermore, as +p → ∞, E[ur] → drˆuk +r. That is, if the robot knows the exact +relative location of the target, it moves towards the target along +the line of sight vector with the maximum possible step size. +Using Eq. (17) and Eq. (18), the expected value of change +in the Lyapunov function (as given by Eq. (16)) between two +consecutive time steps is, +E[∆V ] = d2 +g + d2 +r + 2E +� +∥cg − cr∥ˆl · (ug − ur) +� += d2 +g + d2 +r + 2 ∥cg − cr∥ +� +E[ug] − E[ur] +� +·ˆl += d2 +g + d2 +r − 2 ∥cg − cr∥ dr +sinϕ +ϕ ˆuk +r ·ˆl +(19) +The maximum deviation of the unit vector in the direction of +the kth sensor, ˆuk +r from the LOS vectorˆl is limited to ϕ = π/p +(refer to Fig. 2), that is ˆuk +r · ˆl ≥ cosϕ. Furthermore, when a +robot is in a secondary orbit ∥cg − cr∥ ≥ rencap +g +. Substituting +these bounds in Eq. (19) we have, +E[∆V ] ≤ d2 +g + d2 +r − 2rencap +g +dr +sinϕ +ϕ cosϕ +For stability, we require that E[∆V ] ≤ 0. That is +d2 +r − +� +2rencap +g +sinϕ +ϕ cosϕ +� +dr + d2 +g ≤ 0 +(20) +The necessary condition to satisfy the above inequality is that +cosϕ ≥ 0. That is, the total number of sensors on a robot +must be greater than or equal to three. The roots of the above +quadratic inequality in dr give us an upper bound on the +maximum step-size of a robot dmax +r +which is always larger +than the bound determined in Eq. (6) and hence Eq. (20) is +always satisfied. +(b) Robot is in the primary orbit: Once a robot reaches the +primary orbit, it moves tangentially to the orbit while ensuring +that ∥cg − cr∥ ≥ Orinner +0 +. To analyze this, we look at how the +LOS vector between a target and robot changes between two +time steps, which is given by ∆(cg−cr) = ug−ur. In Eq. (21) +and Eq. (22) we define urad +gr +and utan +gr +representing the polar +coordinates corresponding to the radial and tangent component +of the change in LOS vector in global frame I. +urad +gr = (ug − ur) ·ˆl +(21) +utan +gr = (ug − ur) ·ˆt +(22) +As explained earlier, to ensure a target’s encirclement, it +is necessary that a robot is able to complete a revolution +around the target in the primary orbit Or0. We evaluate +this stochastically by computing the expected value of the +change in the tangential component of the relative LOS vector, +E[utan +gr ] = E[ug]·ˆt−E[ur]·ˆt. For a clockwise orbital rotation, +θr ∈ Θtan,−1 +g +. Since η = 0 for this case, θg ∈ [0, 2π). + +wg +3 +y +IICg,T Cr,Tll +9 +xTransactions on Robotics, VOL. XX, NO. XX, XXXX +9 +Simplifying and substituting Eq. (17) in the above equation +we have +E[utan +gr ] = −dr +� φk+ π +p + π +2 +� +φk− π +p + π +2 +cos(γr + θr) +1 +2π/pdθr +φk+ π +p + π +2 +� +φk− π +p + π +2 +sin(γr + θr) +1 +2π/pdθr +� +·ˆt += −dr +sinϕ +ϕ [−sin(γr + φk) +cos(γr + φk)] ·ˆt += −dr +sinϕ +ϕ ˆuk +r ·ˆl +(23) +Eq. (23) shows that a robot moves clockwise in the primary +orbit with an expected tangential step-size of dr +sinϕ +ϕ cos(ϕ) +with respect to the target. +Case II: ∥cg,T − cr,T ∥ ≤ rescape +g +Since a robot orbiting in Or0 also avoids nearby robots, at a +timestep T, it is possible that a robot is marginally outside the +escape domain of the target but is unable to move in Θavo +g +due +to the presence of other robots in orbit Or1 at a distance of +rsafe +r +. +This behavior could lead to ∥cg,T +1 − cr,T +1∥ < rescape +g +, +resulting in η > 0. The target would then move such that it +can escape from all the robots present in its escape domain. +Let ψg be the angle between a target’s intended heading and +the LOS vector (cg,T −cr,T ), as shown in Fig. 7. If η = 1, then +ψg ∈ [3π/2, π/2]. If η = 2 and α is the angle that these robots +subtend at the center of the target, as shown in Fig. 8, then +ψg ∈ [3π/2 + α, π/2] or ψg ∈ [3π/2, π/2 − α], depending +on which robot ψg is measured with respect to. Without loss +of generality, we can consider one of these. That is, if η > 1, +the available angular range of the target for escaping decreases +from π to (π − (η − 1)α). +To determine α, we use the fact that robots in the primary +orbit disperse such that, on average, they are outside each +other’s influence region. Then, using geometry shown in Fig. 8, +α = cos−1 +� +1− (βr+rr)2 +2(rescape +g +)2 +� +. Note that, the maximum escaping +angular range of a target is limited to π (when η = 1). +Hence, for η > π/α, the target can no longer escape with +the maximum possible step size. For Case II, we have to +Fig. 8: Geometric configuration for computing bounds on the +ratio between target and robot step-sizes. +ensure that (i) robots implementing tangential control law +in the primary orbit are able to encircle the target, that is +E[utan +gr ] ≤ 0 if θr ∈ Θtan,−1 +g +and E[utan +gr ] ≥ 0 if θr ∈ Θtan,1 +g +, +and (ii) robots in secondary orbits are able to move towards +the target, that is E[urad +gr ] ≤ 0 for θr ∈ ΘLOS +g +. +Using Eq. +(22) and the tangential control input (θr ∈ +Θtan,−1 +g +) for robots present in Or0 we have, +E[utan +gr ] = dgE[sinψg] − E[ur] ·ˆt += dg +π +2 +� +3π +2 +(η−1)α +sinψg +1 +π − (η − 1)α dψg − E[ur] ·ˆt += dg +sin((η − 1)α) +(π − (η − 1)α) − dr +sinϕ +ϕ ˆuk +r ·ˆt +To ensure clockwise orbital rotation, E[utan +gr ] ≤ 0, that is, +max +η≤π/α +� +dg +sin((η − 1)α) +(π − (η − 1)α) +� +≤ +min +ˆuk +r·ˆt≥cosϕ +� +dr +sinϕ +ϕ ˆuk +r ·ˆt +� +dg ≤ +α +sin(π − α)dr +sinϕ +ϕ cosϕ +(24) +Intuitively, E[ug · ˆt] is maximal when the target has the least +freedom in choosing its motion, that is η = π/α. Similarly, +E[ur ·ˆt] is minimal when the average heading direction ˆuk +r is +deviated the most from ˆt. +Now, we need to ensure that the robots in the secondary +orbits move toward an escaping target. Using Eq. +(21) and +the LOS control input (θr ∈ ΘLOS +g +) for robots present in Ori>0 +we have, +E[urad +gr ] = E[ug ·ˆl] − E[ur] ·ˆl += dg +π +2 +� +3π +2 +(η−1)α +cosψg +1 +π − (η − 1)α dψg − E[ur] ·ˆl += dg +1 + cos((η − 1)α) +(π − (η − 1)α) +− dr +sinϕ +ϕ ˆuk +r ·ˆl +The distance between a target and robot will decrease if +E[urad +gr ] ≤ 0. That is, +max +η≤π/α +� +dg +1 + cos((η − 1)α) +(π − (η − 1)α) +� +≤ +min +ˆuk +r·ˆl≥cosϕ +� +dr +sinϕ +ϕ ˆuk +r ·ˆl +� +dg ≤ π +2 dr +sinϕ +ϕ cosϕ +(25) +Eq. (24) and Eq. (25) determine an upper bound on the +maximum step size of a target as given by condition (1). +In Fig. 9 we show how the ratio of the step-size be- +tween a target and robot, λ, changes with the number of +sensors p and ∠α (which is proportional to how well the +robots disperses in the primary orbit). As derived above, +λ = min +� +π +2 , +α +sin(π−α) +� +sin(2π/p) +2π/p +. For a given number of +sensors on a robot, λ increases with an increase in ∠α until +π/2 > +α +sin(π−α). We can also see that with an increase in the +number of sensors on a robot, p, the ratio of the step size + +α +escape +π --1)αTransactions on Robotics, VOL. XX, NO. XX, XXXX +10 +Fig. 9: For a given number of sensors on a robot p, λ increases +with an increase in ∠α until π/2 > +α +sin(π−α). The ratio +between the step-size of a target and robot tends to π/2 with +an increasing p indicating that the swarm can encapsulate a +target moving faster than the individual robots in the swarm. +of a target to a robot increases and tends to π/2 > 1, that +is, we can guarantee convergence (encapsulation) even when +the target moves faster than the robots in the swarm. Previous +approaches in the literature typically assume that the target +moves slower than the robots [8]–[12]. +The increased accuracy in the estimation of the relative +location of nearby robots and target enables the robot to +disperse quickly in the primary orbit with less chattering. +Apart from the more accurate estimation, with an increase +in total sensors on a robot, the sensing radius βr of a robot +increases (Eq. (7)), which enables quick dispersion because +of a robot’s behavior of remaining outside other robots’ +influence region. This results in blocking the escaping paths +of the target efficiently. The ratio between the target and +robot step-sizes is zero when the robot has less than three +sensors, for all values of ∠α, implying that a minimum of +three sensors are required to encapsulate a moving target. +Absence of livelocks and encapsulation of the target g: +Similar to Lemma V.3, we can compute the total number of +robots, ni, that can be simultaneously present in an ith orbit. +If at a time step T there are less than ni robots in Ori, empty +spots that could potentially be occupied by nearby robots, will +be present in this orbit. As discussed in Section IV-C, the +robots in the influence of a target either move toward the target +or move, typically, in opposite tangential directions in adjacent +orbits. This ensures that a dynamic empty spot present in an +orbit Ori and the robots present in Ori+1 move so as to align +with each other. As we proved above, the robots present in +the encapsulation ring (or the primary orbit) are guaranteed +to continuously orbit the target. So, when there are less than +ng robots in the encapsulation ring, a dynamic empty spot is +present in the primary orbit which will be eventually occupied +by a robot orbiting in the secondary orbit Or1. When either all +the empty spots in the primary orbit are filled by the robots or +there are at least ng robots in it, a target will be encapsulated. +Once that happens we have from assumption (4) that the target +will stop emitting its signal and set its control parameters to +zero thereafter. All the robots that were in secondary orbits +and in the influence of this target will transition into random +walk behavior. Hence assumption (4) ensures that the robots +would not be stuck in the secondary orbits of an encapsulated +target and can transition into target-searching behavior after +one target is encapsulated. +Lemma V.5. Consider a swarm with a total of n robots and +a target g ∈ G moving randomly in the bounded environment +(as described by motion model 1 in Section III). If, +1) the inner radius of the primary orbit Orinner +0 +≥ rsafe +g ++ +dmax +r +2) the maximum step size of the target, dmax +g +≤ λdmax +r +where +λ = +� +n − +� +2π +cos−1 +� +1 − +(βr+rr)2 +2(Orinner +0 +)2 +� +� ++ 1 +�−1 +the target g will be encapsulated eventually. +Proof. The challenge in encapsulating a randomly moving +target is in ensuring that robots avoid colliding with a target. +For example, say at time T a robot i is in the primary orbit +sandwiched between a target on one side at a distance of +rsafe +g ++ dmax +g +and a robot j at a distance of rsafe +r +on the other +side, along the target’s LOS vector. Furthermore, consider that +for time steps T until T + 2, the randomly moving target acts +adversarial by trying to collide with the robot. That is, at every +time step it moves towards robot i. +To ensure that the ith robot avoids colliding with the target +at T + 1, it must choose a heading direction γi ∈ Θavo +g +. How- +ever, due to the presence of robot j, it cannot move a nonzero +distance at time T in the intended heading direction. Hence +the robot would violate the target-robot safety specification at +T + 1. Now, at T, say the jth robot had moved away from +robot i, implying that at T + 1, the ith robot will not sense +the jth robot and will be free to move away from the target. +That is, it took a minimum of two time steps for the ith robot +to move away from the target. Hence, to ensure safety for +this scenario, dmax +g +≤ dmax +r +/2 and Orinner +0 +≥ rsafe +g ++ dmax +r +(condition 1). +Generalizing this, let the total robots present in the environ- +ment be n and � +n0 = +� +2π +cos−1 +� +1− +(βr+rr)2 +2(Orinner +0 +)2 +� +� +be the number +of robots that can be simultaneously present in the primary +orbit marginally outside Orinner +0 +without repelling each other. +Then, in the worst case scenario, the total time steps for which +a robot present on Orinner +o +may have to remain idle (dr = 0) +is (n − � +n0 + 1). This follows from the fact that only (n − �n0) +number of robots contribute to the idle waiting time of a robot +in the primary orbit present on Orinner +0 +. This constraint on the +idle time of a robot in the primary orbit gives us an upper +bound λ (condition 2) on how slow an adversarial target needs +to be with respect to a robot to ensure safety. The analysis for +stability and encapsulation follows from Lemma V.4. + +1.5 +p=1 +p=2 +p=3 +p= 5 +1I +p= 9 +0.5 +-p = 11 +—p = 13 +0 +0 +50 +100 +150 +200 +α (deg) +sin(2π/p) +π sin(2π/p) +α +入二 +sin(π - α) +2元/p +2 +2元/pTransactions on Robotics, VOL. XX, NO. XX, XXXX +11 +Lemma V.6. Consider a target g ∈ G moving in an unknown +pattern until it senses a robot in its escape domain (as +described by motion model 3 in Section III). If, +1) the maximum step size of the target when moving in an +unknown pattern, dmax +g +< λdmax +r +where +λ = sinϕ +ϕ cos(ϕ), +ϕ = π/p, +for symmetric sensor placement +2) the maximum step-size of the target when escaping +nearby robots, +λ = min +�π +2 , +α +sin(π − α) +�sinϕ +ϕ cosϕ +α = cos−1 +� +1 − (βr + rr)2 +2(rescape +g +)2 +� +ϕ = max(φk − φk+1) +2 +, +k = {1 · · · p} += π/p, +for symmetric sensor placement +the target g will be encapsulated eventually. +Proof. This scenario is comparable to hunting problems [21], +[22] where the target moves at slower speeds in some unknown +motion pattern. But as soon as it detects (target’s sensing +limited to rescape +g +) a predator (robot) in its domain, it escapes +at a faster speed than the predator. To ensure that robots in +the secondary orbits move toward the target, we require that, +E[ug ·ˆl] − E[ur ·ˆl] < 0 +dg(ˆug ·ˆl) < dr +sinϕ +ϕ cosϕ +(26) +Eq. (26) is always satisfied if dg < dr +sinϕ +ϕ cosϕ. It is trivial +to show using Eq. (22) and Eq. (18) that the constraint dg < +dr +sinϕ +ϕ cosϕ also ensures that robot in a primary orbit will +encircle the target. As shown in Lemma V.4, the robots can +successfully encapsulate an escaping target as long as a target +step-size is within the bounds given by Eq. (25) and Eq. (24). +VI. SIMULATION RESULTS +In this section, we study the effect of the total number of +sensors p, target-robot step-size ratio λ, and noisy sensors on +the global behavior of the swarm. For each case, we consider +three different target motion models: (i) target performs +random walk in the environment, (ii) target perform random +walk until there exists a robot such that rgr ≤ rescape +g +, (iii) +target moves with constant velocity until there exists a robot +such that rgr ≤ rescape +g +. The simulation environment consists +of one moving target and ten robots. The total time is capped +at 4000 time-steps. Due to the inherent randomness in the +motion of the robots and targets, we ran 50 simulations for +each data point with the same initial conditions. All the +robots were initialized arbitrarily such that they lie in a sector +of π/4 with respect to the target’s center. This is done to +show the ability of the swarm to successfully encapsulate a +target when it has all the escape paths open. +Fig. 10: For a given p, the ratio between the step-size of +a target and robot decreases with an increase in rescape +g +, +indicating that as the target is able to detect robots sooner, +to ensure encapsulation it must also move slower. +Effect of the total number of sensors: For a given p, with +an increase in the radius of the escape domain of a target, +the bound on the maximum step size of a target decreases +as shown in Fig. 10. This is because, with an increase in +rescape +g +, a target gets a higher margin for escaping. From +Lemmas V.4 - +V.6, the ratio λ, and hence the target’s +step size, is dependent on the total number of sensors on a +robot, p, and rescape +g +. When the escape domain of the target, +rescape +g +≤ rsafe +g ++ π +2 dmax +r +, an increase in the total number +of sensors on a robot enables the swarm to capture a faster +moving target. This can be seen in Fig. 10, where the blue +line corresponds to rescape +g += rsafe +g ++ π +2 rr. As p increases, +dmax +r +tends to rr and λ becomes greater than 1. If the escape +domain is further increased, that is rescape +g +> rsafe +g ++ π +2 dmax +r +, +then dmax +g +decreases proportionally because a robot’s step size +is limited to dmax +r +. For a given ng, rescape +g +and rencap +g +, Fig. 11 +shows how varying the total number of sensors, and hence the +target-robot step-size ratio λ, affects the total time taken for +target encapsulation for each type of target motion model. +Effect of noisy sensors: To study the effect of noise +we +added +Gaussian +noise +to +each +sensor +reading, +zk +s += +(1 − nk +s) � +j∈N k +s Bs(dk +j ), +nk +s +∼ +N(0, σ2) and +nk +s ≤ 1. Similar to the results obtained in our previous work +[5], for all noise levels, we did not observe any collision +within the swarm. However, to ensure that a robot does not +collide with a moving target or the environment boundary, we +increase the radius of Orinner +0 +in proportion to the standard +deviation of the noise. Fig. 12a shows the total time taken by +the swarm to encapsulate a target with p = 7 and λ = 1.1549. +With an increase in noise level, a robot’s estimate of the +target’s location becomes less accurate, leading to an increase +in the total time taken for encapsulation. Furthermore, as can +be seen in Fig. 12b for noise levels greater than 50%, the +probability of success for target encapsulation drops to 40% +when a target moves with constant velocity. +Comparison with algorithm in [5]: The algorithm we +proposed in this paper is more efficient in terms of the total + +1.4 +1.2 +safe ++ πrr/2 +g +safe ++π/2+1 +ax +g +1 +safe ++πrr/2 +2 +g +0 +id +safe +0.8 ++ πrr/2 + 3 +II +0.6 +0.4 +2 +4 +6 +8 +10 +12 +Total Sensors, pTransactions on Robotics, VOL. XX, NO. XX, XXXX +12 +(a) +(b) +(c) +Fig. 11: The total time taken for task completion as a function +of p such that (a) target performs random walk in the environ- +ment (b) target performs random walk while escaping from +nearby robots (c) target moves with a constant velocity while +escaping from nearby robots. The box plot shows median, +25th and 75th percentiles and the min/max values. The line +connects the medians. +time taken by the swarm to encapsulate a static target as +compared with our previous method in [5]. This is due to the +orbiting behavior of the swarm when a robot cannot move +toward the target which results in a faster occupancy of empty +spots in the encapsulation ring. This is shown in Fig. 13. +Scalability: In the supplementary video, we run additional +simulations to show the effect of asymmetric sensor place- +ment, the validity of our algorithm for non-circular robots and +(a) +(b) +Fig. 12: (a) The total time taken for task completion as a +function of noise levels for p = 7, λ = 1.1549 (total time +capped at 4000 time-steps) and different target motion models +(b) probability of success for task completion +Fig. 13: For a given p, the total time taken to encapsulate a +static target is lower for the new approach introduced in this +paper as compared to our previous approach in [5]. + +400 +S +Total time taken +300 +200 +100 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +d300 +S +Total time taken +200 +100 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +p2000 +Total time taken +1500 +1000 +500 +0 +3 +4 +5 +6 +7 +8 +9 +10 1112 +d4000 +-0-RW +-0-RW(escape) +S +-o-const. Vel +Total time taken +3000 +2000 +1000 +T +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Noise levels1.5 +-RW +Probability of success +o-RW(escape) +-const. Vel +0.5 +0 +0 +0.2 +0.4 +0.6 +Noise levels600 +-∞-New Approach +-α- Previous Approach +500 +S +Total time taken ( +400 +300 +- +200 +100 +0 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +pTransactions on Robotics, VOL. XX, NO. XX, XXXX +13 +demonstrate the scalability of our algorithm with a large-scale +simulation of 120 robots and 15 targets moving with different +motion models. +VII. CONCLUSION +In this paper, we propose a decentralized scalable algorithm +for a minimalist swarm to encapsulate dynamic targets with +unknown motion without requiring the exact knowledge of +the relative positions or memory of the previous control +inputs. We consider different scenarios of target motion and +compute bounds on the target-robot step-size ratio to provide +convergence guarantees. We observed the emergence of robots +maintaining an approximate phase difference of 2π/ng in the +encapsulating ring, resulting in uniform distribution around +the target and hence closing off its escaping directions. Fur- +thermore, using extensive simulations we studied the effect of +noisy sensors and showed the validity of our algorithm for non- +circular robots. Our controller can be generalized for robots +equipped with non-isotropic sensors which are not accurate +in measuring the relative distances between two entities. If +the bounds on the measurement error are known, our analysis +can be used to compute bounds on the target-robot step-size +ratio to ensure guaranteed target encapsulations. In the future, +we are planning on implementing our algorithm on physical +robots. We will also study the trade-off between incorporat- +ing the memory of previous states on the desired emergent +behavior and providing timing bounds on task completion. +REFERENCES +[1] L. Bayındır, “A review of swarm robotics tasks,” Neurocomputing, vol. +172, pp. 292–321, 2016. +[2] I. Slavkov, D. Carrillo-Zapata, N. Carranza, X. Diego, F. Jansson, +J. Kaandorp, S. Hauert, and J. Sharpe, “Morphogenesis in robot swarms,” +Science Robotics, vol. 3, no. 25, p. eaau9178, 2018. +[3] M. Rubenstein, C. Ahler, and R. Nagpal, “Kilobot: A low cost scalable +robot system for collective behaviors,” in 2012 IEEE International +Conference on Robotics and Automation, 2012, pp. 3293–3298. +[4] L. Feola and V. Trianni, “Adaptive strategies for team formation in +minimalist robot swarms,” IEEE Robotics and Automation Letters, +vol. 7, no. 2, pp. 4079–4085, 2022. +[5] H. Sinhmar and H. Kress-Gazit, “Decentralized control of minimalistic +robotic swarms for guaranteed target encapsulation,” in 2022 IEEE/RSJ +International Conference on Intelligent Robots and Systems (IROS), +2022, pp. 7251–7258. +[6] A. Babi´c, I. Lonˇcar, B. Arbanas, G. Vasiljevi´c, T. Petrovi´c, S. Bogdan, +and N. Miˇskovi´c, “A novel paradigm for underwater monitoring using +mobile sensor networks,” Sensors, vol. 20, no. 16, p. 4615, 2020. +[7] C. Coquet, C. Aubry, A. Arnold, and P.-j. Bouvet, “A local charged +particle swarm optimization to track an underwater mobile source,” in +OCEANS 2019-Marseille. +IEEE, 2019, pp. 1–7. +[8] Z. Xue, J. Zeng, C. Feng, and Z. Liu, “Swarm target tracking collective +behavior control with formation coverage search agents & globally +asymptotically stable analysis of stochastic swarm.” J. Comput., vol. 6, +no. 8, pp. 1772–1780, 2011. +[9] G. Hollinger, S. Singh, J. Djugash, and A. Kehagias, “Efficient multi- +robot search for a moving target,” The International Journal of Robotics +Research, vol. 28, no. 2, pp. 201–219, 2009. +[10] X. Li, X. Liu, G. Wang, S. Han, C. Shi, and H. Che, “Cooperative target +enclosing and tracking control with obstacles avoidance for multiple +nonholonomic mobile robots,” Applied Sciences, vol. 12, no. 6, p. 2876, +2022. +[11] P. D. Hung, T. Q. Vinh, and T. D. Ngo, “A scalable, decentralised large- +scale network of mobile robots for multi-target tracking,” in Intelligent +Autonomous Systems 13. +Springer, 2016, pp. 621–637. +[12] Q. B. Diep, I. Zelinka, and R. Senkerik, “An algorithm for swarm robot +to avoid multiple dynamic obstacles and to catch the moving target,” +in Artificial Intelligence and Soft Computing, L. Rutkowski, R. Scherer, +M. Korytkowski, W. Pedrycz, R. Tadeusiewicz, and J. M. Zurada, Eds. +Cham: Springer International Publishing, 2019, pp. 666–675. +[13] H. Zhao, D. Straub, and C. A. Rothkopf, “The visual control of +interceptive steering: How do people steer a car to intercept a moving +target?” Journal of Vision, vol. 19, no. 14, pp. 11–11, 12 2019. +[Online]. Available: https://doi.org/10.1167/19.14.11 +[14] F. Belkhouche and B. Belkhouche, “A method for robot navigation +toward a moving goal with unknown maneuvers,” Robotica, vol. 23, +no. 6, p. 709–720, nov 2005. [Online]. Available: https://doi.org/10. +1017/S0263574704001523 +[15] J.-F. Xiong and G.-Z. Tan, “Virtual forces based approach for target +capture with swarm robots,” in 2009 Chinese Control and Decision +Conference. +IEEE, 2009, pp. 642–646. +[16] A. Benzerrouk, L. Adouane, and P. Martinet, “Dynamic obstacle avoid- +ance strategies using limit cycle for the navigation of multi-robot sys- +tem,” in 2012 IEEE/RSJ IROS’12, 4th Workshop on Planning, Perception +and Navigation for Intelligent Vehicles, 2012. +[17] M. Wu, F. Huang, L. Wang, and J. Sun, “A distributed multi-robot co- +operative hunting algorithm based on limit-cycle,” in 2009 International +Asia Conference on Informatics in Control, Automation and Robotics, +2009, pp. 156–160. +[18] Z. Sun, H. Sun, P. Li, and J. Zou, “Cooperative strategy for pursuit- +evasion problem with collision avoidance,” Ocean Engineering, vol. +266, p. 112742, 2022. [Online]. Available: https://www.sciencedirect. +com/science/article/pii/S002980182202025X +[19] M. V. Ramana and M. Kothari, “A cooperative pursuit-evasion game of +a high speed evader,” in 2015 54th IEEE Conference on Decision and +Control (CDC), 2015, pp. 2969–2974. +[20] X. Fang, C. Wang, L. Xie, and J. Chen, “Cooperative pursuit with +multi-pursuer and one faster free-moving evader,” IEEE Transactions +on Cybernetics, vol. 52, no. 3, pp. 1405–1414, 2022. +[21] L. Wu, “Hunting in unknown environments with dynamic deforming +obstacles by swarm robots,” International Journal of Control and +Automation, vol. 8, no. 11, pp. 385–406, 2015. +[22] O. Hamed and M. Hamlich, “Improvised multi-robot cooperation strat- +egy for hunting a dynamic target,” in 2020 International Symposium +on Advanced Electrical and Communication Technologies (ISAECT). +IEEE, 2020, pp. 1–4. +[23] S. Manzoor, S. Lee, and Y. Choi, “A coordinated navigation strategy for +multi-robots to capture a target moving with unknown speed,” Journal +of Intelligent & Robotic Systems, vol. 87, no. 3, pp. 627–641, 2017. +[24] L. Blazovics, K. Csorba, B. Forstner, and C. Hassan, “Target tracking +and surrounding with swarm robots,” in 2012 IEEE 19th International +Conference and Workshops on Engineering of Computer-Based Systems. +IEEE, 2012, pp. 135–141. +[25] K. Xu, Y. Yang, and B. Li, “Brownian cargo capture in mazes via +intelligent colloidal microrobot swarms,” Advanced Intelligent Systems, +vol. 3, no. 11, p. 2100115, 2021. +[26] Y. Yang and M. A. Bevan, “Cargo capture and transport by colloidal +swarms,” Science Advances, vol. 6, no. 4, p. eaay7679, 2020. [Online]. +Available: https://www.science.org/doi/abs/10.1126/sciadv.aay7679 +[27] G. Lee, N. Y. Chong, and H. Christensen, “Tracking multiple moving +targets with swarms of mobile robots,” Intelligent Service Robotics, +vol. 3, no. 2, pp. 61–72, 2010. +[28] A. Franchi, P. Stegagno, and G. Oriolo, “Decentralized multi-robot encir- +clement of a 3d target with guaranteed collision avoidance,” Autonomous +Robots, vol. 40, no. 2, pp. 245–265, 2016. +[29] V. J. Lumelsky and K. Harinarayan, “Decentralized motion planning for +multiple mobile robots: The cocktail party model,” Autonomous Robots, +vol. 4, no. 1, pp. 121–135, 1997. +[30] S. Popov, Recurrence of two-dimensional simple random walk, ser. +Institute of Mathematical Statistics Textbooks. +Cambridge University +Press, 2021, p. 8–32. +[31] Y. Li, W. Zhang, and X. Liu, “Stability of nonlinear stochastic discrete- +time systems,” Journal of Applied Mathematics, vol. 2013, 2013. + diff --git a/cNE5T4oBgHgl3EQfEg7V/content/tmp_files/load_file.txt b/cNE5T4oBgHgl3EQfEg7V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9ff8430014f7475ea347f366862759d3deeb468 --- /dev/null +++ b/cNE5T4oBgHgl3EQfEg7V/content/tmp_files/load_file.txt @@ -0,0 +1,828 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf,len=827 +page_content='Transactions on Robotics, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, XXXX 1 Guaranteed Encapsulation of Targets with Unknown Motion by a Minimalist Robotic Swarm Himani Sinhmar, Hadas Kress-Gazit Abstract—We present a decentralized control algorithm for a robotic swarm given the task of encapsulating static and moving targets in a bounded unknown environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We consider minimalist robots without memory, explicit communication, or lo- calization information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The state-of-the-art approaches generally assume that the robots in the swarm are able to detect the relative position of neighboring robots and targets in order to provide convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In this work, we propose a novel control law for the guaranteed encapsulation of static and moving targets while avoiding all collisions, when the robots do not know the exact relative location of any robot or target in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We make use of the Lyapunov stability theory to prove the convergence of our control algorithm and provide bounds on the ratio between the target and robot speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Furthermore, our proposed approach is able to provide stochastic guarantees under the bounds that we determine on task parameters for scenarios where a target moves faster than a robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Finally, we present an analysis of how the emergent behavior changes with different parameters of the task and noisy sensor readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Index Terms—Collision avoidance, Decentralized control, Min- imalist robot swarm, Lyapunov stability, Target tracking I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' INTRODUCTION T YPICAL approaches to swarm robotics propose simple local behaviors for large numbers of simple robots such that they collectively accomplish a complex task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' many ap- proaches study the properties of the emergent behavior [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In this work, we consider a swarm consisting of homogeneous robots which are minimalist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' they have no memory, cannot broadcast or receive location information from their neighbors and are unable to plan ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Minimalistic robotic swarms [4], [5] have a number of applications, ranging from nanomedicine to underwater monitoring and surveillance [6], [7], where robots might not be able to efficiently communicate with a central controller or with each other, and might not have the ability to self localize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' For example, in an underwater mission, communication may be limited to acoustic signals, which are sensitive to interference and lead to errors in the relative positioning of nearby entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In this paper, we focus on the problem of encapsulating multiple targets, which are moving in unknown motion patterns, by a minimalist robotic swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' A robot in the swarm has no knowledge of the exact relative location of nearby robots, targets, or the boundary of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' This work extends our previous work on encapsulating static targets [5] by addressing moving targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We develop an orbiting behavior for robots to encapsulate the targets, in The authors are with the Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, 14853 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' {hs962,hadaskg}@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' This work is supported by NSF EFMA-1935252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' addition to the searching for targets and avoiding collisions within the swarm, as in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We compare the efficiency of our previous algorithm with the one introduced in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Furthermore, we also show the behavior of our algorithm when applied to non-circular robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Related Work: There has been extensive work on developing various techniques to localize and track a moving target while ensuring collision avoidance [8]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In [14], authors introduced a motion planning strategy for a single robot based on velocity pursuit to intercept a target moving with unknown maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' For target tracking using a multi-robot system, most approaches use artificial potential fields to design a controller consisting of a virtual attraction force to move towards a target and a repulsion force to avoid collision with obstacles [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Another widely used approach to guarantee collision avoidance with dynamic obstacles is using a limit cycle method [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The authors of [17] introduced a hybrid approach where they instead used the limit cycle method to encircle a moving target using a swarm of holonomic robots, and artificial potential fields for collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Since the use of the limit cycle method, either for surrounding a target or avoiding collision with obstacles requires the exact knowledge of the neighbor’s relative position information, we cannot use it for our minimalist robotic swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Pursuit-evasion games [18]–[20] provide guarantees for catching a faster-moving evader by constructing an encircling formation of pursuers composed of a series of Apollonius circles around a target and slowly closing the escape paths of the evader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In this approach, an evader is captured if a pursuer meets the evader at the same point at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Most of the pursuit-evasion methods in the literature assume knowledge of the target’s motion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In this work, we do not assume such knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Existing research [21] in “hunting” of dynamic targets generally makes use of communication within the team and formation-keeping control strategies, while approaching the target, to ensure that all of the escaping routes of the targets are occupied by the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Work in [22] developed a leader- follower strategy based on the behavior of wolves to hunt a randomly moving target with unexpected behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The authors of [23] proposed a limit cycle based algorithm using a neural oscillator to surround a target moving with unknown but constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The authors of [24] utilized rule-based mechanisms using only relative positions of neighbors and no direct communication within the swarm for surrounding an escaping target by introducing a circulating behavior in the swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Recent research in colloidal swarms has shown the capture arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content='05415v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content='RO] 13 Jan 2023 Transactions on Robotics, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, XXXX 2 of multiple randomly moving targets using self-organization control schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In [25], the authors designed a stochastic centralized controller for an intelligent colloidal micro-robotic swarm to capture multiple Brownian targets in a maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In [26], the authors show via simulations, the feedback-controlled reconfigurability of colloidal particles that act as a swarm capable of capturing and transporting microscopic Brownian cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' To implement a distributed approach of searching and encircling targets in an inexpensive and efficient way, in [27] the authors developed a new dual-rotating proximity sensor to obtain relative position information of neighbors for tracking multiple targets with a minimalist swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Authors of [28] proposed a scheme to estimate the global quantities required by the controller in a decentralized way using only local information exchange between robots for the guaranteed encirclement of a 2D or 3D target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' While the above approaches successfully solve the target encirclement while avoiding collisions, most of them rely on the assumption that robots have knowledge of the exact relative location of both their neighbors and the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Furthermore, it is a common assumption that the average speed of the agents in the swarm is greater than that of the moving target to guarantee encapsulation [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In contrast, in this work, we provide guarantees on the encapsulation of dynamic targets without the requirement of accurate (relative) location information and without direct communication within the swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Contributions: This paper’s contributions are: (i) a discrete- time decentralized control law for a minimalist robotic swarm that guarantees the encapsulation of dynamic targets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' for different target motion models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' without accurate detection of the relative location of either the targets or neighboring robots,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' given certain bounds are met (ii) sensor-placement dependent bounds on the ratio between the target and robot speeds to guarantee encapsulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (iii) proof of stochastic convergence of our control law for scenarios when a target is moving faster than a robot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' and (iv) simulations and analysis of emergent behavior of the swarm in the presence of sensor noise and different task parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' DEFINITIONS In this section, we provide definitions from [5] that we use throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Environment: We consider a 2D convex bounded environment E ⊆ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The environment has a fixed global frame I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Robot: We model a robot, R = (cr, γr, rr, p, Z), as a disk of radius rr centered at cr ∈ E with heading γr ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The shape of a robot does not affect the analysis presented in the paper since the robot can always be circumscribed by a circle of radius rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Each robot is reactive, memoryless, has no knowledge of the relative locations of other robots or targets, and cannot communicate with its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The kinematics of a robot is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (1), which is a typical model for a differential drive robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' At each time step, the robot is controlled in a turn-then-move scheme with control inputs θr ∈ S and dr ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The maximum step-size of a robot is dmax r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' γr,T = γr,T −1 + θr cr,T = cr,T −1 + dr[cosγr,T sinγr,T ] (1) A robot has p isotropic sensors arranged on its boundary such that φk ∀k ∈ {1 · · · p} is the angle between the kth sensor and the robot’s heading direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Z is the set of measurements from all sensors on a robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Signal Sources: We consider three types of signal-emitting sources present in the environment that a robot can detect: sg from a point source at the center of a target, sr from a point source at the center of a robot, and se from a line source present on the entire environment boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' For clarity in notation, we hereby denote the signal set {sg, sr, se} by {g, r, e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The strength of any signal s ∈ {g, r, e} located at a distance d from a signal source is given by the function Bs(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The influence distance of a source is limited to βs, such that Bs(d) = 0 ∀d ≥ βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Let N k s be the set of all the sources of type s in the sensing range of the kth sensor and dk j be the distance of this sensor from a source j ∈ N k s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Then the sensor reading zk s = � j∈N k s Bs(dk j ) is the sum of signal strengths from all sources in N k s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' This summation becomes an integral over the boundary segment for a line source present inside the influence region βe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The tuple (zk g, zk r , zk e ) corresponds to the measurements of the kth sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Let Zg = {z1 g · · · zp g}, Zr = {z1 r · · · zp r} and Ze = {z1 e · · · zp e}, then the measurement set is Z = Zg ∪ Zr ∪ Ze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We define rsafe s ∀s ∈ {g, r, e} as the user-specified minimum safety distance that a robot must maintain from a source at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' PROBLEM FORMULATION We model a target g = (cg, rg) as a disk of radius rg centered at cg ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' G is the set of all targets contained in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The kinematics of a target is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' At any time step T, dg ∈ R+ is the distance moved by the target, and γg,T ∈ S is the target heading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' γg,T = γg,T −1 + θg cg,T = cg,T −1 + dg[cosγg,T sinγg,T ] (2) The maximum distance that a target can move is dmax g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Target Motion Models: In this paper, we design controllers and analyze the swarm behavior for different types of target motion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' A target can exhibit one of the following motions: 1) Target moves randomly such that at any time step T, γg,T ∈ [0 2π), dg ∈ [0 dmax g ] and cg,T ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 2) Target moves randomly as in motion model 1 until a robot is in its escape domain = (cg, rescape g ) of radius Transactions on Robotics, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, XXXX 3 rescape g centered at cg, in which case the target chooses a heading direction to escape from all the robots that sat- isfies ∥cg,T − cr,T ∥ ≤ rescape g , and moves the maximum step-size dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 3) Target follows an unknown motion pattern until a robot satisfies ∥cg,T − cr,T ∥ ≤ rescape g , in which case it chooses a heading direction to escape nearby robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Target Encapsulation: For each target g ∈ G, we define an encapsulation ring Ag,T = (cg,T , rsafe g , rencap g ) of inner radius rsafe g and outer radius rencap g centered at cg,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' A robot R is considered to be in Ag,T if, rsafe g < ∥cr,T − cg,T ∥ ≤ rencap g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' A target is encapsulated if the total number of robots present in the encapsulation ring is ng, which is a user-specified input as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 1: A target is encapsulated if ng robots are present simultaneously in the encapsulation ring while maintaining at least a distance of rsafe r from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Problem statement: Consider a bounded environment E ⊆ R2 with m dynamic targets where the initial distribution of the robots and targets is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Given the total number of sensors p on a robot, the user-provided safe distance rsafe s ∀s ∈ {g, r, e}, the encapsulation ring Ag, and the number of robots ng needed to encapsulate each target g such that the total number of robots n ≥ � g∈G ng, our objective is to find a real-time decentralized control law for encapsulating all targets while ensuring safety distances are always maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We make the following assumptions about the environment and the system: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The sensors are arranged on a robot such that when a robot’s center is rsafe s away from a source s, at least one sensor is in the influence region of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' For ease of exposition, we consider circular robots with a symmetric placement of sensors to explain our algorithm, and show in simulations how asymmetric sensor placements and non-circular robots affect swarm behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The distance between any two moving targets is greater than (2βg + 2rr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' That is, a robot can sense at most one target at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We constrain a target to maintain a minimum distance of (rencap g + rsafe e + dmax r ) from the environment boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' This ensures that robots will be able to encapsulate the target without colliding with the environment boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We place no restriction on the target’s knowl- edge of the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' it may be able to perfectly sense the relative location of any robot present in its user-specified escape domain, rescape g thereby knowing the optimal escape route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' However, if a target is encapsulated, we assume it emits a single burst of a shut-off signal and stops emitting any signal subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The influence distance of this signal is limited to Ag, and we assume that thereafter both the robots within the encapsulation ring and the target stop moving, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' dr = 0 and dg = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The signal strength Bs strictly decreases with the radial distance, d from a source and the inverse of the signal function Bs(d) exists and is known to the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' APPROACH Our strategy for designing a local control law is based on geometry and the relative kinematics of the interaction of a robot with its neighboring robots and a dynamic target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We extend our previous work [5] where we only considered static targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' a robot’s behavior there was to either move randomly in the bounded environment when it does not sense any target, or to move towards a target if sensing one while ensuring safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Here, we introduce an additional robot behavior of orbital encirclement of a target, inspired by [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' As we show in Section V, this behavior ensures the encapsulation of an escaping target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In Section IV-A we describe virtual sources as defined in [5] and use them to under-approximate the relative distance between a source and the robot’s center as a function of the sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In Section IV-B we find the bounds on control parameters (dr and θr) for a robot to ensure that it maintains rsafe s distance from a source s ∈ {g, r, e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In Section IV-C we introduce the concept of orbital encirclement of a moving target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' we provide a summary of the overall reactive control law for a robot in the swarm in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Virtual Source Since we assume a robot is equipped with isotropic sensors, a sensor measurement corresponds to the aggregated signal strength from all the nearby sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Hence, the same mea- surement could correspond to a single source nearby or a cluster of sources further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Therefore, for each sensor reading, zk s ∀s ∈ {g, r, e}, we define a virtual source on a circle centered at the sensor k as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' It is shown in [5] that the closest possible location of the virtual source with respect to the robot’s center is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Furthermore, the range of possible directions of the location of the virtual source with respect to the robot’s center is restricted to [φk − π/p, φk + π/p] for symmetric sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' ds = rrcos(π/p) + � (dks)2 − r2rsin2(π/p) (3) For asymmetric sensor placement, we replace π/p with half of the maximum angle that the kth sensor makes with either of its adjacent sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Similarly, for robots that are not circular safe 7 Encapsulation ring, Ag,T Encapsulating 9 robots, ng = 6 e Ds TargetTransactions on Robotics, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, XXXX 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 2: Virtual source for the kth sensor [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' in shape, we replace rr by the distance between the kth sensor and the robot’s center in the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We can see in (3) that as p → ∞, ds → dk s +rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' That is, the error in locating the source is dependent on the total sensors on a robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Collision Avoidance We use the technique introduced in [5] for collision avoid- ance with nearby robots and the environment boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' At each time step, the robot estimates the relative distance be- tween its center and the nearby sources using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (3) for the sensor with the maximum sensor reading zk s ∀s ∈ {g, r, e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' If this distance is less than or equal to (rsafe s +dmax s ), the collision avoidance behavior is triggered for this robot to ensure safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We have shown in [5] that to avoid collisions with static obstacles (such as environment boundary), the robot’s heading direction θr must be chosen from the angular range given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Θavo e = [φk + π/p + π/2, φk − π/p + 3π/2] (4) Whereas to avoid the neighboring moving robots, the distance dr that a robot moves at time step T in a given heading direction γr,T must be chosen such that at T + 1 it maintains at least a distance of rsafe r from the closest neighboring robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 3, let k and l be the indices of the sensors closest to the intended heading direction γr at time T and dk r and dl r are their radii of virtual sources respectively such that dk r > dl r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Then, we can compute the bounds on the step-size dr that the robot can take in the heading direction γr,T using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 3: Computing dr such that collision is avoided with nearby moving robots [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 0 ≤ dr ≤ rrcos(φl − θr) + � (dlr − rsafe r − dmax r )2 − r2rsin2(φl − θr) (5) To ensure that two robots never deadlock, the bounds on the maximum step size a robot can take, and the influence region of a robot’s source, are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (6) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The proof is detailed in Lemma V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content='3 of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' dmax r < rsafe r + rrcos(π/p) 2 − � (rsafe r )2 + r2r − 2rrrsafe r cos(π/p) 2 (6) � (rsafe r )2 + r2r − 2rrrsafe r cos(π/p) + 2dmax r < βr < rsafe r + rrcos(π/p) (7) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Encirclement of a Target In [5], our approach to encapsulate a static target, was for a robot to either move towards the target or move away from an obstacle between itself and the target in the direction of the sensor receiving the minimum reading from nearby moving robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' However, in order to surround a dynamic target, the behavior of a robot should be such that the swarm is able to disperse around the target in order to block off its escaping paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Since we consider minimalist robots that can neither communicate with their neighbors nor know their exact relative position, we can not make use of formation control strategies, such as [17], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Consider a scenario where all the robots in the swarm start on one side of a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Then, for a swarm to disperse around a target, it is necessary that an individual robot be able to catch up with the escaping target, and once the robot reaches the encapsulation ring, it should be able to encircle the target so that the target is prevented from escaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' To ensure encapsulation, we define primary and secondary orbits around each target, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' For each orbit, we define a tie-breaking orbital rotation which can be either clockwise (denoted by a value of -1) or counter- clockwise (denoted by a value of 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The primary orbit, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 4: Primary (purple ring) and secondary (cyan rings) orbits around a target, and the lower bound on the target’s escape domain (red circle) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' A robot moves either clockwise or counter-clockwise in an orbit depending on its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The solid arrows denote the tie-breaking rotation for an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Or0 = (cg,T , Orinner 0 , Orouter 0 , −1) is an annular ring centered at cg,T with an inner radius of Orinner 0 ≥ rsafe g + dmax g , an virtual source actual k sources R S robot kth sensorheading at T+1 closest virtual source at T+1 Ith sensor Cr,T+1 (p kth sensor p SOrbit imax rotations Secondary Primary orbits Ori>0 Target orbit OroTransactions on Robotics, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' XX, XXXX 5 outer radius Orouter 0 = rencap g and a clockwise orbital rotation (chosen arbitrarily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Let w be the width of a secondary orbit, then an ith secondary orbit is given by, Ori = (cg,T , Orouter 0 + (i − 1)w, Orouter 0 + (i)w, (−1)i−1), ∀i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' We consider a robot to be in ith orbit if, Orinner i < ∥cg,T − cr,T ∥ ≤ Orouter i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Each robot in the swarm computes its current orbit using its estimate of ∥cg,T − cr,T ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' At time-step T, let Ori be the current orbit as estimated by a robot, then its control consists of one of the following behaviors: 1) if i > 0, the robot moves towards the target in a heading direction chosen from the line of sight angular range as estimated from the virtual source (ΘLOS g ) while maintaining a safe distance from nearby robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 2) else if i > 0 and the robot cannot move a non-zero distance towards the target, it moves tangentially in its current orbit while maintaining a safe distance from nearby robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The direction of the tangent is chosen such that it maximizes the possible step-size dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In case of symmetry, the robot moves in the orbital rotation of the ith orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 3) else if i > 0 and the robot can neither move in a direc- tion from ΘLOS g nor tangential to the orbit, it chooses a direction of motion that maximizes the possible step-size dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 4) else if i = 0, the robot moves tangentially in its current orbit while maintaining a safe distance from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 5) else if the relative distance between the target and a robot is less than or equal to Orinner 0 , it moves away from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 6) else the robot performs a simple random walk while avoiding nearby moving robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In general, a robot moves toward the target until it reaches the primary orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' If other robots are present between itself and the target, the robot moves tangentially in its current orbit until it can move toward the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' All the robots that place themselves in the primary orbit constantly move tangentially and eventually close off the target’s escape routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The width of a secondary orbit, w, must be less than βr, so that a robot’s neighbors in adjacent orbits lie within its sensing range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' This ensures that a robot doesn’t move towards a target when it senses other robots in the front and instead moves tangentially in its current orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Now, using the sensor readings and their corresponding virtual sources, we find the set of directions that a robot needs to choose from to move towards a target, away from a target, or tangentially in an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Let k be the index of the sensor such that zk g > zl g, ∀l ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Here we have ignored the unlikely scenario where two sensors receive the same maximum intensity from a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Then the angular range, ΘLOS g , for the possible location of the target with respect to the robot’s center is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' ΘLOS g = [φk − π/p, φk + π/p] (8) The angular range, Θavo g (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (9)), to move away from the target can be derived in a similar fashion to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Θavo g = [φk + π/p + π/2, φk − π/p + 3π/2] (9) The angular range to move tangentially in an orbit in a clockwise or counter-clockwise direction is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' (11), respectively, where we define Θtan g = Θtan,+1 g ∪ Θtan,−1 g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Θtan,−1 g = [φk − π/p + π/2, φk + π/p + π/2] (10) Θtan,+1 g = [φk − π/p + 3π/2, φk + π/p + 3π/2] (11) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 5 shows the different angular range sets for a target-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' It is worth mentioning that for noiseless sensors, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 5: The angular range set for a target-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The robot is equipped with 5 sensors placed asymmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' if zk−1 g > zk+1 g then ΘLOS g = [φk − π/p, φk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' This results in a more accurate estimation of the location of a target and reduces the angular resolution error by half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' The estimation of Θtan g and Θavo g also changes accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' As shown in our previous work [5], a heading direction in the angular ranges ΘLOS g and Θavo g is guaranteed to make a robot move towards the target and away from the target, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In contrast, a robot might end up moving towards or away from the target when it moves tangentially in an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Since secondary orbits are at least at a distance of Orouter 0 from a target, a robot moving tangentially in these orbits will always maintain a safe distance from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' However, if a robot is moving tangentially in the primary orbit, we need to make sure that it maintains at least a distance of Orinner 0 from the target after moving dr units in the intended heading direction γr,T such that θr ∈ Θtan g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 6, we can see that at T + 1, the closest possible location of the target is at S∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' If the heading direction θr /∈ ΘLOS g , the closest possible location of the target with respect to the robot’s center at T +1 would be along one of the extremes of the angular range ΘLOS g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' To ensure safety, ∥cr,T +1 − S∥ ≥ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' 6: The distance dr that the robot can move in the intended heading is computed using the geometry of ∆Scr,T cr,T +1 Orinner 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf'} +page_content=' Using the cosine rule of triangle for △Scr,T cr,T +1, g IICg,T - Cr,Tll g SO70 g kth sensor TargetSO70 g Heading of robot at T+1 with a +1 step-size of dr Actual Heading of target 0 we have: +ωg,n = (−1)n · W gBGW +g,n +(x1, · · · , xn)dx1 · · · dxn. +Moreover, the above topological recursion can also be reformulated in terms of +another Bergman kernel (see §5.4 for details): +�B(x1, x2) = +�1 +2 +� +1 +(x1 − x2)2 + +1 +(x1 + x2)2 +� ++ W gBGW +0,2 +(x1, x2) +� +dx1dx2. +In literatures a symmetric bi-differential of the above form is supposed to be the +Bergman kernel for the topological recursion associated to a tau-function of the +BKP hierarchy, see e.g. [5,19,48]. +Remark 1.1. In [1], Alexandrov has conjectured a topological recursion on another +spectral curve for the generalized BGW models (see [1, (128); Conjecture 3.7]): +x2y2 − x − S2 +4 = 0. +His method is to look for an operator annihilating the principal specialization of +the tau-function and then take the semi-classical limit. This curve looks different +from (1), since he made a change of variable x = λ2 in the principal specialization +to represent it in terms of the modified Bessel function (see also [34, §3.2.3]). +Besides the Virasoro constraints and topological recursion, there is an alternative +way to compute the connected n-point functions. Using the Schur Q-expansion +given in [3,4,31], we are able to write down the explicit formulas for the BKP-affine +coordinates for the generalized BGW tau-functions. Then we can apply a formula +proved in [42] to obtain the following (see §3 for details): +Theorem 1.2. The connected n-point functions associated to the generalized BGW +tau-functions are given by: +� +i1,··· ,in>0: odd +∂n log τ (N) +BGW(t/2) +∂ti1 · · · ∂tin +���� +t=0 +· x−i1 +1 +· · · x−in +n += −δn,2 · x1x2(x2 +2 + x2 +1) +2(x2 +1 − x2 +2)2 +− 2n−1 · +� +� +σ: n-cycle +n +� +i=1 +ξ(xσ(i), −xσ(i+1)) +� +odd, +(2) + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +5 +for n ≥ 1, where [·]odd means taking the terms of odd degrees in every xi, and we +use the convention σ(n + 1) := σ(1). The function ξ is: +ξ(xσ(i), −xσ(i+1)) = + + + + + +A(N)(xσ(i), −xσ(i+1)), +σ(i) = σ(i + 1); +�A(N)(xσ(i), −xσ(i+1)), +σ(i) < σ(i + 1); +− �A(N)(−xσ(i+1), xσ(i)), +σ(i) > σ(i + 1), +where A(N) and �A(N) are the generating series of the BKP-affine coordinates (i.e., +the fermionic 2-point functions) whose explicit formulas are: +A(N)(w, x) = w − x + Φ(N) +1 +(−x)Φ(N) +2 +(−w) − Φ(N) +1 +(−w)Φ(N) +2 +(−x) +4(w + x) +, +�A(N)(w, x) = Φ(N) +1 +(−x)Φ(N) +2 +(−w) − Φ(N) +1 +(−w)Φ(N) +2 +(−x) +4(w + x) +, +where Φ(N) +1 +, Φ(N) +2 +are the formal Laurent series: +Φ(N) +1 +(z) = 1 + +∞ +� +k=1 +(−ℏ)k +8k · k! · +k +� +i=1 +� +4N 2 − (2i − 1)2� +· z−k, +Φ(N) +2 +(z) = z + +∞ +� +k=1 +(−ℏ)k +8k · k! · +k +� +i=1 +� +4(1 − N)2 − (2i − 1)2� +· z1−k. +It is interesting to notice that the Eynard-Orantin topological recursion is effi- +cient only at small genus, while the explicit formula (2) is efficient for large genus +and small n. +Finally, we consider the emergence of the quantum spectral curve of type B for +the plane curve (1). In [21] Gukov and Su�lkowski proposed a conjectural construc- +tion of the quantum spectral curve for a classical plane curve using Eynard-Orantin +topological recursion. It is well-known that when the generating series of the E- +O invariants on the classical curve is a tau-function of the KP hierarchy, finding +the quantum spectral curve (i.e., the Schr¨odinger equation [21, (1.10)]) is equiv- +alent to finding an operator which annihilates the principal specialization of the +tau-function. When considering a tau-function of the BKP hierarchy, the principal +specialization is a slightly different with that in case of KP hierarchy, and the defi- +nition of the Baker-Akhiezer function [21, (1.11);(2.4)-(2.7)] may need some simple +modifications. Nevertheless, in what follows we will refer the following two facts as +a quantum spectral curve of type B in the sense of Gukov-Su�lkowski: +1) The connected n-point functions of a tau-function τ(t) of the BKP hierarchy +can be reconstructed from the Eynard-Orantin topological recursion on a +plane curve C; +2) There is an operator P annihilating the following principal specialization: +P +� +τ +� +− 2 +z , − 2 +3z3 , − 2 +5z5, · · · +�� += 0, +such that the semi-classical limit of P gives the defining equation of C. +In [24], a method of deriving Kac-Schwarz operators [26,39] and quantum spec- +tral curve of type B using BKP-affine coordinates has been proposed. Now we apply + +6 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +this method to the case of generalized BGW models using the explicit expressions +of BKP-affine coordinates computed in §3. We show that the operator +P = ℏ3� +(z∂z + 1 +2)2 − N 2�� +∂z − +ℏ +2z2 +� +(z∂z − 1 +2)2 − N 2�� +gives the quantum spectral curve of type B in the sense of Gukov-Su�lkowski for the +classical curve (1), see §6 for details. It is worth pointing out that in [3] Alexandrov +has found that such an operator annihilates the principal specialization (without +knowing the topological recursion) using a different method. +The rest of this paper is arranged as follows. In §2 we recall some preliminaries +of the generalized BGW tau-functions, including the Virasoro constraints. In §3 we +compute the BKP-affine coordinates and derive the formula (2). In §4 we construct +the spectral curve (1) and discuss the quantum deformation theory. In §5 we show +that the Eynard-Orantin topological recursion on the curve (1) emerges naturally +from the Virasoro constraints of the generalized BGW tau-functions. And finally +in §6, we show the emergence of the quantum spectral curve of type B from the +BKP-affine coordinates and the above topological recursion. +2. Preliminaries of Generalized Br´ezin-Gross-Witten Tau-Functions +In this section, we first briefly review some basics of the generalized Br´ezin- +Gross-Witten tau-functions τ (N) +BKP(t), including the Virasoro constraints, recursion +for n-point functions, and Schur Q-function expansion. +2.1. Generalized BGW tau-functions and Virasoro constraints. The gen- +eralized BGW model was introduced by Mironov-Morozov-Semenoff in [34] as a +family of matrix integrations indexed by a parameter N (where N is a complex +number and is not to be confused with the size of the matrices). For each N ∈ C +the partition function τ (N) +BGW(t; ℏ) is a tau-function of the KdV hierarchy with time +variables t = (t1, t3, t5, · · · ). And for N = 0, the partition function τ (0) +BGW is the +original BGW tau-function [9,20]. +In [1], Alexandrov showed that for every N the partition function τ(N) +BGW(t; ℏ) is +uniquely determined by the normalization condition +τ (N) +BGW(0; ℏ) = 1, +together with the Virasoro constraints +(3) +L(N) +n +τ (N) +BGW(t; ℏ) = 0, +∀n ≥ 0, +where the Virasoro operators {L(N) +n +}n≥0 are: +L(N) +n += − 1 +2ℏ +∂ +∂t2n+1 ++ 1 +4 +� +a+b=2n +a,b: odd +∂2 +∂ta∂tb ++ 1 +2 +� +k≥1 +k: odd +ktk +∂ +∂tk+2n ++ 1 − 4N 2 +16 +δn,0. +which satisfy the Virasoro commutation relation: +[L(N) +m , L(N) +n +] = (m − n)L(N) +m+n, +∀m, n ≥ 0. +The free energy associated to τ (N) +BGW is the logarithm +F(N)(t; ℏ) = log τ (N) +BGW(t; ℏ). + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +7 +Denote: +(4) +S = ℏ · N, +then the free energy F(N) has a genus expansion (see e.g. [1, (112)]): +F(N)(ℏ−1t; ℏ) = +∞ +� +g=0 +ℏ2g−2FgBGW +g +(t; S), +where ℏ−1t = (ℏ−1t1, ℏ−1t3, ℏ−1t5, · · · ). In what follows, we will denote: +(5) +FgBGW(t; S) = +∞ +� +g=0 +ℏ2g−2FgBGW +g +(t; S) = F(N)(ℏ−1t; ℏ), +and thus one has τ (N) +BGW(ℏ−1t; ℏ) = eF gBGW(t;S). The Virasoro constraints (3) now +can be rewritten as +(6) +LgBGW +n +� +exp FgBGW(t; S) +� += 0, +∀n ≥ 0, +where: +LgBGW +0 += −1 +2 +∂ +∂t1 ++ 1 +2 +� +k≥1 +k: odd +ktk +∂ +∂tk ++ 1 +16 − ℏ−2 S2 +4 , +LgBGW +n += −1 +2 +∂ +∂t2n+1 ++ ℏ2 +4 +� +a+b=2n +a,b: odd +∂2 +∂ta∂tb ++ 1 +2 +� +k≥1 +k: odd +ktk +∂ +∂tk+2n +, +n ≥ 1. +Remark 2.1. Our notations for the partition function and free energy differ from +those in [1] by a rescaling ti �→ 2ti for every i. +2.2. Recursion for connected n-point functions. In this subsection, we refor- +mulate the Virasoro constraints as recursions for the connected n-point functions. +This has been done by Alexandrov in [1, §3] (in different notations). +Let ⟨pµ1 · · · pµn⟩c +g be the connected correlators defined by: +(7) +⟨pµ1pµ2 · · · pµn⟩c +g = ∂nFgBGW +g +(t; S) +∂tµ1∂tµ2 · · · ∂tµn +��� +t=0, +µ1, · · · , µn > 0 : odd, +then the free energy is of the form: +FgBGW +g +(t; S) = +� +n≥1 +� +µi>0: odd +1 +n!⟨pµ1 · · · pµn⟩c +g · tµ1 · · · tµn. +One can rewrite the Virasoro constraints (6) as the recursion for the correlators: +⟨p2k+1pµ1pµ2 · · · pµn⟩c +g +=1 +2 +� +a+b=2k +a,b>0: odd +� +⟨papbpµ1 · · · pµn⟩c +g−1 + +� +g1+g2=g +I⊔J=[n] +⟨papµI⟩c +g1⟨pbpµJ ⟩c +g2 +� ++ +n +� +i=1 +µi⟨pµi+2kpµ1 · · · ˆpµi · · · pµn⟩c +g, +k ≥ 0, +(8) + +8 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +together with the initial values: +⟨p1⟩c +0 = −1 +2S2, +⟨p1⟩c +1 = 1 +8; +⟨p1⟩c +g = 0, +∀g ≥ 2, +(9) +where ˆpµi means deleting the term pµi. Here [n] = {1, 2, · · · , n}, and for a set of +indices I = {i1, i2, · · · , im} ⊂ [n] we denote pµI = (pµi1 , pµi2 , · · · , pµim). In the +case g = 0, we use the convention ⟨papbpµ1 · · · pµk⟩c +−1 = 0 in the right-hand side. +Now consider the n-point function W gBGW +g,n +of genus g defined by: +W gBGW +g,n +(x1, · · · , xn) = +� +µ1,··· ,µn: odd +⟨pµ1 · · · pµn⟩c +g · x−µ1−1 +1 +· · · x−µn−1 +n += +� +i1,··· ,in≥0 +∂nFgBGW +g +(t; S) +∂t2i1+1 · · · ∂t2in+1 +��� +t=0 · x−2i1−2 +1 +· · · x−2in−2 +n +, +(10) +for g ≥ 0 and n ≥ 1, where x1, · · · , xn are some formal variables. For example, +using the above recursion for correlators one easily finds that: +(11) +⟨p2n+1⟩c +0 = (−1)n+1 +22n+1 +· +1 +n + 1 +�2n +n +� +· S2n+2, +∀n ≥ 0, +and thus: +(12) +W gBGW +0,1 +(x) = +� +n≥0 +(−1)n+1 +22n+1 +· +1 +n + 1 +�2n +n +� +S2n+2x−2n−2 = 1 − +� +1 + S2 +x2 . +Remark 2.2. Here our notation W gBGW +g,n +differs from Wg,n in [1, §3] by: +W gBGW +g,n +(x1, · · · , xn) = 2−n · Wg,n(x2 +1 +4 , · · · , x2 +n +4 ). +The purpose of modifying the notations in this way is to make the discussions fit +into the picture of emergent geometry (see §4 - §5 for details). +In the rest of this subsection, we rewrite the Virasoro constraints as the recursion +for the n-point functions W gBGW +g,n +(x1, · · · , xn). By (8) we have: +W gBGW +g,n+1 (x, x1, · · · , xn) += +� +k≥0 +� +µ1,··· ,µn: odd +⟨p2k+1pµ1 · · · pµn⟩g · x−2k−2x−µ1−1 +1 +· · · x−µn−1 +n += +� +µ +∞ +� +k=0 +� +a+b=2k +1 +2⟨papbpµ⟩g−1 · x−a−1x−b−1x−µ1−1 +1 +· · · x−µn−1 +n ++ +� +µ +∞ +� +k=0 +� +a+b=2k +� +g1+g2=g +I⊔J=[n] +1 +2⟨papµI⟩c +g1⟨pbpµJ⟩c +g2 · x−a−1x−b−1x−µ1−1 +1 +· · · x−µn−1 +n ++ +� +µ +∞ +� +k=0 +n +� +i=1 +µi⟨pµi+2kpµ1 · · · ˆpµi · · · pµn⟩c +g · x−2k−2x−µ1−1 +1 +· · · x−µn−1 +n +. + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +9 +Now we conclude that for every (g, n) ̸= (0, 0), +W gBGW +g,n+1 (x, x1, · · · , xn) = +n +� +i=1 +Dx,xiW gBGW +g,n +(x1, · · · , xn)+ +1 +2Ex,u,v +� +W gBGW +g−1,n+2(u, v, x1, · · · , xn) + +� +g1+g2=g +I⊔J=[n] +W gBGW +g1,|I|+1(u, xI)W gBGW +g2,|J|+1(v, xJ) +� +, +where we denote xI = (xi1, · · · , xim) for I = {i1, · · · , im}, and +(13) +Ex,u,vf(u, v) = +� +lim +v→u f(u, v) +��� +u=x, +and Da,b is an operator such that for every odd m, +Dx,xi(x−m−1 +i +) = +� +2k+µi=m +µi odd; k≥0 +µix−2k−2x−µi−1 +i += +1 +(x2 − x2 +i )2 +� +1 +xm−1 − +1 +xm−1 +i ++ +x2 +i +xm+1 − +x2 +xm+1 +i +� ++ +m + 1 +xm+1 +i +(x2 − x2 +i ) +So one can take the operator Da,b to be: +(14) +Da,bf(b) = +a2 + b2 +(a2 − b2)2 +� +f(a) − f(b) +� +− +b +a2 − b2 ∂bf(b). +Notice that the right-hand side of the above recursion involves terms W gBGW +0,1 +(x). +We may move such terms to the left-hand side and rewrite the recursion as follows: +W gBGW +g,n+1 (x, x1, · · · , xn) = +n +� +i=1 +�Dx,xiW gBGW +g,n +(x1, · · · , xn)+ +1 +2 +�Ex,u,v +� +W gBGW +g−1,n+2(u, v, x[n]) + +s +� +g1+g2=g +I⊔J=[n] +W gBGW +g1,|I|+1(u, xI)W gBGW +g2,|J|+1(v, xJ) +� +, +(15) +where +s� means that we exclude the terms involving W gBGW +0,1 +in this summation, +and we use the notations: +(16) +�Ex,u,v = +1 +1 − W0,1(x)Ex,u,v, +�Dx,xi = +1 +1 − W0,1(x)Dx,xi. +Example 2.1. We can compute W gBGW +0,2 +using the above formula: +W gBGW +0,2 +(x, x1) = +1 +1 − W0,1(x)Dx,x1W0,1(x1) += +1 +(x2 − x2 +1)2 +� +x2 + x2 +1 + 2S2 +� +(1 + S2 +x2 )(1 + S2 +x2 +1 ) +− x2 − x2 +1 +� += − 1 +2S2x−2x−2 +1 ++ 3 +8S4x−2x−4 +1 ++ 3 +8S4x−4x−2 +1 +− 5 +16S6x−2x−6 +1 +− 3 +8S6x−4x−4 +1 +− 5 +16S6x−6x−2 +1 ++ · · · . + +10 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +2.3. Schur Q-function expansion and BKP-affine coordinates. In this sub- +section, we recall the Schur Q-function expansions of the generalized BGW tau- +functions [3,4,31]. +It is known that if τ(t) is a tau-function of the KdV hierarchy with time vari- +ables t = (t1, t3, t5, · · · ), then τ(t/2) is a tau-function of the BKP hierarchy, see [2]. +Therefore for every N, the partition function τ (N) +BGW(t/2) is a tau-function of the +BKP hierarchy. Moreover, every tau-function of the BKP hierarchy can be rep- +resented as a summation of Schur Q-functions, see [47]; and see also [23, 32] for +introductions to Schur Q-functions and the relation to projective representations of +the symmetric groups Sn. Thus τ (N) +BGW(t/2) admits a Schur Q-function expansion +for every N ∈ C. +The following formula was conjectured by Alexandrov [4], and then proved in [31] +and [3] by two different methods: +(17) +τ (N) +BGW(t) = +� +λ∈DP +� ℏ +16 +�|λ| +· 2−l(λ)θλQλ(δk,1)Qλ(t), +where DP is the set of all strict partitions λ = (λ1 > λ2 > · · · > λl(λ) > 0), and +Qµ is the Schur Q-function indexed by µ ∈ DP. And θλ is given by: +(18) +θλ = +l� +j=1 +λj +� +k=1 +θ(k) +for λ = (λ1, λ2, · · · , λl) ∈ DP, where θ is the following function on Z+: +(19) +θ(z) = (2z − 1)2 − 4N 2. +This Schur Q-function expansion shows that τ(N) +BGW(t/2) is a hypergeometric tau- +function [36] of the BKP hierarchy. +Remark 2.3. It is worth noting that in [31], Liu and the second author proved this +Schur Q-function expansion by applying the Virasoro constraints (3). +3. BKP-Affine Coordinates of Generalized BGW Tau-Functions +In this section, we write down an explicit formula for the BKP-affine coordinates +of the generalized BGW tau-functions using the Schur Q-function expansion. Then +we apply the results in [42] to give a formula for the connected n-point functions. +The BKP-affine coordinates will be useful in §6. +3.1. BKP-affine coordinates of τ (N) +BGW(t/2). In this subsection, we write down +the BKP-affine coordinates for the generalized BGW tau-functions. +First we recall some basics about BKP tau-functions and BKP-affine coordinates +[22, 42]. +BKP-affine coordinates are natural coordinates on the big cell of the +isotropic Sato Grassmannian, see [22, §7.3] for an introduction. Let τ(t) be a tau- +function of the BKP hierarchy with BKP-time variables t = (t1, t3, t5, · · · ) satisfying +the initial value condition τ(0) = 1, then the coefficients of the Schur Q-functions +in such an expansion are Pfaffians of the BKP-affine coordinates: +(20) +τ = +� +µ∈DP +(−1)⌈l(µ)/2⌉ · Pf(aµi,µj)1≤i,j≤2⌈l(µ)/2⌉ · Qµ(t/2), + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +11 +where {an,m}n,m≥0 are the BKP-affine coordinates of τ(t), satisfying the anti- +symmetry condition: +(21) +an,m = −am,n, +∀n, m ≥ 0. +For µ = (µ1 > · · · > µl(µ) > 0) ∈ DP, we denote by l(µ) the length of µ, and +here ⌈·⌉ is the ceiling function. For µ ∈ DP with l(µ) odd, we use the convention +µl(µ)+1 = 0. For example, the first a few terms of τ are: +τ =1 + +� +n>0 +a0,n · Q(n)(t/2) + +� +m>n>0 +an,m · Q(m,n)(t/2) ++ +� +m>n>l>0 +(an,ma0,l − al,ma0,n + a0,mal,n)Q(m,n,l)(t/2) ++ +� +m>n>l>k>0 +(an,mak,l − al,mak,n + ak,mal,n)Q(m,n,l,k)(t/2) + · · · . +In particular, the BKP-affine coordinates {an,m}n,m≥0 are exactly the coefficients +of Qµ with l(µ) ≤ 2. +Now consider the case of generalized BGW tau-functions τ (N) +BGW(t). Using the +Schur Q-function expansion (17), we obtain the following: +Proposition 3.1. The BKP-affine coordinates {a(N) +n,m}n,m≥0 for the BKP tau- +functions τ (N) +BGW(t/2) are given by a(N) +0,0 = 0, and: +a(N) +0,n = −a(N) +n,0 = +ℏn +23n+1 · n! · +n +� +k=1 +θ(k), +n > 0; +a(N) +n,m = +ℏm+n +23m+3n+2 · m! · n! · m − n +m + n · +m +� +j=1 +θ(j) · +n +� +k=1 +θ(k), +n, m > 0, +(22) +where θ is the function (19). +Proof. Here we need the following combinatorial identity (see e.g. [33, (56)]): +Qλ(δk,1) = +2|λ| +�l(λ) +i=1 λi! +· +� +i 0; +aBGW +n,m += ℏm+n · +� +(2m − 1)!! · (2n − 1)!! +�2 +23m+3n+2 · m! · n! +· m − n +m + n, +n, m > 0. +3.2. A formula for connected n-point functions. Once knowing the BKP- +affine coordinates, we are able to write down a formula for the connected bosonic +n-point functions (in all genera) by applying [42, Theorem 1.1]: + +12 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +Theorem 3.1 ( [42]). Let A(w, x), � +A(w, x) be the following generating series of the +BKP-affine coordinates {an,m} of a BKP tau-function τ(t) satisfying τ(0) = 1: +A(w, x) = +� +n,m>0 +(−1)m+n+1 · an,mw−nx−m − +� +n>0 +(−1)n +2 +· an,0(w−n − x−n), +�A(w, x) = A(w, x) − 1 +4 − 1 +2 +∞ +� +i=1 +(−1)iw−ixi. +(23) +Then: +� +i>0: odd +∂ log τ(t) +∂ti +���� +t=0 +· x−i = A(−x, x), +and for n ≥ 2, +� +i1,··· ,in>0: odd +∂n log τ(t) +∂ti1 · · · ∂tin +���� +t=0 +· x−i1 +1 +· · · x−in +n += −δn,2 · ix1,x2 +x1x2(x2 +2 + x2 +1) +2(x2 +1 − x2 +2)2 ++ +� +σ: n-cycle +ǫ2,··· ,ǫn∈{±1} +(−ǫ2 · · · ǫn) · +n +� +i=1 +ξ(ǫσ(i)xσ(i), −ǫσ(i+1)xσ(i+1)), +(24) +where +ix1,x2 +x1x2(x2 +2 + x2 +1) +2(x2 +1 − x2 +2)2 += +� +n>0: odd +n +2 x−n +1 xn +2 , +and ξ is given by: +ξ(ǫσ(i)xσ(i), −ǫσ(i+1)xσ(i+1)) = +� �A(ǫσ(i)xσ(i), −ǫσ(i+1)xσ(i+1)), +σ(i) < σ(i + 1); +− �A(−ǫσ(i+1)xσ(i+1), ǫσ(i)xσ(i)), +σ(i) > σ(i + 1), +and we use the conventions ǫ1 := 1 and σ(n + 1) := σ(1). +Remark 3.1. This theorem is the BKP-analogue of a formula for connected bosonic +n-point functions of a KP tau-function proved by Zhou in [50]. +Remark 3.2. The generating series A(w, x) and �A(w, x) are fermionic 2-point func- +tions associated to this BKP-tau-function. See [42, §3] for details. +The above formula for n ≥ 2 can be simplified in the following way. Notice that +for a fixed cycle σ and a fixed j, the variable zj appears only in two terms +ξ(±xi, −ǫjxj) · ξ(ǫjxj, ±xk) +in �n +i=1 ξ(ǫσ(i)xσ(i), −ǫσ(i+1)xσ(i+1)) (where i and k are adjacent to j in this cycle +σ), hence replacing ǫj by −ǫj is equivalent to replacing xj by −xj. +Therefore, +replacing ǫj by −ǫj does not change the terms with odd orders in xj in the product +ǫ2 · · · ǫn+m +n+m +� +i=1 +ξ(ǫσ(i)xσ(i), −ǫσ(i+1)xσ(i+1)). +Moreover, the order of xj in the left-hand side of (24) is always odd, and thus we +can simply take ǫ2 = · · · = ǫn = 1 in the right-hand side and then restrict to terms +of odd degrees. Thus we conclude that: + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +13 +Theorem 3.2. The connected n-point functions associated to the generalized BGW +tau-functions are given by: +� +i1,··· ,in>0: odd +∂n log τ (N) +BGW(t/2) +∂ti1 · · · ∂tin +���� +t=0 +· x−i1 +1 +· · · x−in +n += −δn,2 · x1x2(x2 +2 + x2 +1) +2(x2 +1 − x2 +2)2 +− 2n−1 · +� +� +σ: n-cycle +n +� +i=1 +ξ(xσ(i), −xσ(i+1)) +� +odd, +(25) +for n ≥ 1, where [·]odd means taking the terms of odd degrees in every xi, and ξ is: +ξ(xσ(i), −xσ(i+1)) = + + + + + +A(N)(xσ(i), −xσ(i+1)), +σ(i) = σ(i + 1); +�A(N)(xσ(i), −xσ(i+1)), +σ(i) < σ(i + 1); +− �A(N)(−xσ(i+1), xσ(i)), +σ(i) > σ(i + 1), +and A(N), �A(N) are the generating series of the BKP-affine coordinates: +A(N)(w, x) = +� +n,m>0 +(−1)m+n+1a(N) +n,m · w−nx−m − +� +n>0 +(−1)n +2 +a(N) +n,0 · (w−n − x−n), +�A(N)(w, x) = A(N)(w, x) − 1 +4 − 1 +2 +∞ +� +i=1 +(−1)iw−ixi. +Here we use the convention σ(n + 1) := σ(1). +In the next subsection, we will give a compact formula for the generating series +�A(N)(z, w), which enables us to manipulate the formula (25) more efficiently. +Remark 3.3. In [8], Bertola and Ruzza computed the connected n-point functions +of the generalized BGW models using a different method. +They presented the +connected n-point functions as a summations over all permutations in Sn, with a +correction term at n = 2, see [8, Theorem 1.1]. +3.3. Generating series for the BKP-affine coordinates. Now we give a simple +formula for �A(N)(z, x). In [42, §5.2], we have proved the following: +Proposition 3.2 ( [42]). Let τ(t) be a tau-function of the KdV hierarchy, and let +Φ1(z), Φ2(z) be the first two basis vectors for the corresponding point on the Sato +Grassmannian. Denote: +Φ1(z) = 1 + +� +n≥1 +anz−n, +z−1Φ2(z) = 1 + +� +n≥1 +bnz−n, +and let G(z) be the following 2 × 2 matrix: +G(z) = +� 1 + � +n≥1 a2nz−n +� +n≥0 b2n+1z−n +� +n≥1 a2n−1z−n +1 + � +n≥1 b2nz−n +� +. +If det G(z) = 1 holds, then the generating series (23) of the BKP-affine coordinates +{an,m} of τ(t/2) are given by: +A(w, z) = w − z + Φ1(−z)Φ2(−w) − Φ1(−w)Φ2(−z) +4(w + z) +, +�A(w, z) = Φ1(−z)Φ2(−w) − Φ1(−w)Φ2(−z) +4(w + z) +. + +14 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +Consider the case of generalized BGW tau-functions. It is known that the point +in the Sato Grassmannian corresponding to the KdV tau-function τ (N) +gBGW(t) has a +basis (see Alexandrov [1, (77)]): +(26) +� +Φ(N) +j +(z) = zj−1� +1 + +∞ +� +k=1 +(−ℏ)k · ak(j − N) +8k · k! +· z−k��∞ +j=1, +where +ak(j) = +k +� +i=1 +� +4(j − 1)2 − (2i − 1)2� +. +(Notice here our notations for Φ(N) +j +(z) differs from the notations in [1] by a rescaling +z �→ 2z.) In particular, the first two basis vectors are: +Φ(N) +1 +(z) = 1 + +∞ +� +k=1 +(−ℏ)k +8k · k! · +k +� +i=1 +� +4N 2 − (2i − 1)2� +· z−k, +Φ(N) +2 +(z) = z + +∞ +� +k=1 +(−ℏ)k +8k · k! · +k +� +i=1 +� +4(1 − N)2 − (2i − 1)2� +· z1−k. +(27) +Now we claim that (see [42, §6.4] for the special case N = 0): +Corollary 3.1. The generating series A(N) and �A(N) of the BKP-affine coordinates +in Theorem 3.2 are given by: +A(N)(w, x) = w − x + Φ(N) +1 +(−x)Φ(N) +2 +(−w) − Φ(N) +1 +(−w)Φ(N) +2 +(−x) +4(w + x) +, +�A(N)(w, x) = Φ(N) +1 +(−x)Φ(N) +2 +(−w) − Φ(N) +1 +(−w)Φ(N) +2 +(−x) +4(w + x) +. +(28) +Proof. By Proposition 3.2, we only need to check det G(N)(x) = 1, where: +G(N)(x) = +� +(Φ(N) +1 +(s) + Φ(N) +1 +(−s) +� +/2 +(Φ(N) +2 +(s) + Φ(N) +2 +(−s) +� +/2 +(Φ(N) +1 +(s) − Φ(N) +1 +(−s) +� +/(2s) +(Φ(N) +2 +(s) − Φ(N) +2 +(−s) +� +/(2s) +� +, +and s = x +1 +2 . Or equivalently, we need to check: +(29) +Φ(N) +1 +(−s)Φ(N) +2 +(s) − Φ(N) +1 +(s)Φ(N) +2 +(−s) = 2s. +Here Φ(N) +1 +and Φ(N) +2 +are related by (see [1, (82)]): +(30) +Φ(N) +2 +(s) = ℏs∂sΦ(N) +1 +(s) + sΦ(N) +1 +(s) + ℏ(N − 1 +2)Φ(N) +1 +(s). +Denote Ψ(s) = 1 +s +� +Φ(N) +1 +(−s)Φ(N) +2 +(s) − Φ(N) +1 +(s)Φ(N) +2 +(−s) +� +, then by (30) one has: +Ψ(s) = ℏ(Φ(N) +1 +)′(s)Φ(N) +1 +(−s) + ℏ(Φ(N) +1 +)′(−s)Φ(N) +1 +(s) + 2sΦ(N) +1 +(s)Φ(N) +1 +(−s), +and then: +d +dsΨ(s) = Φ(N) +1 +(−s) · f(s) + Φ(N) +1 +(s) · f(−s), +where f(s) = ℏ(Φ(N) +1 +)′′(s) + 2(Φ(N) +1 +)′(s). Using (27) one can directly check that: +f(s) = ℏ(Φ(N) +1 +)′′(s) + 2(Φ(N) +1 +)′(s) = ℏ +s2 (N 2 − 1 +4)Φ(N) +1 +(s), + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +15 +and thus +d +dsΨ(s) = 0. Then Ψ(s) is a constant, and it is easy to see that this +constant is 2. This completes the proof. +□ +4. Emergence of Spectral Curve and Its Special Deformation +Now we study the generalized BGW tau-functions using Zhou’s formalism of +emergent geometry. In this section, we first give a brief review of this formalism, +and then construct the spectral curve together with its special deformation. They +emerges naturally from the free energy of genus zero (which can be computed using +the Virasoro constraints). We also discuss the quantum deformation theory of this +spectral curve. The emergence of the Eynard-Orantin topological recursion and the +quantum spectral curve will be addressed in the next two sections. +4.1. A brief review of Zhou’s emergent geometry. First we briefly recall +Zhou’s idea of emergent geometry. This is a formalism which provides the associated +geometric structures (such as spectral curve, topological recursion, etc.) from a +given partition function (of a Gromov-Witten type theory). +The first step is the construction of the spectral curve and its special deformation. +The notion of the special deformation was first introduced by Zhou [49] in the +study of topological 2D gravity. In general, given a Gromov-Witten type theory +(A-theory), the spectral curve of this theory together with its special deformation +(as a B-theory) should emerge from the Virasoro constraints. More precisely, let +F0(t) be the genus zero part of the free energy of a Gromov-Witten type theory, +then the special deformation is a formal series of the following form (with some +suitable modification in concrete examples): +y = +� +n≥1 +n˜tnxn−1 + +� +n≥1 +∂F0(t) +∂tn +· x−n−1, +where t = (t1, t2, t3, · · · ) are the coupling constants, and ˜t = (˜t1, ˜t2, ˜t3, · · · ) differs +from t only by a dilaton shift. The spectral curve of this theory is obtained from +this special deformation by restricting to t = 0: +(31) +y = +� � +n≥1 +n˜tnxn−1 + +� +n≥1 +∂F0(t) +∂tn +· x−n−1���� +t=0. +Then one can consider the quantum deformation theory of the spectral curve. +Roughly speaking, if a GW type theory is determined by the Virasoro constraints, +then one hopes that the Virasoro constraints of genus zero can be encoded in the +special deformation, and the Virasoro constraints at all genera can be encoded in +a suitable quantization of the special deformation. +Then one can consider the Eynard-Orantin topological recursion [17] on the +spectral curve. In Zhou’s formalism, the E-O topological recursion on the spectral +curve (31) is supposed to emerge naturally from the Virasoro constraints. Moreover, +the resulting E-O invariants should coincide with the connected n-point functions +of the original GW type theory. It is worth noting that the Bergman kernel for this +emergent topological recursion is of the following from: +(32) +B(x1, x2) = +� +1 +(x1 − x2)2 + W0,2(x1, x2) +� +dx1dx2, + +16 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +where W0,2(x1, x2) is the connected 2-point function of genus zero: +W0,2(x1, x2) = +� +n1,n2≥1 +∂2F0(t) +∂tn1∂tn1 +��� +t=0 · x−n1−1 +1 +x−n2−1 +2 +, +and x is exactly the formal variable x in the expression (31), regarded as a mero- +morphic function on this spectral curve. +Now given a tau-function τ(t) of the KP hierarchy, one can regard it as the +partition function of a formal quantum field theory (see [51, §2]) where the KP- +time variables t = (t1, t2, t3, · · · ) are the coupling constants. The coefficients of +the free energy log τ(t) play the role of the connected correlators. Then these tau- +functions provide a large class of examples of Gromov-Witten type theories. See +[43,44,50,52,53,55] for the emergent geometry for some well-known tau-functions +of the KP hierarchy. In all these examples, the E-O invariants for the emergent +spectral curve (31) and Bergman kernel (32) coincide with the connected n-point +functions associated to the tau-function. In §5, we will see that this ansatz is also +true for the generalized BGW models. +4.2. Special deformation and spectral curve for τ (N) +BGW. In this subsection, we +construct the spectral curve together with its special deformation for the generalized +BGW tau-functions using Zhou’s ansatz. +In this case, the special deformation of the spectral curve is the following formal +series in a formal variable x: +(33) +y = +� +n≥0 +(2n + 1)(t2n+1 − δn,0)x2n + +� +n≥0 +∂FgBGW +0 +(t) +∂t2n+1 +· x−2n−2. +Then the Virasoro constraints of genus zero is encoded in the special deformation +in the following way: +Proposition 4.1. Virasoro constraints for the free energy FgBGW +0 +(t; S) of genus +zero is equivalent to the following condition: +(34) +(y2)− = S2 · x−2, +where we use the notation: +� � +n∈Z +anxn� +− = +� +n<0 +anxn. +Proof. This can be checked by direct computation. By plugging the genus expansion +(5) into (6), we obtain the Virasoro constraints at genus zero: +− 1 +2 +∂FgBGW +0 +∂t1 ++ 1 +2 +� +k: odd +ktk +∂FgBGW +0 +∂tk +− 1 +4S2 = 0; +− 1 +2 +∂FgBGW +0 +∂t2n+1 ++ 1 +4 +� +a+b=2n +a,b: odd +∂FgBGW +0 +∂ta +∂FgBGW +0 +∂tb ++ 1 +2 +� +k: odd +ktk +∂FgBGW +0 +∂tk+2n += 0, + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +17 +where n ≥ 1. On the other hand, we have: +(y2)− = +� +− 2∂FgBGW +0 +∂t1 ++ 2 +� +k: odd +ktk +∂FgBGW +0 +∂tk +� +z−2 ++ +� +n≥1 +� +− 2∂FgBGW +0 +∂t2n+1 ++ +� +a+b=2n +a,b: odd +∂FgBGW +0 +∂ta +∂FgBGW +0 +∂tb ++ 2 +� +k: odd +ktk +∂FgBGW +0 +∂tk+2n +� +z−2n−2, +and thus the conclusion is clear. +□ +The spectral curve is supposed to be obtained from the above special deformation +by taking t = 0. Recall that the connected 1-point correlators of genus zero can be +solved by the Virasoro constraints (see §2.2): +(35) +∂FgBGW +0 +(t) +∂t2n+1 +��� +t=0 = (−1)n+1 +22n+1 +· +1 +n + 1 +�2n +n +� +· S2n+2, +∀n ≥ 0, +Thus we have: +y +�� +t=0 = −1 + 1 − +� +1 + S2 +x2 = − +� +1 + S2 +x2 . +We rewrite this as: +(36) +x2y2 = x2 + S2. +In the framework of emergent geometry, the plane curve defined by this equation +(36) is the spectral curve associated to the generalized BGW tau-functions. +In +§5, we will show that the E-O topological recursion on this spectral curve emerges +naturally from the Virasoro constraints for τ (N) +BGW. +4.3. Quantum deformation theory of the spectral curve. In this subsection, +we discuss the quantum deformation theory of the above spectral curve. We will +see that the Virasoro constraints (at all genera) can be encoded in terms of a +quantization of the special deformation. +Denote by ˆy(x) be the following bosonic field on the spectral curve: +(37) +ˆy(x) = +� +n≥0 +α−(2n+1) · x2n + +� +n≥0 +α2n+1 · x−2n−2, +where {αk}k∈Z: odd are the free bosons: +(38) +αk = ℏ ∂ +∂tk +, +α−k = ℏ−1(ktk − δk,1), +∀k > 0 : odd, +which satisfy the canonical commutation relation +[αi, αj] = iδi+j,0, +∀i, j ∈ Z : odd. +This bosonic field is a quantization of the special deformation (33). The product +of two such fields are: +ˆy(w)ˆy(x) = : ˆy(w)ˆy(x) : + +� +n≥0 +(2n + 1)w−2n−2x2n += : ˆy(w)ˆy(x) : + w2 + x2 +(w2 − x2)2 , + +18 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +where the normal ordered product : αiαj : of two free bosons are given by: +: αiαj := +� +αiαj, +if i ≤ j; +αjαi, +if i > j. +ˆy(x + ǫ)ˆy(x) = : ˆy(x + ǫ)ˆy(x) : +2x2 + 2ǫx + ǫ2 +(2ǫx + ǫ2)2 += : ˆy(x + ǫ)ˆy(x) : + 1 +2ǫ2 + +1 +8x2 − +ǫ +8x3 + 3ǫ2 +32x4 − · · · . +(39) +Notice that there is a singular term +1 +2ǫ2 in the right-hand side. Define the regularized +product of two bosonic fields by deleting this singular term: +(40) +ˆy(x)⊙2 = ˆy(x) ⊙ ˆy(x) = lim +ǫ→0 +� +ˆy(x + ǫ)ˆy(x) − 1 +2ǫ2 +� +, +then from (39) we see that: +(41) +ˆy(x)⊙2 =: ˆy(x)ˆy(x) : +1 +8x−2. +Proposition 4.2. The Virasoro constraints for the generalized BGW models are +equivalent to: +(42) +� +ˆy(x)⊙2 + (1 +8 − ℏ−2S2)x−2� +− +� +exp FgBGW(t; S) +� += 0. +Proof. First we have: +(ˆy(x)⊙2)− = +� +n≥0 +� +� +a+b=2n +a,b≥1: odd +αaαb + 2 +� +k≥1 +k: odd +α−kαk+2n +� +x−2n−2 + 1 +8x−2. +Now compare this with the Virasoro constraints (6), then we see: +� +ˆy(x)⊙2 + (1 +8 − ℏ−2S2)x−2� +− = +� +n≥0 +4LgBGW +n +· x−2n−2. +Thus the conclusion holds. +□ +5. Emergence of Eynard-Orantin Topological Recursion +In this section, we show that the Eynard-Orantin topological recursion for the +generalized BGW tau-functions emerges naturally from the Virasoro constraints. +The spectral curve and Bergman kernel for this topological recursion is given by +(36) and (32) respectively. At the end of this section, we remark that this topo- +logical recursion can be reformulated in terms of another Bergman kernel which is +understood as the Bergman kernel of type B. +5.1. Bergman kernel and Eynard-Orantin invariants on the spectral curve. +Let W gBGW +g,n +be the connected n-point functions defined in §2.2. +Now define B(x1, x2) to be the following symmetric 2-differential on the spectral +curve (36): +(43) +B(x1, x2) = +� +1 +(x1 − x2)2 + W gBGW +0,2 +(x1, x2) +� +dx1dx2. + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +19 +Or more explicitly, +(44) +B(x1, x2) = +1 +(x2 +1 − x2 +2)2 +� +x2 +1 + x2 +2 + 2S2 +� +(1 + S2 +x2 +1 )(1 + S2 +x2 +2 ) ++ 2x1x2 +� +dx1dx2, +by using the explicit expression of W gBGW +0,2 +computed in Example 2.1. +Then +B(x1, x2) is called a fundamental bidifferential on the spectral curve (and is also +called a Bergman kernel in the Eynard-Orantin topological recursion [17]). +Now one can define a family of symmetric differentials {ωg,n}g≥0,n≥1 on the +spectral curve using the E-O topological recursion, for which the input data are +the spectral curve (36), the two meromorphic functions x, y on this curve, and the +Bergman kernel B(x1, x2). Consider the following parametrization of the spectral +curve: +(45) +x = +� +z2 − S2, +y = +z +√ +z2 − S2 , +then x = x(z) has a non-degenerate critical point at z = 0. Near this critical point, +there is an involution σ(z) = −z satisfying x(z) = x(σ(z)). The Eynard-Orantin +invariants {ωg,n} are defined by: +ω0,1(z) = (−1)1 · +� +W gBGW +0,1 +(x(z)) − 1 +� +dx(z) = y(z)dx(z), +ω0,2(z1, z2) = B(x(z1), x(z2)), +(46) +and for 2g − 1 + n > 0, ωg,n+1 is recursively defined by: +ωg,n+1(z0, z1, · · · , zn) = Resz=0 K(z0, z) +� +ωg−1,n+2(z, σ(z), z1, · · · , zn) ++ +s +� +g1+g2=g +I⊔J=[n] +ωg1,|I|+1(z, zI) · ωg2,|J|+1(σ(z), zJ) +� +, +(47) +where the recursion kernel K(z0, z) is defined locally near the critical point by: +(48) +K(z0, z) = +� z +σ(z) B(x(z0), x(z)) +2 +� +y(z) − y(σ(z)) +� +dx(z), +and +s� means excluding all the terms involving ω0,1 in this summation. For every +g ≥ 0 and n ≥ 1, the E-O invariant ωg,n(z1, · · · , zn) is a symmetric n-differential +on the spectral curve (36). +Our main result in this section is the following: +Theorem 5.1. For every pair (g, n) with 2g − 2 + n > 0, we have: +ωg,n(z1, · · · , zn) =(−1)n · W gBGW +g,n +(x1, · · · , xn)dx1 · · · dxn +=(−1)n · +� +µ1,··· ,µn: odd +⟨pµ1 · · · pµn⟩c +g · +dx1 · · · dxn +xµ1+1 +1 +· · · xµn+1 +n +, +(49) +where ⟨pµ1 · · · pµn⟩c +g is the connected correlators of the generalized BGW tau-functions. +The proof of this theorem will be given in §5.3. + +20 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +Remark 5.1. In the special case N = 0 (and thus S = 0), the spectral curve (36) +degenerates and the Eynard-Orantin topological recursion will not work. Neverthe- +less, one can first regard S as a formal variable and compute the n-point functions +W gBGW +g,n +as formal series in S using E-O recursion, and then the n-point functions +for the original BGW model can be obtained by restricting to S = 0. +5.2. Examples of the Eynard-Orantin invariants. Before proving the above +theorem, we give some examples of concrete computations of the above topological +recursion in this subsection. First notice that by the parametrization (45) we have: +dx = +� +1 + S2 +x2 dz, +and thus the Bergman kernel (44) can be rewritten as: +B(x1, x2) = +1 +(z1 − z2)2 dz1dz2, +(50) +and then the recursion kernel is: +K(z0, z) = +1 +2 +� +y(z) − y(−z) +� +dx(z) +� z +−z +1 +(z0 − z)2 dz0dz += (z2 − S2) +2z(z2 +0 − z2) · dz0 +dz . +(51) +Example 5.1. Now we compute ω0,3. By (47) we have: +ω0,3(z0, z1, z2) = Resz=0 K(z0, z) +� +ω0,2(z, z1)ω0,2(−z, z2) + ω0,2(z, z2)ω0,2(−z, z1) +� += +S2 +z2 +0z2 +1z2 +2 +dz0dz1dz2, +or equivalently, +ω0,3(z0, z1, z2) = +S2 · dx0dx1dx2 +x2 +0x2 +1x2 +2 · (1 + S2 +x2 +0 )3/2 · (1 + S2 +x2 +1 )3/2 · (1 + S2 +x2 +2 )3/2 . +And then by Theorem 5.1: +W gBGW +0,3 +(x0, x1, x2) = − +S2 +x2 +0x2 +1x2 +2 ++ +3S4 +2x4 +0x2 +1x2 +2 ++ +3S4 +2x2 +0x4 +1x2 +2 ++ +3S4 +2x2 +0x2 +1x4 +2 +− +15S6 +8x6 +0x2 +1x2 +2 +− +15S6 +8x2 +0x6 +1x2 +2 +− +15S6 +8x2 +0x2 +1x6 +2 +− +9S6 +4x4 +0x4 +1x2 +2 +− +9S6 +4x4 +0x2 +1x4 +2 +− +9S6 +4x2 +0x4 +1x4 +2 ++ · · · . +Example 5.2. Now consider ω0,4. Using (47) we can compute: +ω0,4(z0, z1, z2, z3) = +� +3S4 +z2 +0z2 +1z2 +2z2 +3 +� 1 +z2 +0 ++ 1 +z2 +1 ++ 1 +z2 +2 ++ 1 +z2 +3 +� +− +3S2 +z2 +0z2 +1z2 +2z2 +3 +� +dz0dz2dz2dz3, +and thus by Theorem 5.1 we have: +W gBGW +0,4 +(x0, x1, x2, x3) += +3S2 +�3 +i=0 +� +x2 +i (1 + S2 +x2 +i )3/2� +� +− 1 + +3 +� +i=0 +S2 +x2 +i + S2 +� += − +3S2 +x2 +0x2 +1x2 +2x2 +3 ++ +15S4 +2x4 +0x2 +1x2 +2x2 +3 ++ +15S4 +2x2 +0x4 +1x2 +2x2 +3 ++ +15S4 +2x2 +0x2 +1x4 +2x2 +3 ++ +15S4 +2x2 +0x2 +1x2 +2x4 +3 ++ · · · . + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +21 +Example 5.3. Now we compute ω1,1. By (47) we have: +ω1,1(z0) = Resz=0 K(z0, z)ω0,2(z, −z) = +� +− +1 +8z2 +0 ++ S2 +8z4 +0 +� +dz0, +and then by Theorem 5.1 we have: +W gBGW +1,1 +(x0) = +x3 +0 +8(x2 +0 + S2)5/2 += 1 +8x2 +0 +− 5S2 +16x4 +0 ++ 35S4 +64x6 +0 +− 105S6 +128x8 +0 ++ 1155S8 +1024x10 +0 ++ · · · . +Example 5.4. Now consider ω1,2. By (47) we have: +ω1,2(z0, z1) = Resz=0 K(z0, z) +� +ω0,3(z, −z, z1)+ +ω0,2(z, z1)ω1,1(−z) + ω0,2(−z, z1)ω1,1(z) +� +=z4 +0z4 +1 − 6S2(z4 +0z2 +1 + z2 +0z4 +1) + 3S4z2 +0z2 +1 + 5S4(z4 +0 + z4 +1) +8z6 +0z6 +1 +dz0dz1. +Then by Theorem 5.1 we have: +W gBGW +1,2 +(x0, x1) +=x0x1(2S8 − 3S6(x2 +0 + x2 +1) − 17S4x2 +0x2 +1 − 4S2(x4 +0x2 +1 + x2 +0x4 +1) + x4 +0x4 +1) +8(x2 +0 + S2)7/2 · (x2 +1 + S2)7/2 += +1 +8x2 +0x2 +1 +− +15S2 +16x4 +0x2 +1 +− +15S2 +16x2 +0x4 +1 ++ 175S4 +64x6 +0x2 +1 ++ 175S4 +64x2 +0x6 +1 ++ +93S4 +32x4 +0x4 +1 +− · · · . +5.3. Proof of Theorem 5.1. Now we prove Theorem 5.1. We only need to check +that the E-O topological recursion (47) is equivalent to the recursion (15) derived +from the Virasoro constraints. +From the topological recursion (47) one easily sees that when 2g − 2 + n > 0 the +E-O invariants ωg,n are of the following form: +(52) +ωg,n(z1, · · · zn) = +� +k1,···kn≥0 +Ak1,··· ,kn +g,n +· +�n +i=1(2ki + 1)!! +z2k1+2 +1 +· · · z2kn+2 +n +dz1 · · · dzn. +Take x = +√ +z2 − S2 and z = +√ +x2 + S2, then: +(53) +z−2kdz = x−2k · +� +1 + S2 +x2 +�−k− 1 +2 dx = +∞ +� +m=0 +�−k − 1 +2 +m +� +S2mx−2m−2kdx, +and for 2g − 2 + n > 0 one has: +(54) +ωg,n(z1, · · · zn) = +� +k1,···kn≥0 +Bk1,··· ,kn +g,n +· +�n +i=1(2ki + 1)!! +x2k1+2 +1 +· · · x2kn+2 +n +dx1 · · · dxn, +where the numbers Bk1,··· ,kn +g,n +are related to Ak1,··· ,kn +g,n +by: +Bl1,··· ,ln +g,n += +� +ki+mi=li +n +� +i=1 +(−S2)mi +2mi · mi!Ak1,··· ,kn +g,n +; +Al1,··· ,ln +g,n += +� +ki+mi=li +n +� +i=1 +(S2)mi +2mi · mi!Bk1,··· ,kn +g,n +. +(55) + +22 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +For (g, n) = (0, 1) and (0, 2), we also denote: +ω0,1(z) = dx + +� +k≥0 +Bk +0,1 +(2k + 1)!! +x2k+2 +dx, +ω0,2(z1, z2) = +1 +(x1 − x2)2 + +� +k≥0 +Bk1,k2 +0,2 +(2k1 + 1)!!(2k2 + 1)!! +x2k1+2 +1 +x2k2+2 +2 +dx1dx2, +then direct computation tells that: +Bk +0,1 = − +(−S2)k+1 +2k+1 · (k + 1)! · (2k + 1), +Bk1,k2 +0,2 += +(−S2)k1+k2+1 +2k1+k2+1 · k1! · k2! · (k1 + k2 + 1). +(56) +Now Theorem 5.1 follows from: +Lemma 5.1. For every g ≥ 0 and n ≥ 1, we have: +(57) +Bk1,··· ,kn +g,n += (−1)n · ⟨p2k1+1 · · · p2kn+1⟩c +g,n +�n +i=1(2ki + 1)!! +, +∀k1, · · · , kn ≥ 0. +Proof. The unstable cases (g, n) = (0, 1) and (0, 2) can be checked directly using +the above explicit expressions. Now consider the case (g, n) with 2g − 2 + n > 0. +The recursion formula (47) is: +ωg,n+1(z0, z1, · · · , zn) += Resz=0 K(z0, z) +� +n +� +i=1 +� +ω0,2(z, zi)ωg,n(−z, z[n]\{i}) + ωg,n(z, z[n]\{i})ω0,2(−z, zi) +� ++ ωg−1,n+2(z, ¯z, z[n]) + +g +� +h=0 +stable +� +I+J=[n] +ωh,|I|+1(z, zI)ωg−h,|J|+1(−z, zJ) +� +, +where +stable +� +means we exclude all terms involving ω0,1 or ω0,2 in this summation. +Using the symmetric property of E-O invariants (see [17]): +ωg,n(z1, · · · , zn) = ωg,n(zσ(1), · · · , zσ(n)), +∀σ ∈ Sn, +we know that Ak1,··· ,kn +g,n += A +kσ(1),··· ,kσ(n) +g,n +for every σ ∈ Sn. Then we can rewrite the +above recursion for ωg,n as follows: +� +k0,··· ,kn≥0 +Ak0,··· ,kn +g,n+1 +�n +i=0(2ki + 1)!! +z2k0+2 +0 +· · · z2kn+2 +n += Resz=0 dz +� +S2 − z2 +2z(z2 +0 − z2)× +� +k1,··· ,kn≥0 +�n +i=1(2ki + 1)!! +z2k1+2 +1 +· · · z2kn+2 +n +� +2 +n +� +i=1 +� +a≥0 +z2ki +(2ki − 1)!! +(2a + 1)!! +z2a+2 +A +a,k[n]\{i} +g,n ++ +� +a,b≥0 +(2a + 1)!!(2b + 1)!! +z2a+2b+4 +� +Aa,b,k1,··· ,kn +g−1,n+2 ++ +g +� +h=0 +stable +� +I+J=[n] +Aa,kI +h,|I|+1Ab,kJ +g−h,|J|+1 +��� +. +Notice that near z = 0 we have: +S2 − z2 +z2 +0 − z2 = S2 · +∞ +� +m=0 +z2m +z2m+2 +0 +− +∞ +� +m=0 +z2m+2 +z2m+2 +0 +, + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +23 +thus by taking residues in the above recursion we obtain (after some simplification): +m +� +k0=0 +�m +k0 +� +· +� +− S2 +2 +�m−k0 +· (2k0 + 1)!! +2k0+1 +Ak0,k1,··· ,kn +g,n+1 += − +n +� +i=1 +� +k0≥0 +�m + 1 +k0 +� +· +� +− S2 +2 +�m+1−k0 +· (2ki + 2k0 − 1)!! · A +ki+k0−1,k[n]\{i} +g,n +2k0 · (2ki − 1)!! +− 1 +2 +� +a,b≥0 +� m + 1 +a + b + 2 +�(2a + 1)!!(2b + 1)!! +2a+b+2 +� +− S2 +2 +�m−1−a−b� +A +a,b,k[n] +g−1,n+2 ++ +g +� +h=0 +stable +� +I⊔J=[n] +Aa,kI +h,|I|+1 · Ab,kJ +g−h,|J|+1 +� +. +Then we rewrite this recursion as recursion for Bk1,··· ,kn +g,n +using (55). After some +simplification, the result is: +− +m +� +b=0 +(2m − 2b − 1)!! · (2b + 1)!! · (− S2 +2 )m−b +2m+1 · (m − b)! · (2m − 2b − 1) +· Bb,k1,··· ,kn +g,n+1 += − +n +� +i=1 +� (2m + 2ki + 1)!! +(2ki − 1)!! · 2m+1 · B +m+ki,k[n]\{i} +g,n ++ +� +a+b=m−1 +(2a + 1)!! · (2b + 1)!! · (− S2 +2 )a+ki+1 +2m+1 · a! · ki! · (a + ki + 1) +· B +b,k[n]\{i} +g,n +� +− 1 +2 +� +a′+b′=m−1 +(2a′ + 1)!!(2b′ + 1)!! +2m+1 +� +B +a′,b′,k[n] +g−1,n+2 + +stable +� +g1+g2=g +I+J=[n] +Ba′,kI +g1,|I|+1Bb′,kJ +g2,|J|+1 +� +. +Now using (56) we rewrite this recursion as: +− +m +� +b=0 +�Bm−b−1 +0,1 +�Bb,k1,··· ,kn +g,n+1 += +n +� +i=1 +� +�B +m+ki,k[n]\{i} +g,n ++ +� +a+b=m−1 +�Ba,ki +0,2 �B +b,k[n]\{i} +g,n +� ++ 1 +2 +� +a′+b′=m−1 +� +�B +a′,b′,k[n] +g−1,n+2 + +stable +� +g1+g2=g +I+J=[n] +�Ba′,kI +g1,|I|+1 �Bb′,kJ +g2,|J|+1 +� +. +where we use the notation �B−1 +0,1 = −1, and: +�Bk1,··· ,kn +g,n += (−1)n · +n +� +i=1 +(2ki + 1)!! · Bk1,··· ,kn +g,n +, +∀k1, · · · , kn ≥ 0. +This matches with the Virasoro constraints (8) for the connected correlators (notice +that the summation in the right-hand side of (8) involves unstable terms ⟨pi⟩c +0 and +⟨pipj⟩c +0). Thus we conclude that +�Bk1,··· ,kn +g,n += ⟨p2k1+1 · · · p2kn+1⟩c +g,n, +∀k1, · · · , kn ≥ 0, +since the two sides are determined by the same recursion formula and initial values. +This completes the proof. +□ + +24 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +5.4. A remark on Bergman kernel of type B. In this subsection, we give a +remark on the Bergman kernel of type B which provides a reformulation of the +above topological recursion. +Let C be a rational spectral curve with a complex parameter z, and consider the +following symmetric 2-differential on the spectral curve: +(58) +B(z1, z2) = +z2 +1 + z2 +2 +(z2 +1 − z2 +2)2 dz1dz2 = 1 +2 +� +1 +(z1 − z2)2 + +1 +(z1 + z2)2 +� +dz1dz2. +In literatures the above 2-differential is supposed to serve as the Bergman kernel for +the topological recursion associated to a tau-function of the BKP hierarchy. The +first example of such a Bergman kernel in topological recursion might be the Witten- +Kontsevich tau-function [48], see [48, (26)]. In that work, Zhou proved that the +Virasoro constraints [14,18] for the Witten-Kontsevich tau-function τWK(t) is equiv- +alent the Eynard-Orantin topological recursion on the Airy curve, and Bergman +kernel he chose is exactly (58). Zhou treated τWK(t) as a tau-function of the KdV +hierarchy (due to the Witten Conjecture/Kontsevich Theorem [29, 45]), and now +we may expect that the emergence of such a Bergman kernel in this case is because +τWK(t/2) is a tau-function of the BKP hierarchy. Recently in [5], Alexandrov and +Shadrin proved the blobbed topological recursion for a class of hypergeometric 2- +BKP tau-functions where the Bergman kernel is (58) (see [5, (7.1)]), and their +results confirmed the conjectural topological recursion [19, Conjecture 1.3] for the +spin Hurwitz numbers with completed cycles. It is worth mentioning that the func- +tion +x2 +1+x2 +2 +(x2 +1−x2 +2)2 in this Bergman kernel differs from the correction term at n = 2 in +the right-hand side of (24) only by a factor x1x2 +2 . +Now in this subsection, we make a simple observation that the above Eynard- +Orantin topological recursion for generalized BGW models can also be reformulated +using such a Bergman kernel. Consider the spectral curve (36) with parametrization +(45). Similar to the construction (32), we consider the following Bergman kernel +modified by all connected correlators of type (g, n) = (0, 2): +(59) +�B(x1, x2) = +�1 +2 +� +1 +(x1 − x2)2 + +1 +(x1 + x2)2 +� ++ W gBGW +0,2 +(x1, x2) +� +dx1dx2. +Or more explicitly (by using the result in Example 2.1): +(60) +�B(x1, x2) = +1 +(x2 +1 − x2 +2)2 · +x2 +1 + x2 +2 + 2S2 +� +(1 + S2 +x2 +1 )(1 + S2 +x2 +2 ) +dx1dx2. +Using the parametrization (45), we can rewrite it as: +(61) +�B(x1, x2) = 1 +2 +� +1 +(z1 − z2)2 + +1 +(z1 + z2)2 +� +dz1dz2, +where xi = x(zi). Then we have: +Theorem 5.2. Define a family of symmetric differentials {˜ωg,n}g≥0,n≥1 on the +spectral curve (36) as follows: +˜ω0,1(z) = y(z)dx(z), +˜ω0,2(z1, z2) = �B(x(z1), x(z2)), +˜ω1,1(z0) = +� +− +1 +8z2 +0 ++ S2 +8z4 +0 +� +dz0, + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +25 +and for 2g − 1 + n > 0 and (g, n) ̸= (1, 0), the differential ωg,n+1 is recursively +defined by the Eynard-Orantin topological recursion: +˜ωg,n+1(z0, z1, · · · , zn) = Resz=0 �K(z0, z) +� +˜ωg−1,n+2(z, −z, z1, · · · , zn) ++ +s +� +g1+g2=g +I⊔J=[n] +˜ωg1,|I|+1(z, zI)˜ωg2,|J|+1(−z, zJ) +� +, +(62) +where the recursion kernel �K(z0, z) is defined by: +(63) +�K(z0, z) = +� z +σ(z) �B(x(z0), x(z)) +2 +� +y(z) − y(σ(z)) +� +dx(z). +Then for every pair (g, n) with 2g − 2 + n > 0, we have: +˜ωg,n(z1, · · · , zn) =(−1)n · W gBGW +g,n +(x1, · · · , xn)dx1 · · · dxn. +(64) +Remark 5.2. Here we exclude the case ˜ω1,1 in the recursive definition (62) since in +this case ˜ω0,2(z, −z) in the right-hand side is not well-defined. +Proof. It suffices to check that ˜ωg,n = ωg,n for 2g − 2 + n > 0. +We prove by +induction. Assume that ˜ωg′,n′ = ωg′,n′ holds for all (g′, n′) (where 2g′ − 2 + n′ > 0) +satisfying g′ < g, or g′ = g and n′ < n + 1. We need to prove: +Resz=0 �K(z0, z) +� +˜ωg−1,n+2(z, −z, z[n]) + +s +� +g1+g2=g +I⊔J=[n] +˜ωg1,|I|+1(z, zI)˜ωg2,|J|+1(−z, zJ) +� += Resz=0 K(z0, z) +� +ωg−1,n+2(z, −z, z[n]) + +s +� +g1+g2=g +I⊔J=[n] +ωg1,|I|+1(z, zI)ωg2,|J|+1(−z, zJ) +� +. +First notice that the recursion kernel +�K(z0, z) = +1 +4 +� +y(z) − y(−z) +� +dx(z) +� z +−z +� +1 +(z0 − z)2 + +1 +(z0 + z)2 +� +dz0dz += (z2 − S2) +2z(z2 +0 − z2) · dz0 +dz +coincide with the recursion kernel K(z0, z) (see (51)) for ωg,n. Moreover, +�K(z0, z) = K(z0, z) = −1 +2 +� � +m≥0 +S2 · z2m−1 +z2m+2 +0 +− +� +m≥0 +z2m+1 +z2m+2 +0 +� +is a Laurent series in z which contains only terms of odd degrees in z, thus it suffices +to prove that: +� +˜ωg−1,n+2(z, −z, z[n]) + +s +� +g1+g2=g +I⊔J=[n] +˜ωg1,|I|+1(z, zI)˜ωg2,|J|+1(−z, zJ) +� +even += +� +ωg−1,n+2(z, −z, z[n]) + +s +� +g1+g2=g +I⊔J=[n] +ωg1,|I|+1(z, zI)ωg2,|J|+1(−z, zJ) +� +even +, +(65) + +26 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +where [·]even means taking the terms of even degrees in z. Notice that we have: +[ω0,2]even = [˜ω0,2]even, +and by induction hypothesis and Theorem 5.1 we know that ωg′,n′ = ˜ωg′,n′ contains +only terms of even degrees for g′ < g, or g′ = g and n′ < n+1 (where 2g′−2+n′ > 0), +and thus the identity (65) is clear. +□ +6. Emergence of Quantum Spectral Curve of Type B +In this section, we discuss the emergence of the quantum spectral curve of the +classical curve (36). We first show that there exists a Kac-Schwarz operator P of +type B for the generalized BGW tau-function which annihilates the BKP-wave func- +tion using BKP-affine coordinates. (This operator has been discussed by Alexan- +drov in [3].) Then we show that the semi-classical limit of P gives the classical +curve (36). Since we already know that the generalized BGW models can be re- +constructed from the Eynard-Orantin topological recursion on (36), this operator +P provides a quantum spectral curve in the sense of Gukov-Su�lkowski. +6.1. Emergence of Kac-Schwarz operators from BKP-affine coordinates. +In this subsection we recall the formulation of Kac-Schwarz operators of type B in +terms of BKP-affine coordinates, see [24, §3] for details. +Let τ(t) be a tau-function of the BKP hierarchy satisfying the condition τ(0) = 1, +where t = (t1, t3, t5, · · · ) are the BKP-time variables. The wave function associated +to τ(t) is defined by Sato’s formula [11]: +(66) +wB(t; z) = exp +� ∞ +� +k=0 +t2k+1z2k+1� +· τ(t1 − 2 +z , t3 − +2 +3z3 , t5 − +2 +5z5 , · · · ) +τ(t1, t3, t5, · · · ) +. +In particular, wB(0; z) is a principal specialization of τ: +wB(0; z) = τ +� +− 2 +z , − 2 +3z3 , − 2 +5z5, · · · +� +. +One can associate a linear subspace Uτ of H = C[z] ⊕ z−1C[[z−1]] to τ(t) in the +following way: +(67) +Uτ = span +� +∂k +t1wB(0; z) +� +k≥0. +An operator P (acting on formal Laurent series in z) is called a Kac-Schwarz op- +erator of type B for the tau-function τ(t), if it satisfies: +(68) +P(Uτ) ⊂ Uτ. +The subspace Uτ ⊂ H emerges naturally from the tau-function τ(t) via its BKP- +affine coordinates {an,m}n,m≥0, due to the following: +Theorem 6.1 ( [24]). We have: +(69) +Uτ = span +� +zk + +∞ +� +i=1 +2(−1)i(ak,i − ak,0a0,i)z−i� +k≥0. +In particular, the first basis vector coincide with wB(0; z): +(70) +wB(0; z) = 1 + +∞ +� +i=1 +2(−1)i · a0,iz−i. + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +27 +Remark 6.1. The above linear subspace Uτ ⊂ H and the BKP tau-function τ(t) +are determined by each other, since the BKP-affine coordinates {an,m}n,m≥0 satisfy +the anti-symmetry condition an,m = −am,n. +Remark 6.2. The subspace Uτ ⊂ H can also be spanned by the fermionic 1-point +functions associated to τ(t), see [24, §3.2] for details. +In what follows, we will denote by ΦB +k (z) the k-th basis vector for Uτ: +(71) +ΦB +k (z) = zk + +∞ +� +i=1 +2(−1)i(ak,i − ak,0a0,i)z−i, +k ≥ 0. +Then Uτ = span{ΦB +k (z)}k=0,1,2,···. Similar to the case of KP hierarchy and the +usual Sato Grassmannian, in general we are interested in those BKP tau-functions +for which there are two Kac-Schwarz operators (P, Q) of type B, such that: +P(ΦB +0 ) = 0; +Q(ΦB +k ) − ck+1 · ΦB +k+1 ∈ span{ΦB +0 , ΦB +1 , · · · , ΦB +k }. +(72) +where {ck+1} are some nonzero constants. Moreover, we hope that they satisfy the +canonical commutation relation: +(73) +[P, Q] = 1. +If the equation P(ΦB +0 ) = 0 has a unique solution ΦB +0 +∈ 1 + z−1C[[z−1]], then +{an,m} and hence τ(t) are uniquely determined, since ΦB +k (z) for every k > 0 can +be recursively computed by applying Q to ΦB +k−1(z). +6.2. Kac-Schwarz operators of type B for generalized BGW models. In +last subsection we have reviewed the formulation of Kac-Schwarz operators of type +B in terms of BKP-affine coordinates. From now on, we apply Theorem 6.1 and +the explicit expressions of the BKP-affine coordinates given in §3 to derive the +Kac-Schwarz operators of type B for the generalized BGW tau-functions. +Recall that the BKP-affine coordinates of τ (N) +BGW(t/2) are given by (22). Thus +for each N ∈ C, the linear subspace Uτ (N) +BGW(t/2) ⊂ H is spanned by the following +basis vectors {ΦB,(N) +k +(z)}k≥0: +Φ(N),B +k +(z) =zk + +∞ +� +i=1 +2(−1)i(a(N) +k,i − a(N) +k,0 a(N) +0,i )z−i +=zk + +∞ +� +i=0 +(−1)i · +i +i + k · +ℏk+i +23k+3i · k! · i! +k +� +a=1 +θ(a) +i� +b=1 +θ(b). +(74) +where θ is the function: +θ(z) = (2z − 1)2 − 4N 2, +and we use the convention �0 +a=1 θ(a) = 1. Then we have: +Proposition 6.1. Let P and Q be the following operators: +P = ℏ3� +(z∂z + 1 +2)2 − N 2�� +∂z − +ℏ +2z2 +� +(z∂z − 1 +2)2 − N 2�� +, +Q = ℏ−2� +(z∂z − 1 +2)2 − N 2�−1z. +(75) + +28 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +Then one has [P, Q] = ℏ, and: +P(Φ(N),B +k +) = ℏ3 +4 θ(k) · +� +kΦ(N),B +k−1 +− ℏ +8θ(k − 1)Φ(N),B +k−2 +� +, +Q(Φ(N),B +k +) = ℏ−2 +4 θ(k + 1)−1Φ(N),B +k+1 +− +ℏk−1 · θ(1) · �k +j=1 θ(j) +23k+3 · (k + 1)! · ( 1 +4 − N 2)Φ(N),B +0 +, +for every k ≥ 0, where we use the conventions Φ(N),B +−1 += Φ(N),B +−2 += 0. +Proof. Notice that: +P(zk) = ℏ3 +4 θ(k) +� +kzk−1 − ℏ +8θ(k − 1)zk−2� +, +Q(zk) = ℏ−2 +4 θ(k + 1) · zk+1. +Then the conclusion can be checked directly using these two identities. +□ +This proposition tells that the operators (P, Q) are Kac-Schwarz operators of +type B for the tau-functions τ (N) +BGW(t/2). +6.3. Quantum spectral curve of type B for τ (N) +BGW(t/2). Now we explain that +the Kac-Schwarz operator P defined by (75) gives the quantum spectral curve (in +the sense of Gukov-Su�lkowski) of the plane curve: +(76) +x2y2 = x2 + S2. +We have already seen that P annihilates the BKP-wave function w(N) +B +(t; z) as- +sociated to the tau-function τ (N) +BGW(t) evaluated at t = 0: +P +� +w(N) +B +(0; z) +� += P +� +Φ(N),B +0 +(z) +� += 0, +where w(N) +B +(0; z) coincides with the following principal specialization of the tau- +function due to Sato’s formula (66): +w(N) +B +(0; z) = τ(N) +BGW +� +− 2 +z , − 2 +3z3, − 2 +5z5 , · · · +� +. +Moreover, in §5 we have shown that the free energy log τ(N) +BGW can be reconstructed +from the Eynard-Orantin topological recursion on the spectral curve (76), therefore +we only need to check that (76) is the semi-classical limit of P. Denote: +(77) +ˆx = z·, +ˆy = ℏ∂z, +then [y, x] = ℏ, and the operator P can be rewritten as follows: +P = +� +(ℏz∂z + ℏ +2)2 − S2�� +ℏ∂z − +1 +2z2 +� +(ℏz∂z − ℏ +2)2 − S2�� += +� +(ˆxˆy + ℏ +2)2 − S2�� +ˆy − 1 +2 ˆx−2� +(ˆxˆy − ℏ +2)2 − S2�� +, +(78) +where S = ℏ·N. By taking the semi-classical limit, we obtain the following function +H(x, y) on the (x, y)-plane: +(79) +H(x, y) = (x2y2 − S2) +� +y − +1 +2x2 (x2y2 − S2) +� +. + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +29 +Notice that the spectral curve defined by H(x, y) is reducible, and thus we are not +dealing with the standard case of the topological recursion. Nevertheless, here we +take the second factor in this function which defines a plane curve: +2x2y = x2y2 − S2. +This classical curve coincide with the spectral curve (76) for the Eynard-Orantin +topological recursion after a shift y �→ y+1, and thus we will regard the Schr¨odinger +equation P +� +Φ(N),B +0 +(z) +� += 0 as its quantum spectral curve in the sense of Gukov- +Su�lkowski [21]. +Acknowledgements. The authors thank Prof. Jian Zhou for pointing out that the +correction term in the formula (25) coincide with the Bergman kernel used in [48]. +The authors also thank Prof. Huijun Fan, Prof. Shuai Guo, Prof. Xiaobo Liu for +encouragement, and Dr. Ce Ji for helpful discussions. +References +[1] Alexandrov A. Cut-and-join description of generalized Br´ezin-Gross-Witten model. Advances +in Theoretical and Mathematical Physics, 2018, 22(6). +[2] Alexandrov A. KdV solves BKP. Proceedings of the National Academy of Sciences, 2021, +118(25):e2101917118. +[3] Alexandrov A. Generalized Br´ezin-Gross-Witten tau-function as a hypergeometric solution of +the BKP hierarchy. Advances in Mathematics, 2023, 412:108809. +[4] Alexandrov A. Intersection numbers on Mg,n and BKP hierarchy. Journal of High Energy +Physics, 2021, 2021(9):013. +[5] Alexandrov A, Shadrin A. Elements of spin Hurwitz theory: +closed algebraic formulas, +blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture. +arXiv preprint arXiv:2105.12493, 2021. +[6] Anderson P W. More is different. Science, 1972, 177(4047):393-396. +[7] Balogh F, Yang D. Geometric interpretation of Zhou’s explicit formula for the Witten- +Kontsevich tau function. Letters in Mathematical Physics, 2017, 107(10):1837-1857. +[8] Bertola M, Ruzza G. Brezin-Gross-Witten tau function and isomonodromic deformations. +Commun, Number Theory Phys, 2019, 13:827-883. +[9] Br´ezin E, Gross D J. The external field problem in the large N limit of QCD. Physics Letters +B, 1980, 97(1):120-124. +[10] Chidambaram N K, Garcia-Failde E, Giacchetto A. Relations on Mg,n and the negative +r-spin Witten conjecture. arXiv preprint arXiv:2205.15621, 2022.. +[11] Date E, Jimbo M, Kashiwara M, Miwa T. Transformation groups for soliton equations IV. +A new hierarchy of soliton equations of KP-type. Physica D, 1982, 4(3):343-365. +[12] Date E, Jimbo M, Miwa T. Solitons: Differential equations, symmetries and infinite dimen- +sional algebras. Cambridge University Press, 2000. +[13] Deligne P, Mumford D. The irreducibility of the space of curves of given genus. Publications +Math´ematiques de l’IH´ES, 1969, 36(1): 75-109. +[14] Dijkgraaf R, Verlinde H, Verlinde E. Loop equations and Virasoro constraints in nonpertur- +bative two-dimensional quantum gravity. Nuclear Physics B, 1991, 348(3): 435-456. +[15] Do N, Norbury P. Topological recursion on the Bessel curve. Communications in Number +Theory and Physics, 2018, 12(1):53-73. +[16] Eskin A, Okounkov A, Pandharipande R. The theta characteristic of a branched covering. +Adv. Math., 2008, 217: 873-888. +[17] Eynard B, Orantin N. Invariants of algebraic curves and topological expansion. Communica- +tions in Number Theory & Physics, 2007, 1(2):347-452. +[18] Fukuma M, Kawai H, Nakayama R. Continuum Schwinger-Dyson Equations and Universal +Structures in Two-Dimensional Quantum Gravity. International Journal of Modern Physics +A, 1991, 6(08):1385-1406. +[19] Giacchetto A, Kramer R, Lewa´nski D. A new spin on Hurwitz theory and ELSV via theta +characteristics. arXiv preprint arXiv:2104.05697, 2021. + +30 +ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG +[20] Gross D J, Witten E. Possible third-order phase transition in the large-N lattice gauge theory. +Physical Review D Particles & Fields, 1980, 21(2). +[21] Gukov S, Su�lkowski P. A-polynomial, B-model, and quantization. Journal of High Energy +Physics, 2012, 2012(2):70. +[22] Harnad J, Balogh F. Tau Functions and their Applications. Cambridge Monographs on Math- +ematical Physics. Cambridge University Press, 2021. +[23] Hoffman P N, Humphreys J F. Projective representations of the symmetric groups: +Q- +functions and shifted tableaux. Oxford Mathematical Monographs. Clarendon Press, 1992. +[24] Ji C, Wang Z, Yang C. Kac-Schwarz Operators of Type B, Quantum Spectral Curves, and +Spin Hurwitz Numbers. arXiv preprint arXiv: 2211.08687, 2022. +[25] Jimbo M, Miwa T. Solitons and Infinite-Dimensional Lie Algebras. Publications of the Re- +search Institute for Mathematical Sciences, 1983, 1983(19): 943-1001. +[26] Kac V, Schwarz A. Geometric interpretation of the partition function of 2D gravity. Physics +Letters B, 1991, 257(3-4):329-334. +[27] Kac V, van de Leur J. Polynomial tau-functions of BKP and DKP hierarchies. Journal of +Mathematical Physics, 2019, 60(7):071702. +[28] Knudsen F F. The projectivity of the moduli space of stable curves, II: The stacks Mg,n. +Mathematica Scandinavica, 1983, 52(2): 161-199. +[29] Kontsevich M. Intersection theory on the moduli space of curves and the matrix Airy function. +Communications in Mathematical Physics, 1992, 147(1): 1-23. +[30] Laughlin R B. A Different Universe: Reinventing Physics from the Bottom Down. Basic +Books, New York, 2005. +[31] Liu X, Yang C. Q-Polynomial expansion for Br´ezin-Gross-Witten tau-function. Advances in +Mathematics, 2022, 404: 108456. +[32] MacDonald I G. Symmetric functions and Hall polynomials. 2nd edition. Clarendon Press, +1995. +[33] Mironov A, Morozov A. Superintegrability of Kontsevich matrix model. European Physical +Journal C, 2021, 81(3). +[34] Mironov A, Morozov A, Semenoff G. Unitary matrix integrals in the framework of Generalized +Kontsevich Model. I. Br´ezin-Gross-Witten Model. International Journal of Modern Physics A, +1996, 11(28):5031-5080. +[35] Norbury P. A new cohomology class on the moduli space of curves. arXiv preprint +arXiv:1712.03662, 2017. +[36] Orlov A Y. Hypergeometric functions related to Schur Q-polynomials and BKP equation. +Theoretical & Mathematical Physics, 2003, 137(2):1574-1589. +[37] Sato M. Soliton Equations as Dynamical Systems on an Infinite Dimensional Grassmann +Manifold. RIMS Kokyuroku, 1981, 439:30-46. +[38] Schur J. ¨Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch ge- +brochene lineare Substitutionen. Journal F¨ur Die Reine Und Angewandte Mathematik, 1911, +1911(139):155-250. +[39] Schwarz A. On the solutions to the string equation. Modern Physics Letters A, 2011, 06(29): +2713-2725. +[40] Segal G, Wilson G. Loop Groups and Equations of KdV Type. Publications Math´ematiques +de l’IH´ES, 1985, 61(1):5-65. +[41] van de Leur J. The Adler-Shiota-van Moerbeke formula for the BKP hierarchy. Journal of +Mathematical Physics, 1995, 36:4940-4951. +[42] Wang Z, Yang C. BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic +N-Point Functions. Letters in Mathematical Physics, 2022, 112:62. +[43] Wang Z, Zhou J. A formalism of abstract quantum field theory of summation of fat graphs. +arXiv preprint arXiv: 2108.10498, 2021. +[44] Wang Z, Zhou J. Topological 1D Gravity, KP Hierarchy, and Orbifold Euler Characteristics +of Mg,n. arXiv:2109.03394, 2021. +[45] Witten E. Two-dimensional gravity and intersection theory on moduli space. Surveys in +differential geometry, 1990, 1(1): 243-310. +[46] Yang D, Zhang Q. On The Hodge-BGW Correspondence. arXiv preprint arXiv:2112.12736, +2021. + +EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS +31 +[47] You Y. Polynomial solutions of the BKP hierarchy and projective representations of symmet- +ric groups. Infinite-dimensional Lie algebras and groups, Luminy-Marseille, Adv. Ser. Math. +Phys., Volume 7: 449-464. +[48] Zhou J. Topological recursions of Eynard-Orantin type for intersection numbers on moduli +spaces of curves. Letters in Mathematical Physics, 2013, 103(11): 1191-1206. +[49] Zhou J. Quantum deformation theory of the Airy curve and mirror symmetry of a point. +arXiv preprint arXiv:1405.5296, 2014. +[50] Zhou J. Emergent geometry and mirror symmetry of a point. arXiv preprint arXiv:1507.01679, +2015. +[51] Zhou J. K-Theory of Hilbert schemes as a formal quantum field theory. arXiv preprint +arXiv:1803.06080, 2018. +[52] Zhou J. Fat and thin emergent geometries of Hermitian one-matrix models. arXiv preprint +arXiv:1810.03883, 2018. +[53] Zhou J. Emergent Geometry of Matrix Models with Even Couplings. arXiv preprint +arXiv:1903.10767, 2019. +[54] Zhou J. Grothendieck’s Dessins d’Enfants in a Web of Dualities. arXiv preprint arXiv: +1905.10773, 2019. +[55] Zhou J. Grothendieck’s Dessins d’Enfants in a Web of Dualities. II. arXiv preprint arXiv: +1907.00357, 2019. +School of Mathematical Sciences, Peking University, Beijing, 100871, China +Email address: zhiyuan19@math.pku.edu.cn +Beijing International Center for Mathematical Research, Peking University, Bei- +jing, 100871, China +Email address: yangcl@pku.edu.cn +School of Mathematical Sciences, Peking University, Beijing, 100871, China +Email address: zqs@math.pku.edu.cn + diff --git a/f9AzT4oBgHgl3EQfMfsT/content/tmp_files/load_file.txt b/f9AzT4oBgHgl3EQfMfsT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f9804d478da0d5767ab90d0401e981409bf8818 --- /dev/null +++ b/f9AzT4oBgHgl3EQfMfsT/content/tmp_files/load_file.txt @@ -0,0 +1,1063 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf,len=1062 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='01131v1 [math-ph] 3 Jan 2023 BKP-AFFINE COORDINATES AND EMERGENT GEOMETRY OF GENERALIZED BR´EZIN-GROSS-WITTEN TAU-FUNCTIONS ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Following Zhou’s framework, we consider the emergent geometry of the generalized Br´ezin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges nat- urally from the Virasoro constraints for the generalized BGW tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The BKP-affine coordinates and the topological recursion provide two different approaches towards the con- crete computations of the connected n-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Finally, we show that the quantum spectral curve of type B in the sense of Gukov-Su�lkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Backgrounds and overview 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Description of main results 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Preliminaries of Generalized Br´ezin-Gross-Witten Tau-Functions 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Generalized BGW tau-functions and Virasoro constraints 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Recursion for connected n-point functions 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Schur Q-function expansion and BKP-affine coordinates 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' BKP-Affine Coordinates of Generalized BGW Tau-Functions 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' BKP-affine coordinates of τ (N) BGW(t/2) 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' A formula for connected n-point functions 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Generating series for the BKP-affine coordinates 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Emergence of Spectral Curve and Its Special Deformation 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' A brief review of Zhou’s emergent geometry 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Special deformation and spectral curve for τ (N) BGW 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Quantum deformation theory of the spectral curve 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Emergence of Eynard-Orantin Topological Recursion 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Bergman kernel and Eynard-Orantin invariants on the spectral curve 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Examples of the Eynard-Orantin invariants 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' A remark on Bergman kernel of type B 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Emergence of Quantum Spectral Curve of Type B 26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Emergence of Kac-Schwarz operators from BKP-affine coordinates 26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Kac-Schwarz operators of type B for generalized BGW models 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Quantum spectral curve of type B for τ (N) BGW(t/2) 28 References 29 1 2 ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Backgrounds and overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The reduction/reconstruction and the emer- gence approach are two different philosophies in science, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [6,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The later is now widely used in the study of condensed-matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In recent several years, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Zhou has initiated a series of mathematical works to introduce the emergent point of view into the study of Gromov-Witten type theories and mirror symmetry, see [49, 50, 52, 53, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' His framework is called the emergent geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' A typical question in this framework is that, given a Gromov-Witten type theory, how can one find the associated B-model geometry which plays the role of a mirror?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In his works, Zhou has examined several examples and proposed that a spectral curve (to- gether with a deformation called the special deformation) emerges naturally from the connected one-point functions of the partition function, and the Eynard-Orantin topological recursion [17] emerges naturally from the Virasoro constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' It is worth mentioning that in these examples the partition function can be uniquely de- termined by the Virasoro constraints, and thus Zhou took the Virasoro constraints as the starting point of the emergent geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Moreover, the partition function in Zhou’s examples are all tau-functions of the KP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' There are two main advantages to study theories specified by such tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The first advantage is that a tau-function τ(t) of the KP hierarchy satisfying the initial value condition τ(0) = 1 can be described uniquely by a set of (complex) numbers {an,m}n,m≥0 called the (KP-)affine coordinates [7,50], which are natural coordinates on the big cell of the Sato Grassmannian [12,37,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' One can carry out a lot of explicit computations using the affine coordinates of such a tau-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Another advantage is that, the Sato Grassmannian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=', the set of all tau-functions of the KP hierarchy) is an infinite-dimensional homogeneous space, and thus one can find the duality among different tau-functions (and the theories specified by them) using the action of the infinite-dimensional Lie group � GL(∞), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [54,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Now in this work, we give an example of the emergent geometry associated to a tau-function of the BKP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The BKP hierarchy is an integrable system introduced by the Kyoto School [11,25] which shares a lot of common features with the KP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In particular, tau-functions of the BKP hierarchy can be repre- sented as summations of Schur Q-functions [38], see [47];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' and a tau-function τ(t) satisfying τ(0) = 1 can be described uniquely in terms of its BKP-affine coordi- nates on the big cell of the isotropic Sato Grassmannian, see [22, §7] and [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [2,24,27,36,41] for more about the BKP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Our object of study in this work is the generalized Br´ezin-Gross-Witten (BGW) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The partition functions of the generalized BGW models [34] are known to be a family of tau-functions of the BKP hierarchy indexed by a complex pa- rameter N ∈ C [2, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' When N = 0, this model becomes the original BGW model [9,20], which is known to be related to certain intersection numbers on the Deligne-Mumford moduli spaces of stable curves [13,28], see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [10,35] and [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In this work, we show that the spectral curve together with the Eynard-Orantin topo- logical recursion on it emerges naturally from the Virasoro constraints [1] for these tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Moreover, we will give the explicit expressions of the BKP-affine co- ordinates of these tau-functions, and show that the quantum spectral curve (in the sense of Gukov-Su�lkowski [21]) associated to the spectral curve emerges naturally from the BKP-affine coordinates and the E-O topological recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' We summarize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='our plan of this paper in the following diagram: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='Virasoro constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='for gBGW tau-functions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='(I) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='Schur Q-function expansion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='Spectral curve and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='its special deformation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='BKP-affine coordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='E-O topological recursion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='Computation of connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='n-point functions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='Quantum spectral curve ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='of type B (in the sense ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='of Gukov-Su�lkowski) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='Here the Virasoro constraints for the generalized BGW tau-functions were derived ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='by Alexandrov in [1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' and Step (I) has been accomplished by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Liu and the second author in [31] (see also [3] for another approach to the Schur Q-function expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' All other steps will be addressed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Description of main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Now we briefly summarize our main results in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Denote by τ (N) BGW(t) the generalized BGW tau-function indexed by the parameter N ∈ C, where t = (t1, t3, t5, · · · ) are the coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Then the free energy log τ (N) BGW admits a genus expansion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [1, (112)]): log τ (N) BGW(ℏ−1t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) = ∞ � g=0 ℏ2g−2FgBGW g (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S), where S = ℏ · N is a parameter for the generalized BGW models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Following Zhou’s idea, we define the special deformation of the spectral curve to be the following formal series in x where the coefficients are the times variables t (with a dilaton shift) and the one-point functions of genus zero: y = � n≥0 (2n + 1)(t2n+1 − δn,0)x2n + � n≥0 ∂FgBGW 0 (t) ∂t2n+1 x−2n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' This is a family of curves on the (x, y)-plane parametrized by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' By restricting to t = 0, we obtain a plane curve which plays the role of spectral curve (see §4): (1) x2y2 = x2 + S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' We will also discuss the quantum deformation theory of this spectral curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' More precisely, we show that the Virasoro constraints of genus zero can be encoded in the above special deformation, and the Virasoro constraints of all genera can be encoded in a suitable quantization of the special deformation, see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2-§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Then we consider the Eynard-Orantin topological recursion on this spectral curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Notice that the above spectral curve degenerates when S = 0, and in this case one cannot apply the E-O topological recursion directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Nevertheless, we can treat S as a formal variable and carry out the E-O recursion first, and then the n-point functions for the original BGW model can be obtained by restricting to S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' See also [15] for a topological recursion of the original BGW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 4 ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG In this work, we show that the Eynard-Orantin topological recursion emerges naturally from the Virasoro constraints for the generalized BGW tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Our main theorem is the following (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1 for details): Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Denote by B(x1, x2) be the following symmetric bi-differential on the spectral curve (1): B(x1, x2) = � 1 (x1 − x2)2 + W gBGW 0,2 (x1, x2) � dx1dx2, where W gBGW g,n is the connected n-point function of genus g: W gBGW g,n (x1, · · · , xn) = � i1,··· ,in≥0 ∂nFg(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) ∂t2i1+1 · · · ∂t2in+1 ��� t=0 · x−2i1−2 1 · · x−2in−2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Let ωg,n (where g ≥ 0, n ≥ 1) be the Eynard-Orantin invariants for the spectral curve (1) and Bergman kernel B, then for (g, n) with 2g − 2 + n > 0 we have: ωg,n = (−1)n · W gBGW g,n (x1, · · · , xn)dx1 · · · dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Moreover, the above topological recursion can also be reformulated in terms of another Bergman kernel (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='4 for details): �B(x1, x2) = �1 2 � 1 (x1 − x2)2 + 1 (x1 + x2)2 � + W gBGW 0,2 (x1, x2) � dx1dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In literatures a symmetric bi-differential of the above form is supposed to be the Bergman kernel for the topological recursion associated to a tau-function of the BKP hierarchy, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [5,19,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In [1], Alexandrov has conjectured a topological recursion on another spectral curve for the generalized BGW models (see [1, (128);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='7]): x2y2 − x − S2 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' His method is to look for an operator annihilating the principal specialization of the tau-function and then take the semi-classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' This curve looks different from (1), since he made a change of variable x = λ2 in the principal specialization to represent it in terms of the modified Bessel function (see also [34, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Besides the Virasoro constraints and topological recursion, there is an alternative way to compute the connected n-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Using the Schur Q-expansion given in [3,4,31], we are able to write down the explicit formulas for the BKP-affine coordinates for the generalized BGW tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Then we can apply a formula proved in [42] to obtain the following (see §3 for details): Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The connected n-point functions associated to the generalized BGW tau-functions are given by: � i1,··· ,in>0: odd ∂n log τ (N) BGW(t/2) ∂ti1 · · · ∂tin ���� t=0 x−i1 1 · · x−in n = −δn,2 · x1x2(x2 2 + x2 1) 2(x2 1 − x2 2)2 − 2n−1 · � � σ: n-cycle n � i=1 ξ(xσ(i), −xσ(i+1)) � odd, (2) EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS 5 for n ≥ 1, where [·]odd means taking the terms of odd degrees in every xi, and we use the convention σ(n + 1) := σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The function ξ is: ξ(xσ(i), −xσ(i+1)) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 A(N)(xσ(i), −xσ(i+1)), σ(i) = σ(i + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' �A(N)(xσ(i), −xσ(i+1)), σ(i) < σ(i + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' − �A(N)(−xσ(i+1), xσ(i)), σ(i) > σ(i + 1), where A(N) and �A(N) are the generating series of the BKP-affine coordinates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=', the fermionic 2-point functions) whose explicit formulas are: A(N)(w, x) = w − x + Φ(N) 1 (−x)Φ(N) 2 (−w) − Φ(N) 1 (−w)Φ(N) 2 (−x) 4(w + x) , �A(N)(w, x) = Φ(N) 1 (−x)Φ(N) 2 (−w) − Φ(N) 1 (−w)Φ(N) 2 (−x) 4(w + x) , where Φ(N) 1 , Φ(N) 2 are the formal Laurent series: Φ(N) 1 (z) = 1 + ∞ � k=1 (−ℏ)k 8k · k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · k � i=1 � 4N 2 − (2i − 1)2� z−k, Φ(N) 2 (z) = z + ∞ � k=1 (−ℏ)k 8k · k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · k � i=1 � 4(1 − N)2 − (2i − 1)2� z1−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' It is interesting to notice that the Eynard-Orantin topological recursion is effi- cient only at small genus, while the explicit formula (2) is efficient for large genus and small n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Finally, we consider the emergence of the quantum spectral curve of type B for the plane curve (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In [21] Gukov and Su�lkowski proposed a conjectural construc- tion of the quantum spectral curve for a classical plane curve using Eynard-Orantin topological recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' It is well-known that when the generating series of the E- O invariants on the classical curve is a tau-function of the KP hierarchy, finding the quantum spectral curve (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=', the Schr¨odinger equation [21, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='10)]) is equiv- alent to finding an operator which annihilates the principal specialization of the tau-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' When considering a tau-function of the BKP hierarchy, the principal specialization is a slightly different with that in case of KP hierarchy, and the defi- nition of the Baker-Akhiezer function [21, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='4)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='7)] may need some simple modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Nevertheless, in what follows we will refer the following two facts as a quantum spectral curve of type B in the sense of Gukov-Su�lkowski: 1) The connected n-point functions of a tau-function τ(t) of the BKP hierarchy can be reconstructed from the Eynard-Orantin topological recursion on a plane curve C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 2) There is an operator P annihilating the following principal specialization: P � τ � − 2 z , − 2 3z3 , − 2 5z5, · · · �� = 0, such that the semi-classical limit of P gives the defining equation of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In [24], a method of deriving Kac-Schwarz operators [26,39] and quantum spec- tral curve of type B using BKP-affine coordinates has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Now we apply 6 ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG this method to the case of generalized BGW models using the explicit expressions of BKP-affine coordinates computed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' We show that the operator P = ℏ3� (z∂z + 1 2)2 − N 2�� ∂z − ℏ 2z2 � (z∂z − 1 2)2 − N 2�� gives the quantum spectral curve of type B in the sense of Gukov-Su�lkowski for the classical curve (1), see §6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' It is worth pointing out that in [3] Alexandrov has found that such an operator annihilates the principal specialization (without knowing the topological recursion) using a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The rest of this paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In §2 we recall some preliminaries of the generalized BGW tau-functions, including the Virasoro constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In §3 we compute the BKP-affine coordinates and derive the formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In §4 we construct the spectral curve (1) and discuss the quantum deformation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In §5 we show that the Eynard-Orantin topological recursion on the curve (1) emerges naturally from the Virasoro constraints of the generalized BGW tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' And finally in §6, we show the emergence of the quantum spectral curve of type B from the BKP-affine coordinates and the above topological recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Preliminaries of Generalized Br´ezin-Gross-Witten Tau-Functions In this section, we first briefly review some basics of the generalized Br´ezin- Gross-Witten tau-functions τ (N) BKP(t), including the Virasoro constraints, recursion for n-point functions, and Schur Q-function expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Generalized BGW tau-functions and Virasoro constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The gen- eralized BGW model was introduced by Mironov-Morozov-Semenoff in [34] as a family of matrix integrations indexed by a parameter N (where N is a complex number and is not to be confused with the size of the matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' For each N ∈ C the partition function τ (N) BGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) is a tau-function of the KdV hierarchy with time variables t = (t1, t3, t5, · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' And for N = 0, the partition function τ (0) BGW is the original BGW tau-function [9,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In [1], Alexandrov showed that for every N the partition function τ(N) BGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) is uniquely determined by the normalization condition τ (N) BGW(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) = 1, together with the Virasoro constraints (3) L(N) n τ (N) BGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) = 0, ∀n ≥ 0, where the Virasoro operators {L(N) n }n≥0 are: L(N) n = − 1 2ℏ ∂ ∂t2n+1 + 1 4 � a+b=2n a,b: odd ∂2 ∂ta∂tb + 1 2 � k≥1 k: odd ktk ∂ ∂tk+2n + 1 − 4N 2 16 δn,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' which satisfy the Virasoro commutation relation: [L(N) m , L(N) n ] = (m − n)L(N) m+n, ∀m, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The free energy associated to τ (N) BGW is the logarithm F(N)(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) = log τ (N) BGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS 7 Denote: (4) S = ℏ · N, then the free energy F(N) has a genus expansion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [1, (112)]): F(N)(ℏ−1t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) = ∞ � g=0 ℏ2g−2FgBGW g (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S), where ℏ−1t = (ℏ−1t1, ℏ−1t3, ℏ−1t5, · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In what follows, we will denote: (5) FgBGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) = ∞ � g=0 ℏ2g−2FgBGW g (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) = F(N)(ℏ−1t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ), and thus one has τ (N) BGW(ℏ−1t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ℏ) = eF gBGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The Virasoro constraints (3) now can be rewritten as (6) LgBGW n � exp FgBGW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) � = 0, ∀n ≥ 0, where: LgBGW 0 = −1 2 ∂ ∂t1 + 1 2 � k≥1 k: odd ktk ∂ ∂tk + 1 16 − ℏ−2 S2 4 , LgBGW n = −1 2 ∂ ∂t2n+1 + ℏ2 4 � a+b=2n a,b: odd ∂2 ∂ta∂tb + 1 2 � k≥1 k: odd ktk ∂ ∂tk+2n , n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Our notations for the partition function and free energy differ from those in [1] by a rescaling ti �→ 2ti for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Recursion for connected n-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In this subsection, we refor- mulate the Virasoro constraints as recursions for the connected n-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' This has been done by Alexandrov in [1, §3] (in different notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Let ⟨pµ1 · · · pµn⟩c g be the connected correlators defined by: (7) ⟨pµ1pµ2 · · · pµn⟩c g = ∂nFgBGW g (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) ∂tµ1∂tµ2 · · · ∂tµn ��� t=0, µ1, · · · , µn > 0 : odd, then the free energy is of the form: FgBGW g (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) = � n≥1 � µi>0: odd 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='⟨pµ1 · · · pµn⟩c g · tµ1 · · · tµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' One can rewrite the Virasoro constraints (6) as the recursion for the correlators: ⟨p2k+1pµ1pµ2 · · · pµn⟩c g =1 2 � a+b=2k a,b>0: odd � ⟨papbpµ1 · · · pµn⟩c g−1 + � g1+g2=g I⊔J=[n] ⟨papµI⟩c g1⟨pbpµJ ⟩c g2 � + n � i=1 µi⟨pµi+2kpµ1 · · · ˆpµi · · · pµn⟩c g, k ≥ 0, (8) 8 ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG together with the initial values: ⟨p1⟩c 0 = −1 2S2, ⟨p1⟩c 1 = 1 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ⟨p1⟩c g = 0, ∀g ≥ 2, (9) where ˆpµi means deleting the term pµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Here [n] = {1, 2, · · · , n}, and for a set of indices I = {i1, i2, · · · , im} ⊂ [n] we denote pµI = (pµi1 , pµi2 , · · · , pµim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In the case g = 0, we use the convention ⟨papbpµ1 · · · pµk⟩c −1 = 0 in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Now consider the n-point function W gBGW g,n of genus g defined by: W gBGW g,n (x1, · · · , xn) = � µ1,··· ,µn: odd ⟨pµ1 · · · pµn⟩c g · x−µ1−1 1 · · x−µn−1 n = � i1,··· ,in≥0 ∂nFgBGW g (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' S) ∂t2i1+1 · · · ∂t2in+1 ��� t=0 · x−2i1−2 1 · · x−2in−2 n , (10) for g ≥ 0 and n ≥ 1, where x1, · · · , xn are some formal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' For example, using the above recursion for correlators one easily finds that: (11) ⟨p2n+1⟩c 0 = (−1)n+1 22n+1 1 n + 1 �2n n � S2n+2, ∀n ≥ 0, and thus: (12) W gBGW 0,1 (x) = � n≥0 (−1)n+1 22n+1 1 n + 1 �2n n � S2n+2x−2n−2 = 1 − � 1 + S2 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Here our notation W gBGW g,n differs from Wg,n in [1, §3] by: W gBGW g,n (x1, · · · , xn) = 2−n · Wg,n(x2 1 4 , · · · , x2 n 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The purpose of modifying the notations in this way is to make the discussions fit into the picture of emergent geometry (see §4 - §5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In the rest of this subsection, we rewrite the Virasoro constraints as the recursion for the n-point functions W gBGW g,n (x1, · · · , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' By (8) we have: W gBGW g,n+1 (x, x1, · · · , xn) = � k≥0 � µ1,··· ,µn: odd ⟨p2k+1pµ1 · · · pµn⟩g · x−2k−2x−µ1−1 1 · · x−µn−1 n = � µ ∞ � k=0 � a+b=2k 1 2⟨papbpµ⟩g−1 · x−a−1x−b−1x−µ1−1 1 · · x−µn−1 n + � µ ∞ � k=0 � a+b=2k � g1+g2=g I⊔J=[n] 1 2⟨papµI⟩c g1⟨pbpµJ⟩c g2 · x−a−1x−b−1x−µ1−1 1 · · x−µn−1 n + � µ ∞ � k=0 n � i=1 µi⟨pµi+2kpµ1 · · · ˆpµi · · · pµn⟩c g · x−2k−2x−µ1−1 1 · · x−µn−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS 9 Now we conclude that for every (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' n) ̸= (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' W gBGW g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='n+1 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' xn) = n � i=1 Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='xiW gBGW g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='n (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' xn)+ 1 2Ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='v � W gBGW g−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='n+2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' xn) + � g1+g2=g I⊔J=[n] W gBGW g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='|I|+1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' xI)W gBGW g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='|J|+1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' xJ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' where we denote xI = (xi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' xim) for I = {i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' im},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' and (13) Ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='vf(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' v) = � lim v→u f(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' v) ��� u=x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' and Da,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='b is an operator such that for every odd m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='xi(x−m−1 i ) = � 2k+µi=m µi odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' k≥0 µix−2k−2x−µi−1 i = 1 (x2 − x2 i )2 � 1 xm−1 − 1 xm−1 i + x2 i xm+1 − x2 xm+1 i � + m + 1 xm+1 i (x2 − x2 i ) So one can take the operator Da,b to be: (14) Da,bf(b) = a2 + b2 (a2 − b2)2 � f(a) − f(b) � − b a2 − b2 ∂bf(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Notice that the right-hand side of the above recursion involves terms W gBGW 0,1 (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' We may move such terms to the left-hand side and rewrite the recursion as follows: W gBGW g,n+1 (x, x1, · · · , xn) = n � i=1 �Dx,xiW gBGW g,n (x1, · · · , xn)+ 1 2 �Ex,u,v � W gBGW g−1,n+2(u, v, x[n]) + s � g1+g2=g I⊔J=[n] W gBGW g1,|I|+1(u, xI)W gBGW g2,|J|+1(v, xJ) � , (15) where s� means that we exclude the terms involving W gBGW 0,1 in this summation, and we use the notations: (16) �Ex,u,v = 1 1 − W0,1(x)Ex,u,v, �Dx,xi = 1 1 − W0,1(x)Dx,xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' We can compute W gBGW 0,2 using the above formula: W gBGW 0,2 (x, x1) = 1 1 − W0,1(x)Dx,x1W0,1(x1) = 1 (x2 − x2 1)2 � x2 + x2 1 + 2S2 � (1 + S2 x2 )(1 + S2 x2 1 ) − x2 − x2 1 � = − 1 2S2x−2x−2 1 + 3 8S4x−2x−4 1 + 3 8S4x−4x−2 1 − 5 16S6x−2x−6 1 − 3 8S6x−4x−4 1 − 5 16S6x−6x−2 1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 10 ZHIYUAN WANG, CHENGLANG YANG, AND QINGSHENG ZHANG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Schur Q-function expansion and BKP-affine coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In this sub- section, we recall the Schur Q-function expansions of the generalized BGW tau- functions [3,4,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' It is known that if τ(t) is a tau-function of the KdV hierarchy with time vari- ables t = (t1, t3, t5, · · · ), then τ(t/2) is a tau-function of the BKP hierarchy, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Therefore for every N, the partition function τ (N) BGW(t/2) is a tau-function of the BKP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Moreover, every tau-function of the BKP hierarchy can be rep- resented as a summation of Schur Q-functions, see [47];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' and see also [23, 32] for introductions to Schur Q-functions and the relation to projective representations of the symmetric groups Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Thus τ (N) BGW(t/2) admits a Schur Q-function expansion for every N ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The following formula was conjectured by Alexandrov [4], and then proved in [31] and [3] by two different methods: (17) τ (N) BGW(t) = � λ∈DP � ℏ 16 �|λ| 2−l(λ)θλQλ(δk,1)Qλ(t), where DP is the set of all strict partitions λ = (λ1 > λ2 > · · · > λl(λ) > 0), and Qµ is the Schur Q-function indexed by µ ∈ DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' And θλ is given by: (18) θλ = l� j=1 λj � k=1 θ(k) for λ = (λ1, λ2, · · · , λl) ∈ DP, where θ is the following function on Z+: (19) θ(z) = (2z − 1)2 − 4N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' This Schur Q-function expansion shows that τ(N) BGW(t/2) is a hypergeometric tau- function [36] of the BKP hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' It is worth noting that in [31], Liu and the second author proved this Schur Q-function expansion by applying the Virasoro constraints (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' BKP-Affine Coordinates of Generalized BGW Tau-Functions In this section, we write down an explicit formula for the BKP-affine coordinates of the generalized BGW tau-functions using the Schur Q-function expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Then we apply the results in [42] to give a formula for the connected n-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The BKP-affine coordinates will be useful in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' BKP-affine coordinates of τ (N) BGW(t/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In this subsection, we write down the BKP-affine coordinates for the generalized BGW tau-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' First we recall some basics about BKP tau-functions and BKP-affine coordinates [22, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' BKP-affine coordinates are natural coordinates on the big cell of the isotropic Sato Grassmannian, see [22, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='3] for an introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Let τ(t) be a tau- function of the BKP hierarchy with BKP-time variables t = (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' t5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · · · ) satisfying the initial value condition τ(0) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' then the coefficients of the Schur Q-functions in such an expansion are Pfaffians of the BKP-affine coordinates: (20) τ = � µ∈DP (−1)⌈l(µ)/2⌉ · Pf(aµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='µj)1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='j≤2⌈l(µ)/2⌉ · Qµ(t/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' EMERGENT GEOMETRY OF GENERALIZED BGW TAU-FUNCTIONS 11 where {an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='m}n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='m≥0 are the BKP-affine coordinates of τ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' satisfying the anti- symmetry condition: (21) an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='m = −am,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' ∀n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' For µ = (µ1 > · · · > µl(µ) > 0) ∈ DP, we denote by l(µ) the length of µ, and here ⌈·⌉ is the ceiling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' For µ ∈ DP with l(µ) odd, we use the convention µl(µ)+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' For example, the first a few terms of τ are: τ =1 + � n>0 a0,n · Q(n)(t/2) + � m>n>0 an,m · Q(m,n)(t/2) + � m>n>l>0 (an,ma0,l − al,ma0,n + a0,mal,n)Q(m,n,l)(t/2) + � m>n>l>k>0 (an,mak,l − al,mak,n + ak,mal,n)Q(m,n,l,k)(t/2) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' In particular, the BKP-affine coordinates {an,m}n,m≥0 are exactly the coefficients of Qµ with l(µ) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Now consider the case of generalized BGW tau-functions τ (N) BGW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Using the Schur Q-function expansion (17), we obtain the following: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' The BKP-affine coordinates {a(N) n,m}n,m≥0 for the BKP tau- functions τ (N) BGW(t/2) are given by a(N) 0,0 = 0, and: a(N) 0,n = −a(N) n,0 = ℏn 23n+1 · n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · n � k=1 θ(k), n > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' a(N) n,m = ℏm+n 23m+3n+2 · m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' · m − n m + n · m � j=1 θ(j) · n � k=1 θ(k), n, m > 0, (22) where θ is the function (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' Here we need the following combinatorial identity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' [33, (56)]): Qλ(δk,1) = 2|λ| �l(λ) i=1 λi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfMfsT/content/2301.01131v1.pdf'} +page_content=' � i +1 corresponds to the presence of a CME signal inside +the CME observable ∆γ. +Let us focus on mid-central +collisions where the CME effect is expected to be more +likely to be measured than at the other centrality bins. +For the 20-50% centrality bin, we can see A/a > 1 clearly +except the cases of p=0 and p=2%. Moreover, we observe +that the value of A/a increases as the CME strength +increases. It indicates that A/a can reflect the strength +of the CME signal. +We calculate b using Eq. (16) and compare it with a. +Figure 5 shows the centrality dependence of a and b from +the AMPT model with different strengths of the CME. +0 +10 +20 +30 +40 +50 +60 +70 +80 +-2 +-1 +0 +1 +2 +a or b +Centrality (%) +a = v +2 +{SP}/v +2 +{PP} +b = ( +{PP}(p +0)- +{PP}(p +=0)) / ( +{SP}(p +0)- +{SP}(p +=0)) +Ru + Ru Zr + Zr b + + + p = 10% + + + p = 7.5% + + + p = 5% +Ru + Ru Zr + Zr a + + + p = 10% + + + p = 7.5% + + + p = 5% + + + p = 0 +FIG. 5: (Color online) The centrality dependences of a and +b in isobar collisions at √sNN = 200 GeV from the AMPT +model with different strengths of the CME. The open and +solid symbols represent the results for a and b, respectively. +The data points are shifted along the x axis for clarity. +0 +10 +20 +30 +40 +50 +60 +70 +80 +-2 +-1 +0 +1 +2 + b/a +Centrality (%) +Ru + Ru Zr + Zr + + + p = 10% + + + p = 7.5% + + + p = 5% +FIG. 6: (Color online) The centrality dependences of b/a in +isobar collisions at √sNN = 200 GeV from the AMPT model +with different strengths of the CME. The data points are +shifted along the x axis for clarity. +We find that the values of a and b are different. The value +of b is smaller than the value of a at the 20-50% centrality +bin. The value of a is almost independent of the CME +strength, as same as shown in Fig. 3. Considering statis- +tical errors, the value of b does not vary significantly with +the CME strengths. Note that the case of the 2% CME +strength is not shown due to huge statistical errors. +Figure 6 further shows the centrality dependence of the +b/a ratio from the AMPT model with different strengths +of the CME. For the 20-50% centrality bin in isobar colli- +sions at √sNN = 200 GeV, such an approximate relation +that b = a/2 can be obtained by a constant function fit- +ting. It implies that the relative ratio of the CME signals +with respect to different planes is not equal to the inverse + +6 +-2 +0 +2 +4 +Ru + Ru Zr + Zr +CME + + + + p=10% + + + p=7.5% + + + p=5% + + + p=2% + + + p=0 + + + STAR TPC full-event +CME +Ru + Ru Zr + Zr +CME +{b} + + p=10% + + p=7.5% + + p=5% + + p=2% +0 +10 +20 +30 +40 +50 +60 +70 +80 +-1 +1 +3 +Ru + Ru Zr + Zr + + p=10% + + p=7.5% + + p=5% + + p=2% +Ratio +Centrality (%) +CME +{b}/ +CME +FIG. 7: (Color online) Upper panel: The centrality depen- +dences of fCME{b} and fCME in isobar collisions at √sNN = +200 GeV from the AMPT model with different strengths of +the CME, in comparison with the STAR data [42]. +The +solid and open symbols represent the results for fCME{b} and +fCME, respectively. Lower panel: The centrality dependences +of the ratio of fCME{b} to fCME. The data points are shifted +along the x axis for clarity. +of the elliptic flow ratio with respect to different planes. +Next, we will show that this finding has important im- +plications for extracting the fraction of the CME signal +inside the ∆γ observable. +The upper panel in Fig. 7 shows two kinds of fCME +as a function of centrality from the AMPT model with +different strengths of the CME, where the solid and open +symbols represent fCME calculated from Eq. (12) and +fCME{b} calculated from Eq. (15), respectively. Com- +pared to the STAR experimental data [42], the results of +fCME from p=0% or p=2% are favored. For the 20-50% +centrality bin, when p is not equal to zero, fCME{b} is +found to be less than fCME. The lower panel in Fig. 7 +shows the centrality dependence of the ratio of fCME{b} +to fCME. For the 20-50% centrality bin, the ratio is less +than unity. +It indicates that the fraction of the CME +signal inside the ∆γ observable will be overestimated if +assuming b = a. +0 +10 +20 +30 +40 +50 +60 +70 +80 +0 +1 +2 + p = 5% p = 10% + + +CME + + + +CME +{b} + + +CME +{p} + +CME +Centrality (%) +Ru + Ru +FIG. 8: (Color online) The centrality dependences of fCME, +fCME{b} and fCME{p} from three different methods in Ru ++ Ru collisions at √sNN = 200 GeV from the AMPT model +with two different strengths of the CME. The data points are +shifted along the x axis for clarity. +To discern which of these two above fCME is closer +to the real situation, we can theoretically use another +method to obtain the true fCME, namely fCME{p}, as +defined as follows, +fCME{p} = ∆γCME{PP}(p ̸= 0) +∆γ{PP}(p ̸= 0) +, +(17) +where +∆γCME{PP}(p ̸= 0) = ∆γ{PP}(p ̸= 0) +−∆γ{PP}(p = 0). +(18) +The observables of ∆γ{PP}(p = 0) and ∆γ{PP}(p ̸= 0) +can be obtained from the AMPT model without the CME +and with different strengths of the CME, respectively. In +Fig. 8, compared to the ture fCME, we find that fCME{b} +is closer to and consistent with fCME{p} than fCME for +the 10−50% centrality bin.This suggests that it is neces- +sary to consider b to obtain a more reliable fCME. +From the above results, it is clear that b = a/2 signif- +icantly impacts the final result of fCME in isobar colli- +sions. To understand where this relation comes from, we +calculate a and b for different stages in Ru + Ru collision +at √sNN = 200 GeV in the AMPT model with the CME +strength of p = 10%. We focus on four evolution stages +of heavy-ion collisions: the initial stage, after parton cas- +cade, after coalescence, and after hadron rescatterings. +In the upper panel of Fig. 9, we can find that the value +of a remains constant for the last three stages. Note that +we do not show a for the initial stage, since the elliptic +flow is initially zero. However, the value of b is always +smaller than that of a and decreases stage by stage for the +10-50% centrality bin. In the lower panel of Fig. 9, the +ratio of b/a also decreases with the stage evolution. The + +7 +0.0 +0.2 +0.4 +0.6 +0.8 + Ru + Ru p = 10% + a b + + initial stage + + after parton cascade + + after coalescence + + after hadronic rescatterings +a or b +a = v +2 +{SP}/v +2 +{PP} +b = ( +{PP}(p +0)- +{PP}(p =0)) / ( +{SP}(p +0)- +{SP}(p =0)) +0 +10 +20 +30 +40 +50 +60 +70 +80 +0 +1 + after parton cascade + after coalescence + after hadronic rescatterings +Centrality (%) +b / a +FIG. 9: (Color online) Upper panel: The centrality depen- +dences of a and b in Ru + Ru collisions at √sNN = 200 GeV +for four different stages from the AMPT model with the CME +strength of p = 10%. The open and solid symbols represent +the results for a and b, respectively. Lower panel: The cen- +trality dependences of the ratio of b/a for the different stages +in Ru + Ru collisions. The data points are shifted along the +x axis for clarity. +main reason is due to the change in b. The decreasing of +b indicates the correlation between the CME signals rel- +ative to different planes becomes less and less correlated, +which can be understood as a consequence of the decor- +relation resulting from final state interactions during the +evolution of heavy-ion collisions [49, 67, 68]. +IV. +SUMMARY +Using a multiphase transport model with different +strengths of the CME, we reexamine the proposed two- +plane method to determine the fraction of the CME sig- +nal inside the CME observable of ∆γ in isobar collisions +at √sNN = 200 GeV. We first calculate the elliptic flow +v2 and the CME observable of ∆γ with respect to the +spectator plane and participant plane. The ratio b of the +CME signal relative to the two different planes is found +to be different from the ratio a of background relative to +the two different planes in isobar collisions. If the differ- +ence between a and b is taken into account, we demon- +strate that it will lead to a smaller CME fraction than +the constraint obtained in current experimental way. We +theoretically observe a decrease in the value of b during +the stage evolution in the AMPT model, which indicates +the decorrelation of the chiral magnetic effect relative to +spectator and participant planes is caused by final state +interactions in isobar collisions. Since a and b were as- +sumed to be equal in the current experimental study, the +fraction of the CME signal could be overestimated. We +hope that our study will provide a theoretical reference +for future accurate measurements of the fraction of the +chiral magnetic effect inside the experimental observable +in relativistic heavy-ion collisions. +ACKNOWLEDGMENTS +We thank Prof. Jie Zhao for the helpful discussions. +This work is supported by the National Natural Sci- +ence Foundation of China under Grants No.12147101, +No. +11890714, No. +11835002, No. +11961131011, No. +11421505, No. +12105054, the National Key Research +and Development Program of China under Grant No. +2022YFA1604900, the Strategic Priority Research Pro- +gram of Chinese Academy of Sciences under Grant No. +XDB34030000, and the Guangdong Major Project of +Basic and Applied Basic Research under Grant No. +2020B0301030008. +[1] P. F. Kolb, J. Sollfrank, and U. W. Heinz, Phys. Rev. C +62, 054909 (2000), hep-ph/0006129. +[2] D. Teaney, J. Lauret, and E. V. Shuryak, Phys. Rev. +Lett. 86, 4783 (2001), nucl-th/0011058. +[3] L. Yan, Chin. Phys. C 42, 042001 (2018), 1712.04580. +[4] C. Shen and L. Yan, Nucl. Sci. Tech. 31, 122 (2020), +2010.12377. +[5] H. Song, Y. Zhou, and K. Gajdosova, Nucl. Sci. Tech. +28, 99 (2017), 1703.00670. +[6] S.-W. Lan and S.-S. Shi, Nucl. Sci. Tech. 33, 21 (2022). +[7] S. Wu, C. Shen, and H. Song, Chin. Phys. Lett. 38, +081201 (2021), 2104.13250. +[8] V. Skokov, A. Y. Illarionov, and V. Toneev, Int. J. Mod. +Phys. A 24, 5925 (2009), 0907.1396. +[9] A. Bzdak and V. Skokov, Phys. Lett. B 710, 171 (2012), +1111.1949. +[10] W.-T. Deng and X.-G. Huang, Phys. Rev. C 85, 044907 +(2012), 1201.5108. +[11] X.-L. Zhao, Y.-G. Ma, and G.-L. Ma, Phys. Rev. C 97, +024910 (2018), 1709.05962. +[12] X.-L. Zhao, G.-L. Ma, and Y.-G. Ma, Phys. Lett. B 792, +413 (2019), 1901.04156. + +8 +[13] D. E. Kharzeev and J. Liao, Nature Rev. Phys. 3, 55 +(2021), 2102.06623. +[14] X.-G. Huang, Rept. Prog. Phys. 79, 076302 (2016), +1509.04073. +[15] K. Hattori and X.-G. Huang, Nucl. Sci. Tech. 28, 26 +(2017), 1609.00747. +[16] J.-H. Gao, G.-L. Ma, S. Pu, and Q. Wang, Nucl. Sci. +Tech. 31, 90 (2020), 2005.10432. +[17] D. Kharzeev, Phys. Lett. B 633, 260 (2006), hep- +ph/0406125. +[18] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, +Nucl. Phys. A 803, 227 (2008), 0711.0950. +[19] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, +Phys. Rev. D 78, 074033 (2008), 0808.3382. +[20] S. A. Voloshin, Phys. Rev. C 70, 057901 (2004), hep- +ph/0406311. +[21] B. I. Abelev et al. (STAR), Phys. Rev. Lett. 103, 251601 +(2009), 0909.1739. +[22] B. I. Abelev et al. (STAR), Phys. Rev. C 81, 054908 +(2010), 0909.1717. +[23] L. Adamczyk et al. (STAR), Phys. Rev. C 88, 064911 +(2013), 1302.3802. +[24] L. Adamczyk et al. (STAR), Phys. Rev. Lett. 113, +052302 (2014), 1404.1433. +[25] B. Abelev et al. (ALICE), Phys. Rev. Lett. 110, 012301 +(2013), 1207.0900. +[26] A. Bzdak, V. Koch, and J. Liao, Phys. Rev. C 83, 014905 +(2011), 1008.4919. +[27] J. Liao, V. Koch, and A. Bzdak, Phys. Rev. C 82, 054902 +(2010), 1005.5380. +[28] S. Schlichting and S. Pratt, Phys. Rev. C 83, 014913 +(2011), 1009.4283. +[29] F. Wang, Phys. Rev. C 81, 064902 (2010), 0911.1482. +[30] W.-Y. Wu et al. (2022), 2211.15446. +[31] F.-Q. Wang and J. Zhao, Nucl. Sci. Tech. 29, 179 (2018). +[32] J. Zhao (STAR), Nucl. Phys. A 1005, 121766 (2021), +2002.09410. +[33] J. Zhao and F. Wang, Prog. Part. Nucl. Phys. 107, 200 +(2019), 1906.11413. +[34] F. +Wang, +in +29th +International +Conference +on +Ultra-relativistic +Nucleus-Nucleus +Collisions +(2022), +2207.10044. +[35] W. Li and G. Wang, Ann. Rev. Nucl. Part. Sci. 70, 293 +(2020), 2002.10397. +[36] S. A. Voloshin, Phys. Rev. Lett. 105, 172301 (2010), +1006.1020. +[37] W.-T. Deng, X.-G. Huang, G.-L. Ma, and G. Wang, +Phys. Rev. C 94, 041901 (2016), 1607.04697. +[38] V. Koch, +S. Schlichting, +V. Skokov, +P. Sorensen, +J. Thomas, S. Voloshin, G. Wang, and H.-U. Yee, Chin. +Phys. C 41, 072001 (2017), 1608.00982. +[39] W.-T. Deng, X.-G. Huang, G.-L. Ma, and G. Wang, +Phys. Rev. C 97, 044901 (2018), 1802.02292. +[40] S. Choudhury et al., Chin. Phys. C 46, 014101 (2022), +2105.06044. +[41] J. Adam et al. (STAR), Nucl. Sci. Tech. 32, 48 (2021), +1911.00596. +[42] M. Abdallah et al. (STAR), Phys. Rev. C 105, 014901 +(2022), 2109.00131. +[43] H.-j. Xu, H. Li, X. Wang, C. Shen, and F. Wang, Phys. +Lett. B 819, 136453 (2021), 2103.05595. +[44] H.-j. Xu, W. Zhao, H. Li, Y. Zhou, L.-W. Chen, and +F. Wang (2021), 2111.14812. +[45] C. Zhang and J. Jia, Phys. Rev. Lett. 128, 022301 (2022), +2109.01631. +[46] J. Jia and C.-J. Zhang (2021), 2111.15559. +[47] J. Jia, Phys. Rev. C 105, 014905 (2022), 2106.08768. +[48] J. Jia, Phys. Rev. C 105, 044905 (2022), 2109.00604. +[49] X.-L. Zhao and G.-L. Ma, Phys. Rev. C 106, 034909 +(2022), 2203.15214. +[50] D. E. Kharzeev, J. Liao, and S. Shi, Phys. Rev. C 106, +L051903 (2022), 2205.00120. +[51] H.-j. Xu, J. Zhao, X. Wang, H. Li, Z.-W. Lin, C. Shen, +and F. Wang, +Chin. Phys. C 42, +084103 (2018), +1710.07265. +[52] H.-J. Xu, X. Wang, H. Li, J. Zhao, Z.-W. Lin, C. Shen, +and F. Wang, Phys. Rev. Lett. 121, 022301 (2018), +1710.03086. +[53] X.-L. Zhao, G.-L. Ma, and Y.-G. Ma, Phys. Rev. C 99, +034903 (2019), 1901.04151. +[54] M. S. Abdallah et al. (STAR), Phys. Rev. Lett. 128, +092301 (2022), 2106.09243. +[55] S. Shi, Y. Jiang, E. Lilleskov, and J. Liao, Annals Phys. +394, 50 (2018), 1711.02496. +[56] Y. Feng, J. Zhao, H. Li, H.-j. Xu, and F. Wang, Phys. +Rev. C 105, 024913 (2022), 2106.15595. +[57] Z.-W. Lin, C. M. Ko, B.-A. Li, B. Zhang, and S. Pal, +Phys. Rev. C 72, 064901 (2005), nucl-th/0411110. +[58] G.-L. Ma and Z.-W. Lin, Phys. Rev. C 93, 054911 (2016), +1601.08160. +[59] Z.-W. Lin and L. Zheng, Nucl. Sci. Tech. 32, 113 (2021), +2110.02989. +[60] Z.-W. Lin, Phys. Rev. C 90, 014904 (2014), 1403.6321. +[61] J. D. Orjuela Koop, A. Adare, D. McGlinchey, and J. L. +Nagle, Phys. Rev. C 92, 054903 (2015), 1501.06880. +[62] G.-L. Ma and A. Bzdak, Nucl. Phys. A 956, 745 (2016). +[63] Y. He and Z.-W. Lin, Phys. Rev. C 96, 014910 (2017), +1703.02673. +[64] L. Huang and G.-L. Ma, Chin. Phys. C 45, 074110 +(2021), 2107.09264. +[65] Q. Chen and G.-L. Ma, Phys. Rev. C 106, 014907 (2022), +2207.11736. +[66] Q. Chen, H.-S. Wang, and G.-L. Ma (2022), 2210.17125. +[67] G.-L. Ma and B. Zhang, Phys. Lett. B 700, 39 (2011), +1101.1701. +[68] L. Huang, M.-W. Nie, and G.-L. Ma, Phys. Rev. C 101, +024916 (2020), 1906.11631. + diff --git a/idFLT4oBgHgl3EQfay-h/content/tmp_files/load_file.txt b/idFLT4oBgHgl3EQfay-h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9fb50e02deefbb909eacb259352bb11528e4878 --- /dev/null +++ b/idFLT4oBgHgl3EQfay-h/content/tmp_files/load_file.txt @@ -0,0 +1,811 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf,len=810 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='12076v1 [nucl-th] 28 Jan 2023 On the difference between signal and background of the chiral magnetic effect relative to spectator and participant planes in isobar collisions at √sNN = 200 GeV Bang-Xiang Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2 Xin-Li Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2 and Guo-Liang Ma1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' ∗ 1Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shanghai 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' China 2Shanghai Research Center for Theoretical Nuclear Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' NSFC and Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shanghai 200438,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' China The search for the chiral magnetic effect (CME) in relativistic heavy-ion collisions helps us un- derstand the CP symmetry breaking in strong interactions and the topological nature of the QCD vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Since the background and signal of the CME have different correlations with respect to the spectator and participant planes, a two-plane method has been proposed to extract the fraction of the CME signal inside the CME observable of ∆γ from the experimental measurements relative to the two planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Using a multiphase transport model with different strengths of the CME, we reex- amine the two-plane method in isobar collisions at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The ratios of the CME signal and background relative to the two different planes are found to be different, which is inconsistent with the assumptions made in the current experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The difference arises from the decorrelation of the chiral magnetic effect relative to spectator and participant planes, which originates from final state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Our finding suggests that the current experimental mea- surements may overestimate the fraction of the CME signal in the CME observable in relativistic heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' INTRODUCTION Relativistic heavy-ion collisions not only produce the quark-gluon plasma with strong collectivity [1–7], but also generate the strongest magnetic field known to man as the spectator protons from the target and the pro- jectile pass through each other almost at the speed of light [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This provides a unique experimental way to study the topological properties of the QCD vacuum and the anomalous chiral transport phenomena under the strong magnetic field [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A well-known detection method is by the chiral magnetic effect (CME), which leads to the phenomenon in which the electric charges in a system with a chiral imbalance are separated along the direction of the magnetic field [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The charge-dependent azimuthal correlation was first proposed as a possible observable to detect the CME [20], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=', γαβ = ⟨cos(φα + φβ − 2ΨRP)⟩, where φα(β) is the az- imuthal angle of a charged particle α(β), and ΨRP is the angle of the reaction plane, and ∆γ represents the dif- ference between opposite-charge and same-charge corre- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The first measurements of the charge-dependent azimuthal correlation from the STAR Collaboration [21– 24] at RHIC and the ALICE Collaboration [25] at the LHC are consistent with the expectations of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Unfortunately, some significant background effects con- tribute to the measured correlator due to the strong collective flow, especially from the elliptic flow [26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The recent RHIC-STAR measurements gave a strict con- straint that the CME fraction extracted in Au+Au 200 GeV is quite small, less than 10% [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' To distin- ∗Electronic address: glma@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='cn guish the possible CME signal from the dominant back- ground, many different methods or schemes have been proposed [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' One of the most important schemes is to use isobar collisions because the two isobar systems (96 44Ru+96 44 Ru and 96 40Zr+96 40 Zr) have a same nucleon num- ber but different proton numbers [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' It was expected that there could be a 20% difference in the CME signal with similar elliptic-flow induced backgrounds [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This prompted the RHIC-STAR Collaboration to con- duct the isobar collision experiment for 96 44Ru +96 44 Ru and 96 40Zr +96 40 Zr collisions at √sNN = 200 GeV [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Since the CME signal is positively correlated with the strength of the magnetic field, the ratio of the CME signal from Ru+Ru collisions to that Zr+Zr collisions was theo- retically predicted to be larger than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' However, the newly released experimental results by STAR observed the ratios for various CME observables are all smaller than unity [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' It indicates that the background effects dominate over the CME signal, and the CME signal is either absent or very small in isobar collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Interpret- ing the isobar experimental results has recently become a hot research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For example, different nuclear deformations or nuclear structures have been used to ex- plain the differences in multiplicity and harmonic flows between the two isobar systems [43–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Considering both the halo-type neutron skin structure and CME-like charge separation, we have demonstrated that it is diffi- cult for the CME observables to distinguish the presence or absence of the CME, if the CME strength is weak in isobar collisions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' At the same time, it was found that the STAR results favor a finite CME signal contribution of about (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='6)%, based on the recent finding in the AVFD model [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Many experimental observables have been used to detect the CME in Au+Au and isobar collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A two-plane measurement method that utilizes the charge- 2 dependent azimuthal correlations with respect to the spectator plane (SP) and participant plane (PP) has been proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [51, 52], because the background and the CME signal have different sensitivities or correlations to the two planes [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The STAR collaboration has used the method to detect the fraction of the CME signal in- side the inclusive ∆γ correlation in both Au+Au and iso- bar collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For Au + Au collision at √sNN = 200 GeV, the STAR results show that the fraction of the CME- induced charge separation is consistent with zero in pe- ripheral centrality bins, but possibly finite CME signals exist in mid-central centrality bins [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' And no obvious CME signal is observed in isobar collisions at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This method is believed to remove most of the ef- fect of collective flow in the background effect, but some background effect of non-flow needs further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Re- garding its methodology, it is assumed that the ratio a of elliptic flow with respect to different reaction planes is as same as the ratio b of CME signals with respect to different reaction planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' However, it is possible that the two ratios are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The assumption of equivalence in the current measurements is due to the lack of theo- retical studies on the ratio b [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This open question makes our current interpretation of the experimental re- sults somewhat questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' To solve the problem, it is essential to theoretically study the ratio between the CME signals with respect to different reaction planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This prompted us to calculate the ratios of a and b in isobar collisions at √sNN = 200 GeV by using a mul- tiphase transport (AMPT) model with an initial CME signal, in order to provide some theoretical support to the experimental measurement of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' II, we in- troduce the setting up of the AMPT model with an ini- tial CME signal for isobar collisions and our two-plane method to extract the fraction of the CME signal to the inclusive ∆γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' III, we present and compare our model results with the measurements from the STAR experiment, and discuss the impact of our findings on the interpretation of experimental data and the possible physics sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Finally, a summary is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' MODEL AND METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The AMPT model with initial CME signal The AMPT model is a hybrid transport model with four software packages to simulate four main stages in relativistic heavy-ion collisions [57–59]: (1) The HIJING model provides the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The transverse den- sity profile of the colliding nucleus is taken to as a Woods- Saxon distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The multiple scatterings among par- ticipant nucleons produce the spatial and momentum distributions of minijet partons and soft excited strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' With a string melting mechanism, the quark plasma is produced by melting the parent hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (2) Zhang’s parton cascade model is used to simulate the stage of parton cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The ZPC model describes parton in- teractions with two-body elastic scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The parton cross section is calculated by the leading-order pQCD for gluon-gluon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (3) A quark coalescence model combines two or three nearest partons into hadrons to mimic the hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (4) A relativistic transport (ART) model simulates the stage of hadronic rescatter- ings, including both resonance decays and all hadronic reactions for elastic and inelastic scatterings for baryon- baryon, baryon-meson, and meson-meson interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Many previous studies have shown that the AMPT model can describe well the various experimental observables in both large and small colliding systems at RHIC and the LHC [57–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' To simulate isobar collisions, the spatial distributions of nucleons inside 96 44Ru and 96 40Zr in the rest frame are sampled according to the Woods-Saxon form in spherical coordinates, ρ(r, θ) = ρ0/{1 + exp[(r − R(θ, φ))/a0]}, (1) R(θ, φ) = R0[1 + β2Y2,0(θ, φ) + β3Y3,0(θ, φ)], (2) in which ρ0 is the normal nuclear density, a0 is the sur- face diffuseness parameter, R0 is the nucleus radius, and β2 and β3 are the quadrupole and octupole deformities for the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In our previous work [49], we found that the halo-type neutron skin case is the best one among the eighteen cases which can describe the experimental ratios of charged-particle multiplicity distribution, the average number of charged particles, and elliptic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' There- fore, we also choose the halo-type neutron skin case in this study, where no deformation for both 96 44Ru and 96 40Zr (β2 = β3 = 0), R0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='085 and a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='523 for both protons and neutrons inside 96 44Ru, but R0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='021 and a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='523 for protons and R0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='021 and a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='592 for neutrons inside 96 40Zr due to the possible existence of neutron halo for 96 40Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A CME-like charge separation has been introduced into the initial partonic stage of the AMPT model in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' By tuning the percentage p, which defines what percentage of quarks join the charge separation, we can control the signal strength of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The definition of p is p = N +(−) ↑(↓) − N +(−) ↓(↑) N +(−) ↑(↓) + N +(−) ↓(↑) , (3) where N is the number of quarks of a given species (u or d or s), + and − denote positive and negative charges of quarks, and ↑ and ↓ represent the moving directions of quarks along the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Consid- ering that the magnetic fields for Ru + Ru and Zr + Zr collisions are different [53], we practically make the ini- tial charge separation based on the magnitude and di- rection of the magnetic field by calculating the mag- netic field for each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We denote p as the CME strength in Ru+Ru collisions, for example, p = 2% means pRu+Ru = 2% and pZr+Zr = 2%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='15 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='74% since we keep pRu+Ru/pZr+Zr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Spectator and participant planes In the two-plane method, the elliptic flow-driven back- ground is believed to be more relevant to the participant plane (PP), but the CME signal is more relevant to the spectator plane (SP) [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We reconstruct the spec- tator and participant planes by using the two following equations, respectively, ψSP = atan 2 �� r2 n sin (2φn ) � , � r2 n cos (2φn ) �� 2 , (4) ψPP = atan 2 �� r2 p sin (2φp ) � , � r2 p cos (2φp ) �� + π 2 , (5) where rn and φn are the displacement and azimuthal an- gle of spectator neutrons in the transverse plane, respec- tively, and rp and φp are the displacement and azimuthal angle of participating partons in the transverse plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' All spatial information about the displace- ment and azimuthal angle is obtained from the initial state of the AMPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' With the two different planes, the corresponding elliptic flow, v2{SP} and v2{PP}, can be calculated as follows, respectively, v2{SP} = ⟨cos 2 (φ − ψSP)⟩ , (6) v2{PP} = ⟨cos 2 (φ − ψPP)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (7) The upper panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 1 shows the centrality depen- dence of v2{PP} and v2{SP} of charged hadrons with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='2 < pT < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0 GeV/c and |η| < 1 from the AMPT model with different strengths of the CME in Ru + Ru and Zr + Zr collisions at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We can see that v2{PP} is greater than v2{SP} for all cases, since elliptic flow is more correlated to the participant plane than the spectator plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' On the other hand, for the cen- tral and mid-central centrality of 0−50%, both v2{PP} and v2{SP} decrease slightly with increasing CME sig- nal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For the peripheral centrality of 50−80%, both v2{PP} and v2{SP} are insensitive to the strength of CME signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 1 shows the v2{PP} and v2{SP} ratios of Ru + Ru to Zr + Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The ratio of v2{SP} is greater than that of v2{PP}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The trend in the ratio results has been found to be caused by the nuclear structures of Ru and Zr [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Two-plane method to extract fcme In this subsection, the two-plane method for detecting the CME signal is introduced first, and then how to im- prove the two-plane method using the AMPT model is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The experimentally measured CME observ- able ∆γ includes the CME signal and the background effect mainly arising from the contributions of elliptical flow and non-flow effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Therefore, the experimentally 0 10 20 30 40 50 60 70 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='06 v 2 Ru + Ru Zr + Zr v 2 {PP} p = 0 p = 2% p = 5% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 10% Ru + Ru Zr + Zr v 2 {SP} p = 0 p = 2% p = 5% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 10% Ru + Ru / Zr + Zr Centrality (%) Ratio FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 1: (Color online) Upper panel: AMPT results on central- ity dependences of elliptic flow v2{PP} (solid symbols) and v2{SP} (open symbols) in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lower panel: AMPT results on v2{PP} and v2{SP} ra- tios of Ru + Ru collisions to Zr + Zr collisions as a function of centrality bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' measured CME observable ∆γ with respect to different planes can be divided into two parts as follows, ∆γ{ψ} = ∆γBkg{ψ} + ∆γCME{ψ} (8) where ψ can represent ψPP or ψSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The ratios of the el- liptic flow and the measured observable with two different planes can be defined by a and A as follows, respectively, a = v2{SP}/v2{PP}, (9) A = ∆γ{SP}/∆γ{PP}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (10) Since the CME signal can not be measured directly in an experiment, people usually assume that the ratio of the CME signals of different reaction planes is the inverse of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Thus, the following relation can be obtained, ∆γ{SP} = a∆γBkg{PP} + ∆γCME{PP}/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (11) 4 After a simple transformation, the percentage of the CME signal inside the measured CME observable (fCME) is obtained by, fCME = ∆γCME{PP} ∆γ{PP} = A/a − 1 1/a2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (12) Equation (12) indicates that the percentage of the CME signal inside the measured CME observable can be ob- tained by measuring A and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' However, it has been pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [55, 56] that the ratio of the CME signal and the inverse ratio of the elliptic flow could be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Let us assume the following relation holds, ∆γCME{PP} = b∆γCME{SP}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (13) where b represents the ratio of the CME signals with re- spect to different planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' By replacing the corresponding part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (11), a more realistic relation can be obtained as shown below, ∆γ{SP} = a∆γBkg{PP} + ∆γCME{PP}/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (14) Thus, the more realistic percentage of the CME signal in- side the measured CME observable should be calculated by, fCME{b} = ∆γCME{PP} ∆γ{PP} = A/a − 1 1/ab − 1, (15) after considering b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The left task is calculating b, which can be done theo- retically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In the AMPT model, the CME signal is simu- lated by affecting a certain proportion of partons at the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The proportion of CME particles is denoted by p in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Within our theoretical framework, the value of b can be obtained by using the following equa- tion, b = ∆γ{PP}(p ̸= 0) − ∆γ{PP}(p = 0) ∆γ{SP}(p ̸= 0) − ∆γ{SP}(p = 0) , (16) where the numerator and denominator are the CME sig- nal inside the measured CME observable with respect to participant and spectator planes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS In this section, we present the AMPT results on charge-dependent azimuthal correlations for charged par- ticles relative to spectator and participant planes, and compare them with the results of the STAR isobar ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Our kinetic cuts are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='2 < pT < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0 GeV/c and |η| < 1, as same as the STAR experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The upper panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2 shows the centrality depen- dence of ∆γ{PP} and ∆γ{SP} from the AMPT model with different strengths of the CME in Ru + Ru and Zr + Zr collisions at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Compared to 0 10 20 30 40 50 60 70 80 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0020 Ru + Ru Zr + Zr {PP} p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 2% p = 0 Ru + Ru Zr + Zr {SP} p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 2% p = 0 STAR TPC STAR ZDC {PP} {SP} p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 2% p = 0 Centrality (%) ( /v 2 ) Ru + Ru / ( /v 2 ) Zr + Zr FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2: (Color online) Upper panel: The centrality depen- dences of ∆γ{PP} (solid symbols) and ∆γ{SP} (open sym- bols) in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME, in comparison with the STAR data [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lower panel: The centrality de- pendences of ∆γ/v2 ratios of Ru + Ru collisions to Zr + Zr collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' the STAR data, a small percentage of CME signal is pre- ferred, which is also consistent with our recent study [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The ∆γ{SP} is greater than ∆γ{PP}, which indicates that ∆γ{SP} is more sensitive to the CME than ∆γ{PP} since the spectator plane is more strongly correlated to the direction of the magnetic field than the participant plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2 shows the ∆γ/v2 ratios of Ru + Ru to Zr + Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Within our statistical errors, the ratios are consistent with unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Figure 3 shows the centrality dependences of A = ∆γ{SP}/∆γ{PP} and a = v2{SP}/v2{PP} from the AMPT model with different strengths of the CME, com- pared with the STAR experimental data [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' As the CME strength in the AMPT model increases, the value of a hardly changes and is always less than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' It indi- cates that the CME has the same impact on v2 with re- spect to different planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For p=0 and p=2%, the values of A are almost identical and smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' How- ever, in the other cases, it is greater than unity and in- creases with increasing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This indicates that the CME has different effects on ∆γ relative to different planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Figure 4 shows the A/a ratio as a function of centrality given by the AMPT model for different strengths of the 5 0 10 20 30 40 50 60 70 80 -2 0 2 4 6 8 A or a Centrality (%) A = {SP} / {PP} a = v 2 {SP} / v 2 {PP} Ru + Ru Zr + Zr a p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 2% p = 0 STAR a Ru + Ru Zr + Zr A p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 2% p = 0 STAR A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 3: (Color online) The centrality dependences of A and a in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME, in comparison with the STAR data [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The solid and open symbols rep- resent the results for A and a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 0 1 2 3 4 5 A/a Centrality (%) Ru + Ru Zr + Zr p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 2% p = 0 STAR full-event FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 4: (Color online) The centrality dependences of A/a in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME, in comparison with the STAR data [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (12) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (15), the value of A/a > 1 corresponds to the presence of a CME signal inside the CME observable ∆γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Let us focus on mid-central collisions where the CME effect is expected to be more likely to be measured than at the other centrality bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For the 20-50% centrality bin, we can see A/a > 1 clearly except the cases of p=0 and p=2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Moreover, we observe that the value of A/a increases as the CME strength increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' It indicates that A/a can reflect the strength of the CME signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We calculate b using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (16) and compare it with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Figure 5 shows the centrality dependence of a and b from the AMPT model with different strengths of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 -2 -1 0 1 2 a or b Centrality (%) a = v 2 {SP}/v 2 {PP} b = ( {PP}(p 0)- {PP}(p =0)) / ( {SP}(p 0)- {SP}(p =0)) Ru + Ru Zr + Zr b p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% Ru + Ru Zr + Zr a p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% p = 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 5: (Color online) The centrality dependences of a and b in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The open and solid symbols represent the results for a and b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 -2 -1 0 1 2 b/a Centrality (%) Ru + Ru Zr + Zr p = 10% p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p = 5% FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 6: (Color online) The centrality dependences of b/a in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We find that the values of a and b are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The value of b is smaller than the value of a at the 20-50% centrality bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The value of a is almost independent of the CME strength, as same as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Considering statis- tical errors, the value of b does not vary significantly with the CME strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Note that the case of the 2% CME strength is not shown due to huge statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Figure 6 further shows the centrality dependence of the b/a ratio from the AMPT model with different strengths of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For the 20-50% centrality bin in isobar colli- sions at √sNN = 200 GeV, such an approximate relation that b = a/2 can be obtained by a constant function fit- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' It implies that the relative ratio of the CME signals with respect to different planes is not equal to the inverse 6 -2 0 2 4 Ru + Ru Zr + Zr CME p=10% p=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p=5% p=2% p=0 STAR TPC full-event CME Ru + Ru Zr + Zr CME {b} p=10% p=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p=5% p=2% 0 10 20 30 40 50 60 70 80 -1 1 3 Ru + Ru Zr + Zr p=10% p=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5% p=5% p=2% Ratio Centrality (%) CME {b}/ CME FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 7: (Color online) Upper panel: The centrality depen- dences of fCME{b} and fCME in isobar collisions at √sNN = 200 GeV from the AMPT model with different strengths of the CME, in comparison with the STAR data [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The solid and open symbols represent the results for fCME{b} and fCME, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lower panel: The centrality dependences of the ratio of fCME{b} to fCME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' of the elliptic flow ratio with respect to different planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Next, we will show that this finding has important im- plications for extracting the fraction of the CME signal inside the ∆γ observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The upper panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 7 shows two kinds of fCME as a function of centrality from the AMPT model with different strengths of the CME, where the solid and open symbols represent fCME calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (12) and fCME{b} calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Com- pared to the STAR experimental data [42], the results of fCME from p=0% or p=2% are favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For the 20-50% centrality bin, when p is not equal to zero, fCME{b} is found to be less than fCME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 7 shows the centrality dependence of the ratio of fCME{b} to fCME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' For the 20-50% centrality bin, the ratio is less than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' It indicates that the fraction of the CME signal inside the ∆γ observable will be overestimated if assuming b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 0 1 2 p = 5% p = 10% CME CME {b} CME {p} CME Centrality (%) Ru + Ru FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 8: (Color online) The centrality dependences of fCME, fCME{b} and fCME{p} from three different methods in Ru + Ru collisions at √sNN = 200 GeV from the AMPT model with two different strengths of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' To discern which of these two above fCME is closer to the real situation, we can theoretically use another method to obtain the true fCME, namely fCME{p}, as defined as follows, fCME{p} = ∆γCME{PP}(p ̸= 0) ∆γ{PP}(p ̸= 0) , (17) where ∆γCME{PP}(p ̸= 0) = ∆γ{PP}(p ̸= 0) −∆γ{PP}(p = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (18) The observables of ∆γ{PP}(p = 0) and ∆γ{PP}(p ̸= 0) can be obtained from the AMPT model without the CME and with different strengths of the CME, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 8, compared to the ture fCME, we find that fCME{b} is closer to and consistent with fCME{p} than fCME for the 10−50% centrality bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='This suggests that it is neces- sary to consider b to obtain a more reliable fCME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' From the above results, it is clear that b = a/2 signif- icantly impacts the final result of fCME in isobar colli- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' To understand where this relation comes from, we calculate a and b for different stages in Ru + Ru collision at √sNN = 200 GeV in the AMPT model with the CME strength of p = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We focus on four evolution stages of heavy-ion collisions: the initial stage, after parton cas- cade, after coalescence, and after hadron rescatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 9, we can find that the value of a remains constant for the last three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Note that we do not show a for the initial stage, since the elliptic flow is initially zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' However, the value of b is always smaller than that of a and decreases stage by stage for the 10-50% centrality bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' In the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 9, the ratio of b/a also decreases with the stage evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='8 Ru + Ru p = 10% a b initial stage after parton cascade after coalescence after hadronic rescatterings a or b a = v 2 {SP}/v 2 {PP} b = ( {PP}(p 0)- {PP}(p =0)) / ( {SP}(p 0)- {SP}(p =0)) 0 10 20 30 40 50 60 70 80 0 1 after parton cascade after coalescence after hadronic rescatterings Centrality (%) b / a FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 9: (Color online) Upper panel: The centrality depen- dences of a and b in Ru + Ru collisions at √sNN = 200 GeV for four different stages from the AMPT model with the CME strength of p = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The open and solid symbols represent the results for a and b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lower panel: The cen- trality dependences of the ratio of b/a for the different stages in Ru + Ru collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The data points are shifted along the x axis for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' main reason is due to the change in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The decreasing of b indicates the correlation between the CME signals rel- ative to different planes becomes less and less correlated, which can be understood as a consequence of the decor- relation resulting from final state interactions during the evolution of heavy-ion collisions [49, 67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' SUMMARY Using a multiphase transport model with different strengths of the CME, we reexamine the proposed two- plane method to determine the fraction of the CME sig- nal inside the CME observable of ∆γ in isobar collisions at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We first calculate the elliptic flow v2 and the CME observable of ∆γ with respect to the spectator plane and participant plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' The ratio b of the CME signal relative to the two different planes is found to be different from the ratio a of background relative to the two different planes in isobar collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' If the differ- ence between a and b is taken into account, we demon- strate that it will lead to a smaller CME fraction than the constraint obtained in current experimental way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We theoretically observe a decrease in the value of b during the stage evolution in the AMPT model, which indicates the decorrelation of the chiral magnetic effect relative to spectator and participant planes is caused by final state interactions in isobar collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Since a and b were as- sumed to be equal in the current experimental study, the fraction of the CME signal could be overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' We hope that our study will provide a theoretical reference for future accurate measurements of the fraction of the chiral magnetic effect inside the experimental observable in relativistic heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Jie Zhao for the helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' This work is supported by the National Natural Sci- ence Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='12147101, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 11890714, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 11835002, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 11961131011, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 11421505, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 12105054, the National Key Research and Development Program of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2022YFA1604900, the Strategic Priority Research Pro- gram of Chinese Academy of Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' XDB34030000, and the Guangdong Major Project of Basic and Applied Basic Research under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 2020B0301030008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Kolb, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sollfrank, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Heinz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 62, 054909 (2000), hep-ph/0006129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Teaney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lauret, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shuryak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 86, 4783 (2001), nucl-th/0011058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Yan, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 42, 042001 (2018), 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='04580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shen and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Yan, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 31, 122 (2020), 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='12377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhou, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Gajdosova, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 28, 99 (2017), 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lan and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 33, 21 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Song, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 38, 081201 (2021), 2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='13250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [8] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Skokov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Illarionov, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Toneev, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A 24, 5925 (2009), 0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Bzdak and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Skokov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' B 710, 171 (2012), 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Deng and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 85, 044907 (2012), 1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [11] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 97, 024910 (2018), 1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='05962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' B 792, 413 (2019), 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='04156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 8 [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Kharzeev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Liao, Nature Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 3, 55 (2021), 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='06623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [14] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 79, 076302 (2016), 1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='04073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Hattori and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 28, 26 (2017), 1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Gao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Pu, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 31, 90 (2020), 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='10432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Kharzeev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' B 633, 260 (2006), hep- ph/0406125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Kharzeev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' McLerran, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Warringa, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A 803, 227 (2008), 0711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Fukushima, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Kharzeev, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Warringa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' D 78, 074033 (2008), 0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='3382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Voloshin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 70, 057901 (2004), hep- ph/0406311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Abelev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 103, 251601 (2009), 0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Abelev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 81, 054908 (2010), 0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Adamczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 88, 064911 (2013), 1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='3802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [24] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Adamczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 113, 052302 (2014), 1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [25] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Abelev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (ALICE), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 110, 012301 (2013), 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='0900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Bzdak, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Koch, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Liao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 83, 014905 (2011), 1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='4919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Liao, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Koch, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Bzdak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 82, 054902 (2010), 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='5380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Schlichting and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Pratt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 83, 014913 (2011), 1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='4283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 81, 064902 (2010), 0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [30] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (2022), 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='15446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 29, 179 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao (STAR), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A 1005, 121766 (2021), 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='09410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 107, 200 (2019), 1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='11413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, in 29th International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (2022), 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='10044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [35] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 70, 293 (2020), 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='10397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Voloshin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 105, 172301 (2010), 1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [37] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 94, 041901 (2016), 1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='04697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [38] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Koch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Schlichting, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Skokov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sorensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Thomas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Voloshin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Yee, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 41, 072001 (2017), 1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [39] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 97, 044901 (2018), 1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='02292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Choudhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=', Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 46, 014101 (2022), 2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='06044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Adam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 32, 48 (2021), 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Abdallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 105, 014901 (2022), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [43] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shen, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' B 819, 136453 (2021), 2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='05595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [44] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Chen, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang (2021), 2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='14812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [45] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 128, 022301 (2022), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='01631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Jia and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhang (2021), 2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='15559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [47] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 105, 014905 (2022), 2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='08768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 105, 044905 (2022), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [49] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 106, 034909 (2022), 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='15214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [50] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Kharzeev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Liao, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 106, L051903 (2022), 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='00120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shen, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 42, 084103 (2018), 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='07265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shen, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 121, 022301 (2018), 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='03086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [53] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 99, 034903 (2019), 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='04151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [54] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Abdallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 128, 092301 (2022), 2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='09243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Jiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lilleskov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Liao, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 394, 50 (2018), 1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='02496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [56] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Xu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 105, 024913 (2022), 2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='15595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [57] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ko, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Pal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 72, 064901 (2005), nucl-th/0411110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [58] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 93, 054911 (2016), 1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='08160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [59] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zheng, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' 32, 113 (2021), 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='02989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [60] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 90, 014904 (2014), 1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='6321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [61] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Orjuela Koop, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Adare, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' McGlinchey, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Nagle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 92, 054903 (2015), 1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='06880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [62] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Bzdak, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' A 956, 745 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [63] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' He and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 96, 014910 (2017), 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='02673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [64] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 45, 074110 (2021), 2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='09264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [65] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Chen and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 106, 014907 (2022), 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='11736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [66] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Wang, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma (2022), 2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='17125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [67] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Zhang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' B 700, 39 (2011), 1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' [68] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Nie, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content=' C 101, 024916 (2020), 1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} +page_content='11631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFLT4oBgHgl3EQfay-h/content/2301.12076v1.pdf'} diff --git a/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf b/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6918a288811d19f414d9afc2e9d5dce2360b2ca5 --- /dev/null +++ b/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:772531ecc24c22264f06d83e732d737ef65abec96bfd56262bb1f63967c7ab6d +size 20578409 diff --git a/j9E0T4oBgHgl3EQf7QLf/vector_store/index.faiss b/j9E0T4oBgHgl3EQf7QLf/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2d678eb4c7fbaa213491c361f79e5fbfbc6b9489 --- /dev/null +++ b/j9E0T4oBgHgl3EQf7QLf/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8ebf63b787b2700ee274eb4c34b54d65ba12f252b58cd579b8c3d2adb3492984 +size 10223661 diff --git a/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf b/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bbd9d7969db77ff1c3e26d73b2d1d5c9938f2b91 --- /dev/null +++ b/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f9bef2a7e14601c35e71f3508756f0484a51be4ab8797664b47009e28b4b9d5b +size 3982432 diff --git a/jdAzT4oBgHgl3EQf4_6S/vector_store/index.faiss b/jdAzT4oBgHgl3EQf4_6S/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ff8c7da2ebcd51c18ca100173c3eeeeb335500ff --- /dev/null +++ b/jdAzT4oBgHgl3EQf4_6S/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:571cd7de8b631db21060fee55afcd6b2c8b549a4e24fee95cb93cfe30d4b6238 +size 3080237 diff --git a/kb_46/content/kb_46.pdf b/kb_46/content/kb_46.pdf new file mode 100644 index 0000000000000000000000000000000000000000..44a3123924d54fbe69ee4a1c039e2843b5a2fd06 --- /dev/null +++ b/kb_46/content/kb_46.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c0e3e00b4008656bd4ba06927aa5197dc6679f50053d0ca40d181dec44246a8 +size 4410113 diff --git a/ldE4T4oBgHgl3EQfUAxn/vector_store/index.pkl b/ldE4T4oBgHgl3EQfUAxn/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0eaf504d31c9750daec4213e2cfc77e1293ac5b2 --- /dev/null +++ b/ldE4T4oBgHgl3EQfUAxn/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ac62d4375efc735e2d43265883cb97627bb4869b395aaa8caca4beea6f957140 +size 102235 diff --git a/m9AyT4oBgHgl3EQfyvnQ/content/2301.00691v1.pdf b/m9AyT4oBgHgl3EQfyvnQ/content/2301.00691v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5e996681583c7efcf601134c8f408558ecf7294d --- /dev/null +++ b/m9AyT4oBgHgl3EQfyvnQ/content/2301.00691v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:825e706d8a595607fc16e6531b3ac4828e42d1299fbbfc89aae480b17f880843 +size 1901855 diff --git a/m9AyT4oBgHgl3EQfyvnQ/vector_store/index.faiss b/m9AyT4oBgHgl3EQfyvnQ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2ff360f01b8832515e33cd7a84f317ef1e6d47c2 --- /dev/null +++ b/m9AyT4oBgHgl3EQfyvnQ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:482f9dc54eb673d669a1efc71caf172debc3b32ec4fb4fd9736eb16de8921862 +size 1376301 diff --git a/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf b/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0477c656b355c2dc166b6d7192c0a3681029ee7 --- /dev/null +++ b/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:072a572173dd247aa8cbd6d25ff92a92cfa6cebb78189132844f37f3aa611575 +size 3969214 diff --git a/odAyT4oBgHgl3EQfzPlC/content/2301.00697v1.pdf b/odAyT4oBgHgl3EQfzPlC/content/2301.00697v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..81f80f03736c3be4a667e9ac36f8fc93a65b10d5 --- /dev/null +++ b/odAyT4oBgHgl3EQfzPlC/content/2301.00697v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:17edbc3486ffc88aea5362a7d43c7b87fcd98a716a7a1ac0373bd74941adbca6 +size 643603 diff --git a/odAyT4oBgHgl3EQfzPlC/vector_store/index.faiss b/odAyT4oBgHgl3EQfzPlC/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6f26d749ec4c996a757b64f7b19deb74dfc53721 --- /dev/null +++ b/odAyT4oBgHgl3EQfzPlC/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2a7a3131386427dad09995875e5fce6198ea8b91eb5e81e7896b839f007dfed3 +size 3801133 diff --git a/odAyT4oBgHgl3EQfzPlC/vector_store/index.pkl b/odAyT4oBgHgl3EQfzPlC/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..09de4f275269a229a82f89f89ea8cd23efc1855b --- /dev/null +++ b/odAyT4oBgHgl3EQfzPlC/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e90f836c434769870a885e2332a83459cb0aadd440f529ff39263c9593b3c427 +size 154436 diff --git a/p9E2T4oBgHgl3EQf0ggt/vector_store/index.pkl b/p9E2T4oBgHgl3EQf0ggt/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3055365a132fe4955a9041336ce5ad56906dcdd4 --- /dev/null +++ b/p9E2T4oBgHgl3EQf0ggt/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8e742fe4c67dbfd680c4e45cc611543e55192cc4b55274ffb8ee22df2c0f2051 +size 1210138 diff --git a/q9FPT4oBgHgl3EQf9TVT/content/tmp_files/2301.13211v1.pdf.txt b/q9FPT4oBgHgl3EQf9TVT/content/tmp_files/2301.13211v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3a40ce4d13667433f423653e0b3bcdeba484bad --- /dev/null +++ b/q9FPT4oBgHgl3EQf9TVT/content/tmp_files/2301.13211v1.pdf.txt @@ -0,0 +1,3045 @@ +Prepared for submission to JHEP +Earth mover’s distance as a measure of CP violation +Adam Davis,a Tony Menzo,b Ahmed Youssef,b Jure Zupan,b +aSchool of Physics and Astronomy, University of Manchester, M13 9PL, Manchester, UK +bDepartment of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA +E-mail: adam.davis@manchester.ac.uk, menzoad@mail.uc.edu, +youssead@ucmail.uc.edu, zupanje@ucmail.uc.edu +Abstract: We introduce a new unbinned two sample test statistic sensitive to CP viola- +tion utilizing the optimal transport plan associated with the Wasserstein (earth mover’s) +distance. The efficacy of the test statistic is shown via two examples of CP asymmetric +distributions with varying sample sizes: the Dalitz distributions of B0 → K+π−π0 and of +D0 → π+π−π0 decays. The windowed version of the Wasserstein distance test statistic is +shown to have comparable sensitivity to CP violation as the commonly used energy test +statistic, but also retains information about the localized distributions of CP asymmetry +over the Dalitz plot. For large statistic datasets we introduce two modified Wasserstein +distance based test statistics – the binned and the sliced Wasserstein distance statistics, +which show comparable sensitivity to CP violation, but improved computing time and +memory scalings. Finally, general extensions and applications of the introduced statistics +are discussed. +arXiv:2301.13211v1 [hep-ph] 30 Jan 2023 + +Contents +1 +Introduction +1 +2 +Earth mover’s distance as a measure of CPV +3 +3 +Application to three body B decays +5 +3.1 +Testing for bias in the permutation method +6 +3.2 +Tracing CP violating phase space regions using EMD +7 +3.3 +The windowed EMD +11 +4 +Application to three body D decays +15 +4.1 +Binned Wasserstein test +16 +4.2 +Sliced Wasserstein test +19 +5 +Conclusions +22 +A Public code EMD4CPV +25 +B The optimal transport problem +26 +C EMD analysis of Gaussian distributions +27 +D Details on EMD–test for three body decays +31 +D.1 The p−value error analysis +31 +D.2 Optimizing the q value +33 +D.3 Further results for q = 0.1, 1, 10 +34 +E Energy test +36 +1 +Introduction +The Wasserstein distance or earth mover’s distance (EMD) is a measure of similarity be- +tween two probability distributions, see, e.g., [1] as well as App. B. The value of the EMD +can be visualized as the work required to transport and reshape dirt (weighted samples) in +the form of one distribution into the form of a second distribution. Similar distributions +result in smaller values (≈ zero) of the EMD while dissimilar distributions result in larger +values. The EMD is thus sensitive to density asymmetries between samples and therefore +well suited to be used as a test statistic that quantifies the amount of CP violation (CPV) +in a physical system. +– 1 – + +Taking as an example the B0 decays to a final state f, the direct CP asymmetry Af +is defined as +Af = Br( ¯B0 → f) − Br(B0 → ¯f) +Br( ¯B0 → f) + Br(B0 → ¯f), +(1.1) +where ¯f = CP(f) is the final state CP conjugated to f. For two body B0 decays such as +B0 → K+π−, the direct CPV is fully characterized by Af. In the rest frame of the parent +particle, the two final state particles are back to back and there is no dependence of the +decay rate on their emission angle. Direct CPV is then simply given by the difference of +observed B0 → f and ¯B0 → ¯f decays as in Eq. (1.1). This is not the case, however, for +three-body decays such as, for instance, B0 → K+π−π0 and its CP conjugate mode ¯B0 → +K−π+π0 [2]. In addition to the integrated CPV quantity, Af, there is a continuous set of +CP violating observables, namely the phase space dependent differential CP asymmetries +ACP(s12, s13) = +�dΓ( ¯B0 → ¯f) +dp.s. +− dΓ(B0 → f) +dp.s. +���dΓ( ¯B0 → ¯f) +dp.s. ++ dΓ(B0 → f) +dp.s. +� +, (1.2) +where s12, s13 are the two Dalitz plot variables. To measure ACP one can bin the Dalitz +plot in large enough bins such that they contain reasonably large numbers of events, say +ni, ¯ni ∼ O(20), and define Af,i, Eq. (1.1), for each bin. +In this way one could probe +experimentally, if CP violation is present in the Dalitz plot distributions. +Such an approach is not optimal, however, since the measurements depend on the +choice of the binning. If the primary goal is to test for the presence of phase space dependent +CPV in the Dalitz plot distributions, not just in global Af, two tests were put forward that +improve on the binning method, the SCP test (or the Miranda method) [3, 4] and the +energy test [5–8], both of which have some drawbacks. +The SCP test still relies on a +binning procedure that, like ACP, leads to some loss of sensitivity to CPV and the energy +test, while being quite sensitive to the presence of CPV in the Dalitz plot distributions, is +harder to interpret in terms of the underlying physics. +In this paper we propose an alternative approach – the use of the Wasserstein distance, +or EMD, as a measure of CPV in the Dalitz plot distributions. As we show below, the EMD +based statistic combines the high sensitivity to CPV with easier interpretability, since it +retains information about which part of the Dalitz plot the CPV originates from. The use +of EMD in measuring CPV is reminiscent but distinct from the use of EMD to quantify +the similarities between different LHC events, advocated in [9–12] (see also the related +results in Refs. [13–15]). In particular, the optimal EMD based statistic for CPV involves +reweighting (or filtering) of individual datapoint contributions to the EMD as we discuss +in more detail in Sec. 3.3. +The paper is organized as follows. In Sec. 2 we review the Wasserstein distance and +introduce the relevant notation for its application to three body B and D decays. In Sec. 3 +we analyze three body B0 → K+π−π0 decays and show that the Wasserstein distance +is a sensitive probe of CP violation and introduce an optimized windowed Wasserstein +distance statistic. In Sec. 4 we introduce two further Wasserstein distance based statistics, +the binned Wasserstein distance and the sliced Wasserstein distance, which have improved +computing complexity scalings and may be preferred when dealing with large datasets such +– 2 – + +as the three body D decay data samples. We draw conclusions in Sec. 5, while appendices +contain details about the public code EMD4CPV (App. A), on the computation complexity +of the optimal transport problem (App. B), further examples for EMD using Gaussian +distributions (App. C), further results for probing B → Kππ Dalitz plot CP asymmetries +using Wasserstein distance based statistics (App. D), and a review of the energy test +(App. E). +2 +Earth mover’s distance as a measure of CPV +The Wasserstein distance, Wq(E, ¯E), between the distributions of events, E, in B0 → +K+π−π0, and the distribution ¯E of ¯B0 → K−π+π0 decays is given by, see, e.g., [9, 16–18], +Wq(E, ¯E) = +� +min +{fij≥0} +N +� +i=1 +¯ +N +� +j=1 +fij +� ˆdij +�q +�1/q +, +(2.1) +where q ∈ (0, ∞), with q = 1 defining the EMD.1 The minimization is over the weights +N +� +i=1 +fij = 1 +¯N , +¯ +N +� +j=1 +fij = 1 +N , +N, ¯ +N +� +i,j=1 +fij = 1, +(2.2) +where N( ¯N) are the number of events in sample E( ¯E), and ˆdij is the distance between +the two events, i in E, and j in ¯E. The interpretation of Wq(E, ¯E) is the cost incurred by +moving in an optimal way the probability distribution corresponding to events in E into +the probability distribution of event ¯E, where the penalty is the distance ˆdij between the +events. +Assuming that N = ¯N, so that that there is no integrated CP asymmetry, and that E +and ¯E come from the same distribution (i.e. no CPV in distributions), then Wq(E, ¯E) → 0 +for large N = ¯N. In contrast, if E and ¯E differ (there is CPV), then Wq(E, ¯E) will tend +to a nonzero value. For d−dimensional final phase space the parametric upper bound is +⟨Wp(E, ¯E)⟩ ≲ CN−1/d [20], with C a constant that does not depend on N.2 For the Dalitz +plot we have d = 2 since it is fully described by two Dalitz variables, s12, s13, and thus +⟨Wp(E, ¯E)⟩ ∝ 1/ +√ +N, i.e., it scales in the same way as the variance of the global direct CP +asymmetry δAf ∝ 1/ +√ +N. Since we are mainly interested in CPV in distributions, we will +assume for simplicity that N = ¯N in the rest of the manuscript. However, the analyses +we present below extend trivially to the N ̸= ¯N case, with Wq still probing the CPV in +distributions and Af the integrated CPV. +1In most works on the optimal transport q is restricted to the convex cost functions, q ∈ [1, ∞), such +that its gradient is well defined everywhere, also at the ˆdij = 0 point. An extension to the concave case, +q ∈ (0, 1), requires an introduction of an approximate gradient, however, a unique optimal transport still +exists, see the discussion in chapter 3.3.2 of Ref. [17]. The network simplex algorithm as implemented in the +Wasserstein Python library [9, 19] can then be used without change to solve the optimal transport problem, +in the same way as for q ≥ 1. +2Note that for decays that are dominated by intermediate resonances the effective dimensionality is +lower than the full dimensionality of the phase space. That is for a multibody decays where at most two +resonances overlap we expect the same scaling as for the Dalitz plot ⟨Wp(E, ¯E)⟩ ≲ CN −1/2. +– 3 – + +−50 +0 +50 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Counts +N(¯µ = −20, σ = 10) +N(µ = 20, σ = 10) +E +E +0 +20 +40 +60 +Wq +0 +100 +200 +300 +400 +Counts +q = 1 +N=10, ∆µ =0.0 +N=10, ∆µ =40.0 +Figure 1. The nonzero displacement ∆µ of the two Gaussian distributions (left) can be probed by +using Wq, q = 1, as the test statistic (right), see main text for details. +For the 3D Dalitz plot we use the definition of the (dimensionless) distance ˆdij that is +symmetric in the Dalitz variables, s12, s13, s23, +ˆdij +��� +Dalitz = 1 +m2 +���s12(i) − ¯s12(j) +��r + +��s13(i) − ¯s13(j) +��r + +��s23(i) − ¯s23(j) +��r�1/r +, +(2.3) +where, for example in the B0 → K+π−π0 system, m = mB, and +s12 = (pK+ + pπ−)2, +s13 = (pK+ + pπ0)2, +s23 = (pπ− + pπ0)2, +(2.4) +¯s12 = (pK− + pπ+)2, +¯s13 = (pK− + pπ0)2, +¯s23 = (pπ+ + pπ0)2, +(2.5) +parametrize the B0 → K+π−π0 Dalitz plot and the CP conjugate variables in ¯B0 → +K−π+π0 Dalitz plot, respectively. +The normalization prefactor 1/m2 in Eq. (2.3) was +chosen such that ˆdij < 1. We use the Euclidean distance, i.e., r = 2, in the remainder of +the paper. Other r–values were investigated but no significant changes to the sensitivity +of CP violation were found. +Before discussing the more complicated case of B and D decays, let us first briefly con- +sider a simpler toy example of two displaced Gaussian distributions, G(x) = N(x|∆µ/2, σ) +and ¯G(x) = N(x| − ∆µ/2, σ), i.e., two Gaussian distributions with equal widths, σ, but +with their centers at µ = ∆µ/2 and ¯µ = −∆µ/2 and thus displaced by ∆µ. In this toy +example the question about CPV in multibody B decays is replaced with a test whether +or not ∆µ ̸= 0. Drawing N = 10 events E from G, as well as ¯N = 10 events ¯E from ¯G, +and taking ˆdij in Eq. (2.1) to be the Euclidean distance in 1D, gives a W1 that is clus- +tered around ⟨W1⟩ ≃ ∆µ, see the grey distribution in Fig. 1 (right). This is appreciably +larger than the distribution of W1 values for ∆µ = 0 (blue), even for relatively small event +samples. In App. C we show more illustrations of how the W1 probes a difference between +distributions, including an example of displaced 2D Gaussian distributions. In particular, +we show numerically that W1 can be used as a statistic, and that the CL intervals obtained +from a known ∆µ = 0 probability distribution for W1 coincide with the expected exclusion +intervals from negative log likelihood for ∆µ. +– 4 – + +0 +10 +20 +m2(K−π+) +0 +5 +10 +15 +20 +25 +m2(K−π0) +B0 → K−π+π0 +0 +1 +2 +3 +m2(π−π0) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +m2(π+π0) +D0 → π−π+π0 +Figure 2. The 2–dimensional B0 → K+π−π0 (left) and D0 → π−π+π0 (right) Dalitz plots and +their respective 1–dimensional histograms along the borders for 106 events. +3 +Application to three body B decays +As the first realistic example of using the Wasserstein distance to test for CP violation +we use the B0 → K+π−π0 and the CP conjugate ¯B0 → K−π+π0 decays. The events +are generated from the amplitude model by BaBar [2] implemented in the AmpGen [21] +framework. We create two data samples: the CP conserving (CPC) and the CP violating +datasets. For the CPC datasets we use the central values of amplitudes and phases in the +B0 BaBar isobar model [2] for both B0 and ¯B0 decays. For the CPV datasets, on the +other hand, the amplitudes and phases for B0 and ¯B0 isobar models differ and are set to +the central values of the measurements in Ref. [2]. The B − ¯B meson mixing is ignored in +the generation of the samples. The resulting B0 → K+π−π0 Dalitz plot with 106 events is +shown in Figure 2 (left). +For three-body B decays we highlight the use of Wq( ˆdij) on the low statistic datasets +containing N = 103 events in each of the samples, the B and ¯B decays (N = ¯N). This +choice was made to roughly match the reported experimental sensitivity [22]. The im- +plementation and computation of the Wasserstein distance is done in two steps: first, the +distances ˆdij, Eq. (2.3), are computed using the cdist method within the SciPy framework +[23] which utilizes optimized C code to efficiently compute the distances. The computa- +tions of Wq( ˆdij) and the extraction of optimal transport data is then obtained using the +EMD class within the Wasserstein library [9, 19]. There are two continuous parameters in +the definition of Wq( ˆdij), r and q, cf. Eqs. (2.1), (2.3). These can be chosen such that the +sensitivity to CPV is maximized. The optimal value of q = 0.1 was chosen by finding, for +r = 2, the minimum average CL p–value for which the CPC hypothesis is excluded given +the toy model CPV Dalitz plot distributions, as obtained from an ensemble of Ne = 500 +distinct datasets generated from the BaBar model [2], see further details in App. D. In the +– 5 – + +0 +500 +1000 +PDF +q = 0.1 +Master +Permutation +10−3 +10−1 +SF +Master +Permutation +0.0065 0.0070 0.0075 0.0080 0.0085 0.0090 0.0095 0.0100 +Wq +0 +2 +4 +Ratio +Master/Permutation +Figure 3. The Wq distribution function (PDF, top panel), and the survival factor (SF=1-CDF, +where CDF is the cumulative distribution function, middle panel) obtained from the permutation +method (orange) compared to the true CP conserving distribution (the master method, in blue), +while the bottom panel shows the ratio of the SF obtained using the two methods. Each distribution +consists of 103 Wq values with the solid curves and bands representing the average ±1σ ranges for +the bin counts obtained over 10 distinct distributions. +analyzed examples, changing r in the definition of the distance Eq. (2.3) did not lead to +significant changes in the sensitivity. Thus, in the numerical results below we use the opti- +mized values {r = 2, q = 0.1}, while in App. D we also show the results for the non-optimal +choices, {r = 2, q = 1} and {r = 2, q = 10}. +To determine the p−value with which the CPC hypothesis is excluded for the particular +CPV Dalitz plot sample, one needs the Wq probability distribution functions (PDF) for the +CPC Dalitz plot distributions. In the experiment one can determine the CPC PDF using +the permutation method, which, as we show next, is estimated to lead to only a relatively +small bias compared to the true CP conserving PDF. +3.1 +Testing for bias in the permutation method +In order to assign a p−value with which the CPC hypothesis is excluded, given two samples +of B and ¯B decays, one first calculates the Wasserstein distance between the two, W exp +q +. +This encodes the dissimilarity between the two distributions of events. However, the value +– 6 – + +of W exp +q +by itself is not particularly informative, except that smaller W exp +q +values indicate +more similar distributions. For a quantitative assessment of CPV we need the distribution +of Wq for the CP conserving case. We obtain this using two methods: 1. using the permu- +tation method, i.e., by permuting the original B and ¯B samples (which have non-zero direct +CPV) and then calculating Wq for each such permutation and 2. using the master method, +which is the true CP conserving PDF given our assumptions: we generate an ensemble +of B and ¯B decay event samples, using the B decay model for both, and then calculate +the corresponding Wq probability density function (that is, we assume for simplicity that +all the CP violating phases reside in the ¯B0 decay amplitude). The permutation method +can be implemented with experimental data, since it involves only the measured B and ¯B +event samples. The master method, on the other hand, is only possible given a theoretical +model of the decay amplitudes. +The PDFs for the two methods, the permutation (orange) and master (blue), are shown +in Fig. 3, as obtained from an ensemble of Ne = 10 datasets containing N = ¯N = 103 events +in each sample. We see that the permutation method is a very good approximation of the +true CP conserving PDF for Wq. Such a test of a possible bias in the permutation method +can be performed for any multibody B decay (or any multibody distribution in general) +for which a reasonable description is available in terms of a resonance amplitude model. +One can also test for a potential bias in the permutation method using only experi- +mental data, but in this case only for N that corresponds to a fraction, for instance half, +of the measured sample size. That is, from data one can construct several distinct hy- +potheses for the CP conserving Wq PDF. The first CP conserving Wq PDF hypothesis can +be constructed by randomly splitting the measured B decay sample into two halves and +calculating the corresponding distribution of Wq. An alternative CP conserving Wq PDF +hypothesis is similarly obtained by randomly splitting the measured ¯B decay sample. These +can then be compared to the Wq PDF that is obtained using the permutation method (but +again using only half of the measured B and ¯B decay samples). The differences between +the three PDFs should be a good proxy for the size of the possible bias in the permutation +method when applied to the full dataset. +In the numerical results below we use the master method, i.e., the true CP conserving +PDF for Wq shown in Fig. 4, even though this is not accessible from experimental data. +This choice was done for numerical expediency, and we expect it to introduce only small +bias in the comparisons. +3.2 +Tracing CP violating phase space regions using EMD +A benefit of the Wasserstein distance based statistic is that it traces in a straightforward +fashion the variation of the CP asymmetry across the Dalitz plot. The standard definition +of direct CP asymmetry, Eq. (1.1), also applies to the differential distributions, Eq. (1.2), +repeated here for convenience, +ACP(s12, s13) = d¯Γ(¯s12, ¯s13) − dΓ(s12, s13) +d¯Γ(¯s12, ¯s13) + dΓ(s12, s13), +(3.1) +– 7 – + +0 +500 +1000 +PDF +q = 0.1 +Wq +0.0 +0.5 +1.0 +CDF +0.007 +0.008 +0.009 +0.010 +Wq +10−4 +10−1 +SF +¯p = 0.005+0.004 +−0.038 +Figure 4. The Wq probability distribution function (PDF), the cumulative distribution function +(CDF), and the survival factor (SF=1-CDF) for the CPC case and r = 2, q = 0.1, obtained using +the master method with a fit to the Johnson’s SU distribution. The orange bands (blue band on +top panel) denote the ±1σ fit errors (statistical errors). The vertical red line (band) denotes the +average Wq value (the ±1σ Wq ranges) obtained from 103 CPV datasets. We see that, on average, +the CPC hypothesis is in this example excluded at the ∼ 3σ level, i.e., with a p−value of ∼ 0.005. +where dΓ(s12, s13) is the B0 → K+π−π0 partial decay width into the region of the Dalitz +plot with s12 = (pK+ + pπ−)2 ≡ m2(K+π−), s13 = (pK+ + pπ0)2 ≡ m2(K+π0). Similarly, +d¯Γ(¯s12, ¯s13) is the CP conjugate partial decay width for ¯B0 → K−π+π0, with ¯s12 = (pK− + +pπ+)2 ≡ m2(K−π+), ¯s13 = (pK− + pπ0)2 ≡ m2(K−π0). The binned version of the CP +asymmetry ACP for the CP violating dataset, where we used the central values of the +parameters for the BaBar amplitude model from [2], is shown in the upper-right panel in +Fig. 5. The lower-right panel in Fig. 5 shows the binned ACP for the CP conserving case, +i.e., assuming that the B0 → K+π−π0 inputs in the amplitude model [2] apply to both +the B0 and ¯B0 decays. The panels in Fig. 5 show expected CP asymmetries in each bin, +obtained by averaging over an ensemble of Ne = 100 datasets containing N = ¯N = 103 B +and ¯B pairwise samples. +Next, we define the Wasserstein asymmetry utilizing the Wasserstein statistic Wq, +Eq. (2.1). We denote the contribution to Wq from each datapoint i in the B0 Dalitz plot +as δWq(i), and likewise δ ¯Wq(¯i) denotes the contribution from datapoint ¯i in the ¯B0 Dalitz +plot, such that +W q +q = +� +i +δWq(i) = +� +¯i +δ ¯Wq(¯i). +(3.2) +– 8 – + +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Wq +CP CPV +q = 0.1 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +ACP CPV +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Wq +CP CPC +q = 0.1 +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +ACP CPC +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +Figure 5. Binned Dalitz plot comparison between the Wasserstein asymmetry Wq +CP (left) and +direct CP asymmetry ACP (right), shown for CP violating B0 → K+π−π0 decays (top) and CP +conserving decays (bottom), i.e., decays in which the asymmetries in the amplitude model were +set to zero. The results shown are normalized and averaged over 100 datasets, each containing +2N = 2 × 103 (B and ¯B) events. +We define the binned Wasserstein asymmetry Wq +CP within each bin in Fig. 5 as +Wq +CP(s12, s13) = +� +¯i δ ¯Wq(¯i) − � +i δWq(i) +� +¯i δ ¯Wq(¯i) + � +i δWq(i), +(3.3) +where the summation over i (¯i) is only over the data-points contained in the bin centered at +(s12, s13) (the CP conjugated ¯B0 bin centered at (¯s12, ¯s13)). By construction, Wq +CP vanishes +when summed over the whole Dalitz plot, i.e., when there is only one bin encompassing the +whole Dalitz plot. The Wasserstein asymmetry Wq +CP is also statistically consistent with +zero in the regions of the Dalitz plot that have vanishing CP asymmetry. Comparison of left +and right panels in Fig. 5 shows that Wq +CP faithfully traces the variation of ACP over the +Dalitz plot, including the statistical fluctuations, most readily visible in the CP conserving +datasets shown in the lower panels in Fig. 5. This makes the Wq +CP easily interpretable in +terms of the underlying physics, i.e., which components of the resonant structure contribute +most to the CP violation. +– 9 – + +10−11 +10−6 +10−1 +p(Wq) +10−11 +10−8 +10−5 +10−2 +p(T) +q = 0.1, ϵ = 0.304 +10−11 +10−6 +10−1 +p(Wq) +10−11 +10−8 +10−5 +10−2 +p(T) +q = 1, ϵ = 0.100 +10−11 +10−6 +10−1 +p(Wq) +10−11 +10−8 +10−5 +10−2 +p(T) +q = 10, ϵ = 0.264 +Figure 6. The scatter plot of p–values at which the CPC hypothesis is excluded, for the ensemble +of Ne = 500 samples with N = 103 B0 → K+π−π0 (and ¯N = 103 CP conjugated ¯B0 → K−π+π0) +decays, calculated using either the Wq or T statistics (dots), with 1σ fit error bars shown as lines, +and setting q = {0.1, 1, 10} (from left to right). The fraction ϵ of points above the p(Wq) = p(T) +diagonal line denotes the fraction of ensembles for which Wq is more sensitive to CPV. The dotted +gray lines (solid bands) show the average (1σ ranges of) p−values for the ensemble. +The advantage of Wasserstein distance over direct CP asymmetry, Eq. (3.1), as a +measure of CP violation in the Dalitz plot distributions is that Wq does not require binning. +It is a global quantity that encodes the cumulative differences between the B0 and ¯B0 +Dalitz plots. As such it can be used as a statistic sensitive to the CP violating Dalitz plot +distributions. In Fig. 6 we compare the sensitivity of Wq to CPV relative to another such +unbinned statistic, the energy test statistic T [5, 6, 8], see App. E for further details on the +energy test. The energy test has already been successfully applied to search for CPV in +multibody decays [7]. On the other hand, we do not show comparisons with the SCP test, +a.k.a. the Miranda method [3, 4], which uses optimized bins. In our numerical studies we +found the SCP test to always be less sensitive. +From Fig. 6 we see that the Wasserstein distance and the energy test have comparable +sensitivity to CPV, but with Wq somewhat less sensitive on average. This can be quantified +by introducing +ϵ ≡ 1 +Ne +Ne +� +i=1 +� ++1 +pi(Wq) < pi(T), +0 +otherwise, +(3.4) +where Ne = 500 is the ensemble size for which the CPC exclusion CL p−values were +obtained either using the Wq (giving p(Wq)) or the T statistic (giving p(T)). That is, ϵ +gives the fraction of randomly sampled datasets for which Wq statistic leads to stronger +sensitivity to CPV than the energy test. Since ϵ < 0.5 one may conclude that Wq is on +average less sensitive. However, the average p–values for Wq and T test statistics (dashed +lines) agree within 1σ ranges (gray bands). Similarly, many scatter points in Fig. 6 agree +with the p(Wq) = p(T) line within the error bars that are reflecting the uncertainties with +which the p−values were determined from the fit. That is, for small p−values, p ≲ O(10−4), +we estimate the significance of the exclusion using an extrapolation of a fit to corresponding +PDFs, where the fit distributions are chosen according to the minimization of a χ2. The +– 10 – + +energy test statistic is fit with a gamma distribution while for q = 0.1, we fit the Wq master +distribution with Johnson’s SU distribution. Errors are assigned according to the 1σ bands +on the respective fit parameters, see App. D.1 for further details. The ϵ ratio does not +take into account the error associated with our estimates of the p−value for each statistic. +These errors can be large especially for small p-values, and as such ϵ should only be used +as a cautious measure of performance. +The T statistic has a continuous parameter, σ, which defines the scale of correlations +probed by the energy test. For results shown in Fig. 6 the value of σ was set to its (close +to) optimal value σ = 0.2 GeV2, for which the energy test on average leads to the smallest +expected p−values. Similarly, the parameter q in Wq was optimized, with the results in +Fig. 6 shown for close to optimal value q = 0.1. Note that in the actual experiment the +above optimization should be performed on the mock data, using a model for B → Kππ +decay amplitudes, and not on actual experimental data, in order not to introduce bias. +If the amplitude model does not describe well the data, this would lead to suboptimal +choice for the continuous parameter and reduced sensitivity to CPV, but otherwise is not +problematic. +We expect that the somewhat reduced sensitivity of Wq to CPV compared to the +energy test is because Wq also receives contributions from areas in the Dalitz plot that are +CP conserving. This is in contrast to the energy test statistic T, which has a vanishing +expectation value in those areas regardless of the number of events in the dataset. The +contributions to Wq from these regions, on the other hand, only slowly tend to zero with +increasing sample size N. That is, Wq may be written as the sum of two contributions +Wq = +� +i +δWq(i) = +� +i +� +δW signal +q +(i) + δW noise +q +(i) +� +where +lim +N→∞ +� +i +δW noise +q +(i) = 0. +(3.5) +The term δW noise +q +comes from CP conserving regions of the Dalitz plot, while δW signal +q +is due to the presence of CPV and tends to a nonzero value for N → ∞. If the signal +and noise contributions preferentially occur at different length scales, one can construct a +modified Wasserstein distance test with higher sensitivity to CPV, as shown in the next +subsection. +3.3 +The windowed EMD +As discussed above, the disadvantage of the Wasserstein distance as a CPV test statistic +is that, because all δWq(i) are positive, it includes an abundance of small nonzero con- +tributions even in the absence of CPV, generating a long–tailed CP conserving PDF for +Wq. Within the Dalitz plot, CP violation manifests as local density differences between +the B and ¯B datasets. If this CPV is either localized and/or relatively small, such as in +B0 → K+π−π0 Dalitz plots, this translates into relatively small differences in the δWq(i) +distributions between CPV and CP conserving B0 decays. +This is illustrated in Fig. 7 (top), which shows binned counts of log(δWq), averaged +over the ensemble of Ne = 103 CPC (blue) and Ne = 103 CPV (orange) samples, each +containing N = ¯N = 103 events, with the bands denoting the 1σ ranges for bin counts. +Fig. 7 (bottom) shows the difference between the average CPC and CPV bin counts, as +– 11 – + +0 +20 +40 +Counts +q = 0.1 +CPC +CPV +−8.2 +−8.0 +−7.8 +−7.6 +−7.4 +−7.2 +−7.0 +log(δWq) +−10 +0 +10 +Counts +CPV − CPC +Window +Figure 7. Top: joined points (bands) represent the histogram of average (1σ range) log(δWq) +counts for 100 bins, i.e., the binned counts of the log of pairwise optimal transport distances between +B and ¯B sample events for N = ¯N = 103 sample sizes, averaged over ensemble of Ne = 103 samples, +for CPC (blue) and CPV (orange) datasets. Bottom: joined points in purple (purple band) denote +the average (1σ ranges of) differences between CPC and CPV log(δWq) counts, with the green band +denoting the [δW win +min, δW win +max] range for which sample events are used with positive weights in the +construction of the windowed Wasserstein distance statistic, cf. Eqs. (3.6)–(3.7) (in this example +the range [δW win +min, δW win +max] for negative weights is taken to be zero). +well as the 1σ ranges. We see that the δWq distributions for CPC and CPV cases overlap +significantly in many regions of pairwise δWq values. However, we also expect the CPC +distributions to be more likely to lead to smaller δWq, given that the B0 and ¯B0 Dalitz +plot are more similar than in the CPV cases. Consequently, for the CPV case one would +expect an excess of datapoints with larger δWq and a related excess of CPC bin counts +at smaller δWq values, as shown in Fig. 7. Depending on the details of the Dalitz plot +the δWq distributions could exhibit other differences between the CPC and CPV cases not +present in the example in Fig. 7. For instance, if CPV is localized in a small region of the +Dalitz plot containing n events and of size ˆd, cf. Eq. (2.3), then we would expect an excess +of CPV δWq bin counts over CPC in Fig. 7 at δWq ∼ O( ˆd/n). Once one sums over all +δWq(i), and considers only the global Wasserstein distance Wq = � +i Wq(i) as a measure +of CPV, the information about such differences in the δWq distributions is lost. +Since there is more information in the δWq(i) distributions than in the global Wq +– 12 – + +10−12 +10−7 +10−2 +p(Iq) +10−12 +10−9 +10−6 +10−3 +100 +p(Wq) +CPV +q = 0.1 +ϵ = 0.866 +10−12 +10−7 +10−2 +p(Iq) +10−12 +10−9 +10−6 +10−3 +100 +CPC +q = 0.1 +ϵ = 0.484 +10−15 +10−9 +10−3 +p(Iq) +10−15 +10−11 +10−7 +10−3 +p(T) +CPV +q = 0.1 +ϵ = 0.566 +10−15 +10−9 +10−3 +p(Iq) +10−15 +10−11 +10−7 +10−3 +CPC +q = 0.1 +ϵ = 0.430 +Figure 8. The comparison of estimated p-value exclusions of CP conserving hypothesis for CPV +(left) and CPC (right) B0 → K+π−π0 decays, comparing the windowed Wasserstein distance Iq +with either the global Wasserstein Wq (top) or the energy test T (bottom) statistic, for q = 0.1, on +500 distinct datasets. +observable, we can define an improved statistic Iq +Iq ≡ +� +i +w +� +δW win +min, δW win +max, δW win +min, δW win +max; δWi +� +, +(3.6) +where for the example of B → Kππ decays we define the window function as +w(x) = +� +� +� +� +� +� +� ++1 +x ∈ [δW win +min, δW win +max], +−1 +x ∈ [δW win +min, δW win +max], +0 +otherwise. +(3.7) +The window function w splits datapoints into three categories. The events in the high δWq +values window δWq ∈ [δW win +min, δW win +max], and the events in the anti-window of mid-range +δWq values, δWq(i) ∈ [δW win +min, δW win +max], are included in the windowed Wasserstein distance +statistic Iq, but weighted with opposite signs, thus enhancing the difference between the +CPC and CPV distributions. The remaining events, for which the CPC and CPV δWq +distributions do not differ significantly, are instead not included in Iq. +Keeping these +events would only dilute the sensitivity to CPV. +– 13 – + +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Iq +CP CPV +q = 0.1 +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +ACP CPV +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +Figure 9. The comparison of binned Dalitz plot asymmetry: the windowed Wasserstein asymmetry +Iq +CP (left) and the fractional CP asymmetry ACP (right), cf. +also top panel in Fig. 5. +When +compared with the asymmetry significance’s shown in Fig. 25 we see that the chosen window +is correctly filtering CP conserving δWq values and retaining δWq values in the most significant +regions of CPV. +The optimization of window and anti-window ranges requires a model for B0 and +¯B0 amplitudes. +Importantly, the δWq values depend on the sample size N = ¯N, and +thus the optimization should be performed for the number of events actually measured +in the experiment. One could attempt a data driven optimization of w by splitting the +measured dataset into subsets, correcting for the effect of smaller sample sizes, but we +did not explore this further. For other decay channels, depending on the actual decay +width distributions, other forms of window function could be better suited than the one in +Eq. (3.7). For instance, one could define multiple disjoint window and anti-window regions, +or use weights that are smooth functions of δWq, not just the discrete values {−1, 0, +1}. +For B0 → K+π−π0 Dalitz plot and q = 0.1, N = ¯N = 103, there is on average an excess +of CPC over CPV δWq distributions in the mid-value region log(δWq) ∈ (−7.55, 7.3). +However, it is accompanied with a large variability in bin counts, and thus for this case it +proves advantageous to define Iq using only events in the window shown as the green band +in Fig. 7, and drop all other events (that is, the anti-window range is shrunk to zero). For +other values of q both window and anti-window ranges are nonzero, see App. D. +Fig. 8 shows, for q = 0.1, N = ¯N = 103, the comparison of p-values at which the CPC +hypothesis is excluded, when either the windowed Wasserstein statistic Iq or the global +Wasserstein statistic Wq are used, Fig. 8 (top), or if the energy test statistic, T, is used +instead, Fig. 8 (bottom). We see that the windowed Wasserstein distance statistics, Iq, +is as sensitive, or even slightly more sensitive, to the presence of CPV in the Dalitz plot +distributions than the energy test, while both outperform the global Wasserstein distance +statistic. Fig. 8 also demonstrate that Iq, like Wq and the T test statistic, does not introduce +bias when CPC distributions are considered. As an additional confirmation that the chosen +windows are in fact selecting the relevant areas of the Dalitz plot associated with CPV and +– 14 – + +CPC we plot in Fig. 9 the binned CP and Wasserstein asymmetries, but in the later only +keeping the events that contribute to Iq. That is, we define +Iq +CP(s12, s13) = +� +¯i w(δ ¯Wq(¯i)) − � +i w(δWq(i)) +� +¯i w(δ ¯Wq(¯i)) + � +i w(δWq(i)), +(3.8) +where each event is weighted according to the window function in Eq. (3.7). The summation +over i (¯i) is only over the data-points contained in the bin centered at (s12, s13) (the CP +conjugated ¯B0 bin centered at (¯s12, ¯s13)). The comparison of left and right panels in Fig. 9 +shows that the chosen window from Fig. 7 does indeed correctly select the regions of the +Dalitz plot exhibiting CP violation and acts as a filter to better resolve CP asymmetries. +The shown results could be improved further. First of all, we did not perform a full +optimization of the window function Eq. (3.7), but rather only selected among several +discrete, manually chosen, forms. It would also be interesting to explore if the features ob- +served in the δWq distributions, Fig. 7, can further inform amplitude models, in particular +about the existence of CPV regions with resonances interfering. +4 +Application to three body D decays +Next, we apply the analysis to larger datasets with small but nonzero amount of CP +violation. As a concrete example we consider the three body D decay D0 → π+π−π0 and +its CP conjugated channel, ¯D0 → π−π+π0. The CP violation in D decays is expected +to be small, parametrically suppressed by O(VcbVub/VcdVud) ∼ 10−3 [24–32] and has only +recently been measured to be nonzero [33, 34]. Further searches for CP violation within the +charm sector are highly motivated, since the discovery of enhanced CPV in specific modes, +including multibody decays, could point to a discovery of new physics (for sum rules that +the SM needs to satisfy see [35–37]). +The D0 → π+π−π0 decays have been studied at the LHCb using the energy test, and +found that the CPC hypothesis is excluded at the p = (2.6±0.5)% C.L. [7]. Below, we show +how the Wasserstein distance based statistics could be used as alternative analysis strategies +to search for CPV in this and other multibody charm decays, taking D0 → π+π−π0 as a +toy example. +We generate the two datasets, for D0 → π+π−π0 and ¯D0 → π−π+π0 decays, using the +BaBar amplitude model [38] implemented within the Laura++ framework [39], similarly to +the case of B0 → K+π−π0 decays discussed in Sec. 3. As a toy example of CP violation +in the D0 → π+π−π0 Dalitz plot we follow Ref. [8] (where this was used to explore the +sensitivity of the energy test), and increase for the generation of CPV datasets the fit +fraction of the ρ(770)− by 2% and the phase of the corresponding decay amplitude by 2◦. +The D − ¯D meson mixing is ignored in the generation of the samples. +The present experimental D → πππ decay samples are roughly 102 − 103 times larger +than the B → Kππ decay samples. Because of the current implementation of the Wasser- +stein distance calculation that we use [9, 19], large statistic datasets present a numerical +problem. To solve the optimal transport problem utilizing the current publicly available +linear programming libraries require the full cost matrix ˆdij as the input. The cost matrix +– 15 – + +Figure 10. Pictorial comparison between unbinned (left) and binned (right) Wasserstein statistic +methods. Note how in the binned case, the optimal transport algorithm effectively sets to zero +in the last step the number counts in the bins that have the same counts between the two CP +conjugate datasets (red and blue). +scales as N ¯N ∼ O(N2) and quickly demands more random access memory than available in +an average personal computer. For example, the cost matrix for datasets containing ∼ 106 +events, i.e., comparable to the number of currently experimentally available D0 → π+π−π0 +decays, requires roughly 7 TB of memory space. +There are a number of solutions to the above memory problem. Below we develop two +strategies, both of which use approximate calculations of (variants of) Wasserstein distance +between the D0 and ¯D0 decay samples: a binned Wasserstein test in Sec. 4.1 and a sliced +Wasserstein test in Sec. 4.2. The two approximate approaches to the Wasserstein based +statistic can be applied to large datasets, while continuing to use the publicly available and +optimized software. Alternatively, one could attempt to create a new optimal transport +algorithm geared toward large datasets, such as the D decays, utilizing lazy evaluation and +the sparseness of the transport matrix that does not require the full form of the cost matrix +as an input. The latter, however, goes beyond the scope of the present manuscript. +4.1 +Binned Wasserstein test +Since the resonances in the D0 → π+π−π0 Dalitz plot have typical decay widths of +O(100 MeV) or so, cf. Fig. 2, we expect it is possible to capture well the change of the CP +asymmetry across the Dalitz plot already with relatively modest numbers of bins. One can +then apply the Wasserstein distance statistic to the binned Dalitz plot data in order to ob- +tain a global measure of CPV in the distributions. While there is some loss of information +due to binning compared to the Wasserstein distance statistic applied to full samples, we +expect the loss to be small, if the binning is fine enough. In the limit of infinitely small +bins one of course reverts to the case of unbinned statistic discussed in Sec. 3. +The binned Wasserstein distance is given by +W bin +q +(E, ¯E) = +� +min +{fij≥0} +Nb +� +i,j=1 +fij +� ˆdij +�q +�1/q +, +(4.1) +where Nb is the total number of bins in the D (and ¯D) Dalitz plot, with bin counts wi +( ¯wj) in the i−th (j−th) bin. +In the Dalitz plot we will use equal binning along each +– 16 – + +dimension, with nbins in each direction, so that the number of bins with nonzero entries +equals to Nb ≃ nbins(nbins − 1)/2.3 The minimization of the weights fij ( ¯fij) gives the +optimal transport from bins in D to ¯D Dalitz plot, subject to the constraints +nbins +� +i +fij = ¯wj +¯N , +nbins +� +j +fij = wi +N , +nbins +� +i,j +fij = 1 +(4.2) +with the distances ˆdij taken to be between the centers of the i−th and j−th bins. The +construction of the binned Wasserstein distance statistic W bin +q +is illustrated in Fig. 10. +Since the binned versions of E and ¯E event samples use the same binning, the optimal +transport algorithm will always ‘zero’ out the like counts in each bin between E and ¯E, i.e., +it takes no ‘work’ to transport mass by zero distance. What is left is a representation of +the local bin count density asymmetry between E and ¯E. These count density asymmetries +then get re-distributed by the optimal transport algorithm. Thus, instead of encoding the +CPV information via the distances between events in each dataset, as is done in Wq, the +CPV is now encoded as the excess or overabundance of weight between datasets (as well +as how far these weight overabundances in D Dalitz plot are from overabundances in the +¯D Dalitz plot). +Denoting the contribution to W bin +q +from the i−th bin in the D0 Dalitz plot as δW bin +q +(i), +and likewise by δ ¯W bin +q +(¯i) the contribution to W bin +q +from ¯i−th bin in the ¯D0 Dalitz plot, +such that +(W bin +q +)q = +� +i +δW bin +q +(i) = +� +¯i +δ ¯W bin +q +(¯i), +(4.3) +we define in analogy with Eq. (3.3) the binned Wasserstein asymmetry Wq,bin +CP +as +Wq,bin +CP (i) = δ ¯W bin +q +(¯i) − δW bin +q +(i) +δ ¯W bin +q +(¯i) + δW bin +q +(i), +(4.4) +where the ¯i-th bin in the ¯D Dalitz plot is the CP-conjugate of the i−th bin in the D Dalitz +plot. +Fig. 11 shows a comparison between the binned Wasserstein distance asymmetry Wbin +q +(left panels) and the CP asymmetry ACP (right panels). We find that the binning results +in enhanced asymmetries when data is represented using the Wbin +q +compared to ACP. This +is true both for the CPV dataset, as well as for statistical fluctuations in the CPC example. +Since direct CP violation in D decays is small, it is hard to discern by eye whether or not +there is CP violation in the Dalitz plot distributions, and one is forced to rely on a statistic +sensitive to CPV in distributions such as W bin +q +or the energy test. +Fig. 12 shows that the Wasserstein test statistic is still sensitive to CP violation de- +spite the binning procedure. The three panels show from top to bottom the probability +3The equality sign applies in the mπ → 0 limit or for large enough bins. In our numerical implementation +we use square nb × nb arrays that cover fully the Dalitz plot and take Nb = n2 +bins to be the total number +of bins, including the ones containing zero events. The bins outside the kinematically allowed region are +trivially zero, and do not add any complexity to the calculation of the binned Wasserstein distance, while +this approach simplifies the encoding of the Dalitz plot in the binned array. +– 17 – + +0 +1 +2 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +m2(π+π0) (GeV2) +Wq,bin +CP +CPV +q = 1.0 +0 +1 +2 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ACP CPV +0 +1 +2 +3 +m2(π−π0) (GeV2) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +m2(π+π0) (GeV2) +Wq,bin +CP +CPC +q = 1.0 +0 +1 +2 +3 +m2(π−π0) (GeV2) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ACP CPC +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−0.2 +−0.1 +0.0 +0.1 +0.2 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +Asymmetry +Figure 11. Comparison between the binned Wasserstein asymmetry Wq,bin +CP +(left), cf. Eq. (4.4), +and the binned CP asymmetry ACP (right) for the D0 → π+π−π0 Dalitz plot with N = ¯N = 106 +events in a sample, and nbins = 20. +distribution function (PDF), the cumulative distribution function (CDF), and the survival +factor (SF=1-CDF) as functions of the binned Wasserstein statistic W bin +q +, for r = 2, q = 1, +and using nbins = 50, for CP conserving D0 → π+π−π0 Dalitz plot with N = ¯N = 105 +events in the sample. The average W bin +q +value (red vertical line) for our CPV toy D decay +model example is well above the bulk of the CP conserving W bin +q +PDF. We see that, on +average, the CPC hypothesis is in this example expected to be excluded at the ∼ 2.5σ +level, i.e., with a p−value of ∼ 0.01. +In fact, Fig. 13 shows that the chosen binning size nbins = 50 (which was not optimized) +is already fine enough for N = ¯N = 104 that there is only little loss of sensitivity to +CPV compared to the unbinned Wq. In the scatter plot of p−values at which the CPC +hypothesis is excluded, we see that the exclusion levels obtained by either using the Wq +or the W bin +q +statistic are comparable, and consistent within estimated errors (due to the +systematic and statistical uncertainties in the extrapolation of the fit to the CPC PDF). +The binned Wasserstein statistic does have, however, the additional advantages of less +memory consumption (space complexity) and computational efficiency (time complexity) +due to the reduction of the dataset size from N ¯N to ∼ n2 +bins. Whether nbins = 50 suffices +also for sample sizes 106, or whether fined binning will be required, should be tested when +– 18 – + +0 +500 +PDF +q = 1 +W bin +q +0.0 +0.5 +1.0 +CDF +0.002 +0.003 +0.004 +0.005 +0.006 +W bin +q +10−2 +100 +SF +¯p = 0.010+0.019 +−0.007 +Figure 12. The CPC probability distribution function (PDF), the cumulative distribution function +(CDF), and the survival factor (SF=1-CDF) as functions of the binned Wasserstein statistic W bin +q +for r = 2, q = 1, and nbins = 50, as obtained from the numerical master method result for the +PDF, consequently fit to a gamma distribution, for N = ¯N = 105 events in the sample, using an +ensemble of Ne = 103 samples. The orange bands (blue band in the top panel) denote the ±1σ fit +errors (statistical errors). The vertical red line (band) denotes the average W bin +q +value (the ±1σ +W bin +q +ranges) obtained from an ensemble of Ne = 102 CPV datasets for our toy D0 → π+π−π0 +amplitude model. +the method is applied to the actual D decay data, however, we find the above results quite +encouraging. +4.2 +Sliced Wasserstein test +The Sliced Wasserstein distance (SWq) is a variant of the Wasserstein distance, in which +the optimal transport in d-dimensions is replaced with a set of optimal transport problems +on 1D slices, with the data points projected onto them. That is, the sliced Wasserstein +distance SWq(g, f) between two distributions in d−dimension, g(x) and f(x), is given by +[40] +SWq(g, f) = +� � +Sd−1 Wq(Rg(·, θ), Rf(·, θ)dθ +� 1 +q +, +(4.5) +where Rg(·, θ) is the Radon transform of function g(x), defined to be the projection of +function g(x) onto the line in the direction of the unit vector θ, which then runs over the d−1 +unit sphere Sd−1. The Wq in (4.5) is therefore a 1D Wasserstein distance between functions +Rg(·, θ) and Rf(·, θ). +The 1D Wq has a closed form solution, given by the integrated +distance between the CDFs for the two functions, and can be efficiently calculated through +a simple sorting algorithm. +– 19 – + +10−4 +10−2 +100 +p(Wq) +10−4 +10−3 +10−2 +10−1 +100 +p(W bin +q ) +q = 1, ϵ = 0.658 +Figure 13. The scatter plot of p−values at which the CP conserving hypothesis is excluded for our +toy D decay example. The plot shows an ensemble of Ne = 500 datasets with samples of N = 104 +D0 → π+π−π0 (and ¯N = 104 CP conjugated ¯D0 → π−π+π0) decays, with p−values calculated +either using the unbinned Wq, giving p(Wq), or using the binned W bin +q +statistic with nbin = 50, +giving p(W bin +q +) (dots), where in both cases we set q = 1. +The 1σ fit error bars on p− values +are shown as lines. The fraction ϵ of points above p(Wq) = p(W binned +q +) diagonal line denotes the +fraction of ensemble samples for which Wq is more sensitive to CPV than W binned +q +is. The dotted +gray horizontal and vertical lines (solid bands) show the average (1σ ranges of) p−values for the +ensemble. +0.0004 +0.0005 +0.0006 +0.0007 +SWq +q = 1 +SWq +101 +102 +103 +104 +101 +103 +tratio +tratio[Nslices ≈ 7500] = 1 +Figure 14. Top: The approximate evaluation of SW q, Eq. (4.6), as a function of the number of +slices used, Nslices, for a particular CPC D0 → π+π−π0 sample with N = ¯N = 104 events, setting +q = 1. The solid blue line (blue band) shows the mean (1σ range) of the SW q estimates obtained +from an ensemble of 50 different samplings of Nslices slices for each solid point. Bottom: the speed +up of calculating SWq compared to Wq defined as the ratio of computational times in the two cases, +tratio = tWq/tSWq. +– 20 – + +10−2 +10−1 +100 +p(Wq) +10−2 +10−1 +100 +p(SWq) +Nslices = 100 +q = 1 +ϵ = 0.628 +10−2 +10−1 +100 +p(Wq) +10−2 +10−1 +100 +p(SWq) +Nslices = 1000 +q = 1 +ϵ = 0.600 +10−2 +10−1 +100 +p(Wq) +10−2 +10−1 +100 +p(SWq) +Nslices = 10000 +q = 1 +ϵ = 0.618 +Figure 15. The scatter plot of p–values at which no CPV hypothesis is excluded, calculated using +SWq and Wq for Nslices = 102, 103, 104 (from left to right), for the ensemble of a Ne = 500 datasets +with samples of N = 104 D0 → π+π−π0 (and ¯N = 104 CP conjugated ¯D0 → π−π+π0) decays. The +fraction ϵ of points above p(Wq) = p(SWq) diagonal line denotes the fraction of ensemble samples +for which Wq is more sensitive to CPV than SWq. +The sliced Wasserstein distance can thus be efficiently calculated, at least approxi- +mately, by performing a large enough number of slices, Nslices, +SWq(g, f) ≈ +� +1 +Nslices +Nslices +� +k=1 +Wq(Rq(·, θk), Rfν(·, θk)) +� 1 +q +, +(4.6) +where θk are random unit vectors uniformly distributed over the unit sphere Sd−1. In the +Nslices → ∞ limit the l.h.s. approaches the r.h.s. in the above equation. +Importantly for our purposes, both Wq and SWq(g, f) are distances in the space of +functions and both measure dissimilarity of f and g distributions. The SWq can therefore +also be used as a test statistic, in the same way as we used the Wasserstein distance Wq +in the previous sections. Furthermore, SWq is closely related to the Wasserstein distance, +Wq. For instance, for q = 2 we have SW2(g, f) ≤ W2(g, f)/ +√ +d, and in general SWq(g, f) ≤ +cqWq(g, f) with a known constant cq ≤ 1 (for q ∈ [1, ∞)). +The improved computational efficiency for SWq relative to Wq is shown in Fig. 14. +The ratio of the computing times, tratio = tWq/tSWq, where tWq(tSWq) denotes the time +required to calculate Wq (approximate calculation of SW q using Eq. (4.6)) for a particular +D0 → π+π−π0 sample with N = ¯N = 104 events, where we take q = 1. For small number +of slices, Nslices ∼ O(10) the speed up is several orders of magnitude, however, at that point +also the approximate evaluation of SW q still has a large uncertainty. The latter is denoted +with the blue band, corresponding to 1σ range of SW q values obtained using Eq. (4.6), +cycling through 50 iterations. +We observe that in this example the SWq evaluation is +faster than the Wq one for Nslices ≲ 7500. We also observe that the approximate SWq +evaluation converges to its limiting value for Nslices ≈ 1000, indicating a ∼ 7× speedup +in the calculation of SWq compared to Wq. Beyond the speed-up, and maybe even more +importantly for the scaling to large sample sizes, the evaluation of SWq does not require +large memory resources. We have also checked that as the number of slices increases the +– 21 – + +SWq and Wq distributions, obtained from an ensemble of N = ¯N = 104 event samples, +agree up to a scaling factor as expected. Finally, since we are interested in the sensitivity +to CPV and not in SWq itself, we show next that a high sensitivity to CPV can be achieved +already with relatively approximate estimate of SWq, relying on just a limited number of +slices. +Fig. 15 shows the p−values at which the CPC hypothesis is excluded, either calculated +using Wq (giving p(Wq)) or via approximate evaluation of SWq using Eq. (4.6) (giving +p(SWq)) for three different values of slices, Nslices = 102, 103, 104 (from left to right). The +fraction ϵ of points above p(Wq) = p(SWq) diagonal line denotes the fraction of Ne = 500 +datasets ensemble of N = ¯N = 104 event samples for which Wq is more sensitive to CPV +than SWq. We see that even for Nslices = 102 the obtained p−values are already compa- +rable to the p−values obtained using full Wq, even though at that point the approximate +evaluation of SWq still has a rather large spread, cf. Fig. 14. This is quite encouraging, +and it would be interesting to explore in the future whether this feature remains for larger +sample sizes. Similarly, it would be interesting to explore where a windowed SWq, defined +in analogy with the windowed Wasserstein distance statistic Iq, would lead to a similar +increase in sensitivity to CPV that we saw in the case of full Wq. +5 +Conclusions +The Wasserstein distance based test statistics are potentially powerful tools that can be +used to search for the presence of CP violation in multibody decays. They combine the +benefits of two alternative tests sensitive to CPV in distributions: (i) in a similar way as the +binned CP asymmetry, the Wasserstein distance based test statistics trace asymmetries to +the regions of phase space the CPV resides in, while at the same time (ii) being a sensitive +probe of CPV as an integrated measure, in a similar way as the energy test is. +In this manuscript we introduced several such Wasserstein distance based test statistics, +taking the multibody B0 → K+π−π0 and D0 → π+π−π0 decays as concrete examples for +numerical studies. The simplest one is the Wasserstein distance, Wq, see Eq. (2.1) for the +case of B0 and ¯B0 decays. The use of Wq as a measure of CPV in principle requires no +tuning, though there are optimizations that can be made regarding the exact definition of +the distance in the Dalitz plot one uses, Eq. (2.3), as well as the value of the continuous +parameter q in the definition of the Wasserstein distance, Eq. (2.1). For instance, instead +of the fully symmetric definition of the distance in Eq. (2.3) one could have used a simple +Euclidean distance in the Dalitz plot, or the Euclidean distance in the square Dalitz plot. +One can also tune the value of q using an amplitude model to obtain the highest expected +sensitivity to CPV, as we did in Sect. 3 (see also App. D.2). +However, even without +an amplitude model, origins of CPV across the Dalitz plot can be identified. Such tests +allow for unbinned, model independent tests of CPV in the phase space of distributions, +thereby informing future analyses. Its use with weighted datasets is also straightforward, +as illustrated in Sect. 4.1. +Since Wq measures the cummulative presence of CPV in the Dalitz plot one therefore +needs only two observables to fully quantify the amount of direct CPV in a multibody B +– 22 – + +−0.001 +0.000 +0.001 +0.002 +˜Wq +−0.2 +−0.1 +0.0 +0.1 +0.2 +ACP +q = 0.1 +1σ +2σ +3σ +B0 → K−π+π0 +Figure 16. The expected 1σ, 2σ, 3σ C.L. contours (red shaded regions) in the (ACP, ˜Wq = Wq−W q) +plane, assuming CP conserving B0 → K+π−π0 decays. The black lines show the current global +average for ACP = 0.064 ± 0.050 as well as the 1σ range for ˜Wq that follows from our toy model +of CP violation in distributions, introduced in Sect. 3. The data was generated by first randomly +sampling ACP via a Gaussian with mean µ = 0 and standard deviation σ = 0.05 to mimic the +uncertainty in the global average, selecting B and ¯B decays accordingly with the total number +of events fixed to 2 × 103, and computing Wq. This was repeated with 103 distinct datasets and +contours drawn according to a numerical integration of a 2–dimensional joint PDF where ˜Wq and +ACP were assumed to be independent. +decay: the total direct CP asymmetry, ACP, and the Wasserstein distance Wq (or a related +Wasserstein distance test statistic such as the windowed Wasserstein distance Iq). This is +illustrated in Fig. 16, which shows a contour plot of ACP vs ˜Wq = Wq − W q, where W q +is the median Wq expected for CP conserving B decays (in this case obtained using the +amplitude model, but could be obtained using permutation method). For CP conserving +decays both ACP and ˜Wq are consistent with zero within statistical uncertainties, and would +be nonzero if there is significant CP violation. The two give complementary information: +ACP is nonzero if there is a difference in the partial decay widths between B0 → K+π−π0 +and ¯B0 → K−π+π0 decay channels, while ˜Wq is nonzero, if there is a difference between +the phase space distributions of the two CP conjugated decays. +For its simplicity, Wq does have a drawback — due to the CP conserving noise it +usually results in a lower sensitivity to CPV compared to the energy test with an optimized +regulator function. Applying filters on the optimal transports for each individual B0 and ¯B0 +decay datapoints in the Dalitz plot, however, gives an optimized version of the Wasserstein +test statistic, Iq, with sensitivity to CPV indistinguishable on statistical basis from the +optimized energy test. In Sect. 3.3 we focused on windowed filtering, with Heavyside step +functions abruptly switching on and off (or assigning negative weights) to certain ranges +of optimal transport distances, however, one could have also used other filtering variants +– 23 – + +with smooth versions of the window function Eq. (3.7). +The windowed Wasserstein distance statistic Iq can match the extreme sensitivity of +the energy test to the presence of CPV, a feature which can ultimately be attributed to the +lack of long-tailed CP conserving probability distributions in both cases. That is, the energy +test statistic successfully mitigates superfluous contributions from CP conserving variations +among data samples. This comes at the price of additional N(N −1)+ ¯N( ¯N −1) ∼ O(N2) +computations (cf. the first two terms in Eq. (E.1), encoding the CP conserving distance +variations within each sample), as well as the need for a regulator function ψ, which +restricts contributions to be only within a sphere of influence of radius σ, see App. E. +Such suppression of CP conserving variations is expected to be necessary for any metric +based statistic with enhanced sensitivity to CPV. As we showed in Sect. 3.3 the suppression +of CP conserving variation can be implemented for the case of the Wasserstein distance +based statistics by using windowed filtering. Again, this comes at the cost of additional +computations required for the optimization of the filtering window function. +The computation requirements may become prohibitive when faced with large datasam- +ples, such as the D decays with N ≳ 106 events in a sample. In that case one can use +approximate versions of Wasserstein distance to construct test statistics that scale better +with N, at a rather small cost to sensitivity. In Sect. 4 we discussed two such possibilities, +the binned Wasserstein test statistic, W bin +q +, and the sliced Wasserstein test statistic, SWq. +Both were shown to give similar sensitivities to CPV as Wq, when either the binning is fine +enough (for W bin +q +) or for large enough number of slices (for SWq). +The work presented in this manuscript could be extended in several directions. The +extension to higher dimensional spaces, such as the n-body meson decays, n ≤ 4, is straight- +forward with no changes to the formalism required. The main question in that case is the +scaling with the number of particles in the final state, where the usual curse of dimen- +sionality may be mitigated by the fact that the multi-body decays tend to have large +quasi-two-body resonant decay structure. Less trivial extensions include time dependent +weighting of decay rates in order to probe indirect CP violation. Finally, one could explore +other deviations from the Wasserstein distance. For instance, an interesting direction would +be to explore entropic smoothing of the Wasserstein distance, i.e., an entropic regulariza- +tion of the optimal transportation problem. The resulting Sinkhorn divergence depends on +a hyperparameter λ which interpolates between the Wasserstein distance (λ = ∞) and the +energy test (λ = 0) [18]. +Finally, we provide a public code EMD4CPV that allows a straightforward use of the +introduced Wasserstein based statistics for two-sample tests, with further details about the +code given in App. A, +Acknowledgements +We thank S. Bressler, J. Thaler for discussions on the two sample tests, and M. Gersabeck, +D. White, G. Sarpis, S. Chen and Y. Brodzicka in particular for extended discussions on the +energy test, as well as M. Szewc for comments on the manuscript. We thank T. Latham for +help with the Laura++ framework, T. Evans for support using the AmpGen framework, and +– 24 – + +Figure 17. Pictorial representation of the EMD4CPV program architecture. The largest box repre- +sents the highest level class, delta Wq, followed by lower level sub-classes contained within it. The +arrows represent the inheritance of each sub-class. +J. Brod for providing access to the local computing resources. AD acknowledges support +from STFC grants ST/S000925 and ST/W000601/1. AY, JZ and TM acknowledge support +in part by the DOE grant de-sc0011784 and NSF OAC-2103889. +A +Public code EMD4CPV +The public code and repository for this project may be found at +https://github.com/adamdddave/EMD4CPV. +The program architecture is hierarchically structured, resembling a nested doll of +classes and subclasses, as shown schematically in Fig. 17, +delta Wq(delta Wq statistics(delta Wq fit(delta Wq versus))). +(A.1) +The delta Wq is the highest level class and contains the sub-class delta Wq statistics, +which in turn contains the sub-class delta Wq fit, which finally contains the sub-class +delta Wq versus. This nested structure was implemented for three main reasons. Firstly, +the modularity improves readability and the ease of use, since the programs using the +classes are structured as function calls from a software library. Secondly, this class–subclass +structure follows the natural progression of the analysis pipeline used to compute and +compare different Wq based statistics. For example, a typical usage of the library will +follow a nested call of functions within each class, +delta Wq → delta Wq statistics → delta Wq fit → delta Wq versus. +(A.2) +Finally, since each sub-class inherits all functions from the previous class this allows the +user to work at any level of the program architecture while only needing to initialize one +class instance. While the use case of the program is oriented toward 3–body decays, the +code is generic enough such that it can be used with any n−dimensional dataset. +Below we summarize briefly the software pipeline (see the documentation as well as +the example Python notebook example.ipynb within the repository for more details): +– 25 – + +• The delta Wq class contains functions which allow the user to input two n–dimensional +distributions and obtain the associated binned or unbinned δWq values chosen by the +optimization. Since in most cases the CP conserving distributions (functionals of +δWq) need to be calculated, the class is set up such that the generation of the CP +even distributions via the master or permutation methods can be done efficiently by +randomly selecting a subset of unique datapoints from a larger datapool provided by +the user in the form of a text file. In addition, this class may be used to compute the +sliced Wasserstein distance SWq. +• Once the δWq ensemble is obtained, the subclass delta Wq statistics can be used +to compute the Wq, Iq, or any other user defined statistical distributions. +• Oftentimes, when computing the p−values from the CPV datasets a fit is needed +in order to extrapolate outside the ranges of explicitly calculated CP conserving +distributions. These fits can be performed using the delta Wq subclass which allows +the user to iteratively fit to any distribution within the SciPy.stats library and +return the associated PDFs, CDFs, SFs, χ2–values, as well as the PDFs, CDFs, and +SFs associated with the ±1σ errors on the fit parameters. +• Finally, the delta Wq versus4 subclass may be used to iteratively compare the sen- +sitivity of different statistics on ensembles of like datasets. +• Additionally, for convenience, the script energy test.py provides a Python imple- +mentation of the energy test statistic, i.e., the computation of the test statistic T (see +Eq. (E.1)) between two n–dimensional distributions for use when computing CPV +statistic values in delta Wq versus. The program also includes an interface with +Manet [8] (which utilizes the CUDA API to parallelize the computation on NVIDIA +GPUs) such that the user can efficiently generate large statistic CP conserving T +distributions if desired. +B +The optimal transport problem +Consider two discrete samples P and ¯P, the first consisting of n points sampled from +probability distribution p, each with weight wi such that total weight is W = � +i wi, +and the second consisting of ¯n points sampled from ¯p with weights ¯wi and total weight +¯W = � +i ¯wi. The problem optimal transport consists of transporting the weight W of P +into the weight of ¯W of ¯P, i.e., P → ¯P, as efficiently as possible given some cost function +related to the distances among the points. +This requires minimizing an nׯn transport plan matrix T, which contains information +about the amount of work required to transporting P → ¯P, such that the work required is +minimal. The transport matrix thus requires knowledge of both the distances between i-th +and j-th points as well as the amount of weight to be transported between the points of P +and ¯P. The distances between each point in P and ¯P can be encoded in an n × ¯n matrix +4This subclass requires Python 3.10+ while all other classes require Python 3+. +– 26 – + +C. The information about the transportation of weights can be encoded via the n × ¯n +‘flow’ matrix F subject to � +i Fij = ¯wj, � +j Fij = wi. That is, F specifies the fractional +amount of weight to be transferred from i-th point in P to the j-th point in ¯P, subject to +the condition that the total weight from each point must be conserved.5 +The total work or ‘cost’ of a given configuration is then given by the inner product +of the flow and distance matrices T = ⟨F, C⟩. Finding the most efficient plan amounts to +finding the transport plan F which minimizes the total cost. We denote the optimal flow +matrix as F∗ and define the Wasserstein distance as +Wq = ⟨F∗, Cq⟩1/q ≡ +� +� +n +� +i +¯n +� +j +F ∗ +ijCq +ij +� +� +1/q +(B.1) +Solving for Wq optimally takes super-cubical time complexity with respect to the size of +the input datasets O(N3) [41]. +C +EMD analysis of Gaussian distributions +In this appendix we give further details on how the Wasserstein distance Wq can be used +as a statistic sensitive to dissimilarities between two distributions. We use the toy example +of two displaced Gaussian distributions, either in 1D or 2D, where the difference between +distributions is taken to be controlled by a single “CP violating” parameter. We consider +two limits: i) the two Gaussian peaks do not overlap, but are rather displaced by ∆µ, and +ii) the peaks of the two Gaussian distributions overlap, while their widths differ, ∆σ ̸= 0. +In the main text we showed an example for the first choice where we considered two 1D +Gaussians displaced by ∆µ = 40 and with widths σ = 10, cf. Fig. 1, where ˆdij in Eq. (2.1) +here and below is taken to be the Euclidean distance. Since the optimal transport needs +to move the points in the datasets sampled from the two distributions by a distance ∼ ∆µ, +the Wasserstein distance coincides with ∆µ, W1 → ∆µ, for large N (this is true to quite a +good degree even for rather small values of N, cf. Fig. 1). This is also shown in the top +panels in Fig. 18, where we consider 11 different values ∆µ = 0, . . . , 10, while σ = 1, cf. +Fig. 18 (top left). The distributions of W1 straddle ∆µ for ∆µ sufficiently far away from +zero (for ∆µ = 0, W1 ≥ ∆µ since W1 is nonnegative), shown for N = ¯N = 103 in Fig. 18 +(top middle). Fig. 18 (top right) shows that the average value of the Wasserstein distance, +¯W1, linearly increases with ∆µ, where for small values of ∆µ/σ there is a deviation from +this linear behavior, which however is almost imperceptible on the plot. +The lower panels in Fig. 18 show the dependence of Wasserstein distance on the width +of the distributions. +In this example one Gaussian distribution is held fixed, G(x) = +N(x|µ = 0, σ = 1) while the other is taken to have different widths but a coinciding peak, +¯G(x) = N(x|¯µ = 0, ¯σ), ¯σ = 1, . . . , 7, see Fig. 18 (bottom left). With increasing ¯σ the +typical value of Wasserstein distance ¯W1 increases, since the two distributions differ more +and more, see Fig. 18 (bottom middle). The values of the W1 also form a wider distribution +5Note that a particular transport configuration is not required to be a bijective map, i.e., the weight of +a particular point in P can be partitioned to different points in ¯P. +– 27 – + +−10 +0 +10 +x +0.0 +0.1 +0.2 +0.3 +0.4 +PDF +N(∆µ = 0, . . . , 10; σ = 1) +0 +5 +10 +Wq +0 +5 +10 +PDF +q = 1 +0 +5 +10 +∆µ +0.0 +2.5 +5.0 +7.5 +10.0 +W q +q = 1 +W q = 0.997∆µ + 0.02 +−20 +0 +20 +x +0.0 +0.1 +0.2 +0.3 +0.4 +PDF +N(µ = 0; ∆σ = 1, . . . , 7) +0 +2 +4 +Wq +0.0 +2.5 +5.0 +7.5 +10.0 +PDF +q = 1 +2 +4 +6 +∆σ +0 +2 +4 +W q +q = 1 +W q = 0.789∆σ − 0.75 +Figure 18. The dependence of W1 on the displacement ∆µ (top row) or the width difference ∆σ +(bottom) of the two Gaussian distributions. Middle panels show the corresponding W1 distributions, +while the right panels show the linear dependence of the average value ¯W1 on the displacement ∆µ +and width difference ∆σ. The error bars shown in the right panels denote the 1σ bands of the W1 +distributions shown in the middle panels. +for larger values of ¯σ, since the larger difference between the two Gaussians translates to +a larger scatter of optimal transportation distances. The increase in the average value of +the Wasserstein distance, ¯W1, is linear in ∆σ = ¯σ − σ, cf. Fig. 18 (bottom right). +Next, we check that the W1 statistic leads to the same CL intervals as the negative log +likelihood. Fig. 19 (left) shows the expected 90%, 3σ and 5σ CL for ∆µ as a function of N +(solid contours) that follow from a known ∆µ = 0 probability distribution for W1. These +coincide with the expect CL exclusion intervals obtained from the negative log likelihood for +∆µ (dotted contours). We see that in this case the W1 statistic gives the correct coverage +for all considered values of N and ∆µ. +In Fig. 19 (right) we also show the estimates of the exclusion contours that follow +from a permutation (or re-randomization) test, i.e., where the symmetric “CP-even” W1 +probability distribution is modeled by randomly sampling events from E and ¯E. We see that +the re-randomization estimate of the true ∆µ = 0 probability distribution for W1 results in +a bias and thus in underestimated exclusion p−values. The benefit of the re-randomization +is that such modeling of “CP-even” W1 probability distribution is always possible, however +it also means that the use of Wq statistics is best suited for the cases where one has already +a reasonable model of the distributions and can check potential bias due to the use of +permutation test. The multi-body B and D decays fall in this category since one can use +– 28 – + +0 +1 +2 +3 +∆µ/σ +3 +6 +9 +12 +15 +18 +21 +24 +27 +30 +Nstat +5σnll +3σnll +90%nll +5σW1 +3σW1 +90%W1 +0 +1 +2 +3 +∆µ/σ +3 +6 +9 +12 +15 +18 +21 +24 +27 +30 +5σnll +3σ.nll +90%nll +5σW1 +3σW1 +90%W1 +10−5 +10−3 +10−1 +p +Figure 19. The expected 90% C.L., 3σ, and 5σ exclusion lines for different Gaussian peak displace- +ments, ∆µ/σ, and sizes of statistical samples Nstat using EMD W1 statistic (negative log-likelihood) +are denoted with dashed (solid) lines, where one uses either true (left) or modeled (right) ∆µ = 0 +probability distributions. +the fitted for amplitude models to estimate the potential bias in the permutation method +for Wq statistic. This was found to be small for the two B and D decays considered in the +main text. +A toy example that is closer to the case of three body B and D decay Dalitz plots is the +example of two displaced 2D Gaussian distributions, g(x, y) ∼ N(x|µx−∆µx/2, σ)N(y|µy− +∆µy/2, σ), and ¯g(x, y) ∼ N(x|µx + ∆µx/2, σ)N(y|µy + ∆µy/2, σ). For simplicity we take +the widths of all the Gaussian distributions to be the same, so that there is no CP violation +(the two distributions are the same) if and only if ∆µx = ∆µy = 0. The statistical analysis +of this case is a trivial extension of the case of a 1D Gaussian toy model. Using the true +“CP-even” W1 distribution for ∆µx = ∆µy = 0 leads to the correct coverage, while the +permutation method gives some bias, as in the 1D case. +For two-dimensional distributions there are additional observables and visualization +tools that prove to be useful. First of all, for arbitrary two-sample 2D distributions one +can define a Wasserstein distance asymmetry distribution Wq +CP in the same way as for the +Dalitz plot, Eq. (3.3), +Wq +CP(x, y) = +� +¯i δ ¯Wq(¯i) − � +i δWq(i) +� +¯i δ ¯Wq(¯i) + � +i δWq(i), +(C.1) +where the summation over i (¯i) is only over the data-points from g(x, y) sample contained +in the bin centered at (x, y) (from ¯g(x, y) data in the bin centered at (x, y)). In addition, +we can also define a Wasserstein distance asymmetry heatmap +ωq +CP(x, y) = 1 +si +� � +¯i +δ ¯Wq(¯i) − +� +i +δWq(i) +� +, +(C.2) +where si = ∆xi∆yi is the area of the bin center at (x, y). +Both Wq +CP(x, y) and ωq +CP(x, y) are intensive quantities. That is, in the large Nstat limit +and small bin sizes the Wq +CP(x, y) and ωq +CP(x, y) become independent of the sizes of the +bins (that is as long as bins are small enough such that the variation of Wq +CP(x, y) and +– 29 – + +−4 +−2 +0 +2 +4 +x +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +y +Nbins = 50 × 50 +−4 +−2 +0 +2 +4 +x +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +y +Nbins = 25 × 25 +−1500 +−1000 +−500 +0 +500 +1000 +1500 +ωq +CP +−1500 +−1000 +−500 +0 +500 +1000 +1500 +ωq +CP +Figure 20. +The Wasserstein distance asymmetry heatmap ωq +CP, q = 1, for two 2D Gaussian +distributions with equal widths, σ = 1, but displaced by ∆µx = ∆µy = 3, where the sample sizes +are N = ¯N = 103. Changing the binning size, from 50×50 bins (left) in the range shown to 25×25 +bins (right) does not change the overall size of the asymmetry heatmeap, just coarse-grains it, since +the asymmetry is an intensive quantity normalized to the area. +0 +50 +x +0 +10 +0 +5 +−50 +−25 +0 +25 +y +0 +50 +−50 +−25 +0 +25 +Figure 21. The visualization of optimal transport of dataset sampled from G (blue) to a dataset +sampled from ¯G (red), where G and ¯G are two 2D Gaussians displaced by ∆µx = ∆µy = 40, and +the sample size is N = ¯N = 150. +ωq +CP(x, y) from bin to bin is negligible). A numerical example for ωq +CP(x, y) is shown in +Fig. 20, where we see that changing the size of the bins simply corresponds to averaging +the Wasserstein distance heatmap over a larger area. +For relatively small samples it is also possible to visualize the optimal transport for +– 30 – + +0 +500 +PDF +q = 0.1 +0.0 +0.5 +1.0 +CDF +johnsonsu (χ2 = 49.17) +exponnorm (χ2 = 69.22) +invgauss (χ2 = 98.25) +0.007 +0.008 +0.009 +0.010 +Wq +10−3 +10−1 +SF +0.00 +0.02 +0.04 +PDF +q = 0.1 +0.0 +0.5 +1.0 +CDF +exponnorm (χ2 = 503.12) +johnsonsu (χ2 = 504.32) +skewnorm (χ2 = 505.70) +gamma (χ2 = 510.29) +f (χ2 = 510.19) +80 +100 +120 +140 +160 +Iq +10−3 +10−1 +SF +Figure 22. +The CP conserving PDF, CDF and SF distributions for Wq (left) and Iq (right) +obtained using the master method for B0 → K−π+π0 decays with q = 0.1 and N = ¯N = 103. The +different fit functions are denoted in the legend, along with respective χ2 values. +each individual point, an example of which is shown in Fig. 21 for two-sample 2D Gaussian +distributions displaced by ∆µx = 40, ∆µy = 40, and a sample size of N = ¯N = 150. The +optimal transport moves the datapoints sampled from distribution G (blue) to data sampled +from ¯G (red) shown with lines connecting pairwise the two samples. Since the two samples +are of the same size, the optimal transport is a bijective map between G and ¯G datasets. +We see that the typical shift is of order ∆µ = (∆µ2 +x + ∆µ2 +y)1/2, with datapoints on the +far (near) side of G distribution transported to near (far) side of ¯G distribution, where +near/far is defined with respect to the barycenter of G and ¯G. +D +Details on EMD–test for three body decays +In this appendix we give further details on the implementation of Wasserstein distance as +a measure of CPV in three body B and D decays. In Sect. D.1 we discuss the details of the +error analysis on the quoted p−values, in Sect. D.2 the optimization of the q parameter, +while in Sect. D.3 we collect the additional results for q = {0.1, 1, 10}, supplementing the +results shown in the main text. +D.1 +The p−value error analysis +In the numerical results in the main text we determine the p−value at which the hypothesis +of CP conservation is excluded from the master Wq distribution. +This is numerically +advantageous since it does not need to be recalculated for each dataset of B and ¯B or D +and ¯D events, while given the results in Fig. 3 we do not expect to introduce a significant +difference to the estimates using the permutation method. +– 31 – + +0 +20 +40 +PDF +q = 1 +0.0 +0.5 +1.0 +CDF +johnsonsu (χ2 = 148.25) +f (χ2 = 110.11) +invgauss (χ2 = 87.06) +gamma (χ2 = 222.52) +0.02 +0.04 +0.06 +0.08 +Wq +10−3 +10−1 +SF +0.000 +0.002 +0.004 +PDF +q = 1 +0.0 +0.5 +1.0 +CDF +skewnorm (χ2 = 148.47) +johnsonsu (χ2 = 148.47) +moyal (χ2 = 139.46) +−700 +−600 +−500 +−400 +−300 +−200 +−100 +Iq +10−3 +10−1 +SF +0 +5 +PDF +q = 10 +0.0 +0.5 +1.0 +CDF +johnsonsu (χ2 = 33.23) +gengamma (χ2 = 31.66) +f (χ2 = 33.27) +norminvgauss (χ2 = 35.66) +gamma (χ2 = 33.20) +skewnorm (χ2 = 33.27) +exponnorm (χ2 = 33.35) +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Wq +10−2 +100 +SF +0.000 +0.001 +0.002 +PDF +q = 10 +0.0 +0.5 +1.0 +CDF +skewnorm (χ2 = 82.91) +johnsonsu (χ2 = 159.78) +gamma (χ2 = 103.13) +gausshyper (χ2 = 83.09) +−600 +−400 +−200 +0 +200 +400 +Iq +10−3 +10−1 +SF +Figure 23. Same as Fig. 22 but for q = 1 (top panels) and q = 10 (bottom panels). +For B decays we calculate the master Wq distribution from 104 unique samplings of B +and ¯B datasets, each with 103 events, while for D decays we use 103 unique samplings of +D and ¯D datasets, each containing 105 events. The master Wq PDF is fit with a smooth +curve. First, the data is binned such that each bin is populated with at least one event. An +initial fit is then performed using the SciPy’s non-linear least squares fitter, from which +we obtain the initial values of the fit parameters. These parameters are then passed to +the SciPy’s curve fit function along with statistical errors on the ith bin according to +δNi = +� +Ni(1 − Ni/N). This returns an updated list of fit parameters along with the +parameter covariance matrix. The 1σ parameter fit values are given as the square root +of the diagonal elements of the covariance matrix. Errors on p–values are then estimated +– 32 – + +10−3 +10−2 +10−1 +100 +101 +q +10−2 +10−1 +100 +p(Wq) +W q +Figure 24. The q dependence of expected exclusion C.L., p(Wq) (blue dots joined by a solid line), +at which the CP conserving hypothesis is excluded for N = ¯N = 103 event sample sizes, obtained +by performing ensemble averages over Ne = 500 CPV datasets, with the blue band indicate the 1σ +spread of p−values over the ensemble. +by considering the one sigma confidence bands on the SF distribution as shown in Fig. +7. +From the fit of the PDF we compute the survival factor distribution, SF=1−CDF, +from which one can directly read off the p−value with which the no CPV hypothesis is +excluded, for each value of the measured W exp +q +, as seen in Fig. 4. For many of the CPV +datasets we consider the value of the statistic W exp +q +falls well outside the range for which +the master distribution was computed. For these cases we use the fit to extrapolate to +smaller p−values, where the error on the extrapolation is estimated from errors on the fit +parameters as described above. +The results of the above procedure for B0 → K+π−π0 decay samples of size N = ¯N = +103 are shown in Figs. 22, 23 for the Wasserstein statistic Wq and windowed Wasserstein +statistic Iq for q = 0.1, 1, and 10. The CPC PDFs are iteratively fit to built-in contin- +uous distributions contained in the SciPy statistics library, as listed in the legend of the +corresponding panels. In most cases, the best fit is chosen according to the minimum of a +χ2 statistic, however, in cases where multiple distributions achieve similar χ2 values, the +distribution that best matches the tail of the distribution is chosen. In particular, for Iq +we use the skewnorm fit for the extrapolation to small p−values. +D.2 +Optimizing the q value +The Wasserstein distance weighting exponent parameter q, Eq. (2.1), may be tuned to +maximize the expected sensitivity to CPV in a particular distribution, such as the B0 → +K+π−π0 Dalitz plot. Such an optimization of course depends on the assumed model for +B0 → K+π−π0 decay amplitudes and in particular on the assumed values of the strong +and weak phases that are hard to calculate but can in principle be fit from data. +In the example shown in Fig. 24 we used the nominal toy model for CPV in B0 → +K+π−π0 Dalitz plot that we used throughout Sec. 3, with the amplitudes and phases +– 33 – + +for B0 and ¯B0 isobar models set to the central values of the measurement in Ref [2]. +Similarly, for the CPC datasets we use the central values of amplitudes and phases in the +B0 BaBar isobar model [2] for both B0 and ¯B0 decays. Fig. 24 shows the variation with +q of the expected C.L. p(Wq) for exclusion of the CPC hypothesis, given our CPV model, +for N = ¯N = 103 event sample sizes. The blue bands give a 1σ range of expected C.L. +exclusions as obtained form an ensemble of Ne = 500 CPV samples. We see that for q ≲ 0.1 +the expected p(Wq) remains unchanged when lowering q within the range considered, while +for higher q there is in general diminished sensitivity to CPV, with the exception of the +region around q ∼ O(10). We suspect that these ranges of q correspond to typical scales +in the problem, i.e., the typical widths of the resonances (relative to the mass of B quark), +but did not explore this hypothesis further. +In the numerics in the main text we chose q = 0.1, which roughly optimizes the +sensitivity to CPV, but show in Sect. D.3 below also the results for q = 1, 10. +D.3 +Further results for q = 0.1, 1, 10 +In this appendix we list further results for Wasserstein distances with q = 0.1, 1, 10 both +for B and D decays, supplementing the results discussed in Sects. 3, 4. +Fig. 25 shows the CP asymmetry significance, ACP/δACP, where the error on the CP +asymmetry is given by +δACP = +� +1 − A2 +CP +N + ¯N . +(D.1) +The upper panels in Fig. 25 show the CP asymmetry for the case of B0 → K+π−π0 decay, +with our toy example CPV amplitude model (left panel) leading to clearly identifiable +regions in the Dalitz plot with CPV, and only noise in the Dalitz plot for the CPC case +(right). The difference between CPV and CPC decays is less pronounced in the D0 → +π+π−π0. Even so the Wasserstein distance based test statistics can still lead to exclusions +of CPC hypothesis (cf. Fig. 12, where the analysis was done for a sample size of N = ¯N = +105). +Fig. 26 shows the relative difference between the binned Wasserstein distance asymme- +try, Wq +CP, defined in Eq. (3.3), and the CP asymmetry, ACP, Eq. (3.1). This complements +Fig. 5 and Fig. 27, which show the actual values of the binned Wasserstein distance asym- +metry, Wq +CP, and the CP asymmetry, ACP for q = 0.1 and q = 1, 10, respectively. We see +that for the optimal value of q = 0.1 the binned Wasserstein distance asymmetry, Wq +CP +almost completely matches ACP up to ∼ 10% relative differences, where the differences are +even closer to just a few percent in the regions of the Dalitz plot where the CP asymmetry +significance is large, cf. Fig. 25. The Wq +CP still tracks well the CP asymmetry ACP, however +with exaggerated differences in the regions of the Dalitz plot with lower CP asymmetry +significance. +Fig. 28 shows the binned Wasserstein asymmetry Wq +CP and direct CP asymmetry ACP +for q = {1, 10} both for CPV and CPC B0 → K+π−π0 decays, where the same inputs +for the B0 decays were used as in the main text. The results shown were averaged over +an ensemble of Ne = 100 datasets, each containing N = ¯N = 103 events. This figure +– 34 – + +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +ACP CPV +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +ACP CPC +−0.04 +−0.02 +0.00 +0.02 +0.04 +Asymmetry Significance +−0.0100 +−0.0075 +−0.0050 +−0.0025 +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +Asymmetry Significance +0 +1 +2 +3 +m2(π−π0) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +m2(π+π0) +ACP CPV +0 +1 +2 +3 +m2(π−π0) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ACP CPC +−0.006 +−0.004 +−0.002 +0.000 +0.002 +0.004 +0.006 +Asymmetry Significance +−0.003 +−0.002 +−0.001 +0.000 +0.001 +0.002 +0.003 +Asymmetry Significance +Figure 25. The binned CP asymmetry significance ACP/δACP, with δACP given in Eq. (D.1), for +B0 → K+π−π0 (D0 → π+π−π0) Dalitz plot are shown in the top (bottom) panels for N = ¯N = +103(106) event samples averaged over an ensemble of Ne = 100 (Ne = 1) datasets with CPV toy +example show on the left and CP conserving datasets on the right. +supplements Fig. 5 for q = 0.1 in the main text. We see that in all cases, q = {0.1, 1, 10}, +the Wq +CP faithfully traces ACP for q = {1, 10} throughout the Dalitz plot, especially where +the CP asymmetries are statistically most significant. +Fig. 7 shows log(δWq) distributions and the difference between CPV and CPC log(δWq) +distributions for q = 1, 10. Compared to the q = 0.1 case, shown in Fig. 7, there is a more +pronounced deficit of δWq counts in CPV distribution relative to the CPC one for the +intermediate log(δWq). The window function w(x), Eq. (3.7), is therefore chosen to have +support both for the +1 (green bands in Fig. 7) and −1 (red bands) weights. +Fig. 29 is the q = 1, 10 complement of Fig. 8 in the main text. It shows, 500 distinct +datasets each with N = ¯N = 103 events, the confidence levels with which the CP conserving +hypothesis is excluded when applying different tests, either the energy test, giving CLs +denoted with p(T), the Wasserstein distance statistic test, giving p(Wq), or the windowed +Wasserstein distance statistic, giving p(Iq). The windows and anti-windows for q = 1, 10 +are shown in Fig. 28. For q = 10 the performance of the windowed Wasserstein distance is +comparable yet slightly less sensitive than the energy test, while for q = 1 the sensitivity +of windowed Wasserstein distance statistic is significantly reduced. For q = 10 the use of +windowed Wasserstein distance is comparable to the simple Wasserstein distance statistic, +– 35 – + +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +1 − Wq +CP / ACP CPV +q = 0.1 +−0.100 +−0.075 +−0.050 +−0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +1 − Ratio +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +1 − Wq +CP / ACP CPV +q = 1 +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1 − Ratio +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +1 − Wq +CP / ACP CPV +q = 10 +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1 − Ratio +Figure 26. We show 1 − Wq +CP/ACP for each bin within the two–dimensional Dalitz plot. Smaller +values indicate better agreement. +This is shown for CP violating B0 → K+π−π0 decays with +nbins = 50, q = {0.1, 1, 10}, and averaged over an Ne = 100 dataset ensemble where each dataset +contains N = ¯N = 103 B0 and ¯ +B0 events. +i.e., it does not lead to any real gain, while for q = 1 case, the selected windows and +anti-window reduce sensitivity of Iq compared to Wq. However, additional tuning of the +window and anti-window regions could be conducted to maximize significance. +E +Energy test +The energy test, introduced in [5], is an unbinned two-sample test utilizing a test statistic, +T, to analyze average distances between data points in phase space. The first proposal +to utilize the energy test in searches for CP violation was described in [6] and subsequent +analyses performed in [7, 8]. +The statistic utilizes a weighting (distance) function ψij dependent on the distance dij +between the ith and jth event in the first and second sample, respectively. For searches of +CP violation the two samples are distinguished by flavor (B0 and ¯B0). The test statistic +– 36 – + +is defined as [5, 6] +T = +N +� +i,j>i +ψij +N(N − 1) + +¯ +N +� +i,j>i +ψij +¯N( ¯N − 1) − +N, ¯ +N +� +i,j +ψij +N ¯N , +(E.1) +where N, ¯N denote the total number of events in the B0 and ¯B0 samples, respectively . The +weighting function ψij is chosen such that the weight decreases with increasing distance, +dij, between points. The summations in Eq. (E.1) are, from left to right, over B0 sample, +¯B0 sample and over both B0 (index i) and ¯B0 (index j) samples, respectively. The form +of the test statistic T is motivated by the form of the electrostatic energy for overlapping +distributions of positive and negative charges, in which case ψij ∝ 1/dij. If the two charge +distributions are exactly the same, the net charge distribution is zero, and T vanishes. +The functional form on the weighting function ψij can be freely chosen, for instance +in order to increase the sensitivity to local asymmetries at some typical length-scales, +minimizing dilutions due to averaging over large Dalitz plot areas. We follow Refs. [7, 8] +and choose a Gaussian weighting function +ψij ≡ ψ(dij; σ) = e−d2 +ij/2σ2, +(E.2) +where σ is a tunable parameter describing the effective radius between data points where +asymmetry is measured, while dij is the Euclidean distance in the Dalitz plot. +For events sampled from two identical distributions T is expected to fluctuate close to +zero, while for samples drawn from dissimilar distributions T will tend to a nonzero value. +To obtain the null hypothesis PDF for T we use the master method described in Sect. 3.1. +The labels for B0 and ¯B0 samples are ignored, and the N + ¯N events randomly assigned +to E and ¯E samples, each with N = ¯N events, thus simulating the CP even datasets. +Repeating this n times give a null hypothesis PDF for T, which is then fitted to a gamma +distribution, used finally to obtain the p−values corresponding to the “measured” value of +T. +The computation of CP conserving T distributions was done with the Manet software +package [8] (while for single computations our own implementation was used, see App. A). +The analysis was performed on N = ¯N = 103 B0 and ¯B0 events generated by AmpGen [21]. +The null hypothesis T-distributions were computed with N = 103 permutations, while the +tunable parameter in the weighting function, σ ≈ 0.2 GeV2, was chosen to maximize the +significance (minimize p−value) in the case of a CP violating sample. +– 37 – + +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Wq +CP CPV +q = 1 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +ACP CPV +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Wq +CP CPC +q = 1 +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +ACP CPC +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Wq +CP CPV +q = 10 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +ACP CPV +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +m2(K−π0) (GeV2) +Wq +CP CPC +q = 10 +0 +5 +10 +15 +20 +25 +m2(K−π+) (GeV2) +0 +5 +10 +15 +20 +25 +ACP CPC +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Asymmetry +Figure 27. Binned Dalitz plot comparison between the Wasserstein asymmetry Wq +CP (left) for +q = {1, 10} (shown in top two and bottom two rows) and direct CP asymmetry ACP (right), shown +for CP violating B0 → K+π−π0 decays (1st and 3rd row) and CP conserving decays, i.e., the +decays in which the asymmetries in the amplitude model were set to zero (2nd and 4th row). The +results shown are normalized and averaged over an ensemble of Ne = 100 datasets, each containing +N = ¯N = 103 events. +– 38 – + +0 +20 +40 +Counts +q = 1.0 +CPC +CPV +−20 +−18 +−16 +−14 +−12 +−10 +−8 +log(δWq) +−10 +0 +10 +Counts +CPV − CPC +Window +Window +0 +20 +40 +60 +Counts +q = 10 +CPC +CPV +−120 +−100 +−80 +−60 +−40 +−20 +log(δWq) +−40 +−20 +0 +20 +40 +Counts +CPV − CPC +Window +Window +Figure 28. The same as Fig. 7 but for q = {1, 10}. On the lower panels the green bands denote +the [δW win +min, δW win +max] ranges and the red bands the [δW win +min, δW win +max] range, each for the corresponding +q values. +– 39 – + +10−5 +10−2 +p(Iq) +10−5 +10−3 +10−1 +p(Wq) +CPV +q = 1 +ϵ = 0.080 +10−5 +10−2 +p(Iq) +10−5 +10−3 +10−1 +CPC +q = 1 +ϵ = 0.512 +10−11 +10−6 +10−1 +p(Iq) +10−11 +10−8 +10−5 +10−2 +p(T) +CPV +q = 1, ϵ = 0.024 +10−11 +10−6 +10−1 +p(Iq) +10−11 +10−8 +10−5 +10−2 +CPC +q = 1 +ϵ = 0.490 +10−5 +10−2 +p(Iq) +10−5 +10−3 +10−1 +p(Wq) +CPV +q = 10 +ϵ = 0.440 +10−5 +10−2 +p(Iq) +10−5 +10−3 +10−1 +CPC +q = 10 +ϵ = 0.470 +10−12 +10−7 +10−2 +p(Iq) +10−12 +10−9 +10−6 +10−3 +100 +p(T) +CPV +q = 10, ϵ = 0.214 +10−12 +10−7 +10−2 +p(Iq) +10−12 +10−9 +10−6 +10−3 +100 +CPC +q = 10 +ϵ = 0.504 +Figure 29. The comparison of CL exclusion of the CP conserving hypothesis either using energy +test (p(T)), the Wasserstein distance (p(Wq)), or the windowed Wasserstein distance (p(Iq)), with +the window function w(x) as denoted in Fig. 28. The top (bottom) two rows are for q = 1(10), with +500 distinct CP violating (conserving) B0 → K+π−π0 decay datasets shown on the right (left). +– 40 – + +References +[1] V. M. Panaretos and Y. Zemel, Statistical Aspects of Wasserstein Distances, Annual Review +of Statistics and Its Application 6 (Mar., 2019) 405–431, [1806.05500]. +[2] BaBar collaboration, J. Lees et al., Amplitude Analysis of B0 → K+π−π0 and Evidence of +Direct CP Violation in B → K∗π decays, Phys. Rev. D 83 (2011) 112010, [1105.0125]. +[3] I. Bediaga, I. I. Bigi, A. Gomes, G. Guerrer, J. Miranda and A. C. d. Reis, On a CP +anisotropy measurement in the Dalitz plot, Phys. Rev. D 80 (2009) 096006, [0905.4233]. +[4] BaBar collaboration, B. Aubert et al., Search for CP Violation in Neutral D Meson +Cabibbo-suppressed Three-body Decays, Phys. Rev. D 78 (2008) 051102, [0802.4035]. +[5] B. Aslan and G. Zech, New test for the multivariate two-sample problem based on the concept +of minimum energy, Stat. Comp. Simul. 75 (2004) 109–119, [math/0309164]. +[6] M. Williams, Observing CP Violation in Many-Body Decays, Phys. Rev. D 84 (2011) +054015, [1105.5338]. +[7] LHCb collaboration, R. Aaij et al., Search for CP violation in D0 → π−π+π0 decays with +the energy test, Phys. Lett. B 740 (2015) 158–167, [1410.4170]. +[8] C. Parkes, S. Chen, J. Brodzicka, M. Gersabeck, G. Dujany and W. Barter, On +model-independent searches for direct CP violation in multi-body decays, J. Phys. G 44 +(2017) 085001, [1612.04705]. +[9] P. T. Komiske, E. M. Metodiev and J. Thaler, The Hidden Geometry of Particle Collisions, +JHEP 07 (2020) 006, [2004.04159]. +[10] M. Crispim Rom˜ao, N. F. Castro, J. G. Milhano, R. Pedro and T. Vale, Use of a generalized +energy Mover’s distance in the search for rare phenomena at colliders, Eur. Phys. J. C 81 +(2021) 192, [2004.09360]. +[11] T. Cai, J. Cheng, N. Craig and K. Craig, Linearized optimal transport for collider events, +Phys. Rev. D 102 (2020) 116019, [2008.08604]. +[12] T. Cai, J. Cheng, K. Craig and N. Craig, Which metric on the space of collider events?, +Phys. Rev. D 105 (2022) 076003, [2111.03670]. +[13] P. T. Komiske, S. Kryhin and J. Thaler, Disentangling quarks and gluons in CMS open data, +Phys. Rev. D 106 (2022) 094021, [2205.04459]. +[14] C. E. Mitchell, R. D. Ryne and K. Hwang, Using kernel-based statistical distance to study the +dynamics of charged particle beams in particle-based simulation codes, Phys. Rev. E 106 +(2022) 065302, [2204.04275]. +[15] O. Kitouni, N. Nolte and M. Williams, Finding NEEMo: Geometric Fitting using Neural +Estimation of the Energy Mover’s Distance, 2209.15624. +[16] C. Villani, Optimal transport, old and new. Springer, Berlin, 2008. +[17] F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, +PDEs and Modeling. 2015. +[18] A. Ramdas, N. Garc´ıa Trillos and M. Cuturi, On wasserstein two-sample testing and related +families of nonparametric tests, Entropy 19 (2017) 47. +[19] P. T. Komiske, E. M. Metodiev and J. Thaler, Metric Space of Collider Events, Phys. Rev. +Lett. 123 (2019) 041801, [1902.02346]. +– 41 – + +[20] J. Weed and F. Bach, Sharp asymptotic and finite-sample rates of convergence of empirical +measures in wasserstein distance, 1707.00087. +[21] T. Evans, “Ampgen.” https://github.com/GooFit/AmpGen. +[22] LHCb collaboration, R. Aaij et al., Measurements of CP violation in the three-body phase +space of charmless B± decays, Phys. Rev. D 90 (2014) 112004, [1408.5373]. +[23] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau et al., +SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17 +(2020) 261–272. +[24] Y. Grossman, A. L. Kagan and Y. Nir, New physics and CP violation in singly Cabibbo +suppressed D decays, Phys. Rev. D 75 (2007) 036008, [hep-ph/0609178]. +[25] J. Brod, A. L. Kagan and J. Zupan, Size of direct CP violation in singly Cabibbo-suppressed +D decays, Phys. Rev. D 86 (2012) 014023, [1111.5000]. +[26] H.-n. Li, C.-D. Lu and F.-S. Yu, Branching ratios and direct CP asymmetries in D → PP +decays, Phys. Rev. D 86 (2012) 036012, [1203.3120]. +[27] E. Franco, S. Mishima and L. Silvestrini, The Standard Model confronts CP violation in +D0 → π+π− and D0 → K+K−, JHEP 05 (2012) 140, [1203.3131]. +[28] T. Feldmann, S. Nandi and A. Soni, Repercussions of Flavour Symmetry Breaking on CP +Violation in D-Meson Decays, JHEP 06 (2012) 007, [1202.3795]. +[29] H.-Y. Cheng and C.-W. Chiang, Direct CP violation in two-body hadronic charmed meson +decays, Phys. Rev. D 85 (2012) 034036, [1201.0785]. +[30] B. Bhattacharya, M. Gronau and J. L. Rosner, CP asymmetries in singly-Cabibbo-suppressed +D decays to two pseudoscalar mesons, Phys. Rev. D 85 (2012) 054014, [1201.2351]. +[31] D. Pirtskhalava and P. Uttayarat, CP Violation and Flavor SU(3) Breaking in D-meson +Decays, Phys. Lett. B 712 (2012) 81–86, [1112.5451]. +[32] A. Dery, Y. Grossman, S. Schacht and D. Tonelli, CP violation in b and c quark decays, in +2022 Snowmass Summer Study, 9, 2022. 2209.07429. +[33] LHCb collaboration, R. Aaij et al., Observation of CP Violation in Charm Decays, Phys. +Rev. Lett. 122 (2019) 211803, [1903.08726]. +[34] LHCb collaboration, Measurement of the time-integrated CP asymmetry in D0 → K−K+ +decays, 2209.03179. +[35] Y. Grossman, A. L. Kagan and J. Zupan, Testing for new physics in singly Cabibbo +suppressed D decays, Phys. Rev. D 85 (2012) 114036, [1204.3557]. +[36] A. Dery, Y. Grossman, S. Schacht and A. Soffer, Probing the ∆U = 0 rule in three body +charm decays, JHEP 05 (2021) 179, [2101.02560]. +[37] Y. Grossman and D. J. Robinson, SU(3) Sum Rules for Charm Decay, JHEP 04 (2013) 067, +[1211.3361]. +[38] BaBar collaboration, B. Aubert et al., Measurement of CP Violation Parameters with a +Dalitz Plot Analysis of B± → D(pi+ π−π0 ) K±, Phys. Rev. Lett. 99 (2007) 251801, +[hep-ex/0703037]. +[39] J. Back, T. Gershon, P. Harrison, T. Latham, D. O’Hanlon, W. Qian et al., Laura ++: A +dalitz plot fitter, Computer Physics Communications 231 (2018) 198–242. +– 42 – + +[40] S. Helgason, Integral Geometry and Radon Transforms. Springer, New York, 2015. +[41] K. Atasu and T. Mittelholzer, Linear-complexity data-parallel earth mover’s distance +approximations, in Proceedings of the 36th International Conference on Machine Learning +(K. Chaudhuri and R. Salakhutdinov, eds.), vol. 97 of Proceedings of Machine Learning +Research, pp. 364–373, PMLR, 09–15 Jun, 2019. +– 43 – + diff --git a/q9FPT4oBgHgl3EQf9TVT/content/tmp_files/load_file.txt b/q9FPT4oBgHgl3EQf9TVT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ca6243639281d0c852d6301070f81a5eae2ceaf --- /dev/null +++ b/q9FPT4oBgHgl3EQf9TVT/content/tmp_files/load_file.txt @@ -0,0 +1,1655 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf,len=1654 +page_content='Prepared for submission to JHEP Earth mover’s distance as a measure of CP violation Adam Davis,a Tony Menzo,b Ahmed Youssef,b Jure Zupan,b aSchool of Physics and Astronomy, University of Manchester, M13 9PL, Manchester, UK bDepartment of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA E-mail: adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='davis@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='uk, menzoad@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='edu, youssead@ucmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='edu, zupanje@ucmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='edu Abstract: We introduce a new unbinned two sample test statistic sensitive to CP viola- tion utilizing the optimal transport plan associated with the Wasserstein (earth mover’s) distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The efficacy of the test statistic is shown via two examples of CP asymmetric distributions with varying sample sizes: the Dalitz distributions of B0 → K+π−π0 and of D0 → π+π−π0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The windowed version of the Wasserstein distance test statistic is shown to have comparable sensitivity to CP violation as the commonly used energy test statistic, but also retains information about the localized distributions of CP asymmetry over the Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For large statistic datasets we introduce two modified Wasserstein distance based test statistics – the binned and the sliced Wasserstein distance statistics, which show comparable sensitivity to CP violation, but improved computing time and memory scalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Finally, general extensions and applications of the introduced statistics are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='13211v1 [hep-ph] 30 Jan 2023 Contents 1 Introduction 1 2 Earth mover’s distance as a measure of CPV 3 3 Application to three body B decays 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 Testing for bias in the permutation method 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Tracing CP violating phase space regions using EMD 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 The windowed EMD 11 4 Application to three body D decays 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 Binned Wasserstein test 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Sliced Wasserstein test 19 5 Conclusions 22 A Public code EMD4CPV 25 B The optimal transport problem 26 C EMD analysis of Gaussian distributions 27 D Details on EMD–test for three body decays 31 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 The p−value error analysis 31 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Optimizing the q value 33 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 Further results for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10 34 E Energy test 36 1 Introduction The Wasserstein distance or earth mover’s distance (EMD) is a measure of similarity be- tween two probability distributions, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', [1] as well as App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The value of the EMD can be visualized as the work required to transport and reshape dirt (weighted samples) in the form of one distribution into the form of a second distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Similar distributions result in smaller values (≈ zero) of the EMD while dissimilar distributions result in larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The EMD is thus sensitive to density asymmetries between samples and therefore well suited to be used as a test statistic that quantifies the amount of CP violation (CPV) in a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 1 – Taking as an example the B0 decays to a final state f, the direct CP asymmetry Af is defined as Af = Br( ¯B0 → f) − Br(B0 → ¯f) Br( ¯B0 → f) + Br(B0 → ¯f), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) where ¯f = CP(f) is the final state CP conjugated to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For two body B0 decays such as B0 → K+π−, the direct CPV is fully characterized by Af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the rest frame of the parent particle, the two final state particles are back to back and there is no dependence of the decay rate on their emission angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Direct CPV is then simply given by the difference of observed B0 → f and ¯B0 → ¯f decays as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is not the case, however, for three-body decays such as, for instance, B0 → K+π−π0 and its CP conjugate mode ¯B0 → K−π+π0 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In addition to the integrated CPV quantity, Af, there is a continuous set of CP violating observables, namely the phase space dependent differential CP asymmetries ACP(s12, s13) = �dΓ( ¯B0 → ¯f) dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' − dΓ(B0 → f) dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' ���dΓ( ¯B0 → ¯f) dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' + dΓ(B0 → f) dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) where s12, s13 are the two Dalitz plot variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' To measure ACP one can bin the Dalitz plot in large enough bins such that they contain reasonably large numbers of events, say ni, ¯ni ∼ O(20), and define Af,i, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In this way one could probe experimentally, if CP violation is present in the Dalitz plot distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Such an approach is not optimal, however, since the measurements depend on the choice of the binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' If the primary goal is to test for the presence of phase space dependent CPV in the Dalitz plot distributions, not just in global Af, two tests were put forward that improve on the binning method, the SCP test (or the Miranda method) [3, 4] and the energy test [5–8], both of which have some drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The SCP test still relies on a binning procedure that, like ACP, leads to some loss of sensitivity to CPV and the energy test, while being quite sensitive to the presence of CPV in the Dalitz plot distributions, is harder to interpret in terms of the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In this paper we propose an alternative approach – the use of the Wasserstein distance, or EMD, as a measure of CPV in the Dalitz plot distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' As we show below, the EMD based statistic combines the high sensitivity to CPV with easier interpretability, since it retains information about which part of the Dalitz plot the CPV originates from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The use of EMD in measuring CPV is reminiscent but distinct from the use of EMD to quantify the similarities between different LHC events, advocated in [9–12] (see also the related results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [13–15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In particular, the optimal EMD based statistic for CPV involves reweighting (or filtering) of individual datapoint contributions to the EMD as we discuss in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 2 we review the Wasserstein distance and introduce the relevant notation for its application to three body B and D decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3 we analyze three body B0 → K+π−π0 decays and show that the Wasserstein distance is a sensitive probe of CP violation and introduce an optimized windowed Wasserstein distance statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4 we introduce two further Wasserstein distance based statistics, the binned Wasserstein distance and the sliced Wasserstein distance, which have improved computing complexity scalings and may be preferred when dealing with large datasets such – 2 – as the three body D decay data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We draw conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5, while appendices contain details about the public code EMD4CPV (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' A), on the computation complexity of the optimal transport problem (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' B), further examples for EMD using Gaussian distributions (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' C), further results for probing B → Kππ Dalitz plot CP asymmetries using Wasserstein distance based statistics (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D), and a review of the energy test (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 2 Earth mover’s distance as a measure of CPV The Wasserstein distance, Wq(E, ¯E), between the distributions of events, E, in B0 → K+π−π0, and the distribution ¯E of ¯B0 → K−π+π0 decays is given by, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', [9, 16–18], Wq(E, ¯E) = � min {fij≥0} N � i=1 ¯ N � j=1 fij � ˆdij �q �1/q , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) where q ∈ (0, ∞), with q = 1 defining the EMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 The minimization is over the weights N � i=1 fij = 1 ¯N , ¯ N � j=1 fij = 1 N , N, ¯ N � i,j=1 fij = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) where N( ¯N) are the number of events in sample E( ¯E), and ˆdij is the distance between the two events, i in E, and j in ¯E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The interpretation of Wq(E, ¯E) is the cost incurred by moving in an optimal way the probability distribution corresponding to events in E into the probability distribution of event ¯E, where the penalty is the distance ˆdij between the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Assuming that N = ¯N, so that that there is no integrated CP asymmetry, and that E and ¯E come from the same distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' no CPV in distributions), then Wq(E, ¯E) → 0 for large N = ¯N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In contrast, if E and ¯E differ (there is CPV), then Wq(E, ¯E) will tend to a nonzero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For d−dimensional final phase space the parametric upper bound is ⟨Wp(E, ¯E)⟩ ≲ CN−1/d [20], with C a constant that does not depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 For the Dalitz plot we have d = 2 since it is fully described by two Dalitz variables, s12, s13, and thus ⟨Wp(E, ¯E)⟩ ∝ 1/ √ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', it scales in the same way as the variance of the global direct CP asymmetry δAf ∝ 1/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since we are mainly interested in CPV in distributions, we will assume for simplicity that N = ¯N in the rest of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, the analyses we present below extend trivially to the N ̸= ¯N case, with Wq still probing the CPV in distributions and Af the integrated CPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 1In most works on the optimal transport q is restricted to the convex cost functions, q ∈ [1, ∞), such that its gradient is well defined everywhere, also at the ˆdij = 0 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' An extension to the concave case, q ∈ (0, 1), requires an introduction of an approximate gradient, however, a unique optimal transport still exists, see the discussion in chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The network simplex algorithm as implemented in the Wasserstein Python library [9, 19] can then be used without change to solve the optimal transport problem, in the same way as for q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 2Note that for decays that are dominated by intermediate resonances the effective dimensionality is lower than the full dimensionality of the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is for a multibody decays where at most two resonances overlap we expect the same scaling as for the Dalitz plot ⟨Wp(E, ¯E)⟩ ≲ CN −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 3 – −50 0 50 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 Counts N(¯µ = −20, σ = 10) N(µ = 20, σ = 10) E E 0 20 40 60 Wq 0 100 200 300 400 Counts q = 1 N=10, ∆µ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 N=10, ∆µ =40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The nonzero displacement ∆µ of the two Gaussian distributions (left) can be probed by using Wq, q = 1, as the test statistic (right), see main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For the 3D Dalitz plot we use the definition of the (dimensionless) distance ˆdij that is symmetric in the Dalitz variables, s12, s13, s23, ˆdij ��� Dalitz = 1 m2 ���s12(i) − ¯s12(j) ��r + ��s13(i) − ¯s13(j) ��r + ��s23(i) − ¯s23(j) ��r�1/r , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) where, for example in the B0 → K+π−π0 system, m = mB, and s12 = (pK+ + pπ−)2, s13 = (pK+ + pπ0)2, s23 = (pπ− + pπ0)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4) ¯s12 = (pK− + pπ+)2, ¯s13 = (pK− + pπ0)2, ¯s23 = (pπ+ + pπ0)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5) parametrize the B0 → K+π−π0 Dalitz plot and the CP conjugate variables in ¯B0 → K−π+π0 Dalitz plot, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The normalization prefactor 1/m2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) was chosen such that ˆdij < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We use the Euclidean distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', r = 2, in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Other r–values were investigated but no significant changes to the sensitivity of CP violation were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Before discussing the more complicated case of B and D decays, let us first briefly con- sider a simpler toy example of two displaced Gaussian distributions, G(x) = N(x|∆µ/2, σ) and ¯G(x) = N(x| − ∆µ/2, σ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', two Gaussian distributions with equal widths, σ, but with their centers at µ = ∆µ/2 and ¯µ = −∆µ/2 and thus displaced by ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In this toy example the question about CPV in multibody B decays is replaced with a test whether or not ∆µ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Drawing N = 10 events E from G, as well as ¯N = 10 events ¯E from ¯G, and taking ˆdij in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) to be the Euclidean distance in 1D, gives a W1 that is clus- tered around ⟨W1⟩ ≃ ∆µ, see the grey distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is appreciably larger than the distribution of W1 values for ∆µ = 0 (blue), even for relatively small event samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' C we show more illustrations of how the W1 probes a difference between distributions, including an example of displaced 2D Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In particular, we show numerically that W1 can be used as a statistic, and that the CL intervals obtained from a known ∆µ = 0 probability distribution for W1 coincide with the expected exclusion intervals from negative log likelihood for ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 4 – 0 10 20 m2(K−π+) 0 5 10 15 20 25 m2(K−π0) B0 → K−π+π0 0 1 2 3 m2(π−π0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 m2(π+π0) D0 → π−π+π0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The 2–dimensional B0 → K+π−π0 (left) and D0 → π−π+π0 (right) Dalitz plots and their respective 1–dimensional histograms along the borders for 106 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3 Application to three body B decays As the first realistic example of using the Wasserstein distance to test for CP violation we use the B0 → K+π−π0 and the CP conjugate ¯B0 → K−π+π0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The events are generated from the amplitude model by BaBar [2] implemented in the AmpGen [21] framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We create two data samples: the CP conserving (CPC) and the CP violating datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For the CPC datasets we use the central values of amplitudes and phases in the B0 BaBar isobar model [2] for both B0 and ¯B0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For the CPV datasets, on the other hand, the amplitudes and phases for B0 and ¯B0 isobar models differ and are set to the central values of the measurements in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The B − ¯B meson mixing is ignored in the generation of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The resulting B0 → K+π−π0 Dalitz plot with 106 events is shown in Figure 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For three-body B decays we highlight the use of Wq( ˆdij) on the low statistic datasets containing N = 103 events in each of the samples, the B and ¯B decays (N = ¯N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This choice was made to roughly match the reported experimental sensitivity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The im- plementation and computation of the Wasserstein distance is done in two steps: first, the distances ˆdij, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3), are computed using the cdist method within the SciPy framework [23] which utilizes optimized C code to efficiently compute the distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The computa- tions of Wq( ˆdij) and the extraction of optimal transport data is then obtained using the EMD class within the Wasserstein library [9, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' There are two continuous parameters in the definition of Wq( ˆdij), r and q, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These can be chosen such that the sensitivity to CPV is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The optimal value of q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 was chosen by finding, for r = 2, the minimum average CL p–value for which the CPC hypothesis is excluded given the toy model CPV Dalitz plot distributions, as obtained from an ensemble of Ne = 500 distinct datasets generated from the BaBar model [2], see further details in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the – 5 – 0 500 1000 PDF q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 Master Permutation 10−3 10−1 SF Master Permutation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0100 Wq 0 2 4 Ratio Master/Permutation Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The Wq distribution function (PDF, top panel), and the survival factor (SF=1-CDF, where CDF is the cumulative distribution function, middle panel) obtained from the permutation method (orange) compared to the true CP conserving distribution (the master method, in blue), while the bottom panel shows the ratio of the SF obtained using the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Each distribution consists of 103 Wq values with the solid curves and bands representing the average ±1σ ranges for the bin counts obtained over 10 distinct distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' analyzed examples, changing r in the definition of the distance Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) did not lead to significant changes in the sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Thus, in the numerical results below we use the opti- mized values {r = 2, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1}, while in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D we also show the results for the non-optimal choices, {r = 2, q = 1} and {r = 2, q = 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' To determine the p−value with which the CPC hypothesis is excluded for the particular CPV Dalitz plot sample, one needs the Wq probability distribution functions (PDF) for the CPC Dalitz plot distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the experiment one can determine the CPC PDF using the permutation method, which, as we show next, is estimated to lead to only a relatively small bias compared to the true CP conserving PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 Testing for bias in the permutation method In order to assign a p−value with which the CPC hypothesis is excluded, given two samples of B and ¯B decays, one first calculates the Wasserstein distance between the two, W exp q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This encodes the dissimilarity between the two distributions of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, the value – 6 – of W exp q by itself is not particularly informative, except that smaller W exp q values indicate more similar distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For a quantitative assessment of CPV we need the distribution of Wq for the CP conserving case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We obtain this using two methods: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' using the permu- tation method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', by permuting the original B and ¯B samples (which have non-zero direct CPV) and then calculating Wq for each such permutation and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' using the master method, which is the true CP conserving PDF given our assumptions: we generate an ensemble of B and ¯B decay event samples, using the B decay model for both, and then calculate the corresponding Wq probability density function (that is, we assume for simplicity that all the CP violating phases reside in the ¯B0 decay amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The permutation method can be implemented with experimental data, since it involves only the measured B and ¯B event samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The master method, on the other hand, is only possible given a theoretical model of the decay amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The PDFs for the two methods, the permutation (orange) and master (blue), are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3, as obtained from an ensemble of Ne = 10 datasets containing N = ¯N = 103 events in each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that the permutation method is a very good approximation of the true CP conserving PDF for Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Such a test of a possible bias in the permutation method can be performed for any multibody B decay (or any multibody distribution in general) for which a reasonable description is available in terms of a resonance amplitude model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' One can also test for a potential bias in the permutation method using only experi- mental data, but in this case only for N that corresponds to a fraction, for instance half, of the measured sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, from data one can construct several distinct hy- potheses for the CP conserving Wq PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The first CP conserving Wq PDF hypothesis can be constructed by randomly splitting the measured B decay sample into two halves and calculating the corresponding distribution of Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' An alternative CP conserving Wq PDF hypothesis is similarly obtained by randomly splitting the measured ¯B decay sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These can then be compared to the Wq PDF that is obtained using the permutation method (but again using only half of the measured B and ¯B decay samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The differences between the three PDFs should be a good proxy for the size of the possible bias in the permutation method when applied to the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the numerical results below we use the master method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', the true CP conserving PDF for Wq shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4, even though this is not accessible from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This choice was done for numerical expediency, and we expect it to introduce only small bias in the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Tracing CP violating phase space regions using EMD A benefit of the Wasserstein distance based statistic is that it traces in a straightforward fashion the variation of the CP asymmetry across the Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The standard definition of direct CP asymmetry, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), also applies to the differential distributions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2), repeated here for convenience, ACP(s12, s13) = d¯Γ(¯s12, ¯s13) − dΓ(s12, s13) d¯Γ(¯s12, ¯s13) + dΓ(s12, s13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) – 7 – 0 500 1000 PDF q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 Wq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='010 Wq 10−4 10−1 SF ¯p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='005+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='038 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The Wq probability distribution function (PDF), the cumulative distribution function (CDF), and the survival factor (SF=1-CDF) for the CPC case and r = 2, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, obtained using the master method with a fit to the Johnson’s SU distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The orange bands (blue band on top panel) denote the ±1σ fit errors (statistical errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The vertical red line (band) denotes the average Wq value (the ±1σ Wq ranges) obtained from 103 CPV datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that, on average, the CPC hypothesis is in this example excluded at the ∼ 3σ level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', with a p−value of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' where dΓ(s12, s13) is the B0 → K+π−π0 partial decay width into the region of the Dalitz plot with s12 = (pK+ + pπ−)2 ≡ m2(K+π−), s13 = (pK+ + pπ0)2 ≡ m2(K+π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Similarly, d¯Γ(¯s12, ¯s13) is the CP conjugate partial decay width for ¯B0 → K−π+π0, with ¯s12 = (pK− + pπ+)2 ≡ m2(K−π+), ¯s13 = (pK− + pπ0)2 ≡ m2(K−π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The binned version of the CP asymmetry ACP for the CP violating dataset, where we used the central values of the parameters for the BaBar amplitude model from [2], is shown in the upper-right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The lower-right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 shows the binned ACP for the CP conserving case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', assuming that the B0 → K+π−π0 inputs in the amplitude model [2] apply to both the B0 and ¯B0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 show expected CP asymmetries in each bin, obtained by averaging over an ensemble of Ne = 100 datasets containing N = ¯N = 103 B and ¯B pairwise samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Next, we define the Wasserstein asymmetry utilizing the Wasserstein statistic Wq, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We denote the contribution to Wq from each datapoint i in the B0 Dalitz plot as δWq(i), and likewise δ ¯Wq(¯i) denotes the contribution from datapoint ¯i in the ¯B0 Dalitz plot, such that W q q = � i δWq(i) = � ¯i δ ¯Wq(¯i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) – 8 – 0 5 10 15 20 25 0 5 10 15 20 25 m2(K−π0) (GeV2) Wq CP CPV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0 5 10 15 20 25 0 5 10 15 20 25 ACP CPV 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) Wq CP CPC q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 ACP CPC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Binned Dalitz plot comparison between the Wasserstein asymmetry Wq CP (left) and direct CP asymmetry ACP (right), shown for CP violating B0 → K+π−π0 decays (top) and CP conserving decays (bottom), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', decays in which the asymmetries in the amplitude model were set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The results shown are normalized and averaged over 100 datasets, each containing 2N = 2 × 103 (B and ¯B) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We define the binned Wasserstein asymmetry Wq CP within each bin in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 as Wq CP(s12, s13) = � ¯i δ ¯Wq(¯i) − � i δWq(i) � ¯i δ ¯Wq(¯i) + � i δWq(i), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) where the summation over i (¯i) is only over the data-points contained in the bin centered at (s12, s13) (the CP conjugated ¯B0 bin centered at (¯s12, ¯s13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' By construction, Wq CP vanishes when summed over the whole Dalitz plot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', when there is only one bin encompassing the whole Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The Wasserstein asymmetry Wq CP is also statistically consistent with zero in the regions of the Dalitz plot that have vanishing CP asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Comparison of left and right panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 shows that Wq CP faithfully traces the variation of ACP over the Dalitz plot, including the statistical fluctuations, most readily visible in the CP conserving datasets shown in the lower panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This makes the Wq CP easily interpretable in terms of the underlying physics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', which components of the resonant structure contribute most to the CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 9 – 10−11 10−6 10−1 p(Wq) 10−11 10−8 10−5 10−2 p(T) q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='304 10−11 10−6 10−1 p(Wq) 10−11 10−8 10−5 10−2 p(T) q = 1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='100 10−11 10−6 10−1 p(Wq) 10−11 10−8 10−5 10−2 p(T) q = 10, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='264 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The scatter plot of p–values at which the CPC hypothesis is excluded, for the ensemble of Ne = 500 samples with N = 103 B0 → K+π−π0 (and ¯N = 103 CP conjugated ¯B0 → K−π+π0) decays, calculated using either the Wq or T statistics (dots), with 1σ fit error bars shown as lines, and setting q = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10} (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The fraction ϵ of points above the p(Wq) = p(T) diagonal line denotes the fraction of ensembles for which Wq is more sensitive to CPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The dotted gray lines (solid bands) show the average (1σ ranges of) p−values for the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The advantage of Wasserstein distance over direct CP asymmetry, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), as a measure of CP violation in the Dalitz plot distributions is that Wq does not require binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' It is a global quantity that encodes the cumulative differences between the B0 and ¯B0 Dalitz plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' As such it can be used as a statistic sensitive to the CP violating Dalitz plot distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 6 we compare the sensitivity of Wq to CPV relative to another such unbinned statistic, the energy test statistic T [5, 6, 8], see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E for further details on the energy test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The energy test has already been successfully applied to search for CPV in multibody decays [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' On the other hand, we do not show comparisons with the SCP test, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' the Miranda method [3, 4], which uses optimized bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In our numerical studies we found the SCP test to always be less sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 6 we see that the Wasserstein distance and the energy test have comparable sensitivity to CPV, but with Wq somewhat less sensitive on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This can be quantified by introducing ϵ ≡ 1 Ne Ne � i=1 � +1 pi(Wq) < pi(T), 0 otherwise, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4) where Ne = 500 is the ensemble size for which the CPC exclusion CL p−values were obtained either using the Wq (giving p(Wq)) or the T statistic (giving p(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, ϵ gives the fraction of randomly sampled datasets for which Wq statistic leads to stronger sensitivity to CPV than the energy test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since ϵ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 one may conclude that Wq is on average less sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, the average p–values for Wq and T test statistics (dashed lines) agree within 1σ ranges (gray bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Similarly, many scatter points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 6 agree with the p(Wq) = p(T) line within the error bars that are reflecting the uncertainties with which the p−values were determined from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, for small p−values, p ≲ O(10−4), we estimate the significance of the exclusion using an extrapolation of a fit to corresponding PDFs, where the fit distributions are chosen according to the minimization of a χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The – 10 – energy test statistic is fit with a gamma distribution while for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, we fit the Wq master distribution with Johnson’s SU distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Errors are assigned according to the 1σ bands on the respective fit parameters, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The ϵ ratio does not take into account the error associated with our estimates of the p−value for each statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These errors can be large especially for small p-values, and as such ϵ should only be used as a cautious measure of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The T statistic has a continuous parameter, σ, which defines the scale of correlations probed by the energy test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 6 the value of σ was set to its (close to) optimal value σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 GeV2, for which the energy test on average leads to the smallest expected p−values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Similarly, the parameter q in Wq was optimized, with the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 6 shown for close to optimal value q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Note that in the actual experiment the above optimization should be performed on the mock data, using a model for B → Kππ decay amplitudes, and not on actual experimental data, in order not to introduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' If the amplitude model does not describe well the data, this would lead to suboptimal choice for the continuous parameter and reduced sensitivity to CPV, but otherwise is not problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We expect that the somewhat reduced sensitivity of Wq to CPV compared to the energy test is because Wq also receives contributions from areas in the Dalitz plot that are CP conserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is in contrast to the energy test statistic T, which has a vanishing expectation value in those areas regardless of the number of events in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The contributions to Wq from these regions, on the other hand, only slowly tend to zero with increasing sample size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, Wq may be written as the sum of two contributions Wq = � i δWq(i) = � i � δW signal q (i) + δW noise q (i) � where lim N→∞ � i δW noise q (i) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5) The term δW noise q comes from CP conserving regions of the Dalitz plot, while δW signal q is due to the presence of CPV and tends to a nonzero value for N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' If the signal and noise contributions preferentially occur at different length scales, one can construct a modified Wasserstein distance test with higher sensitivity to CPV, as shown in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 The windowed EMD As discussed above, the disadvantage of the Wasserstein distance as a CPV test statistic is that, because all δWq(i) are positive, it includes an abundance of small nonzero con- tributions even in the absence of CPV, generating a long–tailed CP conserving PDF for Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Within the Dalitz plot, CP violation manifests as local density differences between the B and ¯B datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' If this CPV is either localized and/or relatively small, such as in B0 → K+π−π0 Dalitz plots, this translates into relatively small differences in the δWq(i) distributions between CPV and CP conserving B0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7 (top), which shows binned counts of log(δWq), averaged over the ensemble of Ne = 103 CPC (blue) and Ne = 103 CPV (orange) samples, each containing N = ¯N = 103 events, with the bands denoting the 1σ ranges for bin counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7 (bottom) shows the difference between the average CPC and CPV bin counts, as – 11 – 0 20 40 Counts q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 CPC CPV −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='8 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 log(δWq) −10 0 10 Counts CPV − CPC Window Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Top: joined points (bands) represent the histogram of average (1σ range) log(δWq) counts for 100 bins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', the binned counts of the log of pairwise optimal transport distances between B and ¯B sample events for N = ¯N = 103 sample sizes, averaged over ensemble of Ne = 103 samples, for CPC (blue) and CPV (orange) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bottom: joined points in purple (purple band) denote the average (1σ ranges of) differences between CPC and CPV log(δWq) counts, with the green band denoting the [δW win min, δW win max] range for which sample events are used with positive weights in the construction of the windowed Wasserstein distance statistic, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7) (in this example the range [δW win min, δW win max] for negative weights is taken to be zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' well as the 1σ ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that the δWq distributions for CPC and CPV cases overlap significantly in many regions of pairwise δWq values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, we also expect the CPC distributions to be more likely to lead to smaller δWq, given that the B0 and ¯B0 Dalitz plot are more similar than in the CPV cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Consequently, for the CPV case one would expect an excess of datapoints with larger δWq and a related excess of CPC bin counts at smaller δWq values, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Depending on the details of the Dalitz plot the δWq distributions could exhibit other differences between the CPC and CPV cases not present in the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For instance, if CPV is localized in a small region of the Dalitz plot containing n events and of size ˆd, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3), then we would expect an excess of CPV δWq bin counts over CPC in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7 at δWq ∼ O( ˆd/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Once one sums over all δWq(i), and considers only the global Wasserstein distance Wq = � i Wq(i) as a measure of CPV, the information about such differences in the δWq distributions is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since there is more information in the δWq(i) distributions than in the global Wq – 12 – 10−12 10−7 10−2 p(Iq) 10−12 10−9 10−6 10−3 100 p(Wq) CPV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='866 10−12 10−7 10−2 p(Iq) 10−12 10−9 10−6 10−3 100 CPC q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='484 10−15 10−9 10−3 p(Iq) 10−15 10−11 10−7 10−3 p(T) CPV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='566 10−15 10−9 10−3 p(Iq) 10−15 10−11 10−7 10−3 CPC q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='430 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The comparison of estimated p-value exclusions of CP conserving hypothesis for CPV (left) and CPC (right) B0 → K+π−π0 decays, comparing the windowed Wasserstein distance Iq with either the global Wasserstein Wq (top) or the energy test T (bottom) statistic, for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, on 500 distinct datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' observable, we can define an improved statistic Iq Iq ≡ � i w � δW win min, δW win max, δW win min, δW win max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' δWi � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6) where for the example of B → Kππ decays we define the window function as w(x) = � � � � � � � +1 x ∈ [δW win min, δW win max], −1 x ∈ [δW win min, δW win max], 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7) The window function w splits datapoints into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The events in the high δWq values window δWq ∈ [δW win min, δW win max], and the events in the anti-window of mid-range δWq values, δWq(i) ∈ [δW win min, δW win max], are included in the windowed Wasserstein distance statistic Iq, but weighted with opposite signs, thus enhancing the difference between the CPC and CPV distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The remaining events, for which the CPC and CPV δWq distributions do not differ significantly, are instead not included in Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Keeping these events would only dilute the sensitivity to CPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 13 – 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) Iq CP CPV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 ACP CPV −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The comparison of binned Dalitz plot asymmetry: the windowed Wasserstein asymmetry Iq CP (left) and the fractional CP asymmetry ACP (right), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' also top panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' When compared with the asymmetry significance’s shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 25 we see that the chosen window is correctly filtering CP conserving δWq values and retaining δWq values in the most significant regions of CPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The optimization of window and anti-window ranges requires a model for B0 and ¯B0 amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Importantly, the δWq values depend on the sample size N = ¯N, and thus the optimization should be performed for the number of events actually measured in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' One could attempt a data driven optimization of w by splitting the measured dataset into subsets, correcting for the effect of smaller sample sizes, but we did not explore this further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For other decay channels, depending on the actual decay width distributions, other forms of window function could be better suited than the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For instance, one could define multiple disjoint window and anti-window regions, or use weights that are smooth functions of δWq, not just the discrete values {−1, 0, +1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For B0 → K+π−π0 Dalitz plot and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, N = ¯N = 103, there is on average an excess of CPC over CPV δWq distributions in the mid-value region log(δWq) ∈ (−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='55, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, it is accompanied with a large variability in bin counts, and thus for this case it proves advantageous to define Iq using only events in the window shown as the green band in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7, and drop all other events (that is, the anti-window range is shrunk to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For other values of q both window and anti-window ranges are nonzero, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 8 shows, for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, N = ¯N = 103, the comparison of p-values at which the CPC hypothesis is excluded, when either the windowed Wasserstein statistic Iq or the global Wasserstein statistic Wq are used, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 8 (top), or if the energy test statistic, T, is used instead, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 8 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that the windowed Wasserstein distance statistics, Iq, is as sensitive, or even slightly more sensitive, to the presence of CPV in the Dalitz plot distributions than the energy test, while both outperform the global Wasserstein distance statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 8 also demonstrate that Iq, like Wq and the T test statistic, does not introduce bias when CPC distributions are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' As an additional confirmation that the chosen windows are in fact selecting the relevant areas of the Dalitz plot associated with CPV and – 14 – CPC we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 9 the binned CP and Wasserstein asymmetries, but in the later only keeping the events that contribute to Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, we define Iq CP(s12, s13) = � ¯i w(δ ¯Wq(¯i)) − � i w(δWq(i)) � ¯i w(δ ¯Wq(¯i)) + � i w(δWq(i)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='8) where each event is weighted according to the window function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The summation over i (¯i) is only over the data-points contained in the bin centered at (s12, s13) (the CP conjugated ¯B0 bin centered at (¯s12, ¯s13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The comparison of left and right panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 9 shows that the chosen window from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7 does indeed correctly select the regions of the Dalitz plot exhibiting CP violation and acts as a filter to better resolve CP asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The shown results could be improved further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' First of all, we did not perform a full optimization of the window function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7), but rather only selected among several discrete, manually chosen, forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' It would also be interesting to explore if the features ob- served in the δWq distributions, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7, can further inform amplitude models, in particular about the existence of CPV regions with resonances interfering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4 Application to three body D decays Next, we apply the analysis to larger datasets with small but nonzero amount of CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' As a concrete example we consider the three body D decay D0 → π+π−π0 and its CP conjugated channel, ¯D0 → π−π+π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The CP violation in D decays is expected to be small, parametrically suppressed by O(VcbVub/VcdVud) ∼ 10−3 [24–32] and has only recently been measured to be nonzero [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Further searches for CP violation within the charm sector are highly motivated, since the discovery of enhanced CPV in specific modes, including multibody decays, could point to a discovery of new physics (for sum rules that the SM needs to satisfy see [35–37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The D0 → π+π−π0 decays have been studied at the LHCb using the energy test, and found that the CPC hypothesis is excluded at the p = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5)% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Below, we show how the Wasserstein distance based statistics could be used as alternative analysis strategies to search for CPV in this and other multibody charm decays, taking D0 → π+π−π0 as a toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We generate the two datasets, for D0 → π+π−π0 and ¯D0 → π−π+π0 decays, using the BaBar amplitude model [38] implemented within the Laura++ framework [39], similarly to the case of B0 → K+π−π0 decays discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' As a toy example of CP violation in the D0 → π+π−π0 Dalitz plot we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [8] (where this was used to explore the sensitivity of the energy test), and increase for the generation of CPV datasets the fit fraction of the ρ(770)− by 2% and the phase of the corresponding decay amplitude by 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The D − ¯D meson mixing is ignored in the generation of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The present experimental D → πππ decay samples are roughly 102 − 103 times larger than the B → Kππ decay samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Because of the current implementation of the Wasser- stein distance calculation that we use [9, 19], large statistic datasets present a numerical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' To solve the optimal transport problem utilizing the current publicly available linear programming libraries require the full cost matrix ˆdij as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The cost matrix – 15 – Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Pictorial comparison between unbinned (left) and binned (right) Wasserstein statistic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Note how in the binned case, the optimal transport algorithm effectively sets to zero in the last step the number counts in the bins that have the same counts between the two CP conjugate datasets (red and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' scales as N ¯N ∼ O(N2) and quickly demands more random access memory than available in an average personal computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For example, the cost matrix for datasets containing ∼ 106 events, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', comparable to the number of currently experimentally available D0 → π+π−π0 decays, requires roughly 7 TB of memory space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' There are a number of solutions to the above memory problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Below we develop two strategies, both of which use approximate calculations of (variants of) Wasserstein distance between the D0 and ¯D0 decay samples: a binned Wasserstein test in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 and a sliced Wasserstein test in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The two approximate approaches to the Wasserstein based statistic can be applied to large datasets, while continuing to use the publicly available and optimized software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Alternatively, one could attempt to create a new optimal transport algorithm geared toward large datasets, such as the D decays, utilizing lazy evaluation and the sparseness of the transport matrix that does not require the full form of the cost matrix as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The latter, however, goes beyond the scope of the present manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 Binned Wasserstein test Since the resonances in the D0 → π+π−π0 Dalitz plot have typical decay widths of O(100 MeV) or so, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 2, we expect it is possible to capture well the change of the CP asymmetry across the Dalitz plot already with relatively modest numbers of bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' One can then apply the Wasserstein distance statistic to the binned Dalitz plot data in order to ob- tain a global measure of CPV in the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' While there is some loss of information due to binning compared to the Wasserstein distance statistic applied to full samples, we expect the loss to be small, if the binning is fine enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the limit of infinitely small bins one of course reverts to the case of unbinned statistic discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The binned Wasserstein distance is given by W bin q (E, ¯E) = � min {fij≥0} Nb � i,j=1 fij � ˆdij �q �1/q , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) where Nb is the total number of bins in the D (and ¯D) Dalitz plot, with bin counts wi ( ¯wj) in the i−th (j−th) bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the Dalitz plot we will use equal binning along each – 16 – dimension, with nbins in each direction, so that the number of bins with nonzero entries equals to Nb ≃ nbins(nbins − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 The minimization of the weights fij ( ¯fij) gives the optimal transport from bins in D to ¯D Dalitz plot, subject to the constraints nbins � i fij = ¯wj ¯N , nbins � j fij = wi N , nbins � i,j fij = 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) with the distances ˆdij taken to be between the centers of the i−th and j−th bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The construction of the binned Wasserstein distance statistic W bin q is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since the binned versions of E and ¯E event samples use the same binning, the optimal transport algorithm will always ‘zero’ out the like counts in each bin between E and ¯E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', it takes no ‘work’ to transport mass by zero distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' What is left is a representation of the local bin count density asymmetry between E and ¯E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These count density asymmetries then get re-distributed by the optimal transport algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Thus, instead of encoding the CPV information via the distances between events in each dataset, as is done in Wq, the CPV is now encoded as the excess or overabundance of weight between datasets (as well as how far these weight overabundances in D Dalitz plot are from overabundances in the ¯D Dalitz plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Denoting the contribution to W bin q from the i−th bin in the D0 Dalitz plot as δW bin q (i), and likewise by δ ¯W bin q (¯i) the contribution to W bin q from ¯i−th bin in the ¯D0 Dalitz plot, such that (W bin q )q = � i δW bin q (i) = � ¯i δ ¯W bin q (¯i), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) we define in analogy with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) the binned Wasserstein asymmetry Wq,bin CP as Wq,bin CP (i) = δ ¯W bin q (¯i) − δW bin q (i) δ ¯W bin q (¯i) + δW bin q (i), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4) where the ¯i-th bin in the ¯D Dalitz plot is the CP-conjugate of the i−th bin in the D Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 11 shows a comparison between the binned Wasserstein distance asymmetry Wbin q (left panels) and the CP asymmetry ACP (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We find that the binning results in enhanced asymmetries when data is represented using the Wbin q compared to ACP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is true both for the CPV dataset, as well as for statistical fluctuations in the CPC example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since direct CP violation in D decays is small, it is hard to discern by eye whether or not there is CP violation in the Dalitz plot distributions, and one is forced to rely on a statistic sensitive to CPV in distributions such as W bin q or the energy test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 12 shows that the Wasserstein test statistic is still sensitive to CP violation de- spite the binning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The three panels show from top to bottom the probability 3The equality sign applies in the mπ → 0 limit or for large enough bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In our numerical implementation we use square nb × nb arrays that cover fully the Dalitz plot and take Nb = n2 bins to be the total number of bins, including the ones containing zero events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The bins outside the kinematically allowed region are trivially zero, and do not add any complexity to the calculation of the binned Wasserstein distance, while this approach simplifies the encoding of the Dalitz plot in the binned array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 17 – 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 m2(π+π0) (GeV2) Wq,bin CP CPV q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 ACP CPV 0 1 2 3 m2(π−π0) (GeV2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 m2(π+π0) (GeV2) Wq,bin CP CPC q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0 1 2 3 m2(π−π0) (GeV2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 ACP CPC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='15 Asymmetry Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Comparison between the binned Wasserstein asymmetry Wq,bin CP (left), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4), and the binned CP asymmetry ACP (right) for the D0 → π+π−π0 Dalitz plot with N = ¯N = 106 events in a sample, and nbins = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' distribution function (PDF), the cumulative distribution function (CDF), and the survival factor (SF=1-CDF) as functions of the binned Wasserstein statistic W bin q , for r = 2, q = 1, and using nbins = 50, for CP conserving D0 → π+π−π0 Dalitz plot with N = ¯N = 105 events in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The average W bin q value (red vertical line) for our CPV toy D decay model example is well above the bulk of the CP conserving W bin q PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that, on average, the CPC hypothesis is in this example expected to be excluded at the ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5σ level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', with a p−value of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In fact, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 13 shows that the chosen binning size nbins = 50 (which was not optimized) is already fine enough for N = ¯N = 104 that there is only little loss of sensitivity to CPV compared to the unbinned Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the scatter plot of p−values at which the CPC hypothesis is excluded, we see that the exclusion levels obtained by either using the Wq or the W bin q statistic are comparable, and consistent within estimated errors (due to the systematic and statistical uncertainties in the extrapolation of the fit to the CPC PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The binned Wasserstein statistic does have, however, the additional advantages of less memory consumption (space complexity) and computational efficiency (time complexity) due to the reduction of the dataset size from N ¯N to ∼ n2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Whether nbins = 50 suffices also for sample sizes 106, or whether fined binning will be required, should be tested when – 18 – 0 500 PDF q = 1 W bin q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='006 W bin q 10−2 100 SF ¯p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='010+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='007 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The CPC probability distribution function (PDF), the cumulative distribution function (CDF), and the survival factor (SF=1-CDF) as functions of the binned Wasserstein statistic W bin q for r = 2, q = 1, and nbins = 50, as obtained from the numerical master method result for the PDF, consequently fit to a gamma distribution, for N = ¯N = 105 events in the sample, using an ensemble of Ne = 103 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The orange bands (blue band in the top panel) denote the ±1σ fit errors (statistical errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The vertical red line (band) denotes the average W bin q value (the ±1σ W bin q ranges) obtained from an ensemble of Ne = 102 CPV datasets for our toy D0 → π+π−π0 amplitude model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' the method is applied to the actual D decay data, however, we find the above results quite encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Sliced Wasserstein test The Sliced Wasserstein distance (SWq) is a variant of the Wasserstein distance, in which the optimal transport in d-dimensions is replaced with a set of optimal transport problems on 1D slices, with the data points projected onto them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, the sliced Wasserstein distance SWq(g, f) between two distributions in d−dimension, g(x) and f(x), is given by [40] SWq(g, f) = � � Sd−1 Wq(Rg(·, θ), Rf(·, θ)dθ � 1 q , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5) where Rg(·, θ) is the Radon transform of function g(x), defined to be the projection of function g(x) onto the line in the direction of the unit vector θ, which then runs over the d−1 unit sphere Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The Wq in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5) is therefore a 1D Wasserstein distance between functions Rg(·, θ) and Rf(·, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The 1D Wq has a closed form solution, given by the integrated distance between the CDFs for the two functions, and can be efficiently calculated through a simple sorting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 19 – 10−4 10−2 100 p(Wq) 10−4 10−3 10−2 10−1 100 p(W bin q ) q = 1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='658 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The scatter plot of p−values at which the CP conserving hypothesis is excluded for our toy D decay example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The plot shows an ensemble of Ne = 500 datasets with samples of N = 104 D0 → π+π−π0 (and ¯N = 104 CP conjugated ¯D0 → π−π+π0) decays, with p−values calculated either using the unbinned Wq, giving p(Wq), or using the binned W bin q statistic with nbin = 50, giving p(W bin q ) (dots), where in both cases we set q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The 1σ fit error bars on p− values are shown as lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The fraction ϵ of points above p(Wq) = p(W binned q ) diagonal line denotes the fraction of ensemble samples for which Wq is more sensitive to CPV than W binned q is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The dotted gray horizontal and vertical lines (solid bands) show the average (1σ ranges of) p−values for the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0007 SWq q = 1 SWq 101 102 103 104 101 103 tratio tratio[Nslices ≈ 7500] = 1 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Top: The approximate evaluation of SW q, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6), as a function of the number of slices used, Nslices, for a particular CPC D0 → π+π−π0 sample with N = ¯N = 104 events, setting q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The solid blue line (blue band) shows the mean (1σ range) of the SW q estimates obtained from an ensemble of 50 different samplings of Nslices slices for each solid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bottom: the speed up of calculating SWq compared to Wq defined as the ratio of computational times in the two cases, tratio = tWq/tSWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 20 – 10−2 10−1 100 p(Wq) 10−2 10−1 100 p(SWq) Nslices = 100 q = 1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='628 10−2 10−1 100 p(Wq) 10−2 10−1 100 p(SWq) Nslices = 1000 q = 1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='600 10−2 10−1 100 p(Wq) 10−2 10−1 100 p(SWq) Nslices = 10000 q = 1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='618 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The scatter plot of p–values at which no CPV hypothesis is excluded, calculated using SWq and Wq for Nslices = 102, 103, 104 (from left to right), for the ensemble of a Ne = 500 datasets with samples of N = 104 D0 → π+π−π0 (and ¯N = 104 CP conjugated ¯D0 → π−π+π0) decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The fraction ϵ of points above p(Wq) = p(SWq) diagonal line denotes the fraction of ensemble samples for which Wq is more sensitive to CPV than SWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The sliced Wasserstein distance can thus be efficiently calculated, at least approxi- mately, by performing a large enough number of slices, Nslices, SWq(g, f) ≈ � 1 Nslices Nslices � k=1 Wq(Rq(·, θk), Rfν(·, θk)) � 1 q , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6) where θk are random unit vectors uniformly distributed over the unit sphere Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the Nslices → ∞ limit the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' approaches the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' in the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Importantly for our purposes, both Wq and SWq(g, f) are distances in the space of functions and both measure dissimilarity of f and g distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The SWq can therefore also be used as a test statistic, in the same way as we used the Wasserstein distance Wq in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Furthermore, SWq is closely related to the Wasserstein distance, Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For instance, for q = 2 we have SW2(g, f) ≤ W2(g, f)/ √ d, and in general SWq(g, f) ≤ cqWq(g, f) with a known constant cq ≤ 1 (for q ∈ [1, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The improved computational efficiency for SWq relative to Wq is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The ratio of the computing times, tratio = tWq/tSWq, where tWq(tSWq) denotes the time required to calculate Wq (approximate calculation of SW q using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6)) for a particular D0 → π+π−π0 sample with N = ¯N = 104 events, where we take q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For small number of slices, Nslices ∼ O(10) the speed up is several orders of magnitude, however, at that point also the approximate evaluation of SW q still has a large uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The latter is denoted with the blue band, corresponding to 1σ range of SW q values obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6), cycling through 50 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We observe that in this example the SWq evaluation is faster than the Wq one for Nslices ≲ 7500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We also observe that the approximate SWq evaluation converges to its limiting value for Nslices ≈ 1000, indicating a ∼ 7× speedup in the calculation of SWq compared to Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Beyond the speed-up, and maybe even more importantly for the scaling to large sample sizes, the evaluation of SWq does not require large memory resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We have also checked that as the number of slices increases the – 21 – SWq and Wq distributions, obtained from an ensemble of N = ¯N = 104 event samples, agree up to a scaling factor as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Finally, since we are interested in the sensitivity to CPV and not in SWq itself, we show next that a high sensitivity to CPV can be achieved already with relatively approximate estimate of SWq, relying on just a limited number of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 15 shows the p−values at which the CPC hypothesis is excluded, either calculated using Wq (giving p(Wq)) or via approximate evaluation of SWq using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='6) (giving p(SWq)) for three different values of slices, Nslices = 102, 103, 104 (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The fraction ϵ of points above p(Wq) = p(SWq) diagonal line denotes the fraction of Ne = 500 datasets ensemble of N = ¯N = 104 event samples for which Wq is more sensitive to CPV than SWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that even for Nslices = 102 the obtained p−values are already compa- rable to the p−values obtained using full Wq, even though at that point the approximate evaluation of SWq still has a rather large spread, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is quite encouraging, and it would be interesting to explore in the future whether this feature remains for larger sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Similarly, it would be interesting to explore where a windowed SWq, defined in analogy with the windowed Wasserstein distance statistic Iq, would lead to a similar increase in sensitivity to CPV that we saw in the case of full Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 Conclusions The Wasserstein distance based test statistics are potentially powerful tools that can be used to search for the presence of CP violation in multibody decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' They combine the benefits of two alternative tests sensitive to CPV in distributions: (i) in a similar way as the binned CP asymmetry, the Wasserstein distance based test statistics trace asymmetries to the regions of phase space the CPV resides in, while at the same time (ii) being a sensitive probe of CPV as an integrated measure, in a similar way as the energy test is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In this manuscript we introduced several such Wasserstein distance based test statistics, taking the multibody B0 → K+π−π0 and D0 → π+π−π0 decays as concrete examples for numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The simplest one is the Wasserstein distance, Wq, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) for the case of B0 and ¯B0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The use of Wq as a measure of CPV in principle requires no tuning, though there are optimizations that can be made regarding the exact definition of the distance in the Dalitz plot one uses, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3), as well as the value of the continuous parameter q in the definition of the Wasserstein distance, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For instance, instead of the fully symmetric definition of the distance in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3) one could have used a simple Euclidean distance in the Dalitz plot, or the Euclidean distance in the square Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' One can also tune the value of q using an amplitude model to obtain the highest expected sensitivity to CPV, as we did in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3 (see also App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, even without an amplitude model, origins of CPV across the Dalitz plot can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Such tests allow for unbinned, model independent tests of CPV in the phase space of distributions, thereby informing future analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Its use with weighted datasets is also straightforward, as illustrated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since Wq measures the cummulative presence of CPV in the Dalitz plot one therefore needs only two observables to fully quantify the amount of direct CPV in a multibody B – 22 – −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 ˜Wq −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 ACP q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 1σ 2σ 3σ B0 → K−π+π0 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The expected 1σ, 2σ, 3σ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' contours (red shaded regions) in the (ACP, ˜Wq = Wq−W q) plane, assuming CP conserving B0 → K+π−π0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The black lines show the current global average for ACP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='064 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='050 as well as the 1σ range for ˜Wq that follows from our toy model of CP violation in distributions, introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The data was generated by first randomly sampling ACP via a Gaussian with mean µ = 0 and standard deviation σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='05 to mimic the uncertainty in the global average, selecting B and ¯B decays accordingly with the total number of events fixed to 2 × 103, and computing Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This was repeated with 103 distinct datasets and contours drawn according to a numerical integration of a 2–dimensional joint PDF where ˜Wq and ACP were assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' decay: the total direct CP asymmetry, ACP, and the Wasserstein distance Wq (or a related Wasserstein distance test statistic such as the windowed Wasserstein distance Iq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 16, which shows a contour plot of ACP vs ˜Wq = Wq − W q, where W q is the median Wq expected for CP conserving B decays (in this case obtained using the amplitude model, but could be obtained using permutation method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For CP conserving decays both ACP and ˜Wq are consistent with zero within statistical uncertainties, and would be nonzero if there is significant CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The two give complementary information: ACP is nonzero if there is a difference in the partial decay widths between B0 → K+π−π0 and ¯B0 → K−π+π0 decay channels, while ˜Wq is nonzero, if there is a difference between the phase space distributions of the two CP conjugated decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For its simplicity, Wq does have a drawback — due to the CP conserving noise it usually results in a lower sensitivity to CPV compared to the energy test with an optimized regulator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Applying filters on the optimal transports for each individual B0 and ¯B0 decay datapoints in the Dalitz plot, however, gives an optimized version of the Wasserstein test statistic, Iq, with sensitivity to CPV indistinguishable on statistical basis from the optimized energy test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 we focused on windowed filtering, with Heavyside step functions abruptly switching on and off (or assigning negative weights) to certain ranges of optimal transport distances, however, one could have also used other filtering variants – 23 – with smooth versions of the window function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The windowed Wasserstein distance statistic Iq can match the extreme sensitivity of the energy test to the presence of CPV, a feature which can ultimately be attributed to the lack of long-tailed CP conserving probability distributions in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, the energy test statistic successfully mitigates superfluous contributions from CP conserving variations among data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This comes at the price of additional N(N −1)+ ¯N( ¯N −1) ∼ O(N2) computations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' the first two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), encoding the CP conserving distance variations within each sample), as well as the need for a regulator function ψ, which restricts contributions to be only within a sphere of influence of radius σ, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Such suppression of CP conserving variations is expected to be necessary for any metric based statistic with enhanced sensitivity to CPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' As we showed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 the suppression of CP conserving variation can be implemented for the case of the Wasserstein distance based statistics by using windowed filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Again, this comes at the cost of additional computations required for the optimization of the filtering window function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The computation requirements may become prohibitive when faced with large datasam- ples, such as the D decays with N ≳ 106 events in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In that case one can use approximate versions of Wasserstein distance to construct test statistics that scale better with N, at a rather small cost to sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4 we discussed two such possibilities, the binned Wasserstein test statistic, W bin q , and the sliced Wasserstein test statistic, SWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Both were shown to give similar sensitivities to CPV as Wq, when either the binning is fine enough (for W bin q ) or for large enough number of slices (for SWq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The work presented in this manuscript could be extended in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The extension to higher dimensional spaces, such as the n-body meson decays, n ≤ 4, is straight- forward with no changes to the formalism required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The main question in that case is the scaling with the number of particles in the final state, where the usual curse of dimen- sionality may be mitigated by the fact that the multi-body decays tend to have large quasi-two-body resonant decay structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Less trivial extensions include time dependent weighting of decay rates in order to probe indirect CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Finally, one could explore other deviations from the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For instance, an interesting direction would be to explore entropic smoothing of the Wasserstein distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', an entropic regulariza- tion of the optimal transportation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The resulting Sinkhorn divergence depends on a hyperparameter λ which interpolates between the Wasserstein distance (λ = ∞) and the energy test (λ = 0) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Finally, we provide a public code EMD4CPV that allows a straightforward use of the introduced Wasserstein based statistics for two-sample tests, with further details about the code given in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' A, Acknowledgements We thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bressler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Thaler for discussions on the two sample tests, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Gersabeck, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' White, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Sarpis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Chen and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Brodzicka in particular for extended discussions on the energy test, as well as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Szewc for comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Latham for help with the Laura++ framework, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Evans for support using the AmpGen framework, and – 24 – Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Pictorial representation of the EMD4CPV program architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The largest box repre- sents the highest level class, delta Wq, followed by lower level sub-classes contained within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The arrows represent the inheritance of each sub-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Brod for providing access to the local computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' AD acknowledges support from STFC grants ST/S000925 and ST/W000601/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' AY, JZ and TM acknowledge support in part by the DOE grant de-sc0011784 and NSF OAC-2103889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' A Public code EMD4CPV The public code and repository for this project may be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='com/adamdddave/EMD4CPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The program architecture is hierarchically structured, resembling a nested doll of classes and subclasses, as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 17, delta Wq(delta Wq statistics(delta Wq fit(delta Wq versus))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) The delta Wq is the highest level class and contains the sub-class delta Wq statistics, which in turn contains the sub-class delta Wq fit, which finally contains the sub-class delta Wq versus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This nested structure was implemented for three main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Firstly, the modularity improves readability and the ease of use, since the programs using the classes are structured as function calls from a software library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Secondly, this class–subclass structure follows the natural progression of the analysis pipeline used to compute and compare different Wq based statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For example, a typical usage of the library will follow a nested call of functions within each class, delta Wq → delta Wq statistics → delta Wq fit → delta Wq versus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) Finally, since each sub-class inherits all functions from the previous class this allows the user to work at any level of the program architecture while only needing to initialize one class instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' While the use case of the program is oriented toward 3–body decays, the code is generic enough such that it can be used with any n−dimensional dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Below we summarize briefly the software pipeline (see the documentation as well as the example Python notebook example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='ipynb within the repository for more details): – 25 – The delta Wq class contains functions which allow the user to input two n–dimensional distributions and obtain the associated binned or unbinned δWq values chosen by the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since in most cases the CP conserving distributions (functionals of δWq) need to be calculated, the class is set up such that the generation of the CP even distributions via the master or permutation methods can be done efficiently by randomly selecting a subset of unique datapoints from a larger datapool provided by the user in the form of a text file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In addition, this class may be used to compute the sliced Wasserstein distance SWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Once the δWq ensemble is obtained, the subclass delta Wq statistics can be used to compute the Wq, Iq, or any other user defined statistical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Oftentimes, when computing the p−values from the CPV datasets a fit is needed in order to extrapolate outside the ranges of explicitly calculated CP conserving distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These fits can be performed using the delta Wq subclass which allows the user to iteratively fit to any distribution within the SciPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='stats library and return the associated PDFs, CDFs, SFs, χ2–values, as well as the PDFs, CDFs, and SFs associated with the ±1σ errors on the fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Finally, the delta Wq versus4 subclass may be used to iteratively compare the sen- sitivity of different statistics on ensembles of like datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Additionally, for convenience, the script energy test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='py provides a Python imple- mentation of the energy test statistic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', the computation of the test statistic T (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1)) between two n–dimensional distributions for use when computing CPV statistic values in delta Wq versus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The program also includes an interface with Manet [8] (which utilizes the CUDA API to parallelize the computation on NVIDIA GPUs) such that the user can efficiently generate large statistic CP conserving T distributions if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' B The optimal transport problem Consider two discrete samples P and ¯P, the first consisting of n points sampled from probability distribution p, each with weight wi such that total weight is W = � i wi, and the second consisting of ¯n points sampled from ¯p with weights ¯wi and total weight ¯W = � i ¯wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The problem optimal transport consists of transporting the weight W of P into the weight of ¯W of ¯P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', P → ¯P, as efficiently as possible given some cost function related to the distances among the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This requires minimizing an nׯn transport plan matrix T, which contains information about the amount of work required to transporting P → ¯P, such that the work required is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The transport matrix thus requires knowledge of both the distances between i-th and j-th points as well as the amount of weight to be transported between the points of P and ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The distances between each point in P and ¯P can be encoded in an n × ¯n matrix 4This subclass requires Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='10+ while all other classes require Python 3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 26 – C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The information about the transportation of weights can be encoded via the n × ¯n ‘flow’ matrix F subject to � i Fij = ¯wj, � j Fij = wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, F specifies the fractional amount of weight to be transferred from i-th point in P to the j-th point in ¯P, subject to the condition that the total weight from each point must be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 The total work or ‘cost’ of a given configuration is then given by the inner product of the flow and distance matrices T = ⟨F, C⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Finding the most efficient plan amounts to finding the transport plan F which minimizes the total cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We denote the optimal flow matrix as F∗ and define the Wasserstein distance as Wq = ⟨F∗, Cq⟩1/q ≡ � � n � i ¯n � j F ∗ ijCq ij � � 1/q (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) Solving for Wq optimally takes super-cubical time complexity with respect to the size of the input datasets O(N3) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' C EMD analysis of Gaussian distributions In this appendix we give further details on how the Wasserstein distance Wq can be used as a statistic sensitive to dissimilarities between two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We use the toy example of two displaced Gaussian distributions, either in 1D or 2D, where the difference between distributions is taken to be controlled by a single “CP violating” parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We consider two limits: i) the two Gaussian peaks do not overlap, but are rather displaced by ∆µ, and ii) the peaks of the two Gaussian distributions overlap, while their widths differ, ∆σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the main text we showed an example for the first choice where we considered two 1D Gaussians displaced by ∆µ = 40 and with widths σ = 10, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 1, where ˆdij in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) here and below is taken to be the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since the optimal transport needs to move the points in the datasets sampled from the two distributions by a distance ∼ ∆µ, the Wasserstein distance coincides with ∆µ, W1 → ∆µ, for large N (this is true to quite a good degree even for rather small values of N, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is also shown in the top panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18, where we consider 11 different values ∆µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' , 10, while σ = 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 (top left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The distributions of W1 straddle ∆µ for ∆µ sufficiently far away from zero (for ∆µ = 0, W1 ≥ ∆µ since W1 is nonnegative), shown for N = ¯N = 103 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 (top middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 (top right) shows that the average value of the Wasserstein distance, ¯W1, linearly increases with ∆µ, where for small values of ∆µ/σ there is a deviation from this linear behavior, which however is almost imperceptible on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The lower panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 show the dependence of Wasserstein distance on the width of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In this example one Gaussian distribution is held fixed, G(x) = N(x|µ = 0, σ = 1) while the other is taken to have different widths but a coinciding peak, ¯G(x) = N(x|¯µ = 0, ¯σ), ¯σ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' , 7, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 (bottom left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' With increasing ¯σ the typical value of Wasserstein distance ¯W1 increases, since the two distributions differ more and more, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 (bottom middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The values of the W1 also form a wider distribution 5Note that a particular transport configuration is not required to be a bijective map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', the weight of a particular point in P can be partitioned to different points in ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 27 – −10 0 10 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4 PDF N(∆µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' , 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' σ = 1) 0 5 10 Wq 0 5 10 PDF q = 1 0 5 10 ∆µ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 W q q = 1 W q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='997∆µ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02 −20 0 20 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4 PDF N(µ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' ∆σ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' , 7) 0 2 4 Wq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 PDF q = 1 2 4 6 ∆σ 0 2 4 W q q = 1 W q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='789∆σ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The dependence of W1 on the displacement ∆µ (top row) or the width difference ∆σ (bottom) of the two Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Middle panels show the corresponding W1 distributions, while the right panels show the linear dependence of the average value ¯W1 on the displacement ∆µ and width difference ∆σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The error bars shown in the right panels denote the 1σ bands of the W1 distributions shown in the middle panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' for larger values of ¯σ, since the larger difference between the two Gaussians translates to a larger scatter of optimal transportation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The increase in the average value of the Wasserstein distance, ¯W1, is linear in ∆σ = ¯σ − σ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 18 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Next, we check that the W1 statistic leads to the same CL intervals as the negative log likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 19 (left) shows the expected 90%, 3σ and 5σ CL for ∆µ as a function of N (solid contours) that follow from a known ∆µ = 0 probability distribution for W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These coincide with the expect CL exclusion intervals obtained from the negative log likelihood for ∆µ (dotted contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that in this case the W1 statistic gives the correct coverage for all considered values of N and ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 19 (right) we also show the estimates of the exclusion contours that follow from a permutation (or re-randomization) test, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', where the symmetric “CP-even” W1 probability distribution is modeled by randomly sampling events from E and ¯E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that the re-randomization estimate of the true ∆µ = 0 probability distribution for W1 results in a bias and thus in underestimated exclusion p−values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The benefit of the re-randomization is that such modeling of “CP-even” W1 probability distribution is always possible, however it also means that the use of Wq statistics is best suited for the cases where one has already a reasonable model of the distributions and can check potential bias due to the use of permutation test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The multi-body B and D decays fall in this category since one can use – 28 – 0 1 2 3 ∆µ/σ 3 6 9 12 15 18 21 24 27 30 Nstat 5σnll 3σnll 90%nll 5σW1 3σW1 90%W1 0 1 2 3 ∆µ/σ 3 6 9 12 15 18 21 24 27 30 5σnll 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='nll 90%nll 5σW1 3σW1 90%W1 10−5 10−3 10−1 p Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The expected 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', 3σ, and 5σ exclusion lines for different Gaussian peak displace- ments, ∆µ/σ, and sizes of statistical samples Nstat using EMD W1 statistic (negative log-likelihood) are denoted with dashed (solid) lines, where one uses either true (left) or modeled (right) ∆µ = 0 probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' the fitted for amplitude models to estimate the potential bias in the permutation method for Wq statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This was found to be small for the two B and D decays considered in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' A toy example that is closer to the case of three body B and D decay Dalitz plots is the example of two displaced 2D Gaussian distributions, g(x, y) ∼ N(x|µx−∆µx/2, σ)N(y|µy− ∆µy/2, σ), and ¯g(x, y) ∼ N(x|µx + ∆µx/2, σ)N(y|µy + ∆µy/2, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For simplicity we take the widths of all the Gaussian distributions to be the same, so that there is no CP violation (the two distributions are the same) if and only if ∆µx = ∆µy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The statistical analysis of this case is a trivial extension of the case of a 1D Gaussian toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Using the true “CP-even” W1 distribution for ∆µx = ∆µy = 0 leads to the correct coverage, while the permutation method gives some bias, as in the 1D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For two-dimensional distributions there are additional observables and visualization tools that prove to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' First of all, for arbitrary two-sample 2D distributions one can define a Wasserstein distance asymmetry distribution Wq CP in the same way as for the Dalitz plot, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3), Wq CP(x, y) = � ¯i δ ¯Wq(¯i) − � i δWq(i) � ¯i δ ¯Wq(¯i) + � i δWq(i), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) where the summation over i (¯i) is only over the data-points from g(x, y) sample contained in the bin centered at (x, y) (from ¯g(x, y) data in the bin centered at (x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In addition, we can also define a Wasserstein distance asymmetry heatmap ωq CP(x, y) = 1 si � � ¯i δ ¯Wq(¯i) − � i δWq(i) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) where si = ∆xi∆yi is the area of the bin center at (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Both Wq CP(x, y) and ωq CP(x, y) are intensive quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' That is, in the large Nstat limit and small bin sizes the Wq CP(x, y) and ωq CP(x, y) become independent of the sizes of the bins (that is as long as bins are small enough such that the variation of Wq CP(x, y) and – 29 – −4 −2 0 2 4 x −4 −3 −2 −1 0 1 2 3 4 y Nbins = 50 × 50 −4 −2 0 2 4 x −4 −3 −2 −1 0 1 2 3 4 y Nbins = 25 × 25 −1500 −1000 −500 0 500 1000 1500 ωq CP −1500 −1000 −500 0 500 1000 1500 ωq CP Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The Wasserstein distance asymmetry heatmap ωq CP, q = 1, for two 2D Gaussian distributions with equal widths, σ = 1, but displaced by ∆µx = ∆µy = 3, where the sample sizes are N = ¯N = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Changing the binning size, from 50×50 bins (left) in the range shown to 25×25 bins (right) does not change the overall size of the asymmetry heatmeap, just coarse-grains it, since the asymmetry is an intensive quantity normalized to the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 0 50 x 0 10 0 5 −50 −25 0 25 y 0 50 −50 −25 0 25 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The visualization of optimal transport of dataset sampled from G (blue) to a dataset sampled from ¯G (red), where G and ¯G are two 2D Gaussians displaced by ∆µx = ∆µy = 40, and the sample size is N = ¯N = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' ωq CP(x, y) from bin to bin is negligible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' A numerical example for ωq CP(x, y) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 20, where we see that changing the size of the bins simply corresponds to averaging the Wasserstein distance heatmap over a larger area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For relatively small samples it is also possible to visualize the optimal transport for – 30 – 0 500 PDF q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF johnsonsu (χ2 = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='17) exponnorm (χ2 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='22) invgauss (χ2 = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='010 Wq 10−3 10−1 SF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04 PDF q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF exponnorm (χ2 = 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='12) johnsonsu (χ2 = 504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='32) skewnorm (χ2 = 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='70) gamma (χ2 = 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='29) f (χ2 = 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='19) 80 100 120 140 160 Iq 10−3 10−1 SF Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The CP conserving PDF, CDF and SF distributions for Wq (left) and Iq (right) obtained using the master method for B0 → K−π+π0 decays with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 and N = ¯N = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The different fit functions are denoted in the legend, along with respective χ2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' each individual point, an example of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 21 for two-sample 2D Gaussian distributions displaced by ∆µx = 40, ∆µy = 40, and a sample size of N = ¯N = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The optimal transport moves the datapoints sampled from distribution G (blue) to data sampled from ¯G (red) shown with lines connecting pairwise the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Since the two samples are of the same size, the optimal transport is a bijective map between G and ¯G datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that the typical shift is of order ∆µ = (∆µ2 x + ∆µ2 y)1/2, with datapoints on the far (near) side of G distribution transported to near (far) side of ¯G distribution, where near/far is defined with respect to the barycenter of G and ¯G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D Details on EMD–test for three body decays In this appendix we give further details on the implementation of Wasserstein distance as a measure of CPV in three body B and D decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 we discuss the details of the error analysis on the quoted p−values, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 the optimization of the q parameter, while in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 we collect the additional results for q = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10}, supplementing the results shown in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 The p−value error analysis In the numerical results in the main text we determine the p−value at which the hypothesis of CP conservation is excluded from the master Wq distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is numerically advantageous since it does not need to be recalculated for each dataset of B and ¯B or D and ¯D events, while given the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3 we do not expect to introduce a significant difference to the estimates using the permutation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 31 – 0 20 40 PDF q = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF johnsonsu (χ2 = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25) f (χ2 = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='11) invgauss (χ2 = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='06) gamma (χ2 = 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='52) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='08 Wq 10−3 10−1 SF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='004 PDF q = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF skewnorm (χ2 = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='47) johnsonsu (χ2 = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='47) moyal (χ2 = 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='46) −700 −600 −500 −400 −300 −200 −100 Iq 10−3 10−1 SF 0 5 PDF q = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF johnsonsu (χ2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='23) gengamma (χ2 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='66) f (χ2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='27) norminvgauss (χ2 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='66) gamma (χ2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='20) skewnorm (χ2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='27) exponnorm (χ2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='35) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='35 Wq 10−2 100 SF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 PDF q = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CDF skewnorm (χ2 = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='91) johnsonsu (χ2 = 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='78) gamma (χ2 = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='13) gausshyper (χ2 = 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='09) −600 −400 −200 0 200 400 Iq 10−3 10−1 SF Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 22 but for q = 1 (top panels) and q = 10 (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For B decays we calculate the master Wq distribution from 104 unique samplings of B and ¯B datasets, each with 103 events, while for D decays we use 103 unique samplings of D and ¯D datasets, each containing 105 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The master Wq PDF is fit with a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' First, the data is binned such that each bin is populated with at least one event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' An initial fit is then performed using the SciPy’s non-linear least squares fitter, from which we obtain the initial values of the fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' These parameters are then passed to the SciPy’s curve fit function along with statistical errors on the ith bin according to δNi = � Ni(1 − Ni/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This returns an updated list of fit parameters along with the parameter covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The 1σ parameter fit values are given as the square root of the diagonal elements of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Errors on p–values are then estimated – 32 – 10−3 10−2 10−1 100 101 q 10−2 10−1 100 p(Wq) W q Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The q dependence of expected exclusion C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', p(Wq) (blue dots joined by a solid line), at which the CP conserving hypothesis is excluded for N = ¯N = 103 event sample sizes, obtained by performing ensemble averages over Ne = 500 CPV datasets, with the blue band indicate the 1σ spread of p−values over the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' by considering the one sigma confidence bands on the SF distribution as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' From the fit of the PDF we compute the survival factor distribution, SF=1−CDF, from which one can directly read off the p−value with which the no CPV hypothesis is excluded, for each value of the measured W exp q , as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For many of the CPV datasets we consider the value of the statistic W exp q falls well outside the range for which the master distribution was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For these cases we use the fit to extrapolate to smaller p−values, where the error on the extrapolation is estimated from errors on the fit parameters as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The results of the above procedure for B0 → K+π−π0 decay samples of size N = ¯N = 103 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 22, 23 for the Wasserstein statistic Wq and windowed Wasserstein statistic Iq for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The CPC PDFs are iteratively fit to built-in contin- uous distributions contained in the SciPy statistics library, as listed in the legend of the corresponding panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In most cases, the best fit is chosen according to the minimum of a χ2 statistic, however, in cases where multiple distributions achieve similar χ2 values, the distribution that best matches the tail of the distribution is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In particular, for Iq we use the skewnorm fit for the extrapolation to small p−values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 Optimizing the q value The Wasserstein distance weighting exponent parameter q, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), may be tuned to maximize the expected sensitivity to CPV in a particular distribution, such as the B0 → K+π−π0 Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Such an optimization of course depends on the assumed model for B0 → K+π−π0 decay amplitudes and in particular on the assumed values of the strong and weak phases that are hard to calculate but can in principle be fit from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 24 we used the nominal toy model for CPV in B0 → K+π−π0 Dalitz plot that we used throughout Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3, with the amplitudes and phases – 33 – for B0 and ¯B0 isobar models set to the central values of the measurement in Ref [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Similarly, for the CPC datasets we use the central values of amplitudes and phases in the B0 BaBar isobar model [2] for both B0 and ¯B0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 24 shows the variation with q of the expected C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' p(Wq) for exclusion of the CPC hypothesis, given our CPV model, for N = ¯N = 103 event sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The blue bands give a 1σ range of expected C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' exclusions as obtained form an ensemble of Ne = 500 CPV samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that for q ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 the expected p(Wq) remains unchanged when lowering q within the range considered, while for higher q there is in general diminished sensitivity to CPV, with the exception of the region around q ∼ O(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We suspect that these ranges of q correspond to typical scales in the problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', the typical widths of the resonances (relative to the mass of B quark), but did not explore this hypothesis further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' In the numerics in the main text we chose q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, which roughly optimizes the sensitivity to CPV, but show in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 below also the results for q = 1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3 Further results for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10 In this appendix we list further results for Wasserstein distances with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10 both for B and D decays, supplementing the results discussed in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 25 shows the CP asymmetry significance, ACP/δACP, where the error on the CP asymmetry is given by δACP = � 1 − A2 CP N + ¯N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) The upper panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 25 show the CP asymmetry for the case of B0 → K+π−π0 decay, with our toy example CPV amplitude model (left panel) leading to clearly identifiable regions in the Dalitz plot with CPV, and only noise in the Dalitz plot for the CPC case (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The difference between CPV and CPC decays is less pronounced in the D0 → π+π−π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Even so the Wasserstein distance based test statistics can still lead to exclusions of CPC hypothesis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 12, where the analysis was done for a sample size of N = ¯N = 105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 26 shows the relative difference between the binned Wasserstein distance asymme- try, Wq CP, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3), and the CP asymmetry, ACP, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This complements Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 27, which show the actual values of the binned Wasserstein distance asym- metry, Wq CP, and the CP asymmetry, ACP for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 and q = 1, 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that for the optimal value of q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 the binned Wasserstein distance asymmetry, Wq CP almost completely matches ACP up to ∼ 10% relative differences, where the differences are even closer to just a few percent in the regions of the Dalitz plot where the CP asymmetry significance is large, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The Wq CP still tracks well the CP asymmetry ACP, however with exaggerated differences in the regions of the Dalitz plot with lower CP asymmetry significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 28 shows the binned Wasserstein asymmetry Wq CP and direct CP asymmetry ACP for q = {1, 10} both for CPV and CPC B0 → K+π−π0 decays, where the same inputs for the B0 decays were used as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The results shown were averaged over an ensemble of Ne = 100 datasets, each containing N = ¯N = 103 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This figure – 34 – 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) ACP CPV 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 ACP CPC −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04 Asymmetry Significance −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0075 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0100 Asymmetry Significance 0 1 2 3 m2(π−π0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 m2(π+π0) ACP CPV 0 1 2 3 m2(π−π0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 ACP CPC −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='006 Asymmetry Significance −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='003 Asymmetry Significance Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The binned CP asymmetry significance ACP/δACP, with δACP given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1), for B0 → K+π−π0 (D0 → π+π−π0) Dalitz plot are shown in the top (bottom) panels for N = ¯N = 103(106) event samples averaged over an ensemble of Ne = 100 (Ne = 1) datasets with CPV toy example show on the left and CP conserving datasets on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' supplements Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 5 for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We see that in all cases, q = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10}, the Wq CP faithfully traces ACP for q = {1, 10} throughout the Dalitz plot, especially where the CP asymmetries are statistically most significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7 shows log(δWq) distributions and the difference between CPV and CPC log(δWq) distributions for q = 1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Compared to the q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 case, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7, there is a more pronounced deficit of δWq counts in CPV distribution relative to the CPC one for the intermediate log(δWq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The window function w(x), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='7), is therefore chosen to have support both for the +1 (green bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7) and −1 (red bands) weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 29 is the q = 1, 10 complement of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 8 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' It shows, 500 distinct datasets each with N = ¯N = 103 events, the confidence levels with which the CP conserving hypothesis is excluded when applying different tests, either the energy test, giving CLs denoted with p(T), the Wasserstein distance statistic test, giving p(Wq), or the windowed Wasserstein distance statistic, giving p(Iq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The windows and anti-windows for q = 1, 10 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For q = 10 the performance of the windowed Wasserstein distance is comparable yet slightly less sensitive than the energy test, while for q = 1 the sensitivity of windowed Wasserstein distance statistic is significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For q = 10 the use of windowed Wasserstein distance is comparable to the simple Wasserstein distance statistic, – 35 – 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) 1 − Wq CP / ACP CPV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='075 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='100 1 − Ratio 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) 1 − Wq CP / ACP CPV q = 1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 1 − Ratio 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) 1 − Wq CP / ACP CPV q = 10 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 1 − Ratio Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We show 1 − Wq CP/ACP for each bin within the two–dimensional Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Smaller values indicate better agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' This is shown for CP violating B0 → K+π−π0 decays with nbins = 50, q = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1, 1, 10}, and averaged over an Ne = 100 dataset ensemble where each dataset contains N = ¯N = 103 B0 and ¯ B0 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', it does not lead to any real gain, while for q = 1 case, the selected windows and anti-window reduce sensitivity of Iq compared to Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' However, additional tuning of the window and anti-window regions could be conducted to maximize significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E Energy test The energy test, introduced in [5], is an unbinned two-sample test utilizing a test statistic, T, to analyze average distances between data points in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The first proposal to utilize the energy test in searches for CP violation was described in [6] and subsequent analyses performed in [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The statistic utilizes a weighting (distance) function ψij dependent on the distance dij between the ith and jth event in the first and second sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For searches of CP violation the two samples are distinguished by flavor (B0 and ¯B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The test statistic – 36 – is defined as [5, 6] T = N � i,j>i ψij N(N − 1) + ¯ N � i,j>i ψij ¯N( ¯N − 1) − N, ¯ N � i,j ψij N ¯N , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) where N, ¯N denote the total number of events in the B0 and ¯B0 samples, respectively .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The weighting function ψij is chosen such that the weight decreases with increasing distance, dij, between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The summations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1) are, from left to right, over B0 sample, ¯B0 sample and over both B0 (index i) and ¯B0 (index j) samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The form of the test statistic T is motivated by the form of the electrostatic energy for overlapping distributions of positive and negative charges, in which case ψij ∝ 1/dij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' If the two charge distributions are exactly the same, the net charge distribution is zero, and T vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The functional form on the weighting function ψij can be freely chosen, for instance in order to increase the sensitivity to local asymmetries at some typical length-scales, minimizing dilutions due to averaging over large Dalitz plot areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' We follow Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [7, 8] and choose a Gaussian weighting function ψij ≡ ψ(dij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' σ) = e−d2 ij/2σ2, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2) where σ is a tunable parameter describing the effective radius between data points where asymmetry is measured, while dij is the Euclidean distance in the Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' For events sampled from two identical distributions T is expected to fluctuate close to zero, while for samples drawn from dissimilar distributions T will tend to a nonzero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' To obtain the null hypothesis PDF for T we use the master method described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The labels for B0 and ¯B0 samples are ignored, and the N + ¯N events randomly assigned to E and ¯E samples, each with N = ¯N events, thus simulating the CP even datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Repeating this n times give a null hypothesis PDF for T, which is then fitted to a gamma distribution, used finally to obtain the p−values corresponding to the “measured” value of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The computation of CP conserving T distributions was done with the Manet software package [8] (while for single computations our own implementation was used, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The analysis was performed on N = ¯N = 103 B0 and ¯B0 events generated by AmpGen [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The null hypothesis T-distributions were computed with N = 103 permutations, while the tunable parameter in the weighting function, σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2 GeV2, was chosen to maximize the significance (minimize p−value) in the case of a CP violating sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 37 – 0 5 10 15 20 25 0 5 10 15 20 25 m2(K−π0) (GeV2) Wq CP CPV q = 1 0 5 10 15 20 25 0 5 10 15 20 25 ACP CPV 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) Wq CP CPC q = 1 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 ACP CPC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry 0 5 10 15 20 25 0 5 10 15 20 25 m2(K−π0) (GeV2) Wq CP CPV q = 10 0 5 10 15 20 25 0 5 10 15 20 25 ACP CPV 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 m2(K−π0) (GeV2) Wq CP CPC q = 10 0 5 10 15 20 25 m2(K−π+) (GeV2) 0 5 10 15 20 25 ACP CPC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00 Asymmetry Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Binned Dalitz plot comparison between the Wasserstein asymmetry Wq CP (left) for q = {1, 10} (shown in top two and bottom two rows) and direct CP asymmetry ACP (right), shown for CP violating B0 → K+π−π0 decays (1st and 3rd row) and CP conserving decays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', the decays in which the asymmetries in the amplitude model were set to zero (2nd and 4th row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The results shown are normalized and averaged over an ensemble of Ne = 100 datasets, each containing N = ¯N = 103 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 38 – 0 20 40 Counts q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0 CPC CPV −20 −18 −16 −14 −12 −10 −8 log(δWq) −10 0 10 Counts CPV − CPC Window Window 0 20 40 60 Counts q = 10 CPC CPV −120 −100 −80 −60 −40 −20 log(δWq) −40 −20 0 20 40 Counts CPV − CPC Window Window Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 7 but for q = {1, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' On the lower panels the green bands denote the [δW win min, δW win max] ranges and the red bands the [δW win min, δW win max] range, each for the corresponding q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 39 – 10−5 10−2 p(Iq) 10−5 10−3 10−1 p(Wq) CPV q = 1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='080 10−5 10−2 p(Iq) 10−5 10−3 10−1 CPC q = 1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='512 10−11 10−6 10−1 p(Iq) 10−11 10−8 10−5 10−2 p(T) CPV q = 1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='024 10−11 10−6 10−1 p(Iq) 10−11 10−8 10−5 10−2 CPC q = 1 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='490 10−5 10−2 p(Iq) 10−5 10−3 10−1 p(Wq) CPV q = 10 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='440 10−5 10−2 p(Iq) 10−5 10−3 10−1 CPC q = 10 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='470 10−12 10−7 10−2 p(Iq) 10−12 10−9 10−6 10−3 100 p(T) CPV q = 10, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='214 10−12 10−7 10−2 p(Iq) 10−12 10−9 10−6 10−3 100 CPC q = 10 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='504 Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The comparison of CL exclusion of the CP conserving hypothesis either using energy test (p(T)), the Wasserstein distance (p(Wq)), or the windowed Wasserstein distance (p(Iq)), with the window function w(x) as denoted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' The top (bottom) two rows are for q = 1(10), with 500 distinct CP violating (conserving) B0 → K+π−π0 decay datasets shown on the right (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 40 – References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Panaretos and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Zemel, Statistical Aspects of Wasserstein Distances, Annual Review of Statistics and Its Application 6 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', 2019) 405–431, [1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='05500].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [2] BaBar collaboration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Amplitude Analysis of B0 → K+π−π0 and Evidence of Direct CP Violation in B → K∗π decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 83 (2011) 112010, [1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bediaga, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bigi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Gomes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Guerrer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Miranda and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Reis, On a CP anisotropy measurement in the Dalitz plot, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 80 (2009) 096006, [0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [4] BaBar collaboration, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Aubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Search for CP Violation in Neutral D Meson Cabibbo-suppressed Three-body Decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 78 (2008) 051102, [0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4035].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Aslan and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Zech, New test for the multivariate two-sample problem based on the concept of minimum energy, Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Simul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 75 (2004) 109–119, [math/0309164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Williams, Observing CP Violation in Many-Body Decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 84 (2011) 054015, [1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5338].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [7] LHCb collaboration, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Search for CP violation in D0 → π−π+π0 decays with the energy test, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' B 740 (2015) 158–167, [1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='4170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Parkes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Brodzicka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Gersabeck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Dujany and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Barter, On model-independent searches for direct CP violation in multi-body decays, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' G 44 (2017) 085001, [1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04705].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Komiske, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Metodiev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Thaler, The Hidden Geometry of Particle Collisions, JHEP 07 (2020) 006, [2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Crispim Rom˜ao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Castro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Milhano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Pedro and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Vale, Use of a generalized energy Mover’s distance in the search for rare phenomena at colliders, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' C 81 (2021) 192, [2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='09360].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cheng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Craig and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Craig, Linearized optimal transport for collider events, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 102 (2020) 116019, [2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='08604].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cheng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Craig and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Craig, Which metric on the space of collider events?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 105 (2022) 076003, [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='03670].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Komiske, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Kryhin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Thaler, Disentangling quarks and gluons in CMS open data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 106 (2022) 094021, [2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04459].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Mitchell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Ryne and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Hwang, Using kernel-based statistical distance to study the dynamics of charged particle beams in particle-based simulation codes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E 106 (2022) 065302, [2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='04275].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [15] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Kitouni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Nolte and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Williams, Finding NEEMo: Geometric Fitting using Neural Estimation of the Energy Mover’s Distance, 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='15624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Villani, Optimal transport, old and new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Springer, Berlin, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Santambrogio, Optimal Transport for Applied Mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Calculus of Variations, PDEs and Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Ramdas, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Garc´ıa Trillos and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cuturi, On wasserstein two-sample testing and related families of nonparametric tests, Entropy 19 (2017) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Komiske, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Metodiev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Thaler, Metric Space of Collider Events, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 123 (2019) 041801, [1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 41 – [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Weed and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bach, Sharp asymptotic and finite-sample rates of convergence of empirical measures in wasserstein distance, 1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='00087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Evans, “Ampgen.” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='com/GooFit/AmpGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [22] LHCb collaboration, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Measurements of CP violation in the three-body phase space of charmless B± decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 90 (2014) 112004, [1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5373].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Virtanen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Gommers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Oliphant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Haberland, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Reddy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cournapeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17 (2020) 261–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Grossman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Kagan and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Nir, New physics and CP violation in singly Cabibbo suppressed D decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 75 (2007) 036008, [hep-ph/0609178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Brod, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Kagan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Zupan, Size of direct CP violation in singly Cabibbo-suppressed D decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 86 (2012) 014023, [1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lu and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Yu, Branching ratios and direct CP asymmetries in D → PP decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 86 (2012) 036012, [1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Franco, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Mishima and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Silvestrini, The Standard Model confronts CP violation in D0 → π+π− and D0 → K+K−, JHEP 05 (2012) 140, [1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Feldmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Nandi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Soni, Repercussions of Flavour Symmetry Breaking on CP Violation in D-Meson Decays, JHEP 06 (2012) 007, [1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3795].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Cheng and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Chiang, Direct CP violation in two-body hadronic charmed meson decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 85 (2012) 034036, [1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='0785].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [30] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Bhattacharya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Gronau and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rosner, CP asymmetries in singly-Cabibbo-suppressed D decays to two pseudoscalar mesons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 85 (2012) 054014, [1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='2351].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Pirtskhalava and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Uttayarat, CP Violation and Flavor SU(3) Breaking in D-meson Decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' B 712 (2012) 81–86, [1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='5451].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Dery, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Grossman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Schacht and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Tonelli, CP violation in b and c quark decays, in 2022 Snowmass Summer Study, 9, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='07429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [33] LHCb collaboration, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Observation of CP Violation in Charm Decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 122 (2019) 211803, [1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='08726].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [34] LHCb collaboration, Measurement of the time-integrated CP asymmetry in D0 → K−K+ decays, 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='03179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [35] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Grossman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Kagan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Zupan, Testing for new physics in singly Cabibbo suppressed D decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' D 85 (2012) 114036, [1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3557].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Dery, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Grossman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Schacht and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Soffer, Probing the ∆U = 0 rule in three body charm decays, JHEP 05 (2021) 179, [2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='02560].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [37] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Grossman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Robinson, SU(3) Sum Rules for Charm Decay, JHEP 04 (2013) 067, [1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content='3361].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [38] BaBar collaboration, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Aubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Measurement of CP Violation Parameters with a Dalitz Plot Analysis of B± → D(pi+ π−π0 ) K±, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 99 (2007) 251801, [hep-ex/0703037].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Back, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Gershon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Harrison, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Latham, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' O’Hanlon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=', Laura ++: A dalitz plot fitter, Computer Physics Communications 231 (2018) 198–242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 42 – [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Helgason, Integral Geometry and Radon Transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Springer, New York, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Atasu and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Mittelholzer, Linear-complexity data-parallel earth mover’s distance approximations, in Proceedings of the 36th International Conference on Machine Learning (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Chaudhuri and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' Salakhutdinov, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' ), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 97 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' 364–373, PMLR, 09–15 Jun, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} +page_content=' – 43 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FPT4oBgHgl3EQf9TVT/content/2301.13211v1.pdf'} diff --git a/r9FJT4oBgHgl3EQfbyya/content/tmp_files/2301.11541v1.pdf.txt b/r9FJT4oBgHgl3EQfbyya/content/tmp_files/2301.11541v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0328aa89000352e913fe4fad464e967bdb88509 --- /dev/null +++ b/r9FJT4oBgHgl3EQfbyya/content/tmp_files/2301.11541v1.pdf.txt @@ -0,0 +1,3297 @@ +arXiv:2301.11541v1 [cs.GT] 27 Jan 2023 +Hide-and-Seek Game with Capacitated Locations and +Imperfect Detection +Basti´an Bahamondes, Mathieu Dahan +School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, +{bbahamondes3@gatech.edu, mathieu.dahan@isye.gatech.edu} +We consider a variant of the hide-and-seek game in which a seeker inspects multiple hiding locations to find +multiple items hidden by a hider. Each hiding location has a maximum hiding capacity and a probability +of detecting its hidden items when an inspection by the seeker takes place. The objective of the seeker +(resp. hider) is to minimize (resp. maximize) the expected number of undetected items. This model is +motivated by strategic inspection problems, where a security agency is tasked with coordinating multiple +inspection resources to detect and seize illegal commodities hidden by a criminal organization. To solve this +large-scale zero-sum game, we leverage its structure and show that its mixed strategies Nash equilibria can +be characterized using their unidimensional marginal distributions, which are Nash equilibria of a lower +dimensional continuous zero-sum game. This leads to a two-step approach for efficiently solving our hide- +and-seek game: First, we analytically solve the continuous game and compute the equilibrium marginal +distributions. Second, we derive a combinatorial algorithm to coordinate the players’ resources and compute +equilibrium mixed strategies that satisfy the marginal distributions. We show that this solution approach +computes a Nash equilibrium of the hide-and-seek game in quadratic time with linear support. Our analysis +reveals a complex interplay between the game parameters and allows us to evaluate their impact on the +players’ behaviors in equilibrium and the criticality of each location. +Key words : Hide and seek; resource coordination; imperfect detection; large-scale game +1. +Introduction +In this article, we study a variant of the hide-and-seek game in which two players, the hider and the +seeker, coordinate multiple resources among heterogeneous locations. Specifically, the hider deter- +mines where to allocate multiple items within capacitated locations. Simultaneously, the seeker +inspects a limited number of locations to detect the hidden items. However, detection is supposed to +be imperfect: When inspecting a location, the seeker finds the items with a location-specific prob- +ability that captures the local effects undermining the seeker’s detection capabilities. The seeker +(resp. hider) aims to select a (possibly randomized) strategy that minimizes (resp. maximizes) the +expected number of items that are undetected. The objective of this work is to efficiently solve this +large-scale simultaneous zero-sum game, that is, to compute a mixed strategy Nash equilibrium +(NE), and gather insights on the players’ equilibrium behaviors. +Our model is motivated by security applications involving for instance a security agency inter- +ested in dispatching multiple units to inspect warehouses used by a criminal organization to store +1 + +2 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +illegal commodities such as drugs or weapons (Hochbaum and Fishbain 2011). In such settings, +the security agency aims to schedule the patrolling operations of their units to detect and seize +the illegal commodities (Hess et al. 2013). Another motivating application of our model involves a +utility company tasked with coordinating multiple imperfect sensors to inspect its service network +against failures caused by a malicious cyber-physical attacker who is able to target multiple com- +ponents of the network (Pirani et al. 2021). Interestingly, another application of interest concerns +auditing election results (Blocki et al. 2015, Behnezhad et al. 2018). In such problems, an auditor +allocates a limited number of election officials into several polling locations in order to detect elec- +toral fraud by means of recounts. The fraudster may be a malicious organization who is interested +in manipulating the results by coordinating its members to tamper with the votes. +Previous related works in the hide-and-seek literature have not simultaneously considered mul- +tiple resources for both players, heterogeneous hiding capacities, and imperfect detection (Gal +and Casas 2014, Dziubi´nski and Roy 2018). This may reduce the applicability of the results, par- +ticularly in security settings. However, simultaneously considering these features introduces new +challenges: On one hand, the combinatorial nature of both players’ sets of actions due to the +resource multiplicity prevents us from computationally solving the game using linear programming +techniques or approximation algorithms (Freund and Schapire 1999, Lipton et al. 2003, Hellerstein +et al. 2019). On the other hand, the complex interplay between the game’s features renders the +analytical solutions from previous works inapplicable. Hence, we focus on the following research +questions: (i) How to optimally coordinate multiple imperfect inspection resources to detect multiple +hidden commodities? (ii) How are the optimal inspection and hiding strategies jointly impacted by +the detection, location, and players’ characteristics? +1.1. +Contributions +In this article, we formulate the hide-and-seek game as a simultaneous zero-sum game Γ and +extend previous models in the literature by considering the coordination of multiple resources for +both players in locations with heterogeneous hiding capacities and detection probabilities. We then +leverage the game’s structure to derive equilibrium properties of the NE of Γ. In particular, we show +that a strategy profile is a NE of Γ if and only if the corresponding marginal inspection probabilities +and expected numbers of hidden items at each location form a NE of a lower dimensional continuous +game �Γ (Proposition 1). From this equivalence, we derive a two-step approach for solving the +hide-and-seek game Γ. +First, we analytically solve the continuous game �Γ (Theorem 1). We find that NE can be generally +classified into three main regime patterns determined by complex parameters that account for the +interplay between the players’ resources and the heterogeneity of the locations. To the best of our + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +3 +knowledge, the features of our model lead to new NE regimes that have not been observed in the +literature. In fact, we show that our analytical solutions describe all pure strategies NE of �Γ almost +surely (Proposition 2). +By solving �Γ, we obtain marginal inspection probabilities and expected numbers of hidden items +at each location in equilibrium of Γ. Thus, the second step consists of computing a mixed strategy +profile of Γ that is consistent with these unidimensional marginal distributions. To this end, we +extend the algorithm of Dziubi´nski and Roy (2018) to feasibly coordinate the allocation of multiple +resources (Algorithm 1). We show that the algorithm runs in quadratic time and returns equilibrium +inspection and hiding strategies with linear supports (Theorem 2 and Corollary 1). +Thus, our approach efficiently solves the hide-and-seek game Γ. By providing mixed inspection +strategies with linear support, our solutions can easily be implemented in practice via a randomized +scheduling of inspections that can be performed on a day-to-day basis. Furthermore, our analytical +solution of the continuous game �Γ decodes the complex interplay between the game parameters +and provides insights with respect to their impact on the players’ equilibrium behaviors and the +criticality of each location. Such insights can be leveraged by security agencies to inform their +inspection decisions. +1.2. +Related Work +The hide-and-seek game is a two-person zero-sum game introduced by Von Neumann (1953). In +its original version, the hider and the seeker interact on a square matrix of nonnegative entries: +The hider selects an entry aij and the seeker simultaneously selects either a row or a column of the +matrix. If the row or column selected by the seeker contains the entry chosen by the hider, then the +hider pays the seeker aij; otherwise, the seeker pays the hider aij. This game has been studied as +a general model of strategic mismatch (Crawford and Iriberri 2007) and its equilibrium strategies +are well known (Von Neumann 1953, Flood 1972, Karlin and Peres 2016). It also belongs to the +more general category of search games, in which a searcher is concerned with the optimal way of +looking for a hidden adversary in a search space; see for example Lidbetter (2013), Lidbetter and +Lin (2019), Clarkson et al. (2022), and the surveys of Alpern and Gal (2006) and Hohzaki (2016). +Nonetheless, in practical applications, the seeker may be able to simultaneously inspect multiple +locations, and the hider may be able to hide multiple items across the search space. Furthermore, +the seeker’s inspection resources can be affected by local conditions undermining their detection +capabilities. Thus, in order to achieve a better utilization of their resources, each player may benefit +from efficiently coordinating their allocation across the different locations, a problem that the +original model does not address. +One of such practical applications arises in problems of strategic sensor placement for network +inspection (Miloˇsevi´c et al. 2019, Pirani et al. 2021, Dahan et al. 2022, Bahamondes and Dahan + +4 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +2022), in which the defender of a network positions sensors in a subset of given locations to detect +attacks caused by a strategic attacker, who can target multiple network components. Such models +typically account for the detection range of the sensors: Positioning a sensor at a location allows +the defender to monitor a subset of network components—referred to as a monitoring set—and +potentially detect attacks occurring within it. As a result, attacks may be detected from multiple +locations; this overlapping feature renders such games challenging to solve. Dahan et al. (2022) +studied a two-person zero-sum game version of this model under the assumption of perfect detec- +tion, and derived approximate NE strategies by means of minimum set covers and maximum set +packings. Miloˇsevi´c et al. (2019) and Bahamondes and Dahan (2022) studied variants of this model +by respectively considering the critical values of network components and imperfect detection. +They derive heuristic approaches to compute good quality solutions in the case of a single attack +resource. Pirani et al. (2021) formulated a game in which sensors are positioned in the nodes of a +networked control system to detect attacks on them, and considered imperfect detection through +a linear filter that processes the sensors’ measurements to detect attacks. The authors derived +equilibrium results using tools from structured systems and graph theory. Finally, a different but +related model which features location-specific imperfect detection is the network interdiction prob- +lem by Washburn and Wood (1995), in which an interdictor sets up a single inspection checkpoint +along one of the arcs of a directed graph, with the aim of interdicting an evader who attempts +to traverse a path between two nodes. The authors show that NE strategies can be computed in +polynomial time using network flow techniques. +In fact, our hide-and-seek game can be used to model a class of instances of the strategic sensor +placement problems in which the monitoring sets are mutually disjoint. In such instances, our +results are directly applicable and generalize the equilibrium characterizations from (Washburn and +Wood 1995, Dahan et al. 2022, Bahamondes and Dahan 2022). Instances with disjoint monitoring +arise in situations where it is desirable to reduce sensor interference or the energetic cost of the +network (Cardei and Du 2005, Wang and Shao 2014). In other contexts such as in security games, +disjoint monitoring is naturally satisfied (Powell 2009, Behnezhad et al. 2018, Musegaas et al. +2022). +Among the variants of the hide-and-seek game previously examined in the literature, our game is +most closely related to the ones by Dziubi´nski and Roy (2018) and Gal and Casas (2014). Dziubi´nski +and Roy (2018) consider a version of the game with multiple resources for both players, in which +they interact on a set of unit capacity locations, each one associated with a nonnegative value. +Simultaneously, the hider (resp. seeker) selects a subset of locations to hide his objects (resp. to +inspect). Once the choices are made, the seeker pays the hider the value of each uninspected location + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +5 +containing a hidden object. In contrast, our model considers homogeneous values for all locations, +but incorporates heterogeneous hiding capacities and probabilities of successful inspections. +Gal and Casas (2014) propose a pursuit-evasion model of the interaction between a prey and a +predator. The prey chooses a location to hide from the predator, who is able to inspect multiple +locations. However, if the predator visits the location where the prey is hiding, the capture is +uncertain and occurs with some probability. The predator (resp. prey) seeks to maximize (resp. +minimize) the probability of capture. Our work extends this model by allowing multiple preys to +coordinately hide in heterogeneously capacitated locations. Although the subject of animal behavior +is beyond the scope of our work, our model extension addresses analogous situations arising in +security domains in which a security agency can dispatch multiple inspection units to inspect +heterogeneous locations used by a criminal organization to store multiple illegal commodities. +In both of these games, as in ours, a player’s mixed strategy consists of a probability distribu- +tion over the set of resource allocations that satisfy capacity constraints and the resource budget. +Thus, when players have access to multiple resources, their strategy spaces become exponentially +large. One approach to handle the dimensionality consists in characterizing the players’ strategies +in a lower dimensional space. In Dziubi´nski and Roy (2018) and Gal and Casas (2014), the games’ +structures permit the characterization of NE in terms of their marginal probabilities of inspecting +each location for the seeker, and their marginal probabilities of hiding an item in each location +for the hider. Then, in order to compute the mixed strategies NE, it becomes necessary to con- +struct probability distributions over the feasible resource allocations that are compatible with these +marginal probabilities. +This two-step approach of characterizing equilibrium strategies in terms of marginal distributions +and then computing compatible mixed strategies has been previously proposed in the literature, +e.g., by Korzhyk et al. (2010) and Letchford and Conitzer (2013) to compute Stackelberg equilibria +in security games; by Chan et al. (2016) to compute approximate NE in multilinear games; and by +Ahmadinejad et al. (2019) to compute NE for zero-sum bilinear games, with applications to the +Colonel Blotto game (Borel 1921). In all these cases, the computation of the equilibrium marginal +distributions is carried out via linear programming, and the computation of the mixed strategies +from the marginal distributions follows by either an efficient implementation of Birkhoff-von Neu- +mann’s theorem and its generalizations (Budish et al. 2013), or by the more general algorithm +by Gr¨otschel, Lov´asz, and Schrijver (Gr¨otschel et al. (2012), Theorem 6.5.11) that implements +Carath´eodory’s theorem using linear programming techniques. Finding an efficient implementation +of Carath´eodory’s theorem has also been addressed in other contexts, such as in mechanism design +(Cai et al. 2012, Hoeksma and Uetz 2013), scheduling (Hoeksma et al. 2016), and ranking systems +(Kletti et al. 2022a,b). + +6 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Our implementation of the two-step approach is closely related to that of Dziubi´nski and Roy +(2018). In contrast to the above-mentioned literature, we derive analytical expressions for the equi- +librium marginal distributions, which allows us to fully understand the interplay between the game +parameters and to provide detailed insights regarding their impact on the players’ equilibrium +behaviors. Dziubi´nski and Roy (2018) provide a combinatorial algorithm for constructing a mixed +strategy with linear support in quadratic time with respect to the number of locations. Their algo- +rithm iteratively decomposes a given vector representing the marginal probabilities of allocating +one resource in each location into a convex combination of a linear number of integer resource +allocations, and it can be interpreted as a tailored and more efficient implementation of Gr¨otschel, +Lov´asz, and Schrijver’s algorithm. Our algorithm extends this decomposition to the more general +case in which the marginal distributions represent expected numbers of resources allocated in loca- +tions constrained by capacities, and where the budget of resources does not need to be exhausted, +as opposed to Dziubi´nski and Roy’s setting. +The rest of the article is organized as follows. In Section 2, we formulate our hide-and-seek game. +We then characterize and parametrically analyze its NE in Section 3 using a lower dimensional +continuous game, which we solve analytically. In Section 4, we derive a combinatorial algorithm to +coordinate the players’ resources and compute a NE of the hide-and-seek game. We then provide +some concluding remarks in Section 5. Finally, the proofs of our results are listed in the electronic +companion. +2. +Problem Description +We consider a hide-and-seek game involving a seeker who is looking for multiple homogeneous items +hidden by a hider in a search space consisting of a set of n hiding locations �1,n� := {1,...,n}. The +locations are capacitated; namely, the hider can hide up to ci ∈ Z>0 items in each location i. We +let m := �n +i=1 ci be the total hiding capacity. In addition, the seeker has imperfect location-specific +detection capabilities. Specifically, by inspecting a location i, the seeker effectively finds the hidden +items (if any) in that location with probability pi ∈ (0,1] (and therefore, the inspection fails and +leaves the hidden items undetected with probability 1 − pi). We assume that inspection failures at +different locations occur independently. We refer to pi as the detection rate of location i. +We assume that both the hider and seeker are strategic, and hence we adopt a game-theoretic +framework to study their behaviors. We define a simultaneous two-player strategic zero-sum game +Γ := ⟨{S,H},(∆S,∆H),(−U,U)⟩ where S is the seeker and H is the hider. S can select up to rS ∈ Z>0 +hiding locations to inspect. Simultaneously, H can select up to rH ∈ Z>0 items to hide. To model + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +7 +the players’ action sets, we define a generic set of feasible resource allocations given a vector of +capacities b ∈ Zn +>0 and a budget of resources r ∈ Z>0 as follows: +A(b,r) := {z ∈ Zn : 0 ≤ zi ≤ bi, ∀i ∈ �1,n�, and +n +� +i=1 +zi ≤ r}. +(1) +The set A(b,r) contains all the vectors in Zn representing allocations of up to r resources within +the locations i ∈ �1,n�, respecting their capacities given by bi. Then, the pure action sets for S and +H are given by AS := A(1n,rS) and AH := A(c,rH) respectively, where 1n is the vector of ones in +Zn and c = (ci)i∈�1,n� is the vector of hiding capacities. Thus, for every x ∈ AS and i ∈ �1,n�, xi = 1 +if S inspects location i and xi = 0 otherwise, and for each y ∈ AH and i ∈ �1,n�, yi represents the +number of items that H hides at location i. +We consider that the quantity of interest for the players is the average number of undetected +items. Thus, we define the players’ payoff function as follows: +∀(x,y) ∈ AS × AH, u(x,y) := +n +� +i=1 +(1 − pixi)yi. +(2) +For every i ∈ �1,n�, 1−pixi represents the probability that items in location i are undetected given +the pure inspection action x ∈ AS. +In such combinatorial security settings, players significantly benefit from randomizing their +actions (Washburn and Wood 1995, Pita et al. 2008, Zhu and Basar 2015, Gupta et al. 2016, +Hota et al. 2016, Bertsimas et al. 2016, Miao et al. 2018). Thus, we allow the players to use +mixed strategies, defined as probability distributions over their sets of pure actions. The set of +probability distributions over the set of generic feasible resource allocations A(b,r) is defined as +∆(b,r) := +� +σ ∈ [0,1]A(b,r) : � +z∈A(b,r) σz = 1 +� +. Then, the sets of mixed strategies for S and H are +given by ∆S := ∆(1n,rS) and ∆H := ∆(c,rH), respectively. For every mixed strategy σS ∈ ∆S (resp. +σH ∈ ∆H) and every x ∈ AS (resp. y ∈ AH), σS +x (resp. σH +y ) is the probability that action x ∈ AS (resp. +y ∈ AH) is executed by S (resp. H). +Given a strategy profile (σS,σH) ∈ ∆S × ∆H, the expected payoff is then defined as U(σS,σH) := +E(x,y)∼(σS,σH)[u(x,y)] = � +x∈AS +� +y∈AH σS +xσH +y u(x,y). We assume that S (resp. H) seeks to minimize +(resp. maximize) U. For ease of exposition, we use U(x,σH) (resp. U(σS,y)) to denote the case +where σS +x = 1 (resp. σH +y = 1) for some x ∈ AS (resp. y ∈ AH). +Our game Γ is relevant to settings where a city police department is interested in inspecting +warehouses that are used by a criminal organization to store illegal commodities (e.g., drugs, +weapons). In such settings, �1,n� represents the set of warehouses, and for each i ∈ �1,n�, ci +represents the maximum number of illegal commodities that can be stored in warehouse i. The +police department can coordinate multiple police units to simultaneously inspect a maximum of rS + +8 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +warehouses, while the criminal organization has rH units of illegal commodities to hide within the +warehouses. The detection rates pi for i ∈ �1,n� capture the local effects that might undermine the +detection capabilities of the police units, e.g., warehouse characteristics that can impact the efficacy +of drug-sniffing dogs (Jezierski et al. 2014). The objective of the police department (resp. criminal +organization) is to minimize (resp. maximize) the number of illegal commodities that are undetected +by the police department. Then, a mixed inspection (resp. hiding) strategy represents a randomized +schedule of coordinated operations for the police department (resp. criminal organization). +The standard solution concept for simultaneous noncooperative games is given by Nash equilibria +(NE), that is, strategy profiles for which no player has an incentive to unilaterally deviate in order +to improve their payoff. Thus, a strategy profile (σS∗,σH∗) ∈ ∆S × ∆H is a NE of the game Γ if it +satisfies +∀(σS,σH) ∈ ∆S × ∆H, U(σS∗,σH) ≤ U(σS∗,σH∗) ≤ U(σS,σH∗). +We refer to U(σS∗,σH∗) as the value of the game Γ. Since Γ is a zero-sum game with finite pure +action sets, Von Neumann’s minimax theorem (Von Neumann 1928) implies that the value of the +game is unique and the game can be solved using the following linear program (LP): +minimize +t∈R, σS∈∆S +t +subject to +U(σS,y) ≤ t, +∀y ∈ AH. +(LP) +Specifically, the equilibrium inspection strategies, equilibrium hiding strategies, and value of the +game Γ are given by the optimal primal solutions, optimal dual solutions, and optimal value of (LP), +respectively. A remarkable consequence of this result is that no player can benefit from observing +the mixed strategy of the other player before making a decision. Nonetheless, since the cardinality of +AS (resp. AH) grows combinatorially with rS (resp. rH), (LP) becomes computationally challenging +to solve, even for small-sized instances. Similarly, algorithms for computing approximate NE are +inapplicable for realistic instances of the game Γ (Freund and Schapire 1999, Lipton et al. 2003, +Hellerstein et al. 2019). +Thus, we propose a two-step solution approach for solving the game. First in Section 3, we reduce +the dimensionality of the problem by characterizing NE using the marginal inspection probability +and expected number of hidden items in each location. Then in Section 4, we derive an algorithm +to coordinate the players’ resources and recover NE that are consistent with the characterized +marginal probabilities and expected numbers of hidden items in equilibrium. +3. +Analytical Characterization of Equilibrium Strategies +In this section, we show that the NE of the game Γ can be characterized using the corresponding +marginal inspection probability and expected number of hidden items in each location. We prove + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +9 +that these unidimensional quantities are NE of a smaller-sized continuous game, which we solve +analytically. This analytical characterization permits us to examine the impact of the problem +parameters on the players’ behaviors in equilibrium. +3.1. +Continuous Equivalence +To simplify our analysis of the game Γ, we first derive properties of generic randomized resource +allocations. Given a vector of capacities b ∈ Zn +>0 and a budget of resources r ∈ Z>0, we denote by +� +A(b,r) := {ρ ∈ Rn : 0 ≤ ρi ≤ bi, ∀i ∈ �1,n�, and �n +i=1 ρi ≤ r} the linear programming relaxation of +the set of generic feasible resource allocations A(b,r). Then, for every probability distribution σ ∈ +∆(b,r) over A(b,r), we denote as ρ(σ) = (ρi(σ))i∈�1,n� the vector of expected numbers of resources +allocated at each location, given by: +∀i ∈ �1,n�, ρi(σ) := Ez∼σ[zi] = +� +z∈A(b,r) +ziσz. +(3) +We present the following relation between ∆(b,r) and � +A(b,r): +Lemma 1. Consider a vector of capacities b ∈ Zn +>0, a budget of resources r ∈ Z>0, and a vector +ρ′ ∈ Rn. Then, ρ′ ∈ � +A(b,r) if and only if there exists a probability distribution σ ∈ ∆(b,r) that +satisfies ρi(σ) = ρ′ +i for all i ∈ �1,n�. +Lemma 1 is a consequence of the integrality of the polyhedron � +A(b,r). Thus, � +A(b,r) represents +the set of vectors of expected numbers of allocated resources at each location resulting from a +probability distribution in ∆(b,r). In particular, for every inspection strategy σS ∈ ∆S and hiding +strategy σH ∈ ∆H, ρ(σS) = (ρi(σS))i∈�1,n� and ρ(σH) = (ρi(σH))i∈�1,n� respectively represent the +vectors of marginal inspection probabilities and expected numbers of hidden items across the +locations. +Lemma 1 permits us to relate the game Γ to the continuous zero-sum game +�Γ := +⟨{S,H},( � +AS, � +AH),(−˜u, ˜u)⟩, where S and H respectively select a vector of continuous inspection +effort ρS ∈ � +AS := � +A(1n,rS) and a vector of continuous amount of hidden items ρH ∈ � +AH := � +A(c,rH), +and the players’ payoff in �Γ is given by +∀(ρS,ρH) ∈ � +AS × � +AH, ˜u(ρS,ρH) = +n +� +i=1 +(1 − piρS +i )ρH +i , +(4) +as shown by the following proposition: +Proposition 1. The games Γ and �Γ are related as follows: +– For every strategy profile (σS,σH) ∈ ∆S × ∆H in Γ, U(σS,σH) = ˜u(ρ(σS),ρ(σH)). +– (ρS∗,ρH∗) ∈ � +AS × � +AH is a NE of �Γ if and only if there exists a NE (σS∗,σH∗) ∈ ∆S × ∆H of Γ +that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗. + +10 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +– The values of the games Γ and �Γ are identical. +From this proposition, we deduce that NE of the game Γ can be characterized from NE of the +game �Γ. In particular, in a NE (ρS∗,ρH∗) of �Γ, ρS∗ (resp. ρH∗) represents the marginal inspection +probabilities (resp. expected numbers of hidden items) at every location in a NE of Γ. This provides +a significant computational advantage, as these quantities can be represented with vectors of size n, +while the players’ strategies in Γ require vectors of exponentially large sizes. In addition, marginal +inspection probabilities and expected numbers of hidden items more conveniently quantify the +criticality of locations for each player. +3.2. +Preliminary Analysis +We next derive the intuition behind the strategies for both players in equilibrium. In particular, +we describe the players’ incentives and constraints that result from the features of our model. +This discussion will permit us to introduce and motivate the key quantities that are needed to +analytically solve the game �Γ and characterize the NE of the game Γ. +From Proposition 1, we deduce that the players’ expected payoff U(σS,σH) in Γ can be expressed +as the sum over the hiding locations i ∈ �1,n� of the expected numbers of hidden items that remain +undetected, namely, (1 − piρi(σS))ρi(σH). We refer to 1 − piρi(σS) as the undetection probability +of location i, that is, the probability that the hidden items at location i remain undetected when +S plays σS. We also refer to piρi(σH) as the detection performance at location i ∈ �1,n�, which +represents the expected number of hidden items that S is able to detect by inspecting location i +when H plays σH. These quantities will guide our equilibrium analysis, as H’s incentive is to hide +items in locations with highest undetection probabilities, and S’s incentive is to inspect locations +with highest detection performances. +Due to the players’ incentives, we can easily show that when each player has one resource unit +(i.e., rS = rH = 1), then a NE (σS∗,σH∗) satisfies ρi(σS∗) = ρi(σH∗) = (1/pi)/ +��n +j=1 1/pj +� +for every +i ∈ �1,n�, as in Gal and Casas (2014). In other words, S inspects each location i with marginal +probability proportional to 1/pi so as to equalize the undetection probability of every location. +Similarly, H’s equilibrium strategy equalizes the detection performance of every location. +Now, if we consider a general number of player resources rS ≥ 1 and rH ≥ 1, an analogous intuition +would suggest that ρi(σS∗) = (rS/pi)/ +��n +j=1 1/pj +� +and ρi(σH∗) = (rH/pi)/ +��n +j=1 1/pj +� +for every +i ∈ �1,n�. From Proposition 1, ρ(σS∗) and ρ(σH∗) must belong to � +AS and � +AH, respectively. However, +if rS is large enough and the detection rates are heterogeneous enough, then (rS/pi)/ +��n +j=1 1/pj +� +≤ +1 may be violated for some locations. In such cases, S cannot ensure the desired level of inspection +to these locations, thus rendering them more attractive for H. Analogously, if rH is large enough, +the detection rates are heterogeneous enough, and the hiding capacities are small enough, then + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +11 +(rH/pi)/ +��n +j=1 1/pj +� +≤ ci may also be violated for some locations. H cannot ensure the desired +level of detection performance across such locations, thus rendering them less attractive for S. +Hence, the features of our model (i.e., multiple player resources and heterogeneous locations) +create new challenges that we need to address. For the remainder of this section, we order the +locations such that p1c1 ≤ ··· ≤ pncn. We refer to pici as the detection potential of location i ∈ �1,n�, +that is, the maximum expected number of detected items when S inspects location i. Moreover, for +any given i ∈ �0,n−1�, we define a bijective mapping πi : �1,n−i� → �i+1,n� that satisfies pπi(1) ≤ +··· ≤ pπi(n−i), i.e., that orders the set of locations �i+1,n� by their detection rates. For convenience, +we define p0c0 := 0 and pπi(0) := 0 for every i ∈ �0,n − 1�. We also denote Si +k := �n−i +j=k 1/pπi(j) for +every i ∈ �0,n − 1� and k ∈ �1,n − i + 1�. +We remark that when rS ≥ n, an equilibrium inspection strategy for S is to inspect each location, +and an equilibrium hiding strategy for H is to hide min{rH,m} items in the locations i with smallest +detection rates pi. Similarly, when rH ≥ m, an equilibrium hiding strategy is to exhaust all the +hiding capacities, and an equilibrium inspection strategy is to inspect the min{rS,n} locations i +with the highest detection potentials pici. Henceforth, we study the games Γ and �Γ when 0 < rS < n +and 0 < rH < m. +From the discussion above, we find that H may not be able to ensure the desired level of detection +performance for the locations with lowest detection potentials. If we denote by �1,i� such locations, +then S will not inspect them, as her incentive is to allocate her resources among the remaining +locations �i + 1,n� with higher detection performances. For the remaining locations, S’s incentive +is to equalize the undetection probabilities. However, this may not be possible if the detection +rates pj for j ∈ �i+ 1,n� are heterogeneous. Instead, S can inspect the ki most unreliable locations +{πi(1),...,πi(ki)}, that is, the locations with lowest detection rates, and equalize the undetection +probabilities for the locations {πi(ki + 1),...,πi(n − i)}. For every i ∈ �0,n − 1�, the value of ki is +given by the following expression: +ki := max +� +k ∈ �0,n − i� : k + pπi(k)Si +k+1 < rS +� +. +As mentioned above, locations for which S cannot achieve the desired level of inspection become +more attractive for H. Thus, H’s incentive is to exhaust the capacities of the ℓi most unreliable +locations of �i+1,n�, that is, {πi(1),...,πi(ℓi)}, and equalize the detection performance in locations +{πi(ℓi + 1),...,πi(n − i)}. In addition, H must ensure that the detection performance in the latter +set of locations is no less than that of the locations in �1,i�, so that S does not have an incentive to +reallocate her resources to �1,i� (that were initially uninspected by S). Thus, for every i ∈ �0,n−1�, +the value of ℓi is given by the following expression: +ℓi := max +� +ℓ ∈ �0,n − i� : +i +� +j=1 +cj + +ℓ +� +j=1 +cπi(j) + piciSi +ℓ+1 < rH +� +. + +12 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +We note that ℓi exists when the number of items to hide satisfies rH > �i +j=1 cj + piciSi +1. +Interestingly, the interplay between ki and ℓi will play an important role in solving �Γ and char- +acterizing the NE of Γ. Furthermore, it is crucial to determine the number i of locations with +smallest detection potentials that will be exhausted by H and uninspected by S, given the players’ +equilibrium interactions in the remaining set of locations. +3.3. +Analytical Characterization +From the discussion above, we observe that the players’ behaviors in equilibrium depend on capac- +ities, detection rates, detection potentials, numbers of resources, and the parameters ki and ℓi. To +capture this complex interplay and characterize the NE, we define the following key thresholds: +τ−1 := 0, +τi := +i +� +j=1 +cj + +ki +� +j=1 +cπi(j) + pi+1ci+1Si +ki+1, +∀i ∈ �0,n − 1�, +νi := +i +� +j=1 +cj + +ki +� +j=1 +cπi(j) + piciSi +ki+1, +∀i ∈ �0,n − 1�. +First, we show that these thresholds partition the interval [0,m] in the following manner. +Lemma 2. The thresholds τ−1,...,τn−1, and ν0,...,νn−1, satisfy τ−1 = 0, τn−1 = m, and τi−1 ≤ +νi ≤ τi for all i ∈ �0,n − 1�. +Thus, the interval [0,m] is subdivided by the thresholds as 0 = τ−1 ≤ ν0 ≤ τ0 ≤ ν1 ≤ ··· ≤ τn−2 ≤ +νn−1 ≤ τn−1 = m. In fact, given a number of items to hide rH ∈ �1,m − 1�, the subinterval in +which rH resides corresponds to a precise configuration of the parameters and determines a specific +equilibrium regime, as shown in the following theorem: +Theorem 1. Given the players’ resources rS ∈ �1,n − 1� and rH ∈ �1,m − 1�, let i∗ ∈ �0,n − +1� satisfying τi∗−1 < rH ≤ τi∗. Then, any strategy profile (ρS∗,ρH∗) ∈ � +AS × � +AH that satisfies the +conditions below is a NE of �Γ. Furthermore, any strategy profile (σS∗,σH∗) ∈ ∆S ×∆H that satisfies +ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ is a NE of Γ. +Regime Pattern 1: If νi∗ < rH ≤ τi∗, then ki∗ ≤ ℓi∗ and sufficient equilibrium conditions are given +by: +ρS∗ +i = + + + + + + + + + + + +0 +if i ∈ I, +1 +if i ∈ J , +rS − ki∗ +piSi∗ +ki∗ +1 +if i ∈ K, +(5) +ρH∗ +i += + + + + + +ci +if i ∈ I ∪ J , +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +piSi∗ +ki∗ +1 +if i ∈ K, +(6) + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +13 +where I := {1,...,i∗}, J := {πi∗(1),...,πi∗(ki∗)}, and K := {πi∗(ki∗ +1),...,πi∗(n−i∗)}. The value +of the games Γ and �Γ is given by +rH − +ki∗ +� +j=1 +pπi∗(j)cπi∗(j) − +� +rS − ki∗ +�� +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +� +Si∗ +ki∗+1 +. +Regime Pattern 2: If i∗ = 0 and τ−1 < rH ≤ ν0, then ℓ0∗ < k0∗ and sufficient equilibrium condi- +tions are given by: + + + + + + + + + + + + + +ρS∗ +i = 1 +if i ∈ J ∪ {π0(ℓ0 + 1)}, +pπ0(ℓ0+1) +pi +≤ ρS∗ +i ≤ 1 +if i ∈ K \ {π0(ℓ0 + 1)}, +n +� +i=1 +ρS∗ +i ≤ rS, +(7) +ρH∗ +i += + + + + + + + + + + + +ci +if i ∈ J , +rH − +ℓ0 +� +j=1 +cπ0(j) +if i = π0(ℓ0 + 1), +0 +if i ∈ K \ {π0(ℓ0 + 1)}, +(8) +where J := {π0(1),...,π0(ℓ0)} and K := {π0(ℓ0 + 1),...,π0(n)}. The value of the games Γ and �Γ is +given by +rH − +ℓ0 +� +j=1 +pπ0(j)cπ0(j) − pπ0(ℓ0+1) +� +rH − +ℓ0 +� +j=1 +cπ0(j) +� +. +Regime Pattern 3: If i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗, then ℓi∗ < ki∗ and sufficient equilibrium condi- +tions are given by: +ρS∗ +i = + + + + + + + + + + + + + + + + + +0 +if i ∈ I \ {i∗}, +rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ +ℓi∗+1 +if i = i∗, +1 +if i ∈ J ∪ {πi∗(ℓi∗ + 1)}, +pπi∗(ℓi∗+1) +pi +if i ∈ K \ {πi∗(ℓi∗ + 1)}, +(9) +ρH∗ +i += + + + + + + + + + + + + + +ci +if i ∈ I ∪ J , +rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) − pi∗ci∗Si∗ +ℓi∗ +2 +if i = πi∗(ℓi∗ + 1), +pi∗ci∗ +pi +if i ∈ K \ {πi∗(ℓi∗ + 1)}, +(10) +where I := {1,...,i∗}, J := {πi∗(1),...,πi∗(ℓi∗)}, and K := {πi∗(ℓi∗ +1),...,πi∗(n−i∗)}. The value +of the games Γ and �Γ is given by +rH − pi∗ +� +rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ +ℓi∗+1 +� +ci∗ − +ℓi∗ +� +j=1 +pπi∗(j)cπi∗(j) − pπi∗(ℓi∗+1) +� +rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) +� +. + +14 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +From Theorem 1 and thanks to a carefully selected set of thresholds and parameters, we can +analytically solve the game �Γ and characterize NE of Γ. First, given i∗ ∈ �0,n − 1� satisfying +τi∗−1 < rH ≤ τi∗, we generally find that �1,i∗� represents the collection of locations for which H +cannot equalize the detection performance due to their small detection potentials, as intuited in +Section 3.1. When this occurs, S’s incentive is to utilize her resources to inspect the locations +�i∗ + 1,n� with higher detection performance, thus leaving the locations �1,i∗� uninspected while +being exhausted by H. However, the players’ behaviors in the remaining locations differ depending +on the subinterval in which rH belongs. Indeed, we find that the threshold νi∗ determines the +relation between ki∗ and ℓi∗, which in turn impacts the set of locations J that S deterministically +inspects and that H exhausts. It also dictates how the players should randomize their remaining +resources throughout the locations in equilibrium. +Theorem 1 shows that three major equilibrium regime patterns emerge as a result of the complex +and nonlinear interplay between the game parameters, captured by the selected thresholds τ and +ν. In fact, these regime patterns generalize the equilibrium results from the game studied in Gal +and Casas (2014), in which a prey hides from a predator that can inspect multiple locations with +heterogeneous detection capabilities. That game can be derived from ours by setting ci = 1 for all +i ∈ �1,n� and rH = 1. In such a setting, we can show that i∗ = 0 and ℓ0 = 0, and the NE regimes +observed by the authors correspond to Regime Pattern 1 when k0 = 0 and to Regime Pattern 2 when +k0 ≥ 1. We also note that the equilibrium behaviors described in Theorem 1 in its full generality +have not been observed in previously studied models that considered homogeneous detection rates +(Dziubi´nski and Roy 2018, Dahan et al. 2022) or one unit of resources for one or both players +(Washburn and Wood 1995, Karlin and Peres 2016). +In fact, if we allow the vector of capacities and the players’ resources to be continuous in the +game �Γ, and if we consider the set Ψ of parameters for which �Γ is nontrivial, i.e., +Ψ := +� +(n,p,c,rS,rH) : n ∈ Z>0, p ∈ (0,1]n, c ∈ Rn +>0, rS ∈ (0,n), rH ∈ (0, +n +� +i=1 +ci) +� +, +(11) +then we obtain the following stronger result: +Proposition 2. The set of parameters for which conditions (5)-(10) in Theorem 1 are necessary +and sufficient for a strategy profile (ρS∗,ρH∗) ∈ � +AS × � +AH to be a NE of �Γ is a dense subset of Ψ. +Thus, Proposition 2 shows that the analytical expressions in Theorem 1 describe all pure NE of +the continuous game �Γ almost surely. Similarly, for the original discrete game Γ, conditions (5)-(10) +characterize all NE of Γ, apart from some edge cases that are described in the electronic companion +(Proposition EC.1). Next, we provide further insights on the equilibrium behavior in each regime +pattern and illustrate them with examples. + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +15 +Regime Pattern 1: νi∗ < rH ≤ τi∗. In this regime, S does not inspect the set of locations +I = {1,...,i∗}, and instead focuses her resources on the remaining locations. Due to the het- +erogeneity of the detection rates in {i∗ + 1,...,n}, S deterministically inspects the set of ki∗ +most unreliable locations, J = {πi∗(1),...,πi∗(ki∗)}, and randomizes her rS − ki∗ resources in K = +{πi∗(ki∗ +1),...,πi∗(n−i∗)} so as to equalize the undetection probabilities in K. The feasibility of +S’s strategy is guaranteed by the definition of ki∗. +As a result of S’s inspection strategy, H’s hiding strategy exhausts all locations in I that are not +inspected by S and all locations in J for which S cannot equalize the undetection probabilities. Then +H randomizes his remaining rH − � +i∈I∪J ci resources so as to equalize the detection performance +across locations in K. We note that the feasibility and equilibrium guarantee of H’s strategy is a +consequence of the subinterval in which rH belongs. Indeed, since νi∗ < rH, then ki∗ ≤ ℓi∗, which +implies that H can exhaust the locations in I ∪ J and still provide a detection performance that +is sufficient so as to not incentivize S to reallocate some of her resources towards the locations +in I. Furthermore, since rH is upper bounded by τi∗, then H can feasibly equalize the detection +performances in K. +We illustrate this regime pattern with the following example: +Example 1. Consider the hide-and-seek model represented in Figure 1 and assume that rS = 3 +and rH = 7. +p1 = 1 +8 +1 +p2 = 1 +4 +π1(1) = 2 +p3 = 1 +3 +π1(2) = 3 +p5 = 4 +5 +π1(3) = 5 +p6 = 5 +6 +π1(4) = 6 +p4 = 1 +π1(5) = 4 +Figure 1 +Illustration of a NE for Regime Pattern 1 when rS = 3 and rH = 6. The hiding capacity of each location is +represented by the corresponding number of squares. Marginal inspection probabilities (resp. expected +numbers of hidden items) in equilibrium are represented by the blue (resp. red) colors. + +16 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +In this case, 449/80 = ν0 < rH ≤ τ1 = 289/40 and i∗ = 1. Furthermore, k1 = ℓ1 = 1 and π1(1) = 2, +π1(2) = 3, π1(3) = 5, π1(4) = 6, π1(5) = 4. Therefore, in equilibrium, S selects an inspection strategy +σS∗ ∈ ∆S such that ρ1(σS∗) = 0, ρ2(σS∗) = 1, ρ3(σS∗) = 40/43, ρ4(σS∗) = 40/129, ρ5(σS∗) = 50/129, +and ρ6(σS∗) = 16/43. On the other hand, H selects a hiding strategy σH∗ ∈ ∆H such that ρ1(σH∗) = +ρ2(σH∗) = 2, ρ3(σH∗) = 60/43, ρ4(σH∗) = 20/43, ρ5(σH∗) = 25/43, and ρ6(σH∗) = 24/43. The value +of the game is U(σS∗,σH∗) = 5 + 49/86. +△ +Regime Pattern 2: i∗ = 0 and τ−1 < rH ≤ ν0. In this regime, I = ∅ and every location is +inspected by S with positive probability. In fact, H’s number of resources is too small relative to +S’s capability of inspecting hiding locations. As a consequence, H’s equilibrium strategy consists in +greedily hiding items into the locations with smallest detections rates. This results in exhausting +the locations in J = {π0(1),...,π0(ℓ0)}, and assigning his remaining rH − � +i∈J ci resources in +location π0(ℓ0 + 1). The feasibility H’s strategy is guaranteed by the definition of ℓ0. +Since S has enough resources, as guaranteed by k0 ≥ ℓ0 + 1 > ℓ0, then her inspection strategy +consists in deterministically inspecting locations in J ∪ {π0(ℓ0 + 1)}, and randomizing sufficient +resources to ensure that the undetection probabilities of the remaining locations are no more than +that of location π0(ℓ0 + 1) so as to prevent H from reallocating some items from J . Interestingly, +this can be achieved by S without necessarily utilizing all her resources. +We illustrate this regime pattern with the following example: +Example 2. Consider the hide-and-seek model of Figure 2 and assume that rS = 6 and rH = 6. +p1 = 1 +8 +π0(1) = 1 +p2 = 1 +4 +π0(2) = 2 +p3 = 1 +3 +π0(3) = 3 +p5 = 4 +5 +π0(4) = 5 +p6 = 5 +6 +π0(5) = 6 +p4 = 1 +π0(6) = 4 +Figure 2 +Illustration of a NE for Regime Pattern 2 when rS = 6 and rH = 6. The hiding capacity of each location is +represented by the corresponding number of squares. Marginal inspection probabilities (resp. expected +numbers of hidden items) in equilibrium are represented by the blue (resp. red) colors. + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +17 +In this case, 0 = τ−1 < rH ≤ ν0 = 8 and i∗ = 0. Furthermore, 2 = ℓ0 < k0 = 4 and π0(1) = 1, π0(2) = +2, π0(3) = 3, π0(4) = 5, π0(5) = 6, π0(6) = 4. Therefore, one equilibrium inspection strategy for S +is given by σS∗ ∈ ∆S satisfying ρ1(σS∗) = ρ2(σS∗) = ρ3(σS∗) = 1, ρ4(σS∗) = 1/3, ρ5(σS∗) = 5/12, and +ρ6(σS∗) = 2/5. We note that S can implement this strategy with only 5 < rS units of resources. On +the other hand, H chooses a hiding strategy σH∗ ∈ ∆H such that ρ1(σH∗) = ρ2(σH∗) = ρ3(σH∗) = 2, +and ρ4(σH∗) = ρ5(σH∗) = ρ6(σH∗) = 0. The value of the game is U(σS∗,σH∗) = 4 + 7/12. +△ +Regime Pattern 3: i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗. In this final regime, we observe an interesting +and more complex behavior from the players’ equilibrium strategies. H cannot equalize the detection +performance across locations in I = {1,...,i∗} due to their capacity constraints. Thus, locations in +I are exhausted by H and initially left uninspected by S. Then, since ℓi∗ +1 ≤ ki∗, S cannot achieve +the desired level of undetection probability for locations in J ∪{πi∗(ℓi∗ +1)} = {πi∗(1),...,πi∗(ℓi∗ + +1)}. Therefore, H’s incentive is to exhaust locations in J and randomize some of his resources +to equalize the detection performance across K = {πi∗(ℓi∗ + 1),...,πi∗(n − i∗)} to that of i∗ (i.e., +pi∗ci∗). Interestingly, H is left with rH − � +i∈I∪J ci − pi∗ci∗Si∗ +ℓi∗ +1 > 0 resources that he additionally +allocates to location πi∗(ℓi∗ +1), which S deterministically inspects. The feasibility and equilibrium +guarantees of this strategy follow from the definition of ℓi∗. +As a result of H’s strategy, S deterministically inspects locations in J ∪ {πi∗(ℓi∗ + 1)} and +randomizes some of her resources to equalize the undetection probabilities in K \ {πi∗(ℓi∗ + 1)} +to that of πi∗(ℓi∗ + 1). This is possible since ℓi∗ + 1 ≤ ki∗. Interestingly, S still has rS − ℓi∗ − 1 − +pπi∗(ℓi∗+1)Si∗ +ℓi∗ +2 > 0 resources that she can now allocate among the i∗ ≥ 1 locations in I that were +previously left uninspected. S’s incentive is to allocate her remaining resources on the location in +I with highest detection performance, namely, i∗. Note that feasibility and equilibrium guarantees +for this strategy is a consequence of rH > τi∗−1. In particular, the resulting undetection probability +in i∗ is no less than that of locations in K, thus ensuring that H will not reallocate some of his +items from i∗ to locations in K. +We illustrate this final regime pattern with the following example: +Example 3. Consider the hide-and-seek model of Figure 3 and assume that rS = 4 and rH = 10. +In this case, 389/40 = τ1 < rH ≤ ν2 = 33/2 and i∗ = 2. Furthermore, k2 = 3 > 1 = ℓ2 and +π2(1) = 3, π2(2) = 5, π2(3) = 6, π2(4) = 4. Therefore, in equilibrium, S selects an inspection strat- +egy σS∗ ∈ ∆S such that ρ1(σS∗) = 0, ρ2(σS∗) = 6/25, ρ3(σS∗) = 1, ρ4(σS∗) = 4/5, ρ5(σS∗) = 1 and +ρ6(σS∗) = 24/25. On the other hand, H chooses a hiding strategy σH∗ ∈ ∆H such that ρ1(σH∗) = +ρ2(σH∗) = 2, ρ3(σH∗) = 4, ρ4(σH∗) = 1/2, ρ5(σH∗) = 9/10 and ρ6(σH∗) = 3/5. The value of the game +is U(σS∗,σH∗) = 6 + 71/75. +△ + +18 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +p1 = 1 +8 +1 +p2 = 1 +4 +2 +p3 = 1 +3 +π2(1) = 3 +p5 = 4 +5 +π2(2) = 5 +p6 = 5 +6 +π2(3) = 6 +p4 = 1 +π2(4) = 4 +Figure 3 +Illustration of a NE for Regime Pattern 3 when rS = 4 and rH = 10. The hiding capacity of each location +is represented by the corresponding number of squares. Marginal inspection probabilities (resp. expected +numbers of hidden items) in equilibrium are represented by the blue (resp. red) colors. +3.4. +Parametric Analysis +We continue our analysis by illustrating the impact of the players’ resources on the equilibrium +regimes of the game Γ (and �Γ). To this end, we consider the hide-and-seek instance defined by the +6 locations, 18 hiding capacities, and detection rates from Figures 1-3. Then, we plot the regions +determined by the subintervals [τi−1,νi] and [νi,τi] for each i ∈ �0,n − 1� as a function of rS and +rH. The resulting plot is shown in Figure 4. +We first observe that given a specific regime, the values of rS and rH for which that regime +holds form a complex region that may even be disconnected. Indeed, the borders that represent +the values of the thresholds τi and νi are defined by step functions of rS through the parameter +ki. Furthermore, for certain values of rS, some thresholds coincide, thus making certain regimes +unattainable for any value of rH. +Interestingly, when the number of inspection resources rS is high, very few equilibrium regimes +are possible. In such cases, S has enough resources to not leave any location uninspected, resulting +in i∗ = 0. Conversely, when rS is low, S’s strategy is highly sensitive to the number of items to hide +rH. Indeed, if rH is small, then H can equalize the detection performance across all locations and S +should inspect all locations with positive probability. As rH increases, S must carefully determine +which locations to inspect and prefers leaving i∗ locations uninspected to find more items in the +remaining locations. Furthermore, when rS is low, nearly all the regimes that are achievable follow +Regime Pattern 1 (for different values of i∗) since ki∗ ≤ ℓi∗ is most likely to hold for small amounts + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +19 +2 +4 +6 +8 +10 +12 +14 +16 +18 +1 +2 +3 +4 +5 +6 +rH +rS +τ−1 < rH ≤ ν0 +ν0 +< rH ≤ τ0 +τ0 +< rH ≤ ν1 +ν1 +< rH ≤ τ1 +τ1 +< rH ≤ ν2 +ν2 +< rH ≤ τ2 +τ2 +< rH ≤ ν3 +ν3 +< rH ≤ τ3 +τ3 +< rH ≤ ν4 +ν4 +< rH ≤ τ4 +τ4 +< rH ≤ ν5 = τ5 +Figure 4 +Illustration of equilibrium regions as a function of the number of resources rS and rH for the hide-and- +seek instance from Figures 1-3. +of inspection resources. Similarly, when rH is high, a single unit of inspection resources incentives +S to focus her inspection on the last location 6 (i.e., i∗ = 5). As rS increases, S can allocate more +resources to i∗ according to (9) (Regime Pattern 3) until i∗ is deterministically inspected. At this +point, a new regime following Pattern 3 emerges, with a smaller number of uninspected locations +i∗. The resulting impact of the players’ resources on the value of the game is illustrated in Figure +5. +1 +2 +3 +4 +5 +6 +0.4 +0.6 +0.8 +1 +rS +U(σS∗,σH∗)/rH +rH = 2 +rH = 4 +rH = 6 +rH = 8 +rH = 10 +rH = 12 +rH = 14 +rH = 16 +rH = 18 +Figure 5 +Fraction of undetected items in equilibrium as a function of rS, for different values of rH. +Specifically, Figure 5 compares the fractions of undetected items in equilibrium for different +amounts of players’ resources. This figure first shows the value of focusing inspection resources on + +20 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +locations �i∗ + 1,n� when the number of items to hide rH is high, due to the heterogeneity of the +capacities and detection rates. It also shows that the gain in performance by having additional +inspection resources varies as a function of the other parameters. This analysis can be utilized +in situations when S must determine the appropriate number of inspection resources that would +balance cost and performance. +In general, our analytical study provides us with valuable insights on the impact of the problem +characteristics (e.g., detection rates, hiding capacities, amounts of resources) on the players’ behav- +iors as well as the locations’ importance and criticality in equilibrium, which can be leveraged by +security decision makers. +4. +Equilibrium Computation +In the previous section, Theorem 1 solves the continuous game �Γ, and provides marginal inspection +probabilities and expected numbers of hidden items at each location in equilibrium of the game Γ. +However, to solve Γ, we must determine the coordination of rS inspection resources and rH items +to hide in order to satisfy these quantities. Specifically, given the vectors of marginal inspection +probabilities ρS∗ ∈ � +AS and expected numbers of hidden items ρH∗ ∈ � +AH in Theorem 1, we next seek +to efficiently compute mixed strategies (σS∗,σH∗) ∈ ∆S × ∆H such that ρ(σS∗) = ρS∗ and ρ(σH∗) = +ρH∗. From Proposition 1, this will ensure that (σS∗,σH∗) is a NE of Γ. +Thus, we aim to solve the following generic problem: Given a vector of capacities b ∈ Zn +>0, a +budget of resources r ∈ Z>0, and a vector ρ ∈ � +A(b,r), find a solution to the feasibility problem {σ ∈ +RA(b,r) +≥0 +: � +z∈A(b,r) σzz = ρ, � +z∈A(b,r) σz = 1}, which is guaranteed to exist by Lemma 1. Although +this problem involves an exponential number of variables, Carath´eodory’s theorem guarantees that +a solution exists with a support of size at most n + 1. In fact, this problem can be solved in +polynomial time using the ellipsoid method. However, this method is known to be practically +inefficient (Behnezhad et al. 2017). Thus, we derive another algorithm to efficiently solve the +feasibility problem. +Specifically, we extend the algorithm proposed by Dziubi´nski and Roy (2018) that computes in +time O(n2) a probability distribution with linear support over the set {z ∈ {0,1}n : �n +i=1 zi = r} +consistent with prescribed unidimensional marginal probabilities. However, we cannot apply their +algorithm to construct the equilibrium inspection and hiding strategies in our game Γ, as our +model involves locations with possibly heterogeneous capacities, and probabilities may be assigned +to resource allocations z that do not utilize the whole budget of resources r (e.g., in Theorem 1 – +Regime Pattern 2). +Thus, given a vector of capacities b ∈ Zn +>0, a budget of resources r ∈ Z>0, and a vector ρ ∈ � +A(b,r), +our algorithm returns a probability distribution σ ∈ ∆(b,r) that satisfies � +z∈A(b,r) σzzi = ρi for + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +21 +every i ∈ �1,n�. The general idea of each iteration of the algorithm is to express a given vector +ρ in � +A(b,r) as a convex combination of a vector in A(b,r) (i.e., with only integer components) +and a vector in � +A(b,r) with one more integer component than ρ has. To this end, the algorithm +allocates ⌊ρi⌋ resources at each location i ∈ �1,n�, and then determines where to allocate some of +the remaining ¯r := r −�n +i=1⌊ρi⌋ resources given the fractional part of each component of ρ, defined +as ¯ρi := ρi − ⌊ρi⌋. The algorithm determines the maximum number of locations q to allocate the +remaining resources ¯r. Naturally, q is upper bounded by ¯r and the number of positive components +of ¯ρ. Then, the algorithm carefully assigns positive probability to a resource allocation that first +assigns ⌊ρi⌋ resources at each location, and then assigns one additional resource at each of the q +locations with highest fractional parts ¯ρi. The algorithm then updates the vector ¯ρ so that the +new vector contains at least one more integer component than ¯ρ has. The algorithm iterates until +¯ρ ∈ {0,1}n, at which point the algorithm assigns the remaining probability to ⌊ρ⌋+ ¯ρ ∈ A(b,r). We +refer the reader to Algorithm 1 for the detailed pseudocode. +Algorithm 1: Resource Coordination. +Input +: A vector of capacities b ∈ Zn +>0, a budget of resources r ∈ Z>0, and a vector +ρ ∈ � +A(b,r). +Output: A probability distribution σ ∈ ∆(b,r) satisfying Ez∼σ[z] = ρ. +1 σ ← 0A(b,r), +¯ρ1 ← ρ − ⌊ρ⌋, +¯r ← r − �n +i=1⌊ρi⌋ +2 γ1 ← 1, +k ← 1 +3 while ¯ρk /∈ {0,1}n do +4 +θk ← Permutation of �1,n� such that ¯ρk +θk(1) ≥ ··· ≥ ¯ρk +θk(n) +5 +qk ← min +� +¯r, +��� +i ∈ �1,n� : ¯ρk +i > 0 +���� +6 +if qk < n then +7 +δk ← min{¯ρk +θk(qk),1 − ¯ρk +θk(qk+1)} +8 +else +9 +δk ← ¯ρk +θk(qk) +10 +ek ← 0n +11 +foreach j ∈ {1,...,qk} do +12 +ek +θk(j) ← 1 +13 +σ⌊ρ⌋+ek ← σ⌊ρ⌋+ek + γkδk +14 +¯ρk+1 ← +1 +1−δk (¯ρk − δkek) +15 +γk+1 ← γk(1 − δk) +16 +k ← k + 1 +17 σ⌊ρ⌋+¯ρk ← σ⌊ρ⌋+¯ρk + γk +18 return σ + +22 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +We note that at each iteration k of the while loop (3-16), ¯ρk = δkek + (1 − δk)¯ρk+1. In fact, the +algorithm selects ek and δk so that ¯ρk can be expressed as a convex combination of ek ∈ {0,1}n +and a new vector ¯ρk+1 that has at least one more integer component than ¯ρk does. This guarantees +that the algorithm terminates. At termination, the algorithm expresses ρ as a convex combination +of resource allocations in A(b,r). This can be translated into the following theorem: +Theorem 2. Given a vector of capacities b ∈ Zn +>0, a budget of resources r ∈ Z>0, and a vec- +tor ρ ∈ � +A(b,r), Algorithm 1 returns a probability distribution σ ∈ ∆(b,r) satisfying Ez∼σ[z] = ρ. +Furthermore, σ has a support of size at most n + 1 and is computed in time O(n2). +Thus, Algorithm 1 matches the support size guaranteed by Carath´eodory’s theorem on the +polytope � +A(b,r). In fact, the algorithm performs at most n iterations. Furthermore, by reutilizing +the sorting of ¯ρk of the previous iteration, we can implement each iteration (except for the first +one) in time O(n), which guarantees an overall running time of O(n2). +We can now summarize the overall solution approach for computing a NE of the game Γ. First, +marginal inspection probabilities and expected numbers of hidden items in each location are com- +puted according to Theorem 1: Sorting the detection potentials (pici)i∈�1,n� and the detection rates +(pi)i∈�1,n� in non-decreasing order requires O(nlogn) steps. Then, for every i ∈ �0,n−1�, the map- +ping πi, parameters ki and ℓi, and thresholds τi and νi can be computed in time O(n). Identifying +the subinterval [τi∗−1,νi∗] or [νi∗,τi∗] in which rH belongs can be performed in O(logn) steps, and +evaluating the expressions from Theorem 1 requires O(n) more steps. Thus, computing equilibrium +marginal inspection probabilities and expected numbers of hidden items can be implemented in +time O(n2). Finally, Algorithm 1 computes the mixed strategies consistent with the unidimensional +quantities in time O(n2). Hence, we obtain the final result: +Corollary 1. The game Γ can be solved in time O(n2) with equilibrium strategies of support +size at most n + 1 each. +Thus, we obtain an efficient solution approach for solving the large-scale hide-and-seek game Γ +with multiple resources and heterogeneous locations. Finally, our approach provides solutions with +small supports that can be easily implemented by security decision makers. +5. +Conclusion +In this work, we investigated a hide-and-seek game in which a seeker inspects locations to find +items hidden by a hider. We extended previous models in the literature by considering the coordi- +nation of multiple resources for both players in locations with heterogeneous hiding capacities and +probabilities of detecting hidden items where the search takes place. The objective of the seeker + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +23 +(resp. hider) is to minimize (resp. maximize) the expected number of undetected items. To com- +pute the mixed strategies Nash equilibria of this large-scale zero-sum game, we proposed a solution +approach that first derives analytical equilibrium properties and then efficiently coordinates the +players’ resources. In particular, we showed that the marginal inspection probabilities and expected +numbers of hidden items in each location in equilibrium form a Nash equilibrium of a continuous +zero-sum game. By carefully selecting a set of parameters and thresholds, we analytically solved this +continuous game. Our analysis highlighted a complex interplay between the game parameters and +permitted us to evaluate their impact on the players’ behaviors in equilibrium and the criticality of +each location. Then, we derived a quadratic time algorithm that coordinates the players’ resources +to satisfy the characterized equilibrium marginal distributions and computes a Nash equilibrium +of the hide-and-seek game with linear support. Our insights and solution approach can be used to +inform security agencies that are interested in scheduling multiple units to detect and seize illegal +commodities hidden by a criminal organization. +This work can be extended in multiple directions by considering additional features that would +widen the applicability of our results. One extension is to consider different commodities hidden by +the hider with heterogeneous values, as well as different inspection resources with different detection +characteristics. Another extension is to incorporate heterogeneous valuations of the locations to +extend the original setting by Von Neumann (1953). Finally, an interesting research direction is to +consider a repeated version of the hide-and-seek game with players learning the initially unknown +characteristics of their opponent while they interact. +Acknowledgments +This work was supported by the Georgia Tech Stewart fellowship and new faculty start up grant. We are +grateful to Mohit Singh for his valuable feedback. +References +Ahmadinejad A, Dehghani S, Hajiaghayi M, Lucier B, Mahini H, Seddighin S (2019) From duels to +battlefields: Computing equilibria of Blotto and other games. Mathematics of Operations Research +44(4):1304–1325. +Alpern S, Gal S (2006) The theory of search games and rendezvous, volume 55 (Springer Science & Business +Media). +Bahamondes B, Dahan M (2022) Network inspection from locations with imperfect detection capabilities. +2022 American Control Conference (ACC), 613–620. +Behnezhad S, Blum A, Derakhshan M, HajiAghayi M, Mahdian M, Papadimitriou CH, Rivest RL, Seddighin +S, Stark PB (2018) From battlefields to elections: Winning strategies of Blotto and auditing games. +Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2291–2310 +(SIAM). + +24 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Behnezhad S, Dehghani S, Derakhshan M, HajiAghayi M, Seddighin S (2017) Faster and simpler algorithm +for optimal strategies of blotto game. Thirty-First AAAI Conference on Artificial Intelligence. +Bertsimas D, Nasrabadi E, Orlin JB (2016) On the power of randomization in network interdiction. Opera- +tions Research Letters 44(1):114–120. +Blocki J, Christin N, Datta A, Procaccia A, Sinha A (2015) Audit games with multiple defender resources. +Proceedings of the AAAI Conference on Artificial Intelligence 29(1). +Borel E (1921) La th´eorie du jeu et les ´equations int´egralesa noyau sym´etrique. Comptes Rendus de +l’Acad´emie des Sciences 173(1304-1308):58. +Budish E, Che YK, Kojima F, Milgrom P (2013) Designing random allocation mechanisms: Theory and +applications. American Economic Review 103(2):585–623. +Cai Y, Daskalakis C, Weinberg SM (2012) Optimal multi-dimensional mechanism design: Reducing revenue +to welfare maximization. 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, +130–139 (IEEE). +Cardei M, Du DZ (2005) Improving wireless sensor network lifetime through power aware organization. +Wireless Networks 11(3):333–340. +Chan H, Jiang AX, Leyton-Brown K, Mehta R (2016) Multilinear games. International Conference on Web +and Internet Economics, 44–58 (Springer). +Clarkson J, Lin KY, Glazebrook KD (2022) A classical search game in discrete locations. Mathematics of +Operations Research . +Crawford VP, Iriberri N (2007) Fatal attraction: Salience, naivete, and sophistication in experimental “hide- +and-seek” games. American Economic Review 97(5):1731–1750. +Dahan M, Sela L, Amin S (2022) Network inspection for detecting strategic attacks. Operations Research +70(2):1008–1024. +Dziubi´nski M, Roy J (2018) Hide and seek game with multiple resources. International Symposium on +Algorithmic Game Theory, 82–86 (Springer). +Flood MM (1972) The hide and seek game of von Neumann. Management Science 18(5-part-2):107–109. +Freund Y, Schapire RE (1999) Adaptive game playing using multiplicative weights. Games and Economic +Behavior 29(1):79–103. +Gal S, Casas J (2014) Succession of hide–seek and pursuit–evasion at heterogeneous locations. Journal of +the Royal Society Interface 11(94):20140062. +Gr¨otschel M, Lov´asz L, Schrijver A (2012) Geometric algorithms and combinatorial optimization, volume 2 +(Springer Science & Business Media). +Gupta A, Langbort C, Ba¸sar T (2016) Dynamic games with asymmetric information and resource con- +strained players with applications to security of cyberphysical systems. IEEE Transactions on Control +of Network Systems 4(1):71–81. + +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +25 +Hellerstein L, Lidbetter T, Pirutinsky D (2019) Solving zero-sum games using best-response oracles with +applications to search games. Operations Research 67(3):731–743. +Hess KM, Orthmann CH, Cho HL (2013) Police Operations: Theory and Practice (Cengage Learning). +Hochbaum DS, Fishbain B (2011) Nuclear threat detection with mobile distributed sensor networks. Annals +of Operations Research 187(1):45–63. +Hoeksma R, Manthey B, Uetz M (2016) Efficient implementation of Carath´eodory’s theorem for the single +machine scheduling polytope. Discrete Applied Mathematics 215:136–145. +Hoeksma R, Uetz M (2013) Two dimensional optimal mechanism design for a sequencing problem. Interna- +tional Conference on Integer Programming and Combinatorial Optimization, 242–253 (Springer). +Hohzaki R (2016) Search games: Literature and survey. Journal of the Operations Research Society of Japan +59(1):1–34. +Hota AR, Clements AA, Sundaram S, Bagchi S (2016) Optimal and game-theoretic deployment of security +investments in interdependent assets. International Conference on Decision and Game Theory for +Security, 101–113 (Springer). +Jezierski T, Adamkiewicz E, Walczak M, Sobczy´nska M, G´orecka-Bruzda A, Ensminger J, Papet E (2014) +Efficacy of drug detection by fully-trained police dogs varies by breed, training level, type of drug and +search environment. Forensic Science International 237:112–118. +Karlin A, Peres Y (2016) Game Theory, Alive (AMS, Providence, RI). +Kletti T, Renders JM, Loiseau P (2022a) Introducing the expohedron for efficient pareto-optimal fairness- +utility amortizations in repeated rankings. Proceedings of the Fifteenth ACM International Conference +on Web Search and Data Mining, 498–507. +Kletti T, Renders JM, Loiseau P (2022b) Pareto-optimal fairness-utility amortizations in rankings with a +DBN exposure model. arXiv preprint arXiv:2205.07647 . +Korzhyk D, Conitzer V, Parr R (2010) Complexity of computing optimal Stackelberg strategies in security +resource allocation games. Twenty-Fourth AAAI Conference on Artificial Intelligence. +Letchford J, Conitzer V (2013) Solving security games on graphs via marginal probabilities. Twenty-Seventh +AAAI Conference on Artificial Intelligence. +Lidbetter T (2013) Search games with multiple hidden objects. SIAM Journal on Control and Optimization +51(4):3056–3074. +Lidbetter T, Lin KY (2019) Searching for multiple objects in multiple locations. European Journal of Oper- +ational Research 278(2):709–720. +Lipton RJ, Markakis E, Mehta A (2003) Playing large games using simple strategies. Proceedings of the 4th +ACM Conference on Electronic Commerce, 36–41. + +26 +Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Miao F, Zhu Q, Pajic M, Pappas GJ (2018) A hybrid stochastic game for secure control of cyber-physical +systems. Automatica 93:55–63. +Miloˇsevi´c J, Dahan M, Amin S, Sandberg H (2019) A network monitoring game with heterogeneous compo- +nent criticality levels. 2019 IEEE 58th Conference on Decision and Control (CDC), 4379–4384 (IEEE). +Musegaas M, Schlicher L, Blok H (2022) Stackelberg production-protection games: Defending crop production +against intentional attacks. European Journal of Operational Research 297(1):102–119. +Pirani M, Taylor JA, Sinopoli B (2021) Strategic sensor placement on graphs. Systems & Control Letters +148:104855. +Pita J, Jain M, Marecki J, Ord´o˜nez F, Portway C, Tambe M, Western C, Paruchuri P, Kraus S (2008) +Deployed ARMOR protection: The application of a game theoretic model for security at the Los Angeles +International Airport. Proceedings of the 7th International Joint Conference on Autonomous Agents +and Multiagent Systems: Industrial Track, 125–132. +Powell R (2009) Sequential, nonzero-sum “Blotto”: Allocating defensive resources prior to attack. Games +and Economic Behavior 67(2):611–615. +Von Neumann J (1928) Zur theorie der gesellschaftsspiele. Mathematische Annalen 100(1):295–320. +Von Neumann J (1953) A certain zero-sum two-person game equivalent to the optimal assignment problem. +Contributions to the Theory of Games 2(0):5–12. +Wang WQ, Shao H (2014) Radar-to-radar interference suppression for distributed radar sensor networks. +Remote Sensing 6(1):740–755. +Washburn A, Wood K (1995) Two-person zero-sum games for network interdiction. Operations Research +43(2):243–251. +Zhu Q, Basar T (2015) Game-theoretic methods for robustness, security, and resilience of cyberphysical con- +trol systems: games-in-games principle for optimal cross-layer resilient control systems. IEEE Control +Systems Magazine 35(1):46–65. + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec1 +Proofs of Statements +EC.1. +Proofs of Section 3 +Proof of Lemma 1. Let b ∈ Zn +>0 be a vector of capacities and r ∈ Z>0 be a budget of resources. +First, we observe that � +A(b,r) is the convex hull of A(b,r). Indeed, since the row vector 1⊤ +n is +totally unimodular, then (1) is an ideal formulation of A(b,r) and the polytope � +A(b,r) has integral +extreme points, which we denote z1,...,zI. Consider a vector ρ′ ∈ Rn. If ρ′ ∈ � +A(b,r), then there +exist λ1,...,λI ∈ [0,1] such that ρ′ = �I +i=1 λizi and �I +i=1 λi = 1. Then, σ ∈ [0,1]A(b,r) defined by +σzi = λi for i ∈ �1,I� and σz = 0 if z /∈ {z1,...,zI} is a probability distribution in ∆(b,r) and +satisfies ρi(σ) = ρ′ +i for all i ∈ �1,n�. Conversely, if there exists σ ∈ ∆(b,r) that satisfies ρ′ +i = ρi(σ) = +� +z∈A(b,r) ziσz for all i ∈ �1,n�, then ρ′ is a convex combination of elements on A(b,r), and belongs +to its convex hull, � +A(b,r). +□ +Proof of Proposition 1. +– Consider a strategy profile (σS,σH) ∈ ∆S × ∆H. Then, +U(σS,σH) +(2) += +n +� +i=1 +(1 − piEx∼σS[xi])Ey∼σH[yi] +(3),(4) += +˜u(ρ(σS),ρ(σH)). +(EC.1) +– Let (ρS∗,ρH∗) ∈ � +AS × � +AH be a NE of �Γ. From Lemma 1, let (σS∗,σH∗) ∈ ∆S × ∆H that satisfies +ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗. Using equilibrium conditions in �Γ, we obtain: +∀σS ∈ ∆S, U(σS∗,σH∗) +(EC.1) += +˜u(ρS∗,ρH∗) ≤ ˜u(ρ(σS),ρH∗) +(EC.1) += +U(σS,σH∗), +∀σH ∈ ∆H, U(σS∗,σH∗) +(EC.1) += +˜u(ρS∗,ρH∗) ≥ ˜u(ρS∗,ρ(σH)) +(EC.1) += +U(σS∗,σH). +Thus, (σS∗,σH∗) is a NE of Γ. +Conversely, let (σS∗,σH∗) ∈ ∆S × ∆H be a NE of Γ, then we must show that (ρ(σS∗),ρ(σH∗)) is +a NE of �Γ. First, from Lemma 1, we know that (ρ(σS∗),ρ(σH∗)) ∈ � +AS × � +AH. Then, given (ρS,ρH) ∈ +� +AS× � +AH, let (σS,σH) ∈ ∆S×∆H satisfying ρ(σS) = ρS and ρ(σH) = ρH (Lemma 1). Using equilibrium +conditions in Γ, we obtain: +˜u(ρ(σS∗),ρ(σH∗)) +(EC.1) += +U(σS∗,σH∗) ≤ U(σS,σH∗) +(EC.1) += +˜u(ρS,ρ(σH∗)), +and ˜u(ρ(σS∗),ρ(σH∗)) +(EC.1) += +U(σS∗,σH∗) ≥ U(σS∗,σH) +(EC.1) += +˜u(ρ(σS∗),ρH). +Thus, (ρ(σS∗),ρ(σH∗)) is a NE of �Γ. +– Given a NE (σS∗,σH∗) ∈ ∆S × ∆H of Γ, we know that (ρ(σS∗),ρ(σH∗)) is a NE of �Γ and +U(σS∗,σH∗) +(EC.1) += +˜u(ρ(σS∗),ρ(σH∗)). Thus, the values of the games Γ and �Γ are identical. +□ + +ec2 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +We note that when some detection rates are identical, permutations πi ordering �i+1,n� by their +detection rates may not be unique. To simplify our proofs, we assume without loss of generality that +πi maintains the order between identical detection rates, i.e., πi(j) < πi(k) when 1 ≤ j < k ≤ n − i +and pπi(j) = pπi(k), thus rendering πi unique for every i ∈ �0,n − 1�. +Before proving Lemma 2 we need the following auxiliary lemmas. +Lemma EC.1. Let i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i. Then: +πi−1(j) = + + + + + +πi(j) +if j ∈ �1,j∗ − 1�, +i +if j = j∗, +πi(j − 1) +if j ∈ �j∗ + 1,n − i + 1�. +(EC.2) +Moreover, for every k ∈ �0,n − i + 1�, +k +� +j=1 +cπi−1(j) = + + + + + + + + + + + + + +k +� +j=1 +cπi(j) +if k ∈ �0,j∗ − 1�, +k−1 +� +j=1 +cπi(j) + ci +if k ∈ �j∗,n − i + 1�, +(EC.3) +and +Si−1 +k+1 = +� +Si +k+1 + 1 +pi +if k ∈ �0,j∗ − 1�, +Si +k +if k ∈ �j∗,n − i + 1�. +(EC.4) +Proof of Lemma EC.1. Let i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i. The +permutation πi−1 sorts locations �i,n� in order of non-decreasing detection rates. After removing +pi = pπi−1(j∗) from the chain of inequalities, we obtain pπi−1(1) ≤ ··· ≤ pπi−1(j∗−1) ≤ pπi−1(j∗+1) ≤ +··· ≤ pπi−1(n−i+1), which sorts �i + 1,n� by detection rates, thus providing (EC.2). +As a consequence, for every k ∈ �0,j∗ − 1�, +k +� +j=1 +cπi−1(j) = +k +� +j=1 +cπi(j), +Si−1 +k+1 = +j∗−1 +� +j=k+1 +1 +pπi−1(j) ++ 1 +pi ++ +n−i+1 +� +j=j∗+1 +1 +pπi−1(j) += 1 +pi ++ +j∗−1 +� +j=k+1 +1 +pπi(j) ++ +n−i+1 +� +j=j∗+1 +1 +pπi(j−1) += 1 +pi ++ Si +k+1. +Similarly, for every k ∈ �j∗,n − i + 1�, +k +� +j=1 +cπi−1(j) = +j∗−1 +� +j=1 +cπi(j) + ci + +k +� +j=j∗+1 +cπi(j−1) = +k−1 +� +j=1 +cπi(j) + ci, +Si−1 +k+1 = +n−i+1 +� +j=k+1 +1 +pπi−1(j) += +n−i+1 +� +j=k+1 +1 +pπi(j−1) += Si +k. +□ + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec3 +Lemma EC.2. For every i ∈ �0,n − 1�, ki exists and +Ki := �k ∈ �0,n − i� : k + pπi(k)Si +k+1 < rS +� = �0,ki�. +Furthermore, for every i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� such that πi−1(j∗) = i, +ki−1 ≤ +� +ki +if ki−1 ∈ �0,j∗ − 1�, +ki + 1 +if ki−1 ∈ �j∗,n − i + 1�. +(EC.5) +Proof of Lemma EC.2. Let i ∈ �0,n − 1�. Since pπi(0)Si +1 = 0 < rS, then 0 ∈ Ki and ki exists. +Next, we consider k ∈ �1,n − i�. If k ∈ Ki, then k − 1 ∈ Ki, as shown below: +rS > k + pπi(k)Si +k+1 = k − 1 + pπi(k) +� +Si +k+1 + +1 +pπi(k) +� +≥ k − 1 + pπi(k−1)Si +k, +where we used the fact that k ∈ Ki and πi orders locations in �i + 1,n� by their detection rates. +Therefore, Ki = �0,ki�. +We next analyze ki as a function of i. Let i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� be such that +πi−1(j∗) = i. If ki−1 ∈ �0,j∗ − 1�, we obtain: +rS > ki−1 + pπi−1(ki−1)Si−1 +ki−1+1 +(EC.2),(EC.4) +≥ +ki−1 + pπi(ki−1)Si +ki−1+1. +Since ki−1 ≤ j∗ − 1 ≤ n − i, we deduce that ki−1 ∈ Ki, and ki−1 ≤ ki. +If ki−1 ∈ �j∗ + 1,n − i + 1�, we obtain: +rS +(EC.2),(EC.4) +> +ki−1 + pπi(ki−1−1)Si +ki−1 > ki−1 − 1 + pπi(ki−1−1)Si +ki−1−1+1. +Since ki−1 − 1 ≤ n − i, then ki−1 − 1 ∈ Ki and ki−1 − 1 ≤ ki. +Finally, if ki−1 = j∗, we obtain: +rS +(EC.4) +> +ki−1 + pπi−1(ki−1−1)Si +ki−1 +(EC.2) +> +ki−1 − 1 + pπi(ki−1−1)Si +ki−1−1+1. +Thus, ki−1 − 1 ∈ Ki and ki−1 − 1 ≤ ki. Note that we used throughout that pπi(0) = 0. +□ +Lemma EC.3. For every i ∈ �0,n − 1�, if rH > �i +j=1 cj + piciSi +1, then ℓi exists and +Li := +� +ℓ ∈ �0,n − i� : +i +� +j=1 +cj + +ℓ +� +j=1 +cπi(j) + piciSi +ℓ+1 < rH +� += �0,ℓi�. +Proof of Lemma EC.3. Let i ∈ �0,n − 1� and suppose that rH > �i +j=1 cj + piciSi +1. Then, 0 ∈ Li +and ℓi exists. Next, consider ℓ ∈ �1,n − i�. If ℓ ∈ Li, then ℓ − 1 ∈ Li, as shown below: +rH − +i +� +j=1 +cj > +ℓ−1 +� +j=1 +cπi(j) + pπi(ℓ) +pπi(ℓ) +cπi(ℓ) + piciSi +ℓ+1 ≥ +ℓ−1 +� +j=1 +cπi(j) + piciSi +ℓ, +where we used the fact that ℓ ∈ Li and locations in �1,n� are ordered by their detection potentials. +Therefore, Li = �0,ℓi�. +□ + +ec4 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +We are now ready to prove Lemma 2. +Proof of Lemma 2. First, τ−1 = 0 by definition. Moreover, we observe that kn−1 ∈ {0,1}, which +implies that τn−1 = m: If kn−1 = 0, then τn−1 = �n−1 +j=1 cj + pncn/pn = m. If kn−1 = 1, then τn−1 = +�n−1 +j=1 cj + cn = m. We next show that τi−1 ≤ νi ≤ τi for all i ∈ �0,n − 1�. +We note that the inequality νi ≤ τi follows directly from the fact that pici ≤ pi+1ci+1. Thus, it +only remains to show that τi−1 ≤ νi. This is trivial for i = 0, so we may assume that i ∈ �1,n − 1�. +We note that +νi − τi−1 = ci + +ki +� +j=1 +cπi(j) − +ki−1 +� +j=1 +cπi−1(j) − pici +� +Si−1 +ki−1+1 − Si +ki+1 +� +. +(EC.6) +Let j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i. If ki−1 ∈ �0,j∗ − 1�, we obtain: +νi − τi−1 +(EC.3),(EC.4) += +ci + +ki +� +j=1 +cπi(j) − +ki−1 +� +j=1 +cπi(j) − pici +� +Si +ki−1+1 − Si +ki+1 + 1 +pi +� +(EC.5) += +ki +� +j=ki−1+1 +pπi(j)cπi(j) − pici +pπi(j) +≥ 0. +If ki−1 ∈ �j∗,n − i + 1�, we obtain: +νi − τi−1 +(EC.3),(EC.4) += +ki +� +j=1 +cπi(j) − +ki−1−1 +� +j=1 +cπi(j) − pici +� +Si +ki−1 − Si +ki+1 +� (EC.5) += +ki +� +j=ki−1 +pπi(j)cπi(j) − pici +pπi(j) +≥ 0. +□ +The following lemma derives properties satisfied by our auxiliary parameters: +Lemma EC.4. Let i ∈ �0,n − 1� be such that rH > τi−1. Then, the following statements hold: +– ℓi exists. Furthermore, when i ≥ 1, let j∗ ∈ �1,n − i + 1� satisfying πi−1(j∗) = i. Then, +ki−1 ≤ +� +ℓi +if ki−1 ∈ �0,j∗ − 1� +ℓi + 1 +if ki−1 ∈ �j∗,n − i + 1�. +(EC.7) +– If νi < rH, then ki ≤ ℓi. If rH ≤ νi, then ki > ℓi. +Proof of Lemma EC.4. +– Let i ∈ �0,n − 1�. If i = 0 and rH > τ−1 = 0, then 0 ∈ L0 and ℓ0 exists. We now assume that +i ∈ �1,n − 1� and rH > τi−1. Let j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i. If ki−1 ∈ �0,j∗ − 1�, +then: +rH > τi−1 +(EC.3),(EC.4) += +i +� +j=1 +cj + +ki−1 +� +j=1 +cπi(j) + piciSi +ki−1+1. +Since ki−1 ≤ j∗ − 1 ≤ n − i, then ki−1 ∈ Li, ℓi exists, and ki−1 ≤ ℓi. + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec5 +If ki−1 ∈ �j∗,n − i + 1�, then: +rH +(EC.3),(EC.4) +> +i +� +j=1 +cj + +ki−1−1 +� +j=1 +cπi(j) + piciSi +ki−1−1+1. +Since ki−1 − 1 ≤ n − i, then ki−1 − 1 ∈ Li, ℓi exists, and ki−1 − 1 ≤ ℓi. +– Let i ∈ �0,n − 1� be such that rH > τi−1. If rH > νi, then by definition of νi, ki ∈ Li and ki ≤ ℓi. +On the other hand, if rH ≤ νi, then ki /∈ Li. Since ℓi exists and Li = �0,ℓi� (Lemma EC.3), then +ki > ℓi. +□ +We can now prove the first theorem of this article. +Proof of Theorem 1. Let rS ∈ �1,n − 1� and rH ∈ �1,m − 1�. Let i∗ ∈ �0,n − 1� satisfying +τi∗−1 < rH ≤ τi∗. +Regime Pattern 1: Suppose that νi∗ < rH ≤ τi∗. From Lemma EC.4, we know that ki∗ ≤ ℓi∗. Let +ρS∗ ∈ Rn and ρH∗ ∈ Rn satisfying (5) and (6), respectively. We will show that (ρS∗,ρH∗) ∈ � +AS × � +AH +and is a NE of �Γ. +First, we note that ki∗ < n − i∗. Indeed, if ki∗ = n − i∗, then rH > νi∗ = �n +j=1 cj = m, which is a +contradiction. Therefore, K ̸= ∅ and Si∗ +ki∗ +1 > 0. By definition of ki∗, we obtain +rS > ki∗ + pπi∗(ki∗ )Si∗ +ki∗+1 ≥ ki∗, +(EC.8) +which implies that ρS∗ +i +≥ 0 for every i ∈ K. Furthermore, since ki∗ + 1 ≤ n − i∗ and ki∗ + 1 /∈ Ki∗, +then: +rS ≤ ki∗ + 1 + pπi∗ (ki∗+1) +� +Si∗ +ki∗ +1 − +1 +pπi∗(ki∗ +1) +� += ki∗ + pπi∗(ki∗ +1)Si∗ +ki∗ +1. +(EC.9) +Thus, for every i ∈ K, ρS∗ +i ≤ 1. Finally, +n +� +i=1 +ρS∗ +i = |J | + rS − ki∗ +Si∗ +ki∗+1 +Si∗ +ki∗+1 = rS. +Therefore, ρS∗ ∈ � +AS. Next, we show that ρH∗ ∈ � +AH. Since ki∗ ≤ ℓi∗, then ki∗ ∈ Li∗ and +rH > +i∗ +� +j=1 +cj + +ki∗ +� +j=1 +cπi∗(j) + pi∗ci∗Si∗ +ki∗+1 ≥ +i∗ +� +j=1 +cj + +ki∗ +� +j=1 +cπi∗(j). +(EC.10) +Thus, ρH∗ +i +≥ 0 for every i ∈ K. Furthermore, since rH ≤ τi∗ and pi∗+1ci∗+1 ≤ pici for every i ∈ J ∪K, +we obtain: +∀i ∈ K, ρH∗ +i += +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +piSi∗ +ki∗ +1 +≤ pi∗+1ci∗+1 +pi +≤ ci. +(EC.11) + +ec6 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Finally, +n +� +i=1 +ρH∗ +i += +i∗ +� +i=1 +ci + +ki∗ +� +j=1 +cπi∗(j) + +� +rH − +i∗ +� +j=1 +cj − +ki∗ +� +j=1 +cπi∗(j) +� +Si∗ +ki∗ +1 +Si∗ +ki∗ +1 += rH. +Therefore, ρH∗ ∈ � +AH. +Next, we show that (ρS∗,ρH∗) is a NE of �Γ. To this end, we first note the following: +∀ρH ∈ � +AH, min +ρS∈ � +AS +˜u(ρS,ρH) +(4) += +n +� +i=1 +ρH +i +− max +ρS +n +� +i=1 +piρH +i ρS +i +s.t. +n +� +i=1 +ρS +i ≤ rS +(EC.12) +0 ≤ ρS +i ≤ 1, +∀i ∈ �1,n�. +Thus, a best response to ρH ∈ � +AH is an optimal solution to a continuous knapsack problem with +n different (fractional) objects of unitary weights and a knapsack capacity equal to rS. Each object +i ∈ �1,n� has a profit equal to piρH +i . An optimal solution consists in filling the capacity of the +knapsack with the objects with highest profits. +Then, given ρH∗ satisfying (6), the “profit” of each object is given by: +piρH∗ +i += + + + + + +pici +if i ∈ I ∪ J , +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +Si∗ +ki∗ +1 +if i ∈ K. +(EC.13) +We know that for every i ∈ I and every l ∈ J : +pici ≤ pi∗ci∗ +(EC.10) +< +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +Si∗ +ki∗+1 +(EC.11) +≤ +pi∗+1ci∗+1 ≤ plcl. +(EC.14) +Therefore, the objects in J are the most profitable, followed by the objects in K that have equal +profit, followed by the objects in I. We now must determine bounds on rS. We showed in (EC.8) +that rS > ki∗ = |J |. An upper bound is given as follows: +rS +(EC.9) +≤ +ki∗ + 1 + +n−i∗ +� +j=ki∗ +2 +pπi∗(ki∗+1) +pπi∗(j) +≤ n − i∗ = |J | + |K|. +Thus, one best response to ρH∗ selects all the objects in J and any fraction of the objects in K +until the knapsack is full. Hence, ρS∗ defined in (5) is a best response to ρH∗. +To show that ρH∗ is a best response to ρS∗, we similarly observe the following: +∀ρS ∈ � +AS, max +ρH∈ � +AH +˜u(ρS,ρH) +(4) += max +ρH +n +� +i=1 +(1 − piρS +i )ρH +i +s.t. +n +� +i=1 +ρH +i ≤ rH +(EC.15) +0 ≤ ρH +i ≤ ci, +∀i ∈ �1,n�. + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec7 +Thus, a best response to ρS ∈ � +AS is an optimal solution to another continuous knapsack problem +with n different (fractional) objects of unitary weights and a knapsack capacity equal to rH. Each +object i ∈ �1,n� is available ci times and has a profit equal to (1 − piρS +i ). An optimal solution +consists in selecting as many copies as possible of the objects with highest profits until filling the +capacity of the knapsack. +Then, given ρS∗ satisfying (5), the “profit” of each object is given by: +1 − piρS∗ +i = + + + + + + + + + + + +1 +if i ∈ I, +1 − pi +if i ∈ J , +1 − rS − ki∗ +Si∗ +ki∗ +1 +if i ∈ K. +(EC.16) +We have the following inequalities: +∀i ∈ J , 1 − rS − ki∗ +Si∗ +ki∗ +1 +(EC.8) +< +1 − pπi∗(ki∗) ≤ 1 − pi < 1. +(EC.17) +Therefore, the objects in I are the most profitable, followed by the objects in J , followed by +the objects in K that have equal profit. To determine which objects will be selected given rH, we +recall that rH < m = �n +i=1 ci. Furthermore, (EC.10) implies that rH > � +i∈I∪J ci. +Thus, one best response to ρS∗ consists in selecting all copies of the objects in I and J , and in +selecting any fraction of the objects in K until the knapsack is full. Hence, ρH∗ defined in (6) is a +best response to ρS∗. +As a consequence, (ρS∗,ρH∗) is a NE of �Γ. From Proposition 1, we deduce that any strategy profile +(σS∗,σH∗) ∈ ∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ (which exists as a consequence +of Lemma 1) is a NE of Γ. +Furthermore, the value of the games Γ and �Γ is given by: +U(σS∗,σH∗) = ˜u(ρS∗,ρH∗) = rH − +ki∗ +� +j=1 +pπi∗ (j)cπi∗(j) − +� +rS − ki∗ +�� +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +� +Si∗ +ki∗ +1 +. +Regime Pattern 2: We now consider the case when i∗ = 0 and τ−1 < rH ≤ ν0. From Lemma EC.4, +we know that k0 > ℓ0. Let ρS∗ ∈ Rn and ρH∗ ∈ Rn satisfying (7) and (8), respectively. We will +analogously show that (ρS∗,ρH∗) ∈ � +AS × � +AH and is a NE of �Γ. +First, we note that ℓ0 < k0 ≤ n, which implies that ℓ0 + 1 ≤ n. Thus, S0 +ℓ0+2 is well defined and +K ̸= ∅. For every i ∈ K \ {π0(ℓ0 + 1)}, pπ0(ℓ0+1) ≤ pi. Furthermore, since ℓ0 + 1 ∈ K0, we obtain: +n +� +i=1 +ρS∗ +i = ℓ0 + 1 + pπ0(ℓ0+1)S0 +ℓ0+2 < rS. +(EC.18) + +ec8 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Thus, ρS∗ ∈ � +AS. Next, we show that ρH∗ ∈ � +AH. Since ℓ0 ∈ L0, ℓ0 + 1 /∈ L0, and ℓ0 + 1 ≤ n, then: +0 < rH − +ℓ0 +� +j=1 +cπ0(j) = ρH∗ +π0(ℓ0+1) = rH − +ℓ0+1 +� +j=1 +cπ0(j) + cπ0(ℓ0+1) ≤ cπ0(ℓ0+1). +(EC.19) +Since �n +i=1 ρH∗ +i += rH, we can then conclude that ρH∗ ∈ � +AH. +Next, we show that (ρS∗,ρH∗) is a NE of �Γ. Given ρH∗ satisfying (8), the profit of each object in +the knapsack problem (EC.12) is given by: +piρH∗ +i += + + + + + + + + + + + +pici +if i ∈ J , +pπ0(ℓ0+1) +� +rH − +ℓ0 +� +j=1 +cπ0(j) +� +if i = π0(ℓ0 + 1), +0 +if i ∈ K \ {π0(ℓ0 + 1)}. +(EC.20) +Since rS +(EC.18) +> +ℓ0 + 1 = |J | + 1, then a best response to ρH∗ will select all the objects in J ∪ +{π0(ℓ0 +1)} (and might not entirely fill the knapsack). Hence, ρS∗ defined in (7) is a best response +to ρH∗. Then, given ρS∗ satisfying (7), the profit of each object in the knapsack problem (EC.15) +is given by: +1 − piρS∗ +i = +�1 − pi +if i ∈ J ∪ {π0(ℓ0 + 1)}, +1 − pπ0(ℓ0+1) +if i ∈ K \ {π0(ℓ0 + 1)}. +By definition of π0, we have the following inequalities: 1 − pπ0(1) ≥ ··· ≥ 1 − pπ0(ℓ0+1). Since +(EC.19) implies that rH > �ℓ0 +j=1 cπ0(j), then one best response to ρS∗ can select all copies of the +objects in J and can select any fraction of the objects in K until the knapsack is full. Hence, ρH∗ +defined in (8) is a best response to ρS∗. +Thus, (ρS∗,ρH∗) is a NE of �Γ. From Proposition 1, we deduce that any strategy profile (σS∗,σH∗) ∈ +∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ is a NE of Γ. Furthermore, the value of the +games Γ and �Γ is given by: +U(σS∗,σH∗) = ˜u(ρS∗,ρH∗) = rH − +ℓ0 +� +j=1 +pπ0(j)cπ0(j) − pπ0(ℓ0+1) +� +rH − +ℓ0 +� +j=1 +cπ0(j) +� +. +Regime Pattern 3: Finally, we consider the case when i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗. From Lemma +EC.4, we know that ki∗ > ℓi∗. Let ρS∗ ∈ Rn and ρH∗ ∈ Rn satisfying (9) and (10), respectively. We +will analogously show that (ρS∗,ρH∗) ∈ � +AS × � +AH and is a NE of �Γ. +Similarly, we note that ℓi∗ < ki∗ ≤ n − i∗, which implies that ℓi∗ + 1 ≤ n − i∗. Thus, Si∗ +ℓi∗+2 is well +defined and K ̸= ∅. Since ℓi∗ + 1 ∈ Ki∗, then +rS > pπi∗(ℓi∗+1)Si∗ +ℓi∗+2 + ℓi∗ + 1 = pπi∗(ℓi∗+1)Si∗ +ℓi∗+1 + ℓi∗. +(EC.21) + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec9 +Thus, ρS∗ +i∗ > 0. Next, we will show by contradiction the following upper bound: +rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ +ℓi∗+1 ≤ min +�pπi∗ (ℓi∗+1) +pi∗ +, 1 +� +. +(EC.22) +Let us assume that (EC.22) does not hold, and let j∗ ∈ �1,n − i∗ + 1� satisfying πi∗−1(j∗) = i∗. If +ℓi∗ + 1 ≤ j∗ − 1, then: +pπi∗(ℓi∗+1) +pi∗ += min +�pπi∗(ℓi∗+1) +pi∗ +, 1 +� +< rS − ℓi∗ − 1 − pπi∗(ℓi∗+1)Si∗ +ℓi∗+2 +(EC.2),(EC.4) += +rS − ℓi∗ − 1 − pπi∗−1(ℓi∗+1)Si∗−1 +ℓi∗+2 + +pπi∗(ℓi∗+1) +pi∗ +, +which implies that ℓi∗ + 1 ≤ ki∗−1. However, by Lemma EC.4, this can only occur when ki∗−1 ≥ j∗, +for which we obtain the following contradiction j∗ ≤ ki∗−1 +(EC.7) +≤ +ℓi∗ + 1 ≤ j∗ − 1. +If on the other hand ℓi∗ + 1 ≥ j∗, then j∗ < ℓi∗ + 2 ≤ n − i∗ + 1 and +1 = min +�pπi∗−1(ℓi∗+2) +pi∗ +, 1 +� += min +�pπi∗ (ℓi∗+1) +pi∗ +, 1 +� +(EC.2),(EC.4) +< +rS − ℓi∗ − 1 − pπi∗−1(ℓi∗+2)Si∗−1 +ℓi∗ +3, +which implies that ℓi∗ + 2 ≤ ki∗−1, thus contradicting (EC.7). Therefore, (EC.22) holds. Finally, +n +� +i=1 +ρS∗ +i = rS − ℓi∗ − pπ∗(ℓi∗+1)Si∗ +ℓi∗+1 + |J | + 1 + pπ∗(ℓi∗+1)Si∗ +ℓi∗+2 = rS, +which implies that ρS∗ ∈ � +AS. +Next, we show that ρH∗ ∈ � +AH. Since locations in �1,n� are ordered by their detection potentials, +then for every i ∈ K, pici ≥ pi∗ci∗. Since ℓi∗ ∈ Li∗, then: +ρH∗ +πi∗(ℓi∗ +1) = rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) − pi∗ci∗Si∗ +ℓi∗+1 + +pi∗ci∗ +pπi∗(ℓi∗+1) +> +pi∗ci∗ +pπi∗(ℓi∗+1) +≥ 0. +(EC.23) +Furthermore, since ℓi∗ + 1 /∈ Li∗ and ℓi∗ + 1 ≤ n − i∗, then: +ρH∗ +πi∗ (ℓi∗+1) = rH − +i∗ +� +j=1 +cj − +ℓi∗+1 +� +j=1 +cπi∗(j) − pi∗ci∗Si∗ +ℓi∗+2 + cπi∗(ℓi∗+1) ≤ cπi∗(ℓi∗+1). +(EC.24) +Finally, +n +� +i=1 +ρH∗ +i += +� +i∈I∪J +ci + rH − +i∗ +� +i=1 +cj − +ℓi∗ +� +j=1 +cπ∗(j) − pi∗ci∗Si∗ +ℓi∗+2 + pi∗ci∗Si∗ +ℓi∗+2 = rH. +Therefore, ρH∗ ∈ � +AH. +Next, we show that (ρS∗,ρH∗) is a NE of �Γ. Given ρH∗ satisfying (10), the profit of each object +in the knapsack problem (EC.12) is given by: +piρH∗ +i += + + + + + + + + + + + +pici +if i ∈ I ∪ J , +pπi∗(ℓi∗+1) +� +rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) − pi∗ci∗Si∗ +ℓi∗+2 +� +if i = πi∗(ℓi∗ + 1), +pi∗ci∗ +if i ∈ K \ {πi∗(ℓi∗ + 1)}. +(EC.25) + +ec10 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Since the locations in �1,n� are ordered by their detection potentials, then we know that p1c1 ≤ +··· ≤ pi∗ci∗ ≤ pjcj for every j ∈ J . Furthermore, we have the following inequality: +pi∗ci∗ +(EC.23) +< +pπi∗ (ℓi∗+1) +� +rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) − pi∗ci∗Si∗ +ℓi∗+2 +� +. +(EC.26) +Thus, the objects in J ∪ {πi∗(ℓi∗ + 1)} are the most profitable, followed by the objects in {i∗} ∪ +K \ {πi∗(ℓi∗ + 1)} that have equal profit, followed by the objects in I \ {i∗}. +From (EC.21), we know that rS > ℓi∗ + 1 = |J ∪ {πi∗(ℓi∗ + 1)}|. Furthermore, an upper bound is +given as follows: +rS +(EC.22) +≤ +ℓi∗ + 1 + +n−i∗ +� +j=ℓi∗ +1 +pπi∗(ℓi∗+1) +pπi∗ (j) +≤ n − i∗ + 1 = |{i∗} ∪ J ∪ K|. +Therefore, one best response to ρH∗ will select all the objects in J ∪{πi∗(ℓi∗ +1)} and will select +any fraction of the objects in {i∗}∪ K \{πi∗(ℓi∗ + 1)} until the knapsack is full. Hence, ρS∗ defined +in (9) is a best response to ρH∗. +Then, given ρS∗ satisfying (9), the profit of each object in the knapsack problem (EC.15) is given +by: +1 − piρS∗ +i = + + + + + + + + + + + + + + + +1 +if i ∈ I \ {i∗} +1 − pi∗(rS − ℓi∗ − pπ∗(ℓi∗+1)Si∗ +ℓi∗+1) +if i = i∗ +1 − pi +if i ∈ J ∪ {πi∗(ℓi∗ + 1)}, +1 − pπi∗(ℓi∗+1) +if i ∈ K \ {πi∗(ℓi∗ + 1)}. +(EC.27) +By definition of πi∗, we have the following inequalities: 1 − pπi∗(1) ≥ ··· ≥ 1 − pπi∗(ℓi∗+1). Further- +more, (EC.22) implies that 1 − pi∗(rS − ℓi∗ − pπ∗(ℓi∗+1)Si∗ +ℓi∗+1) ≥ 1 − pπi∗(ℓi∗+1). +Since (EC.23) implies that rH > �i∗ +j=1 cj + �ℓi∗ +j=1 cπi∗(j), then one best response to ρS∗ selects all +copies of the objects in I ∪J and selects any fraction of the objects in K until the knapsack is full. +Hence, ρH∗ defined in (10) is a best response to ρS∗. +Thus, (ρS∗,ρH∗) is a NE of �Γ. From Proposition 1, we deduce that any strategy profile (σS∗,σH∗) ∈ +∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ is a NE of Γ. Furthermore, the value of the +games Γ and �Γ is given by: +U(σS∗,σH∗) = ˜u(ρS∗,ρH∗) =rH − pi∗ +� +rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ +ℓi∗+1 +� +ci∗ − +ℓi∗ +� +j=1 +pπi∗ (j)cπi∗(j) +− pπi∗(ℓi∗ +1) +� +rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) +� +. +□ + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec11 +Proposition EC.1. Let rS ∈ �1,n − 1� and rH ∈ �1,m − 1�. Let i∗ ∈ �0,n − 1� satisfying τi∗−1 < +rH ≤ τi∗. +Regime Pattern 1: νi∗ < rH ≤ τi∗. Suppose also that rH < τi∗ and rS < ki∗ + 1 + pπi∗(ki∗ +1)Si∗ +ki∗ +2. +Then, (5)-(6) are necessary and sufficient conditions for a strategy profile (ρS∗,ρH∗) ∈ � +AS × � +AH to +be a NE of �Γ. +Regime Pattern 2: i∗ = 0 and τ−1 < rH ≤ ν0. Suppose +also +that +pπ0(ℓ0) < pπ0(ℓ0+1) < 1 +and +pπ0(ℓ0+1) < pπ0(ℓ0+2) if ℓ0 ≤ n−2. Then, (7)-(8) are necessary and sufficient conditions for a strategy +profile (ρS∗,ρH∗) ∈ � +AS × � +AH to be a NE of �Γ. +Regime Pattern 3: i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗. Suppose also that pi∗−1ci∗−1 < pi∗ci∗ < pi∗+1ci∗+1, +rH < �i∗ +j=1 cj + �ℓi∗+1 +j=1 +cπi∗(j) + pi∗ci∗Si∗ +ℓi∗+2, rS < ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 +ki∗−1+2, pπi∗(ℓi∗) < +pπi∗(ℓi∗+1) < 1, and pπi∗ (ℓi∗+1) < pπi∗(ℓi∗+2) if ℓi∗ ≤ n − i∗ − 2. Then, (9)-(10) are necessary and +sufficient conditions for a strategy profile (ρS∗,ρH∗) ∈ � +AS × � +AH to be a NE of �Γ. +Proof of Proposition EC.1. +Regime Pattern 1: νi∗ < rH ≤ τi∗. We additionally consider the following non-edge case assump- +tions: rH < τi∗ and rS < ki∗ + 1 + pπi∗(ki∗ +1)Si∗ +ki∗ +2. +Let (ρS∗,ρH∗) ∈ � +AS × � +AH satisfying (5)-(6) and let (ρS′,ρH′) ∈ � +AS × � +AH be any NE of the game +�Γ. Since �Γ is a zero-sum game, then (ρS′,ρH∗) is also a NE of �Γ. +Since ρS′ is a best response to ρH∗, it is an optimal solution to the continuous knapsack prob- +lem (EC.12). The profits of each object are given by (EC.13) and satisfy inequalities (EC.14). +Furthermore, rH < τi∗ implies that: +rH − �i∗ +j=1 cj − �ki∗ +j=1 cπi∗(j) +Si∗ +ki∗+1 +< pi∗+1ci∗+1. +Since |J | < rS ≤ |J |+|K|, then any best response to ρH∗ must select all the objects in J , must not +select any object in I, and must entirely fill the knapsack. Thus, ρS′ +i = 0 for every i ∈ I, ρS′ +i = 1 for +every i ∈ J , and �n +i=1 ρS′ +i = rS. +Next, since ρH∗ is a best response to ρS′, then it is an optimal solution to the continuous knapsack +problem (EC.15). Next, we write the dual of (EC.15) associated with ρS′: +min +α,β rHα + +n +� +i=1 +ciβi +s.t. α + βi ≥ 1 − piρS′ +i , +∀i ∈ �1,n� +(EC.28) +α ≥ 0 +βi ≥ 0, +∀i ∈ �1,n�. + +ec12 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Let (α∗,β∗) be an optimal solution of the dual problem (EC.28). Since 0 < ρH∗ +i +< ci for every i ∈ K, +then by complementary slackness, β∗ +i = 0 and ρS′ +i = (1 − α∗)/pi for every i ∈ K. Since ρS′ must fill +the knapsack (EC.12) entirely, then: +rS = +� +i∈I∪J ∪K +ρS′ +i = ki∗ + (1 − α∗)Si∗ +ki∗ +1. +Thus, for every i ∈ K, ρS′ +i = (rS − ki∗)/(piSi∗ +ki∗ +1). In conclusion, ρS′ satisfies (5). +Similarly, (ρS∗,ρH′) is a NE of �Γ. Then, ρH′ is a best response to ρS∗ and is an optimal solution +to the continuous knapsack problem (EC.15). The profits of each object are given by (EC.16) and +satisfy inequalities (EC.17). Since � +i∈I∪J ci < rH < m, then any best response to ρS∗ must select all +copies of the objects in I and J . Therefore, ρH′ +i = ci for every i ∈ I ∪J . Furthermore, ki∗ +1 /∈ Ki∗ +and the non-edge case assumptions imply that: +1 − rS − ki∗ +Si∗ +ki∗+1 +(EC.9) +> +1 − pπi∗(ki∗ +1) ≥ 0. +(EC.29) +Therefore, ρH′ must entirely fill the knapsack and �n +i=1 ρH′ +i = rH. +Next, since ρS∗ is a best response to ρH′, then it is an optimal solution to the continuous knapsack +problem in (EC.12). The dual of (EC.12) associated with ρH′ is given by: +min +η,ξ rSη + +n +� +i=1 +ξi +s.t. η + ξi ≥ piρH′ +i , +∀i ∈ �1,n� +(EC.30) +η ≥ 0, +ξi ≥ 0, +∀i ∈ �1,n�. +Let (η∗,ξ∗) be an optimal solution of the dual problem (EC.30). Since (EC.8) and (EC.29) imply +that 0 < ρS∗ +i < 1 for every i ∈ K, then by complementary slackness, ξ∗ +i = 0 and ρH′ +i = η∗/pi for every +i ∈ K. Since ρH′ must fill the knapsack (EC.15) entirely, then: +rH = +� +i∈I∪J ∪K +ρH′ +i = +i∗ +� +j=1 +cj + +ki∗ +� +j=1 +cπi∗(j) + ηSi∗ +ki∗ +1. +Therefore, ρH′ satisfies (6). +Regime Pattern 2: i∗ = 0 and τ−1 < rH ≤ ν0. We additionally consider the following non-edge +case assumptions: pπ0(ℓ0) < pπ0(ℓ0+1) < 1 and pπ0(ℓ0+1) < pπ0(ℓ0+2) if ℓ0 ≤ n − 2. +Let ρH∗ ∈ � +AH satisfying (8) and let ρS′ be an equilibrium strategy for S in �Γ. Since ρS′ is a best +response to ρH∗, it is an optimal solution to the continuous knapsack problem (EC.12). The profits +of each object are given by (EC.20). Furthermore, (EC.19) and the inequality rS > |J | + 1 imply + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec13 +that any best response to ρH∗ must select all the objects in J ∪ {π0(ℓ0 + 1)}. Therefore, ρS′ +i = 1 for +every i ∈ J ∪ {π0(ℓ0 + 1)}. +Since ρH∗ is a best response to ρS′, then it is an optimal solution to the continuous knap- +sack problem (EC.15). Since 0 +(EC.19) +< +ρH∗ +π0(ℓ0+1), then at optimality of the dual (EC.28), α∗ = 1 − +pπ0(ℓ0+1)ρS′ +π0(ℓ0+1) − β∗ +π0(ℓ0+1) = 1 − pπ0(ℓ0+1) − β∗ +π0(ℓ0+1) ≤ 1 − pπ0(ℓ0+1). Finally, for every i ∈ K \ +{π0(ℓ0 + 1)}, ρH∗ +π0(ℓ0+1) = 0 < cπ0(ℓ0+1). Thus, by complementary slackness, β∗ +i = 0 and ρS′ +i ≥ (1 − +α∗)/pi ≥ pπ0(ℓ0+1)/pi for every i ∈ K \ {π0(ℓ0 + 1)}. In conclusion, ρS′ satisfies (7). +Let ρH′ be an equilibrium strategy for H in �Γ, and consider ρS∗ ∈ AS satisfying +ρS∗ +i = 1 +if i ∈ J ∪ {π0(ℓ0 + 1)}, +pπ0(ℓ0+1) +pi +< ρS∗ +i < 1 +if i ∈ K \ {π0(ℓ0 + 1)}, +n +� +i=1 +ρS∗ +i < rS. +Such a vector exists as a consequence of (EC.18) and since pπ0(ℓ0+1) < pi for every i ∈ K\{π0(ℓ0+1)} +under the non-edge case assumptions. Then, ρH′ is a best response to ρS∗ and is an optimal solution +to the continuous knapsack problem (EC.15). The profits of each object are given by: +∀i ∈ J ∪ {π0(ℓ0 + 1)}, 1 − piρS∗ +i = 1 − pi +∀i ∈ K \ {π0(ℓ0 + 1)}, 1 − piρS∗ +i < 1 − pπ0(ℓ0+1). +Under the non-edge case assumptions, 1 − pπ0(1) ≥ ··· ≥ 1 − pπ0(ℓ0) > 1 − pπ0(ℓ0+1) > 0. Since +�ℓ0 +j=1 cπ0(j) < rH < �ℓ0+1 +j=1 cπ0(j), then any best response to ρS∗ selects all copies of the objects in J +and fills the remaining of the knapsack with objects in π0(ℓ0 + 1). Therefore ρH′ satisfies (8). +Regime Pattern 3: i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗. We additionally consider the following non-edge +case assumptions: pi∗−1ci∗−1 < pi∗ci∗ < pi∗+1ci∗+1, rH < �i∗ +j=1 cj + �ℓi∗+1 +j=1 +cπi∗(j) + pi∗ci∗Si∗ +ℓi∗+2, rS < +ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 +ki∗−1+2, pπi∗ (ℓi∗) < pπi∗(ℓi∗+1) < 1, and pπi∗(ℓi∗+1) < pπi∗(ℓi∗+2) if ℓi∗ ≤ n − +i∗ − 2. +Let (ρS∗,ρH∗) ∈ � +AS × � +AH satisfying (9)-(10) and let (ρS′,ρH′) ∈ � +AS × � +AH be any NE of the game +�Γ. Since ρS′ is a best response to ρH∗, it is an optimal solution to the continuous knapsack problem +(EC.12). The profits of each object are given by (EC.25). Under the non-edge case assumptions, +p1c1 ≤ ··· ≤ pi∗−1ci∗−1 < pi∗ci∗ < pi∗+1ci∗+1 ≤ ··· ≤ pncn. From (EC.26) and the fact that |J | + 1 < +rS ≤ |J | + |K| + 1, we deduce that any best response to ρH∗ selects all the objects in J ∪ {πi∗(ℓi∗ + +1)}, does not select any object in I \{i∗}, and fills the knapsack (EC.12) entirely. Therefore, ρS′ +i = 0 +for every i ∈ I \ {i∗}, ρS′ +i = 1 for every i ∈ J ∪ {πi∗(ℓi∗ + 1)}. and �n +i=1 ρS′ +i = rS. +Since ρH∗ is a best response to ρS′, then it is an optimal solution to the continuous knapsack prob- +lem (EC.15). Under the non-edge case assumptions, 0 < ρH∗ +πi∗ (ℓi∗+1) < cπi∗(ℓi∗+1). Thus, at optimality + +ec14 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +of the dual (EC.28), α∗ = 1 − pπi∗(ℓi∗+1)ρS′ +πi∗ (ℓi∗+1) = 1 − pπi∗(ℓi∗+1). Furthermore, since 0 < ρH∗ +i +< ci +for every i ∈ K \ {πi∗(ℓi∗ + 1)} under the non-edge case assumptions, then ρS′ +i = (1 − α∗)/pi = +pπi∗(ℓi∗+1)/pi. Since ρH′ must fill the knapsack (EC.12) entirely, then: +ρS′ +i∗ = rS − ℓi∗ − 1 − pπi∗(ℓi∗+1)Si∗ +ℓi∗+2 = rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ +ℓi∗+1. +Therefore, ρS′ satisfies (9). +Similarly, (ρS∗,ρH′) is a NE of �Γ. Then, ρH′ is a best response to ρS∗ and is an optimal solution to +the continuous knapsack problem (EC.15). The profits of each object are given by (EC.27). Under +the non-edge case assumptions, 1 − pπi∗(1) ≥ ··· ≥ 1 − pπi∗(ℓi∗ ) > 1 − pπi∗(ℓi∗+1) > 0. We next show +that: +rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ +ℓi∗ +1 < min +�pπi∗ (ℓi∗+1) +pi∗ +, 1 +� +. +(EC.31) +Let us assume that (EC.31) does not hold and let j∗ ∈ �1,n − i∗ + 1� satisfying πi∗−1(j∗) = i∗. +If ℓi∗ + 1 ≤ j∗ − 1, then (EC.22) implies that rS = ℓi∗ + 1 + pπi∗−1(ℓi∗+1)Si∗−1 +ℓi∗+2, which contradicts +the non-edge case assumption. If on the other hand ℓi∗ + 1 ≥ j∗, then j∗ < ℓi∗ + 2 ≤ n − i∗ + 1 and +rS = ℓi∗ + 2 + pπi∗−1(ℓi∗+2)Si∗−1 +ℓi∗ +3, which also contradicts the non-edge case assumptions. Therefore, +(EC.31) holds. +Since � +i∈I∪J ci < rH, then any best response to ρS∗ must select all copies of the objects in +I and J , and must fill the knapsack (EC.15) entirely. Therefore, ρH′ +i += ci for every i ∈ I ∪ J +and �n +i=1 ρH′ +i += rH. Since ρS∗ is a best response to ρH′, then it is an optimal solution to the +continuous knapsack problem (EC.12). Under the non-edge case assumptions, 0 < ρS∗ +i < 1 for every +i ∈ {i∗} ∪ K \ {πi∗(ℓi∗ + 1)}. Therefore, at optimality of the dual (EC.30), η∗ = pi∗ρH′ +i∗ = pi∗ci∗, and +ρH′ +i∗ = η∗/pi = pi∗ci∗/pi for every i ∈ K \ {πi∗(ℓi∗ + 1)}. Finally, since ρH′ fills the knapsack (EC.15) +entirely, then: +ρH′ +πi∗(ℓi∗+1) = rH − +i∗ +� +j=1 +cj − +ℓi∗ +� +j=1 +cπi∗(j) − pi∗ci∗Si∗ +ℓi∗+2. +In conclusion, ρH′ satisfies (10). +□ +Proof of Proposition 2. In this proof, we allow the vector of capacities c and the players’ +resources rS and rH to be continuous in the game �Γ. Let Ψ be the set of parameters given by (11) +for which �Γ is nontrivial. First, we note that +Ψ′ := {(n,p,c,rS,rH) ∈ Ψ : pi < 1 ∀i ∈ �1,n�, pi ̸= pj and pici ̸= pjcj ∀i ̸= j ∈ �1,n�} +is a dense subset of Ψ. Next, we consider an instantiation of the game parameters (n,p,c,rS,rH) ∈ +Ψ′. We order the indices such that pici < pi+1ci+1 for every i ∈ �1,n − 1�. Let i∗ ∈ �0,n − 1� such +that τi∗−1 < rH ≤ τi∗. + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec15 +We first consider the case of Regime Pattern 1, i.e., νi∗ < rH ≤ τi∗. We then consider new player +resources ˆrS = rS − ε and ˆrH = rH − ε for ε > 0 arbitrarily small. To avoid confusion, we denote the +corresponding parameters that depend on ˆrS and ˆrH as ˆki, ˆℓi, ˆτi, ˆνi, and ˆi∗. +By definition of ki∗, and for arbitrarily small ε, we obtain: +ki∗ + pπi∗(ki∗)Si∗ +ki∗+1 < ˆrS < rS ≤ ki∗ + 1 + pπi∗(ki∗+1)Si∗ +ki∗+2. +Thus, ˆki∗ = ki∗. This implies that ˆτi∗ = τi∗ and ˆνi∗ = νi∗. We then deduce the following inequalities +for arbitrarily small ε: ˆνi∗ = νi∗ < ˆrH < rH ≤ τi∗ = ˆτi∗. Thus, ˆi∗ = i∗. +Since νi∗ < ˆrH < τi∗ and ˆrS < ki∗ + 1 + pπi∗(ki∗ +1)Si∗ +ki∗ +2, then Proposition EC.1 implies that +all pure NE of the game �Γ with the parameters (n,p,c, ˆrS, ˆrH) for arbitrarily small ε > 0 satisfy +the corresponding equilibrium conditions (5)-(6). Furthermore (n,p,c, ˆrS, ˆrH) is arbitrarily close to +(n,p,c,rS,rH). +We next consider the case of Regime Pattern 2, i.e., i∗ = 0 and rH ≤ ν0. Proposition EC.1 implies +that all pure NE of the game �Γ with the parameters (n,p,c, ˆrS, ˆrH) ∈ Ψ′ satisfy the corresponding +equilibrium conditions (7)-(8). +Finally, we consider the case of Regime Pattern 3, i.e., i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗. We then +consider new player resources ˆrS = rS − ε and ˆrH = rH − ε for ε > 0 arbitrarily small. Similarly, we +denote the corresponding auxiliary parameters as ˆki, ˆℓi, ˆτi, ˆνi, and ˆi∗. +Using a similar derivation as above, we deduce that for arbitrarily small ε, ˆki∗ = ki∗ and ˆνi∗ = νi∗. +Then, by definition of ki∗−1, we obtain: +ki∗−1 + pπi∗−1(ki∗−1)Si∗−1 +ki∗−1+1 < ˆrS < rS ≤ ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 +ki∗−1+2. +Thus, ˆki∗−1 = ki∗−1 and ˆτi∗−1 = τi∗−1. Then, we obtain that ˆi∗ = i∗ since ˆτi∗−1 = τi∗−1 < ˆrH < rH ≤ +νi∗ = ˆνi∗. Finally, by definition of ℓi∗, we obtain: +i∗ +� +j=1 +cj + +ℓi∗ +� +j=1 +cπi∗(j) + pi∗ci∗Si∗ +ℓi∗+1 < ˆrH < rH ≤ +i∗ +� +j=1 +cj + +ℓi∗+1 +� +j=1 +cπi∗(j) + pi∗ci∗Si∗ +ℓi∗+2. +Thus, ˆℓi∗ = ℓi∗. Since rH < �i∗ +j=1 cj + �ℓi∗+1 +j=1 +cπi∗(j) + pi∗ci∗Si∗ +ℓi∗+2 and rS < ki∗−1 + 1 + +pπi∗−1(ki∗−1+1)Si∗−1 +ki∗−1+2, then Proposition EC.1 implies that all pure NE of the game �Γ with the +parameters (n,p,c, ˆrS, ˆrH) for arbitrarily small ε > 0 satisfy the corresponding equilibrium condi- +tions (9)-(10). Furthermore (n,p,c, ˆrS, ˆrH) is arbitrarily close to (n,p,c,rS,rH). +□ +EC.2. +Proofs of Section 4 +Before proving Theorem 2, we show that Algorithm 1 is well defined and terminates. We denote +as κ∗ ∈ Z≥0 ∪ {+∞} the number of iterations of the while loop (3-16) in Algorithm 1. + +ec16 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +Proposition EC.2. Each iteration of Algorithm 1 is well defined. In particular, +∀k ∈ �1,κ∗ + 1�, ¯ρk ∈ [0,1]n and +n +� +i=1 +¯ρk +i ≤ ¯r, +(EC.32) +∀k ∈ �1,κ∗�, qk ∈ �1,n� and δk ∈ [0,1). +(EC.33) +Proof of Proposition EC.2. We show (EC.32) and (EC.33) by induction. We first consider +k = 1. By construction, ¯ρ1 = ρ − ⌊ρ⌋ ∈ [0,1]n. Furthermore, by definition of � +A(b,r), we obtain: +¯r = r − +n +� +i=1 +⌊ρi⌋ = r − +n +� +i=1 +ρi + +n +� +i=1 +¯ρ1 +i ≥ +n +� +i=1 +¯ρ1 +i. +(EC.34) +Next, q1 is constructed when the algorithm initiates the while loop (3-16), that is, when ¯ρ1 /∈ +{0,1}n. Since ¯ρ1 ≥ 0n and ¯r ∈ Z, then 1 ≤ |{i ∈ �1,n� : ¯ρ1 +i > 0}| ≤ n and 1 ≤ ⌈�n +i=1 ¯ρ1 +i ⌉ +(EC.34) +≤ +¯r. +Therefore, 1 ≤ q1 ≤ n. Finally, δ1 is well defined since q1 ∈ �1,n�, and δ1 ∈ [0,1] as a consequence +of ¯ρ1 ∈ [0,1]n. We next show by contradiction that δ1 < 1. Indeed, if δ1 = 1, then we first deduce +that for every j ∈ �1,q1�, 1 ≥ ¯ρ1 +θ1(j) ≥ ¯ρ1 +θ1(q1) ≥ δ1 = 1. If q1 = n, then this contradicts ¯ρ1 /∈ {0,1}n. If +q1 < n, then, we derive the following inequalities: +q1 = +q1 +� +j=1 +¯ρ1 +θ1(j) ≤ +n +� +i=1 +¯ρ1 +i +(EC.34) +≤ +¯r, +(EC.35) +q1 ≤ +��� +i ∈ �1,n� : ¯ρ1 +i = 1 +��� ≤ +��� +i ∈ �1,n� : ¯ρ1 +i > 0 +���. +(EC.36) +If q1 = ¯r, (resp. q1 = |{i ∈ �1,n� : ¯ρ1 +i > 0}|) then (EC.35) (resp. (EC.36)) implies that ¯ρ1 +θ1(j) = 0 for +every j ∈ �q1 + 1,n�. This also contradicts ¯ρ1 /∈ {0,1}n. Thus, δ1 < 1. +Next, we assume that (EC.32) and (EC.33) hold for k ∈ �1,κ∗�. Since δk < 1, then we obtain: +∀j ∈ �1,qk�, 0 ≤ +¯ρk +θk(j) − ¯ρk +θk(qk) +1 − δk +≤ +¯ρk +θk(j) − δk +1 − δk += ¯ρk+1 +θk(j) ≤ 1 − δk +1 − δk = 1, +and if qk < n, then ∀j ∈ �qk + 1,n�, 0 ≤ +¯ρk +θk(j) +1 − δk = ¯ρk+1 +θk(j) ≤ +¯ρk +θk(j) +¯ρk +θk(qk+1) +≤ 1. +Therefore, for every i ∈ �1,n�, ¯ρk+1 +i +∈ [0,1]. Next, we show that �n +i=1 ¯ρk+1 +i +≤ ¯r: +n +� +i=1 +¯ρk+1 +i += +qk +� +j=1 +¯ρk +θk(j) − δk +1 − δk ++ +n +� +j=qk+1 +¯ρk +θk(j) +1 − δk = +1 +1 − δk +� n +� +i=1 +¯ρk +i − qkδk +� +. +If qk = ¯r, then: +n +� +i=1 +¯ρk+1 +i +≤ +1 +1 − δk +� +¯r − ¯rδk� += ¯r. + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec17 +If on the other hand qk = +��� +i ∈ �1,n� : ¯ρk +i > 0 +���, then ¯ρk ∈ [0,1]n implies that: +n +� +i=1 +¯ρk+1 +i += +1 +1 − δk +� n +� +i=1 +¯ρk +i − +���i ∈ �1,n� : ¯ρk +i > 0���δk +� +≤ +1 +1 − δk + + +n +� +i=1 +¯ρk +i − δk +� +{i∈�1,n� : ¯ρk +i >0} +¯ρk +i + + += +n +� +i=1 +¯ρk +i ≤ ¯r. +Therefore, �n +i=1 ¯ρk+1 +i +≤ ¯r. Since ¯ρk+1 ∈ [0,1]n, then the same argument as the one derived for k = 1 +can be applied to conclude that if k < κ∗ and ¯ρk+1 /∈ {0,1}n, then qk+1 ∈ �1,n� and δk+1 ∈ [0,1). In +conclusion, (EC.32) and (EC.33) hold by induction. +□ +Proposition EC.3. Algorithm 1 terminates after κ∗ ≤ n iterations of the while loop (3-16). In +particular, for every k ∈ �1,κ∗�, δk > 0, and +��� +i ∈ �1,n� : ¯ρk+1 +i +∈ {0,1} +��� > +��� +i ∈ �1,n� : ¯ρk +i ∈ {0,1} +���. +Proof of Proposition EC.3. Let k ∈ �1,κ∗�. First, we show that δk > 0. Since qk ≤ +��� +i ∈ �1,n� : ¯ρk +i > 0 +���, then ¯ρk +θk(qk) > 0. Next, we show by contradiction that if qk < n, then +¯ρk +θk(qk+1) < 1: If instead qk < n and ¯ρk +θk(qk+1) = 1, then we first deduce that qk < +��� +i ∈ �1,n� : ¯ρk +i > 0 +���. +Furthermore, +¯r +(EC.32) +≥ +n +� +i=1 +¯ρk +i ≥ +qk+1 +� +j=1 +¯ρk +θk(j) = qk + 1 > qk. +This contradicts the definition of qk. Therefore if qk < n, then ¯ρk +θk(qk+1) < 1, which in turn implies +that δk > 0. +We now show that +��� +i ∈ �1,n� : ¯ρk+1 +i +∈ {0,1} +��� > +��� +i ∈ �1,n� : ¯ρk +i ∈ {0,1} +���. Let j′ ∈ �1,n� be such +that ¯ρk +θk(j′) = 0. Necessarily, j′ > +��� +i ∈ �1,n� : ¯ρk +i > 0 +��� ≥ qk, which implies that ¯ρk+1 +θk(j′) = ¯ρk +θk(j′)/(1 − +δk) = 0. +Next, we consider j′ ∈ �1,n� such that ¯ρk +θk(j′) = 1. Then, j′ ≤ qk, as implied by the following +inequalities: +j′ = +j′ +� +j=1 +¯ρk +θk(j) ≤ +n +� +i=1 +¯ρk +i +(EC.32) +≤ +¯r, +j′ ≤ +��� +i ∈ �1,n� : ¯ρk +i = 1 +��� ≤ +��� +i ∈ �1,n� : ¯ρk +i > 0 +���. +Thus, j′ ≤ qk and ¯ρk+1 +θk(j′) = (¯ρk +θk(j′) −δk)/(1−δk) = 1. This shows that +���i ∈ �1,n� : ¯ρk+1 +i +∈ {0,1}��� ≥ +���i ∈ �1,n� : ¯ρk +i ∈ {0,1}���. To ensure a strict inequality, we must show that one fractional component +of ¯ρk becomes 0 or 1 in ¯ρk+1. + +ec18 +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +We know that 0 < δk < 1. If δk = ¯ρk +θk(qk), then ¯ρk+1 +θk(qk) = 0. If qk < n and δk = 1 − ¯ρk +θk(qk+1), then +¯ρk+1 +θk(qk+1) = 1. In both cases, a fractional component of ¯ρk becomes 0 or 1 in ¯ρk+1. In conclusion +��� +i ∈ �1,n� : ¯ρk+1 +i +∈ {0,1} +��� > +��� +i ∈ �1,n� : ¯ρk +i ∈ {0,1} +���. +Since for every k ∈ �1,κ∗ + 1�, +��� +i ∈ �1,n� : ¯ρk +i ∈ {0,1} +��� ≤ n, then the algorithm must terminate +after at most n iterations of the while loop (3-16). Therefore κ∗ ≤ n. +□ +Now that we proved that Algorithm 1 is well defined and terminates, we can show Theorem 2. +Proof of Theorem 2. Consider a vector of capacities b ∈ Zn +>0, a budget of resources r ∈ Z>0, +and a vector ρ ∈ � +A(b,r). For convenience, we denote eκ∗+1 := ¯ρκ∗+1 ∈ {0,1}n. We first show that for +every k ∈ �1,κ∗ + 1�, ⌊ρ⌋ + ek ∈ A(b,r). +In Proposition EC.3, we showed that for every k ∈ �1,κ∗�, if a component i ∈ �1,n� satisfies +¯ρk +i = 0, then ¯ρk+1 +i += 0. Thus, for every k ∈ �1,κ∗ + 1�, ¯ρk +i > 0 only if ¯ρ1 +i > 0. By definition of � +A(b,r) +and since bi ∈ Z, we deduce that if ¯ρ1 +i > 0, then bi ≥ ⌈ρi⌉ = ⌊ρi⌋ + ⌈¯ρ1 +i⌉ = ⌊ρi⌋ + 1. Furthermore, +∀k ∈ �1,κ∗�, +n +� +i=1 +(⌊ρi⌋ + ek +i ) = qk + +n +� +i=1 +⌊ρi⌋ ≤ ¯r + +n +� +i=1 +⌊ρi⌋ = r, +and +n +� +i=1 +(⌊ρi⌋ + eκ∗+1 +i +) = +n +� +i=1 +¯ρκ∗+1 +i ++ +n +� +i=1 +⌊ρi⌋ +(EC.32) +≤ +¯r + +n +� +i=1 +⌊ρi⌋ = r. +Therefore, for every k ∈ �1,κ∗ + 1�, ⌊ρ⌋ + ek ∈ A(b,r). +Next, we show that σ returned by the algorithm is a probability distribution. We first note that +for every k ∈ �1,κ∗ + 1�, γk ≥ 0. Furthermore, +� +z∈A(b,r) +σz = γκ∗+1 + +κ∗ +� +k=1 +γkδk = γκ∗+1 + +κ∗ +� +k=1 +(γk − γk+1) = γκ∗+1 + γ1 − γκ∗+1 = 1. +Therefore, σ ∈ ∆(b,r). We now show that σ returned by the algorithm is consistent with the vector +ρ. To this end, we note the following equality: +∀k ∈ �1,κ∗�, ∀i ∈ �1,n�, γk ¯ρk +i − γk+1¯ρk+1 +i += γk(¯ρk +i − (1 − δk)¯ρk+1 +i +) = γkδkek +i . +Then, we obtain: +∀i ∈ �1,n�, Ez∼σ[zi] = γκ∗+1(⌊ρi⌋ + ¯ρκ∗+1 +i +) + +κ∗ +� +k=1 +γkδk(⌊ρi⌋ + ek +i ) += ⌊ρi⌋ +� +z∈A(b,r) +σz + γκ∗+1¯ρκ∗+1 +i ++ γ1¯ρ1 +i − γκ∗+1¯ρκ∗+1 +i += ⌊ρi⌋ + ¯ρ1 +i = ρi. +Thus, σ returned by Algorithm 1 is consistent with the vector ρ. +Since κ∗ ≤ n, then the support of σ is of size at most n + 1. Finally, we argue that Algorithm +1 runs in time O(n2). Indeed, the first iteration of the while loop (3-16) can be implemented in + +e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection +ec19 +time O(nlogn) by using an efficient sorting algorithm (e.g. merge sort) to sort ¯ρ1 and create the +permutation θ1. Fortunately, the subsequent iterations can be implemented in time O(n). Indeed, +we note that for every k ∈ �1,κ∗ −1�, ¯ρk+1 +θk(1) ≥ ··· ≥ ¯ρk+1 +θk(qk) and ¯ρk+1 +θk(qk+1) ≥ ··· ≥ ¯ρk+1 +θk(n). Therefore, we +can sort ¯ρk+1 and create the permutation θk+1 by merging and sorting the lists (¯ρk+1 +θk(1),..., ¯ρk+1 +θk(qk)) +and (¯ρk+1 +θk(qk+1),..., ¯ρk+1 +θk(n)) that are already sorted. This operation can be carried out in time O(n). +Since the number of iterations of the while loop (3-16) is upper bounded by n, then the overall +running time of Algorithm 1 is O(n2). +□ + diff --git a/r9FJT4oBgHgl3EQfbyya/content/tmp_files/load_file.txt b/r9FJT4oBgHgl3EQfbyya/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b8582f571e94f819147dba72cb9ae13e2d32107 --- /dev/null +++ b/r9FJT4oBgHgl3EQfbyya/content/tmp_files/load_file.txt @@ -0,0 +1,1306 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf,len=1305 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='11541v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='GT] 27 Jan 2023 Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Basti´an Bahamondes, Mathieu Dahan School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, {bbahamondes3@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='edu, mathieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='dahan@isye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='edu} We consider a variant of the hide-and-seek game in which a seeker inspects multiple hiding locations to find multiple items hidden by a hider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Each hiding location has a maximum hiding capacity and a probability of detecting its hidden items when an inspection by the seeker takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The objective of the seeker (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' hider) is to minimize (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' maximize) the expected number of undetected items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This model is motivated by strategic inspection problems, where a security agency is tasked with coordinating multiple inspection resources to detect and seize illegal commodities hidden by a criminal organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To solve this large-scale zero-sum game, we leverage its structure and show that its mixed strategies Nash equilibria can be characterized using their unidimensional marginal distributions, which are Nash equilibria of a lower dimensional continuous zero-sum game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This leads to a two-step approach for efficiently solving our hide- and-seek game: First, we analytically solve the continuous game and compute the equilibrium marginal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Second, we derive a combinatorial algorithm to coordinate the players’ resources and compute equilibrium mixed strategies that satisfy the marginal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We show that this solution approach computes a Nash equilibrium of the hide-and-seek game in quadratic time with linear support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our analysis reveals a complex interplay between the game parameters and allows us to evaluate their impact on the players’ behaviors in equilibrium and the criticality of each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Key words : Hide and seek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' resource coordination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' imperfect detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' large-scale game 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Introduction In this article, we study a variant of the hide-and-seek game in which two players, the hider and the seeker, coordinate multiple resources among heterogeneous locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Specifically, the hider deter- mines where to allocate multiple items within capacitated locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Simultaneously, the seeker inspects a limited number of locations to detect the hidden items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, detection is supposed to be imperfect: When inspecting a location, the seeker finds the items with a location-specific prob- ability that captures the local effects undermining the seeker’s detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The seeker (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' hider) aims to select a (possibly randomized) strategy that minimizes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' maximizes) the expected number of items that are undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The objective of this work is to efficiently solve this large-scale simultaneous zero-sum game, that is, to compute a mixed strategy Nash equilibrium (NE), and gather insights on the players’ equilibrium behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our model is motivated by security applications involving for instance a security agency inter- ested in dispatching multiple units to inspect warehouses used by a criminal organization to store 1 2 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection illegal commodities such as drugs or weapons (Hochbaum and Fishbain 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such settings, the security agency aims to schedule the patrolling operations of their units to detect and seize the illegal commodities (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Another motivating application of our model involves a utility company tasked with coordinating multiple imperfect sensors to inspect its service network against failures caused by a malicious cyber-physical attacker who is able to target multiple com- ponents of the network (Pirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interestingly, another application of interest concerns auditing election results (Blocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2015, Behnezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such problems, an auditor allocates a limited number of election officials into several polling locations in order to detect elec- toral fraud by means of recounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The fraudster may be a malicious organization who is interested in manipulating the results by coordinating its members to tamper with the votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Previous related works in the hide-and-seek literature have not simultaneously considered mul- tiple resources for both players, heterogeneous hiding capacities, and imperfect detection (Gal and Casas 2014, Dziubi´nski and Roy 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This may reduce the applicability of the results, par- ticularly in security settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, simultaneously considering these features introduces new challenges: On one hand, the combinatorial nature of both players’ sets of actions due to the resource multiplicity prevents us from computationally solving the game using linear programming techniques or approximation algorithms (Freund and Schapire 1999, Lipton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2003, Hellerstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' On the other hand, the complex interplay between the game’s features renders the analytical solutions from previous works inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, we focus on the following research questions: (i) How to optimally coordinate multiple imperfect inspection resources to detect multiple hidden commodities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (ii) How are the optimal inspection and hiding strategies jointly impacted by the detection, location, and players’ characteristics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Contributions In this article, we formulate the hide-and-seek game as a simultaneous zero-sum game Γ and extend previous models in the literature by considering the coordination of multiple resources for both players in locations with heterogeneous hiding capacities and detection probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We then leverage the game’s structure to derive equilibrium properties of the NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, we show that a strategy profile is a NE of Γ if and only if the corresponding marginal inspection probabilities and expected numbers of hidden items at each location form a NE of a lower dimensional continuous game �Γ (Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From this equivalence, we derive a two-step approach for solving the hide-and-seek game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we analytically solve the continuous game �Γ (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We find that NE can be generally classified into three main regime patterns determined by complex parameters that account for the interplay between the players’ resources and the heterogeneity of the locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To the best of our Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 3 knowledge, the features of our model lead to new NE regimes that have not been observed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, we show that our analytical solutions describe all pure strategies NE of �Γ almost surely (Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By solving �Γ, we obtain marginal inspection probabilities and expected numbers of hidden items at each location in equilibrium of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, the second step consists of computing a mixed strategy profile of Γ that is consistent with these unidimensional marginal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To this end, we extend the algorithm of Dziubi´nski and Roy (2018) to feasibly coordinate the allocation of multiple resources (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We show that the algorithm runs in quadratic time and returns equilibrium inspection and hiding strategies with linear supports (Theorem 2 and Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, our approach efficiently solves the hide-and-seek game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By providing mixed inspection strategies with linear support, our solutions can easily be implemented in practice via a randomized scheduling of inspections that can be performed on a day-to-day basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, our analytical solution of the continuous game �Γ decodes the complex interplay between the game parameters and provides insights with respect to their impact on the players’ equilibrium behaviors and the criticality of each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Such insights can be leveraged by security agencies to inform their inspection decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Related Work The hide-and-seek game is a two-person zero-sum game introduced by Von Neumann (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In its original version, the hider and the seeker interact on a square matrix of nonnegative entries: The hider selects an entry aij and the seeker simultaneously selects either a row or a column of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If the row or column selected by the seeker contains the entry chosen by the hider, then the hider pays the seeker aij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' otherwise, the seeker pays the hider aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This game has been studied as a general model of strategic mismatch (Crawford and Iriberri 2007) and its equilibrium strategies are well known (Von Neumann 1953, Flood 1972, Karlin and Peres 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' It also belongs to the more general category of search games, in which a searcher is concerned with the optimal way of looking for a hidden adversary in a search space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' see for example Lidbetter (2013), Lidbetter and Lin (2019), Clarkson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2022), and the surveys of Alpern and Gal (2006) and Hohzaki (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Nonetheless, in practical applications, the seeker may be able to simultaneously inspect multiple locations, and the hider may be able to hide multiple items across the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, the seeker’s inspection resources can be affected by local conditions undermining their detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, in order to achieve a better utilization of their resources, each player may benefit from efficiently coordinating their allocation across the different locations, a problem that the original model does not address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' One of such practical applications arises in problems of strategic sensor placement for network inspection (Miloˇsevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2019, Pirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2021, Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2022, Bahamondes and Dahan 4 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 2022), in which the defender of a network positions sensors in a subset of given locations to detect attacks caused by a strategic attacker, who can target multiple network components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Such models typically account for the detection range of the sensors: Positioning a sensor at a location allows the defender to monitor a subset of network components—referred to as a monitoring set—and potentially detect attacks occurring within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As a result, attacks may be detected from multiple locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' this overlapping feature renders such games challenging to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2022) studied a two-person zero-sum game version of this model under the assumption of perfect detec- tion, and derived approximate NE strategies by means of minimum set covers and maximum set packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Miloˇsevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2019) and Bahamondes and Dahan (2022) studied variants of this model by respectively considering the critical values of network components and imperfect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' They derive heuristic approaches to compute good quality solutions in the case of a single attack resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Pirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2021) formulated a game in which sensors are positioned in the nodes of a networked control system to detect attacks on them, and considered imperfect detection through a linear filter that processes the sensors’ measurements to detect attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The authors derived equilibrium results using tools from structured systems and graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, a different but related model which features location-specific imperfect detection is the network interdiction prob- lem by Washburn and Wood (1995), in which an interdictor sets up a single inspection checkpoint along one of the arcs of a directed graph, with the aim of interdicting an evader who attempts to traverse a path between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The authors show that NE strategies can be computed in polynomial time using network flow techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, our hide-and-seek game can be used to model a class of instances of the strategic sensor placement problems in which the monitoring sets are mutually disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such instances, our results are directly applicable and generalize the equilibrium characterizations from (Washburn and Wood 1995, Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2022, Bahamondes and Dahan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Instances with disjoint monitoring arise in situations where it is desirable to reduce sensor interference or the energetic cost of the network (Cardei and Du 2005, Wang and Shao 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In other contexts such as in security games, disjoint monitoring is naturally satisfied (Powell 2009, Behnezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2018, Musegaas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Among the variants of the hide-and-seek game previously examined in the literature, our game is most closely related to the ones by Dziubi´nski and Roy (2018) and Gal and Casas (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Dziubi´nski and Roy (2018) consider a version of the game with multiple resources for both players, in which they interact on a set of unit capacity locations, each one associated with a nonnegative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Simultaneously, the hider (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' seeker) selects a subset of locations to hide his objects (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' to inspect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Once the choices are made, the seeker pays the hider the value of each uninspected location Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 5 containing a hidden object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In contrast, our model considers homogeneous values for all locations, but incorporates heterogeneous hiding capacities and probabilities of successful inspections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Gal and Casas (2014) propose a pursuit-evasion model of the interaction between a prey and a predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The prey chooses a location to hide from the predator, who is able to inspect multiple locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, if the predator visits the location where the prey is hiding, the capture is uncertain and occurs with some probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The predator (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' prey) seeks to maximize (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' minimize) the probability of capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our work extends this model by allowing multiple preys to coordinately hide in heterogeneously capacitated locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Although the subject of animal behavior is beyond the scope of our work, our model extension addresses analogous situations arising in security domains in which a security agency can dispatch multiple inspection units to inspect heterogeneous locations used by a criminal organization to store multiple illegal commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In both of these games, as in ours, a player’s mixed strategy consists of a probability distribu- tion over the set of resource allocations that satisfy capacity constraints and the resource budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, when players have access to multiple resources, their strategy spaces become exponentially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' One approach to handle the dimensionality consists in characterizing the players’ strategies in a lower dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In Dziubi´nski and Roy (2018) and Gal and Casas (2014), the games’ structures permit the characterization of NE in terms of their marginal probabilities of inspecting each location for the seeker, and their marginal probabilities of hiding an item in each location for the hider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, in order to compute the mixed strategies NE, it becomes necessary to con- struct probability distributions over the feasible resource allocations that are compatible with these marginal probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This two-step approach of characterizing equilibrium strategies in terms of marginal distributions and then computing compatible mixed strategies has been previously proposed in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', by Korzhyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2010) and Letchford and Conitzer (2013) to compute Stackelberg equilibria in security games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' by Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2016) to compute approximate NE in multilinear games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' and by Ahmadinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2019) to compute NE for zero-sum bilinear games, with applications to the Colonel Blotto game (Borel 1921).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In all these cases, the computation of the equilibrium marginal distributions is carried out via linear programming, and the computation of the mixed strategies from the marginal distributions follows by either an efficient implementation of Birkhoff-von Neu- mann’s theorem and its generalizations (Budish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2013), or by the more general algorithm by Gr¨otschel, Lov´asz, and Schrijver (Gr¨otschel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2012), Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='11) that implements Carath´eodory’s theorem using linear programming techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finding an efficient implementation of Carath´eodory’s theorem has also been addressed in other contexts, such as in mechanism design (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2012, Hoeksma and Uetz 2013), scheduling (Hoeksma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2016), and ranking systems (Kletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 6 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Our implementation of the two-step approach is closely related to that of Dziubi´nski and Roy (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In contrast to the above-mentioned literature, we derive analytical expressions for the equi- librium marginal distributions, which allows us to fully understand the interplay between the game parameters and to provide detailed insights regarding their impact on the players’ equilibrium behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Dziubi´nski and Roy (2018) provide a combinatorial algorithm for constructing a mixed strategy with linear support in quadratic time with respect to the number of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Their algo- rithm iteratively decomposes a given vector representing the marginal probabilities of allocating one resource in each location into a convex combination of a linear number of integer resource allocations, and it can be interpreted as a tailored and more efficient implementation of Gr¨otschel, Lov´asz, and Schrijver’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our algorithm extends this decomposition to the more general case in which the marginal distributions represent expected numbers of resources allocated in loca- tions constrained by capacities, and where the budget of resources does not need to be exhausted, as opposed to Dziubi´nski and Roy’s setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In Section 2, we formulate our hide-and-seek game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We then characterize and parametrically analyze its NE in Section 3 using a lower dimensional continuous game, which we solve analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In Section 4, we derive a combinatorial algorithm to coordinate the players’ resources and compute a NE of the hide-and-seek game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We then provide some concluding remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, the proofs of our results are listed in the electronic companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Problem Description We consider a hide-and-seek game involving a seeker who is looking for multiple homogeneous items hidden by a hider in a search space consisting of a set of n hiding locations �1,n� := {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The locations are capacitated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' namely, the hider can hide up to ci ∈ Z>0 items in each location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We let m := �n i=1 ci be the total hiding capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In addition, the seeker has imperfect location-specific detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Specifically, by inspecting a location i, the seeker effectively finds the hidden items (if any) in that location with probability pi ∈ (0,1] (and therefore, the inspection fails and leaves the hidden items undetected with probability 1 − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We assume that inspection failures at different locations occur independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We refer to pi as the detection rate of location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We assume that both the hider and seeker are strategic, and hence we adopt a game-theoretic framework to study their behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We define a simultaneous two-player strategic zero-sum game Γ := ⟨{S,H},(∆S,∆H),(−U,U)⟩ where S is the seeker and H is the hider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' S can select up to rS ∈ Z>0 hiding locations to inspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Simultaneously, H can select up to rH ∈ Z>0 items to hide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To model Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 7 the players’ action sets, we define a generic set of feasible resource allocations given a vector of capacities b ∈ Zn >0 and a budget of resources r ∈ Z>0 as follows: A(b,r) := {z ∈ Zn : 0 ≤ zi ≤ bi, ∀i ∈ �1,n�, and n � i=1 zi ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (1) The set A(b,r) contains all the vectors in Zn representing allocations of up to r resources within the locations i ∈ �1,n�, respecting their capacities given by bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, the pure action sets for S and H are given by AS := A(1n,rS) and AH := A(c,rH) respectively, where 1n is the vector of ones in Zn and c = (ci)i∈�1,n� is the vector of hiding capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, for every x ∈ AS and i ∈ �1,n�, xi = 1 if S inspects location i and xi = 0 otherwise, and for each y ∈ AH and i ∈ �1,n�, yi represents the number of items that H hides at location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We consider that the quantity of interest for the players is the average number of undetected items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, we define the players’ payoff function as follows: ∀(x,y) ∈ AS × AH, u(x,y) := n � i=1 (1 − pixi)yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (2) For every i ∈ �1,n�, 1−pixi represents the probability that items in location i are undetected given the pure inspection action x ∈ AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such combinatorial security settings, players significantly benefit from randomizing their actions (Washburn and Wood 1995, Pita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2008, Zhu and Basar 2015, Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2016, Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2016, Bertsimas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2016, Miao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, we allow the players to use mixed strategies, defined as probability distributions over their sets of pure actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The set of probability distributions over the set of generic feasible resource allocations A(b,r) is defined as ∆(b,r) := � σ ∈ [0,1]A(b,r) : � z∈A(b,r) σz = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, the sets of mixed strategies for S and H are given by ∆S := ∆(1n,rS) and ∆H := ∆(c,rH), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For every mixed strategy σS ∈ ∆S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' σH ∈ ∆H) and every x ∈ AS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' y ∈ AH), σS x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' σH y ) is the probability that action x ∈ AS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' y ∈ AH) is executed by S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Given a strategy profile (σS,σH) ∈ ∆S × ∆H, the expected payoff is then defined as U(σS,σH) := E(x,y)∼(σS,σH)[u(x,y)] = � x∈AS � y∈AH σS xσH y u(x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We assume that S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' H) seeks to minimize (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' maximize) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For ease of exposition, we use U(x,σH) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' U(σS,y)) to denote the case where σS x = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' σH y = 1) for some x ∈ AS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' y ∈ AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our game Γ is relevant to settings where a city police department is interested in inspecting warehouses that are used by a criminal organization to store illegal commodities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', drugs, weapons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such settings, �1,n� represents the set of warehouses, and for each i ∈ �1,n�, ci represents the maximum number of illegal commodities that can be stored in warehouse i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The police department can coordinate multiple police units to simultaneously inspect a maximum of rS 8 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection warehouses, while the criminal organization has rH units of illegal commodities to hide within the warehouses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The detection rates pi for i ∈ �1,n� capture the local effects that might undermine the detection capabilities of the police units, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', warehouse characteristics that can impact the efficacy of drug-sniffing dogs (Jezierski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The objective of the police department (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' criminal organization) is to minimize (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' maximize) the number of illegal commodities that are undetected by the police department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, a mixed inspection (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' hiding) strategy represents a randomized schedule of coordinated operations for the police department (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' criminal organization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The standard solution concept for simultaneous noncooperative games is given by Nash equilibria (NE), that is, strategy profiles for which no player has an incentive to unilaterally deviate in order to improve their payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, a strategy profile (σS∗,σH∗) ∈ ∆S × ∆H is a NE of the game Γ if it satisfies ∀(σS,σH) ∈ ∆S × ∆H, U(σS∗,σH) ≤ U(σS∗,σH∗) ≤ U(σS,σH∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We refer to U(σS∗,σH∗) as the value of the game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since Γ is a zero-sum game with finite pure action sets, Von Neumann’s minimax theorem (Von Neumann 1928) implies that the value of the game is unique and the game can be solved using the following linear program (LP): minimize t∈R, σS∈∆S t subject to U(σS,y) ≤ t, ∀y ∈ AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (LP) Specifically, the equilibrium inspection strategies, equilibrium hiding strategies, and value of the game Γ are given by the optimal primal solutions, optimal dual solutions, and optimal value of (LP), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' A remarkable consequence of this result is that no player can benefit from observing the mixed strategy of the other player before making a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Nonetheless, since the cardinality of AS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' AH) grows combinatorially with rS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' rH), (LP) becomes computationally challenging to solve, even for small-sized instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, algorithms for computing approximate NE are inapplicable for realistic instances of the game Γ (Freund and Schapire 1999, Lipton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2003, Hellerstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, we propose a two-step solution approach for solving the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First in Section 3, we reduce the dimensionality of the problem by characterizing NE using the marginal inspection probability and expected number of hidden items in each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then in Section 4, we derive an algorithm to coordinate the players’ resources and recover NE that are consistent with the characterized marginal probabilities and expected numbers of hidden items in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Analytical Characterization of Equilibrium Strategies In this section, we show that the NE of the game Γ can be characterized using the corresponding marginal inspection probability and expected number of hidden items in each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We prove Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 9 that these unidimensional quantities are NE of a smaller-sized continuous game, which we solve analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This analytical characterization permits us to examine the impact of the problem parameters on the players’ behaviors in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Continuous Equivalence To simplify our analysis of the game Γ, we first derive properties of generic randomized resource allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Given a vector of capacities b ∈ Zn >0 and a budget of resources r ∈ Z>0, we denote by � A(b,r) := {ρ ∈ Rn : 0 ≤ ρi ≤ bi, ∀i ∈ �1,n�, and �n i=1 ρi ≤ r} the linear programming relaxation of the set of generic feasible resource allocations A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, for every probability distribution σ ∈ ∆(b,r) over A(b,r), we denote as ρ(σ) = (ρi(σ))i∈�1,n� the vector of expected numbers of resources allocated at each location, given by: ∀i ∈ �1,n�, ρi(σ) := Ez∼σ[zi] = � z∈A(b,r) ziσz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (3) We present the following relation between ∆(b,r) and � A(b,r): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Consider a vector of capacities b ∈ Zn >0, a budget of resources r ∈ Z>0, and a vector ρ′ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, ρ′ ∈ � A(b,r) if and only if there exists a probability distribution σ ∈ ∆(b,r) that satisfies ρi(σ) = ρ′ i for all i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lemma 1 is a consequence of the integrality of the polyhedron � A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, � A(b,r) represents the set of vectors of expected numbers of allocated resources at each location resulting from a probability distribution in ∆(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, for every inspection strategy σS ∈ ∆S and hiding strategy σH ∈ ∆H, ρ(σS) = (ρi(σS))i∈�1,n� and ρ(σH) = (ρi(σH))i∈�1,n� respectively represent the vectors of marginal inspection probabilities and expected numbers of hidden items across the locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lemma 1 permits us to relate the game Γ to the continuous zero-sum game �Γ := ⟨{S,H},( � AS, � AH),(−˜u, ˜u)⟩, where S and H respectively select a vector of continuous inspection effort ρS ∈ � AS := � A(1n,rS) and a vector of continuous amount of hidden items ρH ∈ � AH := � A(c,rH), and the players’ payoff in �Γ is given by ∀(ρS,ρH) ∈ � AS × � AH, ˜u(ρS,ρH) = n � i=1 (1 − piρS i )ρH i , (4) as shown by the following proposition: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The games Γ and �Γ are related as follows: – For every strategy profile (σS,σH) ∈ ∆S × ∆H in Γ, U(σS,σH) = ˜u(ρ(σS),ρ(σH)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' – (ρS∗,ρH∗) ∈ � AS × � AH is a NE of �Γ if and only if there exists a NE (σS∗,σH∗) ∈ ∆S × ∆H of Γ that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 10 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection – The values of the games Γ and �Γ are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From this proposition, we deduce that NE of the game Γ can be characterized from NE of the game �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, in a NE (ρS∗,ρH∗) of �Γ, ρS∗ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' ρH∗) represents the marginal inspection probabilities (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' expected numbers of hidden items) at every location in a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This provides a significant computational advantage, as these quantities can be represented with vectors of size n, while the players’ strategies in Γ require vectors of exponentially large sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In addition, marginal inspection probabilities and expected numbers of hidden items more conveniently quantify the criticality of locations for each player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Preliminary Analysis We next derive the intuition behind the strategies for both players in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, we describe the players’ incentives and constraints that result from the features of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This discussion will permit us to introduce and motivate the key quantities that are needed to analytically solve the game �Γ and characterize the NE of the game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Proposition 1, we deduce that the players’ expected payoff U(σS,σH) in Γ can be expressed as the sum over the hiding locations i ∈ �1,n� of the expected numbers of hidden items that remain undetected, namely, (1 − piρi(σS))ρi(σH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We refer to 1 − piρi(σS) as the undetection probability of location i, that is, the probability that the hidden items at location i remain undetected when S plays σS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We also refer to piρi(σH) as the detection performance at location i ∈ �1,n�, which represents the expected number of hidden items that S is able to detect by inspecting location i when H plays σH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' These quantities will guide our equilibrium analysis, as H’s incentive is to hide items in locations with highest undetection probabilities, and S’s incentive is to inspect locations with highest detection performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Due to the players’ incentives, we can easily show that when each player has one resource unit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', rS = rH = 1), then a NE (σS∗,σH∗) satisfies ρi(σS∗) = ρi(σH∗) = (1/pi)/ ��n j=1 1/pj � for every i ∈ �1,n�, as in Gal and Casas (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In other words, S inspects each location i with marginal probability proportional to 1/pi so as to equalize the undetection probability of every location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, H’s equilibrium strategy equalizes the detection performance of every location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Now, if we consider a general number of player resources rS ≥ 1 and rH ≥ 1, an analogous intuition would suggest that ρi(σS∗) = (rS/pi)/ ��n j=1 1/pj � and ρi(σH∗) = (rH/pi)/ ��n j=1 1/pj � for every i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Proposition 1, ρ(σS∗) and ρ(σH∗) must belong to � AS and � AH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, if rS is large enough and the detection rates are heterogeneous enough, then (rS/pi)/ ��n j=1 1/pj � ≤ 1 may be violated for some locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such cases, S cannot ensure the desired level of inspection to these locations, thus rendering them more attractive for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Analogously, if rH is large enough, the detection rates are heterogeneous enough, and the hiding capacities are small enough, then Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 11 (rH/pi)/ ��n j=1 1/pj � ≤ ci may also be violated for some locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' H cannot ensure the desired level of detection performance across such locations, thus rendering them less attractive for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, the features of our model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', multiple player resources and heterogeneous locations) create new challenges that we need to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For the remainder of this section, we order the locations such that p1c1 ≤ ··· ≤ pncn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We refer to pici as the detection potential of location i ∈ �1,n�, that is, the maximum expected number of detected items when S inspects location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Moreover, for any given i ∈ �0,n−1�, we define a bijective mapping πi : �1,n−i� → �i+1,n� that satisfies pπi(1) ≤ ··· ≤ pπi(n−i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', that orders the set of locations �i+1,n� by their detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For convenience, we define p0c0 := 0 and pπi(0) := 0 for every i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We also denote Si k := �n−i j=k 1/pπi(j) for every i ∈ �0,n − 1� and k ∈ �1,n − i + 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We remark that when rS ≥ n, an equilibrium inspection strategy for S is to inspect each location, and an equilibrium hiding strategy for H is to hide min{rH,m} items in the locations i with smallest detection rates pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, when rH ≥ m, an equilibrium hiding strategy is to exhaust all the hiding capacities, and an equilibrium inspection strategy is to inspect the min{rS,n} locations i with the highest detection potentials pici.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Henceforth, we study the games Γ and �Γ when 0 < rS < n and 0 < rH < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From the discussion above, we find that H may not be able to ensure the desired level of detection performance for the locations with lowest detection potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If we denote by �1,i� such locations, then S will not inspect them, as her incentive is to allocate her resources among the remaining locations �i + 1,n� with higher detection performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For the remaining locations, S’s incentive is to equalize the undetection probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, this may not be possible if the detection rates pj for j ∈ �i+ 1,n� are heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Instead, S can inspect the ki most unreliable locations {πi(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi(ki)}, that is, the locations with lowest detection rates, and equalize the undetection probabilities for the locations {πi(ki + 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi(n − i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For every i ∈ �0,n − 1�, the value of ki is given by the following expression: ki := max � k ∈ �0,n − i� : k + pπi(k)Si k+1 < rS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As mentioned above, locations for which S cannot achieve the desired level of inspection become more attractive for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, H’s incentive is to exhaust the capacities of the ℓi most unreliable locations of �i+1,n�, that is, {πi(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi(ℓi)}, and equalize the detection performance in locations {πi(ℓi + 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi(n − i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In addition, H must ensure that the detection performance in the latter set of locations is no less than that of the locations in �1,i�, so that S does not have an incentive to reallocate her resources to �1,i� (that were initially uninspected by S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, for every i ∈ �0,n−1�, the value of ℓi is given by the following expression: ℓi := max � ℓ ∈ �0,n − i� : i � j=1 cj + ℓ � j=1 cπi(j) + piciSi ℓ+1 < rH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 12 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection We note that ℓi exists when the number of items to hide satisfies rH > �i j=1 cj + piciSi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interestingly, the interplay between ki and ℓi will play an important role in solving �Γ and char- acterizing the NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, it is crucial to determine the number i of locations with smallest detection potentials that will be exhausted by H and uninspected by S, given the players’ equilibrium interactions in the remaining set of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Analytical Characterization From the discussion above, we observe that the players’ behaviors in equilibrium depend on capac- ities, detection rates, detection potentials, numbers of resources, and the parameters ki and ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To capture this complex interplay and characterize the NE, we define the following key thresholds: τ−1 := 0, τi := i � j=1 cj + ki � j=1 cπi(j) + pi+1ci+1Si ki+1, ∀i ∈ �0,n − 1�, νi := i � j=1 cj + ki � j=1 cπi(j) + piciSi ki+1, ∀i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we show that these thresholds partition the interval [0,m] in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The thresholds τ−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',τn−1, and ν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',νn−1, satisfy τ−1 = 0, τn−1 = m, and τi−1 ≤ νi ≤ τi for all i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, the interval [0,m] is subdivided by the thresholds as 0 = τ−1 ≤ ν0 ≤ τ0 ≤ ν1 ≤ ··· ≤ τn−2 ≤ νn−1 ≤ τn−1 = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, given a number of items to hide rH ∈ �1,m − 1�, the subinterval in which rH resides corresponds to a precise configuration of the parameters and determines a specific equilibrium regime, as shown in the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Given the players’ resources rS ∈ �1,n − 1� and rH ∈ �1,m − 1�, let i∗ ∈ �0,n − 1� satisfying τi∗−1 < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, any strategy profile (ρS∗,ρH∗) ∈ � AS × � AH that satisfies the conditions below is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, any strategy profile (σS∗,σH∗) ∈ ∆S ×∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ is a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 1: If νi∗ < rH ≤ τi∗, then ki∗ ≤ ℓi∗ and sufficient equilibrium conditions are given by: ρS∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 if i ∈ I, 1 if i ∈ J , rS − ki∗ piSi∗ ki∗ +1 if i ∈ K, (5) ρH∗ i = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ci if i ∈ I ∪ J , rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) piSi∗ ki∗ +1 if i ∈ K, (6) Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 13 where I := {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',i∗}, J := {πi∗(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(ki∗)}, and K := {πi∗(ki∗ +1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(n−i∗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The value of the games Γ and �Γ is given by rH − ki∗ � j=1 pπi∗(j)cπi∗(j) − � rS − ki∗ �� rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) � Si∗ ki∗+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 2: If i∗ = 0 and τ−1 < rH ≤ ν0, then ℓ0∗ < k0∗ and sufficient equilibrium condi- tions are given by: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρS∗ i = 1 if i ∈ J ∪ {π0(ℓ0 + 1)}, pπ0(ℓ0+1) pi ≤ ρS∗ i ≤ 1 if i ∈ K \\ {π0(ℓ0 + 1)}, n � i=1 ρS∗ i ≤ rS, (7) ρH∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ci if i ∈ J , rH − ℓ0 � j=1 cπ0(j) if i = π0(ℓ0 + 1), 0 if i ∈ K \\ {π0(ℓ0 + 1)}, (8) where J := {π0(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',π0(ℓ0)} and K := {π0(ℓ0 + 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',π0(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The value of the games Γ and �Γ is given by rH − ℓ0 � j=1 pπ0(j)cπ0(j) − pπ0(ℓ0+1) � rH − ℓ0 � j=1 cπ0(j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 3: If i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' then ℓi∗ < ki∗ and sufficient equilibrium condi- tions are given by: ρS∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 if i ∈ I \\ {i∗},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ ℓi∗+1 if i = i∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 1 if i ∈ J ∪ {πi∗(ℓi∗ + 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' pπi∗(ℓi∗+1) pi if i ∈ K \\ {πi∗(ℓi∗ + 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (9) ρH∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ci if i ∈ I ∪ J ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) − pi∗ci∗Si∗ ℓi∗ +2 if i = πi∗(ℓi∗ + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' pi∗ci∗ pi if i ∈ K \\ {πi∗(ℓi∗ + 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (10) where I := {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',i∗}, J := {πi∗(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(ℓi∗)}, and K := {πi∗(ℓi∗ +1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(n−i∗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The value of the games Γ and �Γ is given by rH − pi∗ � rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ ℓi∗+1 � ci∗ − ℓi∗ � j=1 pπi∗(j)cπi∗(j) − pπi∗(ℓi∗+1) � rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 14 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection From Theorem 1 and thanks to a carefully selected set of thresholds and parameters, we can analytically solve the game �Γ and characterize NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, given i∗ ∈ �0,n − 1� satisfying τi∗−1 < rH ≤ τi∗, we generally find that �1,i∗� represents the collection of locations for which H cannot equalize the detection performance due to their small detection potentials, as intuited in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' When this occurs, S’s incentive is to utilize her resources to inspect the locations �i∗ + 1,n� with higher detection performance, thus leaving the locations �1,i∗� uninspected while being exhausted by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, the players’ behaviors in the remaining locations differ depending on the subinterval in which rH belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, we find that the threshold νi∗ determines the relation between ki∗ and ℓi∗, which in turn impacts the set of locations J that S deterministically inspects and that H exhausts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' It also dictates how the players should randomize their remaining resources throughout the locations in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Theorem 1 shows that three major equilibrium regime patterns emerge as a result of the complex and nonlinear interplay between the game parameters, captured by the selected thresholds τ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, these regime patterns generalize the equilibrium results from the game studied in Gal and Casas (2014), in which a prey hides from a predator that can inspect multiple locations with heterogeneous detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' That game can be derived from ours by setting ci = 1 for all i ∈ �1,n� and rH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such a setting, we can show that i∗ = 0 and ℓ0 = 0, and the NE regimes observed by the authors correspond to Regime Pattern 1 when k0 = 0 and to Regime Pattern 2 when k0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We also note that the equilibrium behaviors described in Theorem 1 in its full generality have not been observed in previously studied models that considered homogeneous detection rates (Dziubi´nski and Roy 2018, Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2022) or one unit of resources for one or both players (Washburn and Wood 1995, Karlin and Peres 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, if we allow the vector of capacities and the players’ resources to be continuous in the game �Γ, and if we consider the set Ψ of parameters for which �Γ is nontrivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', Ψ := � (n,p,c,rS,rH) : n ∈ Z>0, p ∈ (0,1]n, c ∈ Rn >0, rS ∈ (0,n), rH ∈ (0, n � i=1 ci) � , (11) then we obtain the following stronger result: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The set of parameters for which conditions (5)-(10) in Theorem 1 are necessary and sufficient for a strategy profile (ρS∗,ρH∗) ∈ � AS × � AH to be a NE of �Γ is a dense subset of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, Proposition 2 shows that the analytical expressions in Theorem 1 describe all pure NE of the continuous game �Γ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, for the original discrete game Γ, conditions (5)-(10) characterize all NE of Γ, apart from some edge cases that are described in the electronic companion (Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we provide further insights on the equilibrium behavior in each regime pattern and illustrate them with examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 15 Regime Pattern 1: νi∗ < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In this regime, S does not inspect the set of locations I = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',i∗}, and instead focuses her resources on the remaining locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Due to the het- erogeneity of the detection rates in {i∗ + 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',n}, S deterministically inspects the set of ki∗ most unreliable locations, J = {πi∗(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(ki∗)}, and randomizes her rS − ki∗ resources in K = {πi∗(ki∗ +1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(n−i∗)} so as to equalize the undetection probabilities in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The feasibility of S’s strategy is guaranteed by the definition of ki∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As a result of S’s inspection strategy, H’s hiding strategy exhausts all locations in I that are not inspected by S and all locations in J for which S cannot equalize the undetection probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then H randomizes his remaining rH − � i∈I∪J ci resources so as to equalize the detection performance across locations in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We note that the feasibility and equilibrium guarantee of H’s strategy is a consequence of the subinterval in which rH belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, since νi∗ < rH, then ki∗ ≤ ℓi∗, which implies that H can exhaust the locations in I ∪ J and still provide a detection performance that is sufficient so as to not incentivize S to reallocate some of her resources towards the locations in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, since rH is upper bounded by τi∗, then H can feasibly equalize the detection performances in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We illustrate this regime pattern with the following example: Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Consider the hide-and-seek model represented in Figure 1 and assume that rS = 3 and rH = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' p1 = 1 8 1 p2 = 1 4 π1(1) = 2 p3 = 1 3 π1(2) = 3 p5 = 4 5 π1(3) = 5 p6 = 5 6 π1(4) = 6 p4 = 1 π1(5) = 4 Figure 1 Illustration of a NE for Regime Pattern 1 when rS = 3 and rH = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The hiding capacity of each location is represented by the corresponding number of squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Marginal inspection probabilities (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' expected numbers of hidden items) in equilibrium are represented by the blue (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' red) colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 16 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection In this case, 449/80 = ν0 < rH ≤ τ1 = 289/40 and i∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, k1 = ℓ1 = 1 and π1(1) = 2, π1(2) = 3, π1(3) = 5, π1(4) = 6, π1(5) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, in equilibrium, S selects an inspection strategy σS∗ ∈ ∆S such that ρ1(σS∗) = 0, ρ2(σS∗) = 1, ρ3(σS∗) = 40/43, ρ4(σS∗) = 40/129, ρ5(σS∗) = 50/129, and ρ6(σS∗) = 16/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' On the other hand, H selects a hiding strategy σH∗ ∈ ∆H such that ρ1(σH∗) = ρ2(σH∗) = 2, ρ3(σH∗) = 60/43, ρ4(σH∗) = 20/43, ρ5(σH∗) = 25/43, and ρ6(σH∗) = 24/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The value of the game is U(σS∗,σH∗) = 5 + 49/86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' △ Regime Pattern 2: i∗ = 0 and τ−1 < rH ≤ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In this regime, I = ∅ and every location is inspected by S with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, H’s number of resources is too small relative to S’s capability of inspecting hiding locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As a consequence, H’s equilibrium strategy consists in greedily hiding items into the locations with smallest detections rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This results in exhausting the locations in J = {π0(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',π0(ℓ0)}, and assigning his remaining rH − � i∈J ci resources in location π0(ℓ0 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The feasibility H’s strategy is guaranteed by the definition of ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since S has enough resources, as guaranteed by k0 ≥ ℓ0 + 1 > ℓ0, then her inspection strategy consists in deterministically inspecting locations in J ∪ {π0(ℓ0 + 1)}, and randomizing sufficient resources to ensure that the undetection probabilities of the remaining locations are no more than that of location π0(ℓ0 + 1) so as to prevent H from reallocating some items from J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interestingly, this can be achieved by S without necessarily utilizing all her resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We illustrate this regime pattern with the following example: Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Consider the hide-and-seek model of Figure 2 and assume that rS = 6 and rH = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' p1 = 1 8 π0(1) = 1 p2 = 1 4 π0(2) = 2 p3 = 1 3 π0(3) = 3 p5 = 4 5 π0(4) = 5 p6 = 5 6 π0(5) = 6 p4 = 1 π0(6) = 4 Figure 2 Illustration of a NE for Regime Pattern 2 when rS = 6 and rH = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The hiding capacity of each location is represented by the corresponding number of squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Marginal inspection probabilities (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' expected numbers of hidden items) in equilibrium are represented by the blue (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' red) colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 17 In this case, 0 = τ−1 < rH ≤ ν0 = 8 and i∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, 2 = ℓ0 < k0 = 4 and π0(1) = 1, π0(2) = 2, π0(3) = 3, π0(4) = 5, π0(5) = 6, π0(6) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, one equilibrium inspection strategy for S is given by σS∗ ∈ ∆S satisfying ρ1(σS∗) = ρ2(σS∗) = ρ3(σS∗) = 1, ρ4(σS∗) = 1/3, ρ5(σS∗) = 5/12, and ρ6(σS∗) = 2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We note that S can implement this strategy with only 5 < rS units of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' On the other hand, H chooses a hiding strategy σH∗ ∈ ∆H such that ρ1(σH∗) = ρ2(σH∗) = ρ3(σH∗) = 2, and ρ4(σH∗) = ρ5(σH∗) = ρ6(σH∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The value of the game is U(σS∗,σH∗) = 4 + 7/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' △ Regime Pattern 3: i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In this final regime, we observe an interesting and more complex behavior from the players’ equilibrium strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' H cannot equalize the detection performance across locations in I = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',i∗} due to their capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, locations in I are exhausted by H and initially left uninspected by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, since ℓi∗ +1 ≤ ki∗, S cannot achieve the desired level of undetection probability for locations in J ∪{πi∗(ℓi∗ +1)} = {πi∗(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(ℓi∗ + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, H’s incentive is to exhaust locations in J and randomize some of his resources to equalize the detection performance across K = {πi∗(ℓi∗ + 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',πi∗(n − i∗)} to that of i∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', pi∗ci∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interestingly, H is left with rH − � i∈I∪J ci − pi∗ci∗Si∗ ℓi∗ +1 > 0 resources that he additionally allocates to location πi∗(ℓi∗ +1), which S deterministically inspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The feasibility and equilibrium guarantees of this strategy follow from the definition of ℓi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As a result of H’s strategy, S deterministically inspects locations in J ∪ {πi∗(ℓi∗ + 1)} and randomizes some of her resources to equalize the undetection probabilities in K \\ {πi∗(ℓi∗ + 1)} to that of πi∗(ℓi∗ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This is possible since ℓi∗ + 1 ≤ ki∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interestingly, S still has rS − ℓi∗ − 1 − pπi∗(ℓi∗+1)Si∗ ℓi∗ +2 > 0 resources that she can now allocate among the i∗ ≥ 1 locations in I that were previously left uninspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' S’s incentive is to allocate her remaining resources on the location in I with highest detection performance, namely, i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Note that feasibility and equilibrium guarantees for this strategy is a consequence of rH > τi∗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, the resulting undetection probability in i∗ is no less than that of locations in K, thus ensuring that H will not reallocate some of his items from i∗ to locations in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We illustrate this final regime pattern with the following example: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Consider the hide-and-seek model of Figure 3 and assume that rS = 4 and rH = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In this case, 389/40 = τ1 < rH ≤ ν2 = 33/2 and i∗ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, k2 = 3 > 1 = ℓ2 and π2(1) = 3, π2(2) = 5, π2(3) = 6, π2(4) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, in equilibrium, S selects an inspection strat- egy σS∗ ∈ ∆S such that ρ1(σS∗) = 0, ρ2(σS∗) = 6/25, ρ3(σS∗) = 1, ρ4(σS∗) = 4/5, ρ5(σS∗) = 1 and ρ6(σS∗) = 24/25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' On the other hand, H chooses a hiding strategy σH∗ ∈ ∆H such that ρ1(σH∗) = ρ2(σH∗) = 2, ρ3(σH∗) = 4, ρ4(σH∗) = 1/2, ρ5(σH∗) = 9/10 and ρ6(σH∗) = 3/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The value of the game is U(σS∗,σH∗) = 6 + 71/75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' △ 18 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection p1 = 1 8 1 p2 = 1 4 2 p3 = 1 3 π2(1) = 3 p5 = 4 5 π2(2) = 5 p6 = 5 6 π2(3) = 6 p4 = 1 π2(4) = 4 Figure 3 Illustration of a NE for Regime Pattern 3 when rS = 4 and rH = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The hiding capacity of each location is represented by the corresponding number of squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Marginal inspection probabilities (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' expected numbers of hidden items) in equilibrium are represented by the blue (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' red) colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Parametric Analysis We continue our analysis by illustrating the impact of the players’ resources on the equilibrium regimes of the game Γ (and �Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To this end, we consider the hide-and-seek instance defined by the 6 locations, 18 hiding capacities, and detection rates from Figures 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, we plot the regions determined by the subintervals [τi−1,νi] and [νi,τi] for each i ∈ �0,n − 1� as a function of rS and rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The resulting plot is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We first observe that given a specific regime, the values of rS and rH for which that regime holds form a complex region that may even be disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, the borders that represent the values of the thresholds τi and νi are defined by step functions of rS through the parameter ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, for certain values of rS, some thresholds coincide, thus making certain regimes unattainable for any value of rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interestingly, when the number of inspection resources rS is high, very few equilibrium regimes are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In such cases, S has enough resources to not leave any location uninspected, resulting in i∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Conversely, when rS is low, S’s strategy is highly sensitive to the number of items to hide rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, if rH is small, then H can equalize the detection performance across all locations and S should inspect all locations with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As rH increases, S must carefully determine which locations to inspect and prefers leaving i∗ locations uninspected to find more items in the remaining locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' when rS is low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' nearly all the regimes that are achievable follow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='Regime Pattern 1 (for different values of i∗) since ki∗ ≤ ℓi∗ is most likely to hold for small amounts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='rH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='rS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='τ−1 < rH ≤ ν0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='ν0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ τ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='τ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ ν1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='ν1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ τ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='τ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ ν2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='ν2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ τ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='τ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ ν3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='ν3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ τ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='τ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ ν4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='ν4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ τ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='τ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='< rH ≤ ν5 = τ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='Figure 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='Illustration of equilibrium regions as a function of the number of resources rS and rH for the hide-and- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='seek instance from Figures 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' of inspection resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, when rH is high, a single unit of inspection resources incentives S to focus her inspection on the last location 6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', i∗ = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As rS increases, S can allocate more resources to i∗ according to (9) (Regime Pattern 3) until i∗ is deterministically inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' At this point, a new regime following Pattern 3 emerges, with a smaller number of uninspected locations i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The resulting impact of the players’ resources on the value of the game is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='8 1 rS U(σS∗,σH∗)/rH rH = 2 rH = 4 rH = 6 rH = 8 rH = 10 rH = 12 rH = 14 rH = 16 rH = 18 Figure 5 Fraction of undetected items in equilibrium as a function of rS, for different values of rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Specifically, Figure 5 compares the fractions of undetected items in equilibrium for different amounts of players’ resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This figure first shows the value of focusing inspection resources on 20 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection locations �i∗ + 1,n� when the number of items to hide rH is high, due to the heterogeneity of the capacities and detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' It also shows that the gain in performance by having additional inspection resources varies as a function of the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This analysis can be utilized in situations when S must determine the appropriate number of inspection resources that would balance cost and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In general, our analytical study provides us with valuable insights on the impact of the problem characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', detection rates, hiding capacities, amounts of resources) on the players’ behav- iors as well as the locations’ importance and criticality in equilibrium, which can be leveraged by security decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Equilibrium Computation In the previous section, Theorem 1 solves the continuous game �Γ, and provides marginal inspection probabilities and expected numbers of hidden items at each location in equilibrium of the game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, to solve Γ, we must determine the coordination of rS inspection resources and rH items to hide in order to satisfy these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Specifically, given the vectors of marginal inspection probabilities ρS∗ ∈ � AS and expected numbers of hidden items ρH∗ ∈ � AH in Theorem 1, we next seek to efficiently compute mixed strategies (σS∗,σH∗) ∈ ∆S × ∆H such that ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Proposition 1, this will ensure that (σS∗,σH∗) is a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, we aim to solve the following generic problem: Given a vector of capacities b ∈ Zn >0, a budget of resources r ∈ Z>0, and a vector ρ ∈ � A(b,r), find a solution to the feasibility problem {σ ∈ RA(b,r) ≥0 : � z∈A(b,r) σzz = ρ, � z∈A(b,r) σz = 1}, which is guaranteed to exist by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Although this problem involves an exponential number of variables, Carath´eodory’s theorem guarantees that a solution exists with a support of size at most n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, this problem can be solved in polynomial time using the ellipsoid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, this method is known to be practically inefficient (Behnezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, we derive another algorithm to efficiently solve the feasibility problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Specifically, we extend the algorithm proposed by Dziubi´nski and Roy (2018) that computes in time O(n2) a probability distribution with linear support over the set {z ∈ {0,1}n : �n i=1 zi = r} consistent with prescribed unidimensional marginal probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, we cannot apply their algorithm to construct the equilibrium inspection and hiding strategies in our game Γ, as our model involves locations with possibly heterogeneous capacities, and probabilities may be assigned to resource allocations z that do not utilize the whole budget of resources r (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', in Theorem 1 – Regime Pattern 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, given a vector of capacities b ∈ Zn >0, a budget of resources r ∈ Z>0, and a vector ρ ∈ � A(b,r), our algorithm returns a probability distribution σ ∈ ∆(b,r) that satisfies � z∈A(b,r) σzzi = ρi for Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 21 every i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The general idea of each iteration of the algorithm is to express a given vector ρ in � A(b,r) as a convex combination of a vector in A(b,r) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', with only integer components) and a vector in � A(b,r) with one more integer component than ρ has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To this end, the algorithm allocates ⌊ρi⌋ resources at each location i ∈ �1,n�, and then determines where to allocate some of the remaining ¯r := r −�n i=1⌊ρi⌋ resources given the fractional part of each component of ρ, defined as ¯ρi := ρi − ⌊ρi⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The algorithm determines the maximum number of locations q to allocate the remaining resources ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Naturally, q is upper bounded by ¯r and the number of positive components of ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, the algorithm carefully assigns positive probability to a resource allocation that first assigns ⌊ρi⌋ resources at each location, and then assigns one additional resource at each of the q locations with highest fractional parts ¯ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The algorithm then updates the vector ¯ρ so that the new vector contains at least one more integer component than ¯ρ has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The algorithm iterates until ¯ρ ∈ {0,1}n, at which point the algorithm assigns the remaining probability to ⌊ρ⌋+ ¯ρ ∈ A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We refer the reader to Algorithm 1 for the detailed pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Algorithm 1: Resource Coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Input : A vector of capacities b ∈ Zn >0, a budget of resources r ∈ Z>0, and a vector ρ ∈ � A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Output: A probability distribution σ ∈ ∆(b,r) satisfying Ez∼σ[z] = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 1 σ ← 0A(b,r), ¯ρ1 ← ρ − ⌊ρ⌋, ¯r ← r − �n i=1⌊ρi⌋ 2 γ1 ← 1, k ← 1 3 while ¯ρk /∈ {0,1}n do 4 θk ← Permutation of �1,n� such that ¯ρk θk(1) ≥ ··· ≥ ¯ρk θk(n) 5 qk ← min � ¯r, ��� i ∈ �1,n� : ¯ρk i > 0 ���� 6 if qk < n then 7 δk ← min{¯ρk θk(qk),1 − ¯ρk θk(qk+1)} 8 else 9 δk ← ¯ρk θk(qk) 10 ek ← 0n 11 foreach j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',qk} do 12 ek θk(j) ← 1 13 σ⌊ρ⌋+ek ← σ⌊ρ⌋+ek + γkδk 14 ¯ρk+1 ← 1 1−δk (¯ρk − δkek) 15 γk+1 ← γk(1 − δk) 16 k ← k + 1 17 σ⌊ρ⌋+¯ρk ← σ⌊ρ⌋+¯ρk + γk 18 return σ 22 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection We note that at each iteration k of the while loop (3-16), ¯ρk = δkek + (1 − δk)¯ρk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, the algorithm selects ek and δk so that ¯ρk can be expressed as a convex combination of ek ∈ {0,1}n and a new vector ¯ρk+1 that has at least one more integer component than ¯ρk does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This guarantees that the algorithm terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' At termination, the algorithm expresses ρ as a convex combination of resource allocations in A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This can be translated into the following theorem: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Given a vector of capacities b ∈ Zn >0, a budget of resources r ∈ Z>0, and a vec- tor ρ ∈ � A(b,r), Algorithm 1 returns a probability distribution σ ∈ ∆(b,r) satisfying Ez∼σ[z] = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, σ has a support of size at most n + 1 and is computed in time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, Algorithm 1 matches the support size guaranteed by Carath´eodory’s theorem on the polytope � A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In fact, the algorithm performs at most n iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, by reutilizing the sorting of ¯ρk of the previous iteration, we can implement each iteration (except for the first one) in time O(n), which guarantees an overall running time of O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We can now summarize the overall solution approach for computing a NE of the game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, marginal inspection probabilities and expected numbers of hidden items in each location are com- puted according to Theorem 1: Sorting the detection potentials (pici)i∈�1,n� and the detection rates (pi)i∈�1,n� in non-decreasing order requires O(nlogn) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, for every i ∈ �0,n−1�, the map- ping πi, parameters ki and ℓi, and thresholds τi and νi can be computed in time O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Identifying the subinterval [τi∗−1,νi∗] or [νi∗,τi∗] in which rH belongs can be performed in O(logn) steps, and evaluating the expressions from Theorem 1 requires O(n) more steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, computing equilibrium marginal inspection probabilities and expected numbers of hidden items can be implemented in time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, Algorithm 1 computes the mixed strategies consistent with the unidimensional quantities in time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, we obtain the final result: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The game Γ can be solved in time O(n2) with equilibrium strategies of support size at most n + 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, we obtain an efficient solution approach for solving the large-scale hide-and-seek game Γ with multiple resources and heterogeneous locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, our approach provides solutions with small supports that can be easily implemented by security decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Conclusion In this work, we investigated a hide-and-seek game in which a seeker inspects locations to find items hidden by a hider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We extended previous models in the literature by considering the coordi- nation of multiple resources for both players in locations with heterogeneous hiding capacities and probabilities of detecting hidden items where the search takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The objective of the seeker Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 23 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' hider) is to minimize (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' maximize) the expected number of undetected items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To com- pute the mixed strategies Nash equilibria of this large-scale zero-sum game, we proposed a solution approach that first derives analytical equilibrium properties and then efficiently coordinates the players’ resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, we showed that the marginal inspection probabilities and expected numbers of hidden items in each location in equilibrium form a Nash equilibrium of a continuous zero-sum game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By carefully selecting a set of parameters and thresholds, we analytically solved this continuous game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our analysis highlighted a complex interplay between the game parameters and permitted us to evaluate their impact on the players’ behaviors in equilibrium and the criticality of each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, we derived a quadratic time algorithm that coordinates the players’ resources to satisfy the characterized equilibrium marginal distributions and computes a Nash equilibrium of the hide-and-seek game with linear support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Our insights and solution approach can be used to inform security agencies that are interested in scheduling multiple units to detect and seize illegal commodities hidden by a criminal organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This work can be extended in multiple directions by considering additional features that would widen the applicability of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' One extension is to consider different commodities hidden by the hider with heterogeneous values, as well as different inspection resources with different detection characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Another extension is to incorporate heterogeneous valuations of the locations to extend the original setting by Von Neumann (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, an interesting research direction is to consider a repeated version of the hide-and-seek game with players learning the initially unknown characteristics of their opponent while they interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Acknowledgments This work was supported by the Georgia Tech Stewart fellowship and new faculty start up grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We are grateful to Mohit Singh for his valuable feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' References Ahmadinejad A, Dehghani S, Hajiaghayi M, Lucier B, Mahini H, Seddighin S (2019) From duels to battlefields: Computing equilibria of Blotto and other games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Mathematics of Operations Research 44(4):1304–1325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Alpern S, Gal S (2006) The theory of search games and rendezvous, volume 55 (Springer Science & Business Media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Bahamondes B, Dahan M (2022) Network inspection from locations with imperfect detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2022 American Control Conference (ACC), 613–620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Behnezhad S, Blum A, Derakhshan M, HajiAghayi M, Mahdian M, Papadimitriou CH, Rivest RL, Seddighin S, Stark PB (2018) From battlefields to elections: Winning strategies of Blotto and auditing games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2291–2310 (SIAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 24 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Behnezhad S, Dehghani S, Derakhshan M, HajiAghayi M, Seddighin S (2017) Faster and simpler algorithm for optimal strategies of blotto game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thirty-First AAAI Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Bertsimas D, Nasrabadi E, Orlin JB (2016) On the power of randomization in network interdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Opera- tions Research Letters 44(1):114–120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Blocki J, Christin N, Datta A, Procaccia A, Sinha A (2015) Audit games with multiple defender resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence 29(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Borel E (1921) La th´eorie du jeu et les ´equations int´egralesa noyau sym´etrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Comptes Rendus de l’Acad´emie des Sciences 173(1304-1308):58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Budish E, Che YK, Kojima F, Milgrom P (2013) Designing random allocation mechanisms: Theory and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' American Economic Review 103(2):585–623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Cai Y, Daskalakis C, Weinberg SM (2012) Optimal multi-dimensional mechanism design: Reducing revenue to welfare maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 130–139 (IEEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Cardei M, Du DZ (2005) Improving wireless sensor network lifetime through power aware organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Wireless Networks 11(3):333–340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Chan H, Jiang AX, Leyton-Brown K, Mehta R (2016) Multilinear games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' International Conference on Web and Internet Economics, 44–58 (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Clarkson J, Lin KY, Glazebrook KD (2022) A classical search game in discrete locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Mathematics of Operations Research .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Crawford VP, Iriberri N (2007) Fatal attraction: Salience, naivete, and sophistication in experimental “hide- and-seek” games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' American Economic Review 97(5):1731–1750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Dahan M, Sela L, Amin S (2022) Network inspection for detecting strategic attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Operations Research 70(2):1008–1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Dziubi´nski M, Roy J (2018) Hide and seek game with multiple resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' International Symposium on Algorithmic Game Theory, 82–86 (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Flood MM (1972) The hide and seek game of von Neumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Management Science 18(5-part-2):107–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Freund Y, Schapire RE (1999) Adaptive game playing using multiplicative weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Games and Economic Behavior 29(1):79–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Gal S, Casas J (2014) Succession of hide–seek and pursuit–evasion at heterogeneous locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Journal of the Royal Society Interface 11(94):20140062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Gr¨otschel M, Lov´asz L, Schrijver A (2012) Geometric algorithms and combinatorial optimization, volume 2 (Springer Science & Business Media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Gupta A, Langbort C, Ba¸sar T (2016) Dynamic games with asymmetric information and resource con- strained players with applications to security of cyberphysical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' IEEE Transactions on Control of Network Systems 4(1):71–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection 25 Hellerstein L, Lidbetter T, Pirutinsky D (2019) Solving zero-sum games using best-response oracles with applications to search games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Operations Research 67(3):731–743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hess KM, Orthmann CH, Cho HL (2013) Police Operations: Theory and Practice (Cengage Learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hochbaum DS, Fishbain B (2011) Nuclear threat detection with mobile distributed sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Annals of Operations Research 187(1):45–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hoeksma R, Manthey B, Uetz M (2016) Efficient implementation of Carath´eodory’s theorem for the single machine scheduling polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Discrete Applied Mathematics 215:136–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hoeksma R, Uetz M (2013) Two dimensional optimal mechanism design for a sequencing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Interna- tional Conference on Integer Programming and Combinatorial Optimization, 242–253 (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hohzaki R (2016) Search games: Literature and survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Journal of the Operations Research Society of Japan 59(1):1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hota AR, Clements AA, Sundaram S, Bagchi S (2016) Optimal and game-theoretic deployment of security investments in interdependent assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' International Conference on Decision and Game Theory for Security, 101–113 (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Jezierski T, Adamkiewicz E, Walczak M, Sobczy´nska M, G´orecka-Bruzda A, Ensminger J, Papet E (2014) Efficacy of drug detection by fully-trained police dogs varies by breed, training level, type of drug and search environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Forensic Science International 237:112–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Karlin A, Peres Y (2016) Game Theory, Alive (AMS, Providence, RI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Kletti T, Renders JM, Loiseau P (2022a) Introducing the expohedron for efficient pareto-optimal fairness- utility amortizations in repeated rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining, 498–507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Kletti T, Renders JM, Loiseau P (2022b) Pareto-optimal fairness-utility amortizations in rankings with a DBN exposure model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='07647 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Korzhyk D, Conitzer V, Parr R (2010) Complexity of computing optimal Stackelberg strategies in security resource allocation games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Twenty-Fourth AAAI Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Letchford J, Conitzer V (2013) Solving security games on graphs via marginal probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Twenty-Seventh AAAI Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lidbetter T (2013) Search games with multiple hidden objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' SIAM Journal on Control and Optimization 51(4):3056–3074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lidbetter T, Lin KY (2019) Searching for multiple objects in multiple locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' European Journal of Oper- ational Research 278(2):709–720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lipton RJ, Markakis E, Mehta A (2003) Playing large games using simple strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proceedings of the 4th ACM Conference on Electronic Commerce, 36–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 26 Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Miao F, Zhu Q, Pajic M, Pappas GJ (2018) A hybrid stochastic game for secure control of cyber-physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Automatica 93:55–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Miloˇsevi´c J, Dahan M, Amin S, Sandberg H (2019) A network monitoring game with heterogeneous compo- nent criticality levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' 2019 IEEE 58th Conference on Decision and Control (CDC), 4379–4384 (IEEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Musegaas M, Schlicher L, Blok H (2022) Stackelberg production-protection games: Defending crop production against intentional attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' European Journal of Operational Research 297(1):102–119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Pirani M, Taylor JA, Sinopoli B (2021) Strategic sensor placement on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Systems & Control Letters 148:104855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Pita J, Jain M, Marecki J, Ord´o˜nez F, Portway C, Tambe M, Western C, Paruchuri P, Kraus S (2008) Deployed ARMOR protection: The application of a game theoretic model for security at the Los Angeles International Airport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proceedings of the 7th International Joint Conference on Autonomous Agents and Multiagent Systems: Industrial Track, 125–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Powell R (2009) Sequential, nonzero-sum “Blotto”: Allocating defensive resources prior to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Games and Economic Behavior 67(2):611–615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Von Neumann J (1928) Zur theorie der gesellschaftsspiele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Mathematische Annalen 100(1):295–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Von Neumann J (1953) A certain zero-sum two-person game equivalent to the optimal assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Contributions to the Theory of Games 2(0):5–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Wang WQ, Shao H (2014) Radar-to-radar interference suppression for distributed radar sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Remote Sensing 6(1):740–755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Washburn A, Wood K (1995) Two-person zero-sum games for network interdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Operations Research 43(2):243–251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Zhu Q, Basar T (2015) Game-theoretic methods for robustness, security, and resilience of cyberphysical con- trol systems: games-in-games principle for optimal cross-layer resilient control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' IEEE Control Systems Magazine 35(1):46–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec1 Proofs of Statements EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proofs of Section 3 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let b ∈ Zn >0 be a vector of capacities and r ∈ Z>0 be a budget of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we observe that � A(b,r) is the convex hull of A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, since the row vector 1⊤ n is totally unimodular, then (1) is an ideal formulation of A(b,r) and the polytope � A(b,r) has integral extreme points, which we denote z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',zI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Consider a vector ρ′ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ρ′ ∈ � A(b,r), then there exist λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',λI ∈ [0,1] such that ρ′ = �I i=1 λizi and �I i=1 λi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, σ ∈ [0,1]A(b,r) defined by σzi = λi for i ∈ �1,I� and σz = 0 if z /∈ {z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=',zI} is a probability distribution in ∆(b,r) and satisfies ρi(σ) = ρ′ i for all i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Conversely, if there exists σ ∈ ∆(b,r) that satisfies ρ′ i = ρi(σ) = � z∈A(b,r) ziσz for all i ∈ �1,n�, then ρ′ is a convex combination of elements on A(b,r), and belongs to its convex hull, � A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' – Consider a strategy profile (σS,σH) ∈ ∆S × ∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, U(σS,σH) (2) = n � i=1 (1 − piEx∼σS[xi])Ey∼σH[yi] (3),(4) = ˜u(ρ(σS),ρ(σH)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) – Let (ρS∗,ρH∗) ∈ � AS × � AH be a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Lemma 1, let (σS∗,σH∗) ∈ ∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Using equilibrium conditions in �Γ, we obtain: ∀σS ∈ ∆S, U(σS∗,σH∗) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = ˜u(ρS∗,ρH∗) ≤ ˜u(ρ(σS),ρH∗) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = U(σS,σH∗), ∀σH ∈ ∆H, U(σS∗,σH∗) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = ˜u(ρS∗,ρH∗) ≥ ˜u(ρS∗,ρ(σH)) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = U(σS∗,σH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, (σS∗,σH∗) is a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Conversely, let (σS∗,σH∗) ∈ ∆S × ∆H be a NE of Γ, then we must show that (ρ(σS∗),ρ(σH∗)) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, from Lemma 1, we know that (ρ(σS∗),ρ(σH∗)) ∈ � AS × � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, given (ρS,ρH) ∈ � AS× � AH, let (σS,σH) ∈ ∆S×∆H satisfying ρ(σS) = ρS and ρ(σH) = ρH (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Using equilibrium conditions in Γ, we obtain: ˜u(ρ(σS∗),ρ(σH∗)) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = U(σS∗,σH∗) ≤ U(σS,σH∗) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = ˜u(ρS,ρ(σH∗)), and ˜u(ρ(σS∗),ρ(σH∗)) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = U(σS∗,σH∗) ≥ U(σS∗,σH) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = ˜u(ρ(σS∗),ρH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, (ρ(σS∗),ρ(σH∗)) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' – Given a NE (σS∗,σH∗) ∈ ∆S × ∆H of Γ, we know that (ρ(σS∗),ρ(σH∗)) is a NE of �Γ and U(σS∗,σH∗) (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1) = ˜u(ρ(σS∗),ρ(σH∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, the values of the games Γ and �Γ are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ ec2 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection We note that when some detection rates are identical, permutations πi ordering �i+1,n� by their detection rates may not be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To simplify our proofs, we assume without loss of generality that πi maintains the order between identical detection rates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', πi(j) < πi(k) when 1 ≤ j < k ≤ n − i and pπi(j) = pπi(k), thus rendering πi unique for every i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Before proving Lemma 2 we need the following auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then: πi−1(j) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 πi(j) if j ∈ �1,j∗ − 1�, i if j = j∗, πi(j − 1) if j ∈ �j∗ + 1,n − i + 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2) Moreover, for every k ∈ �0,n − i + 1�, k � j=1 cπi−1(j) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 k � j=1 cπi(j) if k ∈ �0,j∗ − 1�, k−1 � j=1 cπi(j) + ci if k ∈ �j∗,n − i + 1�, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3) and Si−1 k+1 = � Si k+1 + 1 pi if k ∈ �0,j∗ − 1�, Si k if k ∈ �j∗,n − i + 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) Proof of Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The permutation πi−1 sorts locations �i,n� in order of non-decreasing detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' After removing pi = pπi−1(j∗) from the chain of inequalities, we obtain pπi−1(1) ≤ ··· ≤ pπi−1(j∗−1) ≤ pπi−1(j∗+1) ≤ ··· ≤ pπi−1(n−i+1), which sorts �i + 1,n� by detection rates, thus providing (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As a consequence, for every k ∈ �0,j∗ − 1�, k � j=1 cπi−1(j) = k � j=1 cπi(j), Si−1 k+1 = j∗−1 � j=k+1 1 pπi−1(j) + 1 pi + n−i+1 � j=j∗+1 1 pπi−1(j) = 1 pi + j∗−1 � j=k+1 1 pπi(j) + n−i+1 � j=j∗+1 1 pπi(j−1) = 1 pi + Si k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, for every k ∈ �j∗,n − i + 1�, k � j=1 cπi−1(j) = j∗−1 � j=1 cπi(j) + ci + k � j=j∗+1 cπi(j−1) = k−1 � j=1 cπi(j) + ci, Si−1 k+1 = n−i+1 � j=k+1 1 pπi−1(j) = n−i+1 � j=k+1 1 pπi(j−1) = Si k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec3 Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For every i ∈ �0,n − 1�, ki exists and Ki := �k ∈ �0,n − i� : k + pπi(k)Si k+1 < rS � = �0,ki�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, for every i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� such that πi−1(j∗) = i, ki−1 ≤ � ki if ki−1 ∈ �0,j∗ − 1�, ki + 1 if ki−1 ∈ �j∗,n − i + 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='5) Proof of Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since pπi(0)Si 1 = 0 < rS, then 0 ∈ Ki and ki exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we consider k ∈ �1,n − i�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If k ∈ Ki, then k − 1 ∈ Ki, as shown below: rS > k + pπi(k)Si k+1 = k − 1 + pπi(k) � Si k+1 + 1 pπi(k) � ≥ k − 1 + pπi(k−1)Si k, where we used the fact that k ∈ Ki and πi orders locations in �i + 1,n� by their detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, Ki = �0,ki�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We next analyze ki as a function of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i ∈ �1,n − 1� and j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ki−1 ∈ �0,j∗ − 1�, we obtain: rS > ki−1 + pπi−1(ki−1)Si−1 ki−1+1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) ≥ ki−1 + pπi(ki−1)Si ki−1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ki−1 ≤ j∗ − 1 ≤ n − i, we deduce that ki−1 ∈ Ki, and ki−1 ≤ ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ki−1 ∈ �j∗ + 1,n − i + 1�, we obtain: rS (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) > ki−1 + pπi(ki−1−1)Si ki−1 > ki−1 − 1 + pπi(ki−1−1)Si ki−1−1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ki−1 − 1 ≤ n − i, then ki−1 − 1 ∈ Ki and ki−1 − 1 ≤ ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, if ki−1 = j∗, we obtain: rS (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) > ki−1 + pπi−1(ki−1−1)Si ki−1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2) > ki−1 − 1 + pπi(ki−1−1)Si ki−1−1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, ki−1 − 1 ∈ Ki and ki−1 − 1 ≤ ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Note that we used throughout that pπi(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For every i ∈ �0,n − 1�, if rH > �i j=1 cj + piciSi 1, then ℓi exists and Li := � ℓ ∈ �0,n − i� : i � j=1 cj + ℓ � j=1 cπi(j) + piciSi ℓ+1 < rH � = �0,ℓi�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i ∈ �0,n − 1� and suppose that rH > �i j=1 cj + piciSi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, 0 ∈ Li and ℓi exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, consider ℓ ∈ �1,n − i�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ℓ ∈ Li, then ℓ − 1 ∈ Li, as shown below: rH − i � j=1 cj > ℓ−1 � j=1 cπi(j) + pπi(ℓ) pπi(ℓ) cπi(ℓ) + piciSi ℓ+1 ≥ ℓ−1 � j=1 cπi(j) + piciSi ℓ, where we used the fact that ℓ ∈ Li and locations in �1,n� are ordered by their detection potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, Li = �0,ℓi�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ ec4 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection We are now ready to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, τ−1 = 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Moreover, we observe that kn−1 ∈ {0,1}, which implies that τn−1 = m: If kn−1 = 0, then τn−1 = �n−1 j=1 cj + pncn/pn = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If kn−1 = 1, then τn−1 = �n−1 j=1 cj + cn = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We next show that τi−1 ≤ νi ≤ τi for all i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We note that the inequality νi ≤ τi follows directly from the fact that pici ≤ pi+1ci+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, it only remains to show that τi−1 ≤ νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This is trivial for i = 0, so we may assume that i ∈ �1,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We note that νi − τi−1 = ci + ki � j=1 cπi(j) − ki−1 � j=1 cπi−1(j) − pici � Si−1 ki−1+1 − Si ki+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='6) Let j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ki−1 ∈ �0,j∗ − 1�, we obtain: νi − τi−1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) = ci + ki � j=1 cπi(j) − ki−1 � j=1 cπi(j) − pici � Si ki−1+1 − Si ki+1 + 1 pi � (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='5) = ki � j=ki−1+1 pπi(j)cπi(j) − pici pπi(j) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ki−1 ∈ �j∗,n − i + 1�, we obtain: νi − τi−1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) = ki � j=1 cπi(j) − ki−1−1 � j=1 cπi(j) − pici � Si ki−1 − Si ki+1 � (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='5) = ki � j=ki−1 pπi(j)cπi(j) − pici pπi(j) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ The following lemma derives properties satisfied by our auxiliary parameters: Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i ∈ �0,n − 1� be such that rH > τi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, the following statements hold: – ℓi exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, when i ≥ 1, let j∗ ∈ �1,n − i + 1� satisfying πi−1(j∗) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, ki−1 ≤ � ℓi if ki−1 ∈ �0,j∗ − 1� ℓi + 1 if ki−1 ∈ �j∗,n − i + 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='7) – If νi < rH, then ki ≤ ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If rH ≤ νi, then ki > ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' – Let i ∈ �0,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If i = 0 and rH > τ−1 = 0, then 0 ∈ L0 and ℓ0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We now assume that i ∈ �1,n − 1� and rH > τi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let j∗ ∈ �1,n − i + 1� be such that πi−1(j∗) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ki−1 ∈ �0,j∗ − 1�, then: rH > τi−1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) = i � j=1 cj + ki−1 � j=1 cπi(j) + piciSi ki−1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ki−1 ≤ j∗ − 1 ≤ n − i, then ki−1 ∈ Li, ℓi exists, and ki−1 ≤ ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec5 If ki−1 ∈ �j∗,n − i + 1�, then: rH (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) > i � j=1 cj + ki−1−1 � j=1 cπi(j) + piciSi ki−1−1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ki−1 − 1 ≤ n − i, then ki−1 − 1 ∈ Li, ℓi exists, and ki−1 − 1 ≤ ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' – Let i ∈ �0,n − 1� be such that rH > τi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If rH > νi, then by definition of νi, ki ∈ Li and ki ≤ ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' On the other hand, if rH ≤ νi, then ki /∈ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ℓi exists and Li = �0,ℓi� (Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3), then ki > ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ We can now prove the first theorem of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let rS ∈ �1,n − 1� and rH ∈ �1,m − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i∗ ∈ �0,n − 1� satisfying τi∗−1 < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 1: Suppose that νi∗ < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4, we know that ki∗ ≤ ℓi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let ρS∗ ∈ Rn and ρH∗ ∈ Rn satisfying (5) and (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We will show that (ρS∗,ρH∗) ∈ � AS × � AH and is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we note that ki∗ < n − i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, if ki∗ = n − i∗, then rH > νi∗ = �n j=1 cj = m, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, K ̸= ∅ and Si∗ ki∗ +1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By definition of ki∗, we obtain rS > ki∗ + pπi∗(ki∗ )Si∗ ki∗+1 ≥ ki∗, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='8) which implies that ρS∗ i ≥ 0 for every i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, since ki∗ + 1 ≤ n − i∗ and ki∗ + 1 /∈ Ki∗, then: rS ≤ ki∗ + 1 + pπi∗ (ki∗+1) � Si∗ ki∗ +1 − 1 pπi∗(ki∗ +1) � = ki∗ + pπi∗(ki∗ +1)Si∗ ki∗ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='9) Thus, for every i ∈ K, ρS∗ i ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, n � i=1 ρS∗ i = |J | + rS − ki∗ Si∗ ki∗+1 Si∗ ki∗+1 = rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρS∗ ∈ � AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that ρH∗ ∈ � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ki∗ ≤ ℓi∗, then ki∗ ∈ Li∗ and rH > i∗ � j=1 cj + ki∗ � j=1 cπi∗(j) + pi∗ci∗Si∗ ki∗+1 ≥ i∗ � j=1 cj + ki∗ � j=1 cπi∗(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='10) Thus, ρH∗ i ≥ 0 for every i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, since rH ≤ τi∗ and pi∗+1ci∗+1 ≤ pici for every i ∈ J ∪K, we obtain: ∀i ∈ K, ρH∗ i = rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) piSi∗ ki∗ +1 ≤ pi∗+1ci∗+1 pi ≤ ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='11) ec6 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Finally, n � i=1 ρH∗ i = i∗ � i=1 ci + ki∗ � j=1 cπi∗(j) + � rH − i∗ � j=1 cj − ki∗ � j=1 cπi∗(j) � Si∗ ki∗ +1 Si∗ ki∗ +1 = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρH∗ ∈ � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that (ρS∗,ρH∗) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To this end, we first note the following: ∀ρH ∈ � AH, min ρS∈ � AS ˜u(ρS,ρH) (4) = n � i=1 ρH i − max ρS n � i=1 piρH i ρS i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' n � i=1 ρS i ≤ rS (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) 0 ≤ ρS i ≤ 1, ∀i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, a best response to ρH ∈ � AH is an optimal solution to a continuous knapsack problem with n different (fractional) objects of unitary weights and a knapsack capacity equal to rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Each object i ∈ �1,n� has a profit equal to piρH i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' An optimal solution consists in filling the capacity of the knapsack with the objects with highest profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, given ρH∗ satisfying (6), the “profit” of each object is given by: piρH∗ i = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 pici if i ∈ I ∪ J , rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) Si∗ ki∗ +1 if i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='13) We know that for every i ∈ I and every l ∈ J : pici ≤ pi∗ci∗ (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='10) < rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) Si∗ ki∗+1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='11) ≤ pi∗+1ci∗+1 ≤ plcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='14) Therefore, the objects in J are the most profitable, followed by the objects in K that have equal profit, followed by the objects in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We now must determine bounds on rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We showed in (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='8) that rS > ki∗ = |J |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' An upper bound is given as follows: rS (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='9) ≤ ki∗ + 1 + n−i∗ � j=ki∗ +2 pπi∗(ki∗+1) pπi∗(j) ≤ n − i∗ = |J | + |K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, one best response to ρH∗ selects all the objects in J and any fraction of the objects in K until the knapsack is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, ρS∗ defined in (5) is a best response to ρH∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To show that ρH∗ is a best response to ρS∗, we similarly observe the following: ∀ρS ∈ � AS, max ρH∈ � AH ˜u(ρS,ρH) (4) = max ρH n � i=1 (1 − piρS i )ρH i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' n � i=1 ρH i ≤ rH (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) 0 ≤ ρH i ≤ ci, ∀i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec7 Thus, a best response to ρS ∈ � AS is an optimal solution to another continuous knapsack problem with n different (fractional) objects of unitary weights and a knapsack capacity equal to rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Each object i ∈ �1,n� is available ci times and has a profit equal to (1 − piρS i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' An optimal solution consists in selecting as many copies as possible of the objects with highest profits until filling the capacity of the knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, given ρS∗ satisfying (5), the “profit” of each object is given by: 1 − piρS∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 if i ∈ I, 1 − pi if i ∈ J , 1 − rS − ki∗ Si∗ ki∗ +1 if i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='16) We have the following inequalities: ∀i ∈ J , 1 − rS − ki∗ Si∗ ki∗ +1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='8) < 1 − pπi∗(ki∗) ≤ 1 − pi < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='17) Therefore, the objects in I are the most profitable, followed by the objects in J , followed by the objects in K that have equal profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To determine which objects will be selected given rH, we recall that rH < m = �n i=1 ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='10) implies that rH > � i∈I∪J ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, one best response to ρS∗ consists in selecting all copies of the objects in I and J , and in selecting any fraction of the objects in K until the knapsack is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, ρH∗ defined in (6) is a best response to ρS∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' As a consequence, (ρS∗,ρH∗) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Proposition 1, we deduce that any strategy profile (σS∗,σH∗) ∈ ∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ (which exists as a consequence of Lemma 1) is a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, the value of the games Γ and �Γ is given by: U(σS∗,σH∗) = ˜u(ρS∗,ρH∗) = rH − ki∗ � j=1 pπi∗ (j)cπi∗(j) − � rS − ki∗ �� rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) � Si∗ ki∗ +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 2: We now consider the case when i∗ = 0 and τ−1 < rH ≤ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4, we know that k0 > ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let ρS∗ ∈ Rn and ρH∗ ∈ Rn satisfying (7) and (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We will analogously show that (ρS∗,ρH∗) ∈ � AS × � AH and is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we note that ℓ0 < k0 ≤ n, which implies that ℓ0 + 1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, S0 ℓ0+2 is well defined and K ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For every i ∈ K \\ {π0(ℓ0 + 1)}, pπ0(ℓ0+1) ≤ pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, since ℓ0 + 1 ∈ K0, we obtain: n � i=1 ρS∗ i = ℓ0 + 1 + pπ0(ℓ0+1)S0 ℓ0+2 < rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='18) ec8 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Thus, ρS∗ ∈ � AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that ρH∗ ∈ � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ℓ0 ∈ L0, ℓ0 + 1 /∈ L0, and ℓ0 + 1 ≤ n, then: 0 < rH − ℓ0 � j=1 cπ0(j) = ρH∗ π0(ℓ0+1) = rH − ℓ0+1 � j=1 cπ0(j) + cπ0(ℓ0+1) ≤ cπ0(ℓ0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='19) Since �n i=1 ρH∗ i = rH, we can then conclude that ρH∗ ∈ � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that (ρS∗,ρH∗) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Given ρH∗ satisfying (8), the profit of each object in the knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) is given by: piρH∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 pici if i ∈ J , pπ0(ℓ0+1) � rH − ℓ0 � j=1 cπ0(j) � if i = π0(ℓ0 + 1), 0 if i ∈ K \\ {π0(ℓ0 + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='20) Since rS (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='18) > ℓ0 + 1 = |J | + 1, then a best response to ρH∗ will select all the objects in J ∪ {π0(ℓ0 +1)} (and might not entirely fill the knapsack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, ρS∗ defined in (7) is a best response to ρH∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, given ρS∗ satisfying (7), the profit of each object in the knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) is given by: 1 − piρS∗ i = �1 − pi if i ∈ J ∪ {π0(ℓ0 + 1)}, 1 − pπ0(ℓ0+1) if i ∈ K \\ {π0(ℓ0 + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By definition of π0, we have the following inequalities: 1 − pπ0(1) ≥ ··· ≥ 1 − pπ0(ℓ0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='19) implies that rH > �ℓ0 j=1 cπ0(j), then one best response to ρS∗ can select all copies of the objects in J and can select any fraction of the objects in K until the knapsack is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, ρH∗ defined in (8) is a best response to ρS∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, (ρS∗,ρH∗) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Proposition 1, we deduce that any strategy profile (σS∗,σH∗) ∈ ∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ is a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, the value of the games Γ and �Γ is given by: U(σS∗,σH∗) = ˜u(ρS∗,ρH∗) = rH − ℓ0 � j=1 pπ0(j)cπ0(j) − pπ0(ℓ0+1) � rH − ℓ0 � j=1 cπ0(j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 3: Finally, we consider the case when i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4, we know that ki∗ > ℓi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let ρS∗ ∈ Rn and ρH∗ ∈ Rn satisfying (9) and (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We will analogously show that (ρS∗,ρH∗) ∈ � AS × � AH and is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, we note that ℓi∗ < ki∗ ≤ n − i∗, which implies that ℓi∗ + 1 ≤ n − i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, Si∗ ℓi∗+2 is well defined and K ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ℓi∗ + 1 ∈ Ki∗, then rS > pπi∗(ℓi∗+1)Si∗ ℓi∗+2 + ℓi∗ + 1 = pπi∗(ℓi∗+1)Si∗ ℓi∗+1 + ℓi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='21) e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec9 Thus, ρS∗ i∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we will show by contradiction the following upper bound: rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ ℓi∗+1 ≤ min �pπi∗ (ℓi∗+1) pi∗ , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='22) Let us assume that (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='22) does not hold, and let j∗ ∈ �1,n − i∗ + 1� satisfying πi∗−1(j∗) = i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ℓi∗ + 1 ≤ j∗ − 1, then: pπi∗(ℓi∗+1) pi∗ = min �pπi∗(ℓi∗+1) pi∗ , 1 � < rS − ℓi∗ − 1 − pπi∗(ℓi∗+1)Si∗ ℓi∗+2 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) = rS − ℓi∗ − 1 − pπi∗−1(ℓi∗+1)Si∗−1 ℓi∗+2 + pπi∗(ℓi∗+1) pi∗ , which implies that ℓi∗ + 1 ≤ ki∗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' However, by Lemma EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4, this can only occur when ki∗−1 ≥ j∗, for which we obtain the following contradiction j∗ ≤ ki∗−1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='7) ≤ ℓi∗ + 1 ≤ j∗ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If on the other hand ℓi∗ + 1 ≥ j∗, then j∗ < ℓi∗ + 2 ≤ n − i∗ + 1 and 1 = min �pπi∗−1(ℓi∗+2) pi∗ , 1 � = min �pπi∗ (ℓi∗+1) pi∗ , 1 � (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2),(EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='4) < rS − ℓi∗ − 1 − pπi∗−1(ℓi∗+2)Si∗−1 ℓi∗ +3, which implies that ℓi∗ + 2 ≤ ki∗−1, thus contradicting (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, n � i=1 ρS∗ i = rS − ℓi∗ − pπ∗(ℓi∗+1)Si∗ ℓi∗+1 + |J | + 1 + pπ∗(ℓi∗+1)Si∗ ℓi∗+2 = rS, which implies that ρS∗ ∈ � AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that ρH∗ ∈ � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since locations in �1,n� are ordered by their detection potentials, then for every i ∈ K, pici ≥ pi∗ci∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ℓi∗ ∈ Li∗, then: ρH∗ πi∗(ℓi∗ +1) = rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) − pi∗ci∗Si∗ ℓi∗+1 + pi∗ci∗ pπi∗(ℓi∗+1) > pi∗ci∗ pπi∗(ℓi∗+1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='23) Furthermore, since ℓi∗ + 1 /∈ Li∗ and ℓi∗ + 1 ≤ n − i∗, then: ρH∗ πi∗ (ℓi∗+1) = rH − i∗ � j=1 cj − ℓi∗+1 � j=1 cπi∗(j) − pi∗ci∗Si∗ ℓi∗+2 + cπi∗(ℓi∗+1) ≤ cπi∗(ℓi∗+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='24) Finally, n � i=1 ρH∗ i = � i∈I∪J ci + rH − i∗ � i=1 cj − ℓi∗ � j=1 cπ∗(j) − pi∗ci∗Si∗ ℓi∗+2 + pi∗ci∗Si∗ ℓi∗+2 = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρH∗ ∈ � AH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that (ρS∗,ρH∗) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Given ρH∗ satisfying (10), the profit of each object in the knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) is given by: piρH∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 pici if i ∈ I ∪ J , pπi∗(ℓi∗+1) � rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) − pi∗ci∗Si∗ ℓi∗+2 � if i = πi∗(ℓi∗ + 1), pi∗ci∗ if i ∈ K \\ {πi∗(ℓi∗ + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='25) ec10 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Since the locations in �1,n� are ordered by their detection potentials, then we know that p1c1 ≤ ··· ≤ pi∗ci∗ ≤ pjcj for every j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, we have the following inequality: pi∗ci∗ (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='23) < pπi∗ (ℓi∗+1) � rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) − pi∗ci∗Si∗ ℓi∗+2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='26) Thus, the objects in J ∪ {πi∗(ℓi∗ + 1)} are the most profitable, followed by the objects in {i∗} ∪ K \\ {πi∗(ℓi∗ + 1)} that have equal profit, followed by the objects in I \\ {i∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='21), we know that rS > ℓi∗ + 1 = |J ∪ {πi∗(ℓi∗ + 1)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, an upper bound is given as follows: rS (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='22) ≤ ℓi∗ + 1 + n−i∗ � j=ℓi∗ +1 pπi∗(ℓi∗+1) pπi∗ (j) ≤ n − i∗ + 1 = |{i∗} ∪ J ∪ K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, one best response to ρH∗ will select all the objects in J ∪{πi∗(ℓi∗ +1)} and will select any fraction of the objects in {i∗}∪ K \\{πi∗(ℓi∗ + 1)} until the knapsack is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, ρS∗ defined in (9) is a best response to ρH∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, given ρS∗ satisfying (9), the profit of each object in the knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) is given by: 1 − piρS∗ i = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 if i ∈ I \\ {i∗} 1 − pi∗(rS − ℓi∗ − pπ∗(ℓi∗+1)Si∗ ℓi∗+1) if i = i∗ 1 − pi if i ∈ J ∪ {πi∗(ℓi∗ + 1)}, 1 − pπi∗(ℓi∗+1) if i ∈ K \\ {πi∗(ℓi∗ + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='27) By definition of πi∗, we have the following inequalities: 1 − pπi∗(1) ≥ ··· ≥ 1 − pπi∗(ℓi∗+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Further- more, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='22) implies that 1 − pi∗(rS − ℓi∗ − pπ∗(ℓi∗+1)Si∗ ℓi∗+1) ≥ 1 − pπi∗(ℓi∗+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='23) implies that rH > �i∗ j=1 cj + �ℓi∗ j=1 cπi∗(j), then one best response to ρS∗ selects all copies of the objects in I ∪J and selects any fraction of the objects in K until the knapsack is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Hence, ρH∗ defined in (10) is a best response to ρS∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, (ρS∗,ρH∗) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From Proposition 1, we deduce that any strategy profile (σS∗,σH∗) ∈ ∆S × ∆H that satisfies ρ(σS∗) = ρS∗ and ρ(σH∗) = ρH∗ is a NE of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, the value of the games Γ and �Γ is given by: U(σS∗,σH∗) = ˜u(ρS∗,ρH∗) =rH − pi∗ � rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ ℓi∗+1 � ci∗ − ℓi∗ � j=1 pπi∗ (j)cπi∗(j) − pπi∗(ℓi∗ +1) � rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec11 Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let rS ∈ �1,n − 1� and rH ∈ �1,m − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i∗ ∈ �0,n − 1� satisfying τi∗−1 < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 1: νi∗ < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Suppose also that rH < τi∗ and rS < ki∗ + 1 + pπi∗(ki∗ +1)Si∗ ki∗ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, (5)-(6) are necessary and sufficient conditions for a strategy profile (ρS∗,ρH∗) ∈ � AS × � AH to be a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 2: i∗ = 0 and τ−1 < rH ≤ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Suppose also that pπ0(ℓ0) < pπ0(ℓ0+1) < 1 and pπ0(ℓ0+1) < pπ0(ℓ0+2) if ℓ0 ≤ n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, (7)-(8) are necessary and sufficient conditions for a strategy profile (ρS∗,ρH∗) ∈ � AS × � AH to be a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 3: i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Suppose also that pi∗−1ci∗−1 < pi∗ci∗ < pi∗+1ci∗+1, rH < �i∗ j=1 cj + �ℓi∗+1 j=1 cπi∗(j) + pi∗ci∗Si∗ ℓi∗+2, rS < ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 ki∗−1+2, pπi∗(ℓi∗) < pπi∗(ℓi∗+1) < 1, and pπi∗ (ℓi∗+1) < pπi∗(ℓi∗+2) if ℓi∗ ≤ n − i∗ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, (9)-(10) are necessary and sufficient conditions for a strategy profile (ρS∗,ρH∗) ∈ � AS × � AH to be a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 1: νi∗ < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We additionally consider the following non-edge case assump- tions: rH < τi∗ and rS < ki∗ + 1 + pπi∗(ki∗ +1)Si∗ ki∗ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let (ρS∗,ρH∗) ∈ � AS × � AH satisfying (5)-(6) and let (ρS′,ρH′) ∈ � AS × � AH be any NE of the game �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since �Γ is a zero-sum game, then (ρS′,ρH∗) is also a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρS′ is a best response to ρH∗, it is an optimal solution to the continuous knapsack prob- lem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The profits of each object are given by (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='13) and satisfy inequalities (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, rH < τi∗ implies that: rH − �i∗ j=1 cj − �ki∗ j=1 cπi∗(j) Si∗ ki∗+1 < pi∗+1ci∗+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since |J | < rS ≤ |J |+|K|, then any best response to ρH∗ must select all the objects in J , must not select any object in I, and must entirely fill the knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, ρS′ i = 0 for every i ∈ I, ρS′ i = 1 for every i ∈ J , and �n i=1 ρS′ i = rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, since ρH∗ is a best response to ρS′, then it is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we write the dual of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) associated with ρS′: min α,β rHα + n � i=1 ciβi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' α + βi ≥ 1 − piρS′ i , ∀i ∈ �1,n� (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='28) α ≥ 0 βi ≥ 0, ∀i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' ec12 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Let (α∗,β∗) be an optimal solution of the dual problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since 0 < ρH∗ i < ci for every i ∈ K, then by complementary slackness, β∗ i = 0 and ρS′ i = (1 − α∗)/pi for every i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρS′ must fill the knapsack (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) entirely, then: rS = � i∈I∪J ∪K ρS′ i = ki∗ + (1 − α∗)Si∗ ki∗ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, for every i ∈ K, ρS′ i = (rS − ki∗)/(piSi∗ ki∗ +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In conclusion, ρS′ satisfies (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, (ρS∗,ρH′) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, ρH′ is a best response to ρS∗ and is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The profits of each object are given by (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='16) and satisfy inequalities (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since � i∈I∪J ci < rH < m, then any best response to ρS∗ must select all copies of the objects in I and J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρH′ i = ci for every i ∈ I ∪J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, ki∗ +1 /∈ Ki∗ and the non-edge case assumptions imply that: 1 − rS − ki∗ Si∗ ki∗+1 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='9) > 1 − pπi∗(ki∗ +1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='29) Therefore, ρH′ must entirely fill the knapsack and �n i=1 ρH′ i = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, since ρS∗ is a best response to ρH′, then it is an optimal solution to the continuous knapsack problem in (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The dual of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) associated with ρH′ is given by: min η,ξ rSη + n � i=1 ξi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' η + ξi ≥ piρH′ i , ∀i ∈ �1,n� (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='30) η ≥ 0, ξi ≥ 0, ∀i ∈ �1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let (η∗,ξ∗) be an optimal solution of the dual problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='8) and (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='29) imply that 0 < ρS∗ i < 1 for every i ∈ K, then by complementary slackness, ξ∗ i = 0 and ρH′ i = η∗/pi for every i ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρH′ must fill the knapsack (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) entirely, then: rH = � i∈I∪J ∪K ρH′ i = i∗ � j=1 cj + ki∗ � j=1 cπi∗(j) + ηSi∗ ki∗ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρH′ satisfies (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 2: i∗ = 0 and τ−1 < rH ≤ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We additionally consider the following non-edge case assumptions: pπ0(ℓ0) < pπ0(ℓ0+1) < 1 and pπ0(ℓ0+1) < pπ0(ℓ0+2) if ℓ0 ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let ρH∗ ∈ � AH satisfying (8) and let ρS′ be an equilibrium strategy for S in �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρS′ is a best response to ρH∗, it is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The profits of each object are given by (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='19) and the inequality rS > |J | + 1 imply e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec13 that any best response to ρH∗ must select all the objects in J ∪ {π0(ℓ0 + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρS′ i = 1 for every i ∈ J ∪ {π0(ℓ0 + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρH∗ is a best response to ρS′, then it is an optimal solution to the continuous knap- sack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since 0 (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='19) < ρH∗ π0(ℓ0+1), then at optimality of the dual (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='28), α∗ = 1 − pπ0(ℓ0+1)ρS′ π0(ℓ0+1) − β∗ π0(ℓ0+1) = 1 − pπ0(ℓ0+1) − β∗ π0(ℓ0+1) ≤ 1 − pπ0(ℓ0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, for every i ∈ K \\ {π0(ℓ0 + 1)}, ρH∗ π0(ℓ0+1) = 0 < cπ0(ℓ0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, by complementary slackness, β∗ i = 0 and ρS′ i ≥ (1 − α∗)/pi ≥ pπ0(ℓ0+1)/pi for every i ∈ K \\ {π0(ℓ0 + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In conclusion, ρS′ satisfies (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let ρH′ be an equilibrium strategy for H in �Γ, and consider ρS∗ ∈ AS satisfying ρS∗ i = 1 if i ∈ J ∪ {π0(ℓ0 + 1)}, pπ0(ℓ0+1) pi < ρS∗ i < 1 if i ∈ K \\ {π0(ℓ0 + 1)}, n � i=1 ρS∗ i < rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Such a vector exists as a consequence of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='18) and since pπ0(ℓ0+1) < pi for every i ∈ K\\{π0(ℓ0+1)} under the non-edge case assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, ρH′ is a best response to ρS∗ and is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The profits of each object are given by: ∀i ∈ J ∪ {π0(ℓ0 + 1)}, 1 − piρS∗ i = 1 − pi ∀i ∈ K \\ {π0(ℓ0 + 1)}, 1 − piρS∗ i < 1 − pπ0(ℓ0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Under the non-edge case assumptions, 1 − pπ0(1) ≥ ··· ≥ 1 − pπ0(ℓ0) > 1 − pπ0(ℓ0+1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since �ℓ0 j=1 cπ0(j) < rH < �ℓ0+1 j=1 cπ0(j), then any best response to ρS∗ selects all copies of the objects in J and fills the remaining of the knapsack with objects in π0(ℓ0 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore ρH′ satisfies (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Regime Pattern 3: i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We additionally consider the following non-edge case assumptions: pi∗−1ci∗−1 < pi∗ci∗ < pi∗+1ci∗+1, rH < �i∗ j=1 cj + �ℓi∗+1 j=1 cπi∗(j) + pi∗ci∗Si∗ ℓi∗+2, rS < ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 ki∗−1+2, pπi∗ (ℓi∗) < pπi∗(ℓi∗+1) < 1, and pπi∗(ℓi∗+1) < pπi∗(ℓi∗+2) if ℓi∗ ≤ n − i∗ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let (ρS∗,ρH∗) ∈ � AS × � AH satisfying (9)-(10) and let (ρS′,ρH′) ∈ � AS × � AH be any NE of the game �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρS′ is a best response to ρH∗, it is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The profits of each object are given by (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Under the non-edge case assumptions, p1c1 ≤ ··· ≤ pi∗−1ci∗−1 < pi∗ci∗ < pi∗+1ci∗+1 ≤ ··· ≤ pncn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' From (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='26) and the fact that |J | + 1 < rS ≤ |J | + |K| + 1, we deduce that any best response to ρH∗ selects all the objects in J ∪ {πi∗(ℓi∗ + 1)}, does not select any object in I \\{i∗}, and fills the knapsack (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρS′ i = 0 for every i ∈ I \\ {i∗}, ρS′ i = 1 for every i ∈ J ∪ {πi∗(ℓi∗ + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' and �n i=1 ρS′ i = rS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρH∗ is a best response to ρS′, then it is an optimal solution to the continuous knapsack prob- lem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Under the non-edge case assumptions, 0 < ρH∗ πi∗ (ℓi∗+1) < cπi∗(ℓi∗+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, at optimality ec14 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection of the dual (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='28), α∗ = 1 − pπi∗(ℓi∗+1)ρS′ πi∗ (ℓi∗+1) = 1 − pπi∗(ℓi∗+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, since 0 < ρH∗ i < ci for every i ∈ K \\ {πi∗(ℓi∗ + 1)} under the non-edge case assumptions, then ρS′ i = (1 − α∗)/pi = pπi∗(ℓi∗+1)/pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρH′ must fill the knapsack (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12) entirely, then: ρS′ i∗ = rS − ℓi∗ − 1 − pπi∗(ℓi∗+1)Si∗ ℓi∗+2 = rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ ℓi∗+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρS′ satisfies (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, (ρS∗,ρH′) is a NE of �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, ρH′ is a best response to ρS∗ and is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' The profits of each object are given by (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Under the non-edge case assumptions, 1 − pπi∗(1) ≥ ··· ≥ 1 − pπi∗(ℓi∗ ) > 1 − pπi∗(ℓi∗+1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We next show that: rS − ℓi∗ − pπi∗(ℓi∗+1)Si∗ ℓi∗ +1 < min �pπi∗ (ℓi∗+1) pi∗ , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='31) Let us assume that (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='31) does not hold and let j∗ ∈ �1,n − i∗ + 1� satisfying πi∗−1(j∗) = i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If ℓi∗ + 1 ≤ j∗ − 1, then (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='22) implies that rS = ℓi∗ + 1 + pπi∗−1(ℓi∗+1)Si∗−1 ℓi∗+2, which contradicts the non-edge case assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If on the other hand ℓi∗ + 1 ≥ j∗, then j∗ < ℓi∗ + 2 ≤ n − i∗ + 1 and rS = ℓi∗ + 2 + pπi∗−1(ℓi∗+2)Si∗−1 ℓi∗ +3, which also contradicts the non-edge case assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='31) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since � i∈I∪J ci < rH, then any best response to ρS∗ must select all copies of the objects in I and J , and must fill the knapsack (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, ρH′ i = ci for every i ∈ I ∪ J and �n i=1 ρH′ i = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ρS∗ is a best response to ρH′, then it is an optimal solution to the continuous knapsack problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Under the non-edge case assumptions, 0 < ρS∗ i < 1 for every i ∈ {i∗} ∪ K \\ {πi∗(ℓi∗ + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, at optimality of the dual (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='30), η∗ = pi∗ρH′ i∗ = pi∗ci∗, and ρH′ i∗ = η∗/pi = pi∗ci∗/pi for every i ∈ K \\ {πi∗(ℓi∗ + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, since ρH′ fills the knapsack (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='15) entirely, then: ρH′ πi∗(ℓi∗+1) = rH − i∗ � j=1 cj − ℓi∗ � j=1 cπi∗(j) − pi∗ci∗Si∗ ℓi∗+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In conclusion, ρH′ satisfies (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In this proof, we allow the vector of capacities c and the players’ resources rS and rH to be continuous in the game �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let Ψ be the set of parameters given by (11) for which �Γ is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we note that Ψ′ := {(n,p,c,rS,rH) ∈ Ψ : pi < 1 ∀i ∈ �1,n�, pi ̸= pj and pici ̸= pjcj ∀i ̸= j ∈ �1,n�} is a dense subset of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we consider an instantiation of the game parameters (n,p,c,rS,rH) ∈ Ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We order the indices such that pici < pi+1ci+1 for every i ∈ �1,n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let i∗ ∈ �0,n − 1� such that τi∗−1 < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec15 We first consider the case of Regime Pattern 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', νi∗ < rH ≤ τi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We then consider new player resources ˆrS = rS − ε and ˆrH = rH − ε for ε > 0 arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To avoid confusion, we denote the corresponding parameters that depend on ˆrS and ˆrH as ˆki, ˆℓi, ˆτi, ˆνi, and ˆi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By definition of ki∗, and for arbitrarily small ε, we obtain: ki∗ + pπi∗(ki∗)Si∗ ki∗+1 < ˆrS < rS ≤ ki∗ + 1 + pπi∗(ki∗+1)Si∗ ki∗+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, ˆki∗ = ki∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This implies that ˆτi∗ = τi∗ and ˆνi∗ = νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We then deduce the following inequalities for arbitrarily small ε: ˆνi∗ = νi∗ < ˆrH < rH ≤ τi∗ = ˆτi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, ˆi∗ = i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since νi∗ < ˆrH < τi∗ and ˆrS < ki∗ + 1 + pπi∗(ki∗ +1)Si∗ ki∗ +2, then Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1 implies that all pure NE of the game �Γ with the parameters (n,p,c, ˆrS, ˆrH) for arbitrarily small ε > 0 satisfy the corresponding equilibrium conditions (5)-(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore (n,p,c, ˆrS, ˆrH) is arbitrarily close to (n,p,c,rS,rH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We next consider the case of Regime Pattern 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', i∗ = 0 and rH ≤ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1 implies that all pure NE of the game �Γ with the parameters (n,p,c, ˆrS, ˆrH) ∈ Ψ′ satisfy the corresponding equilibrium conditions (7)-(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, we consider the case of Regime Pattern 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', i∗ ≥ 1 and τi∗−1 < rH ≤ νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We then consider new player resources ˆrS = rS − ε and ˆrH = rH − ε for ε > 0 arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Similarly, we denote the corresponding auxiliary parameters as ˆki, ˆℓi, ˆτi, ˆνi, and ˆi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Using a similar derivation as above, we deduce that for arbitrarily small ε, ˆki∗ = ki∗ and ˆνi∗ = νi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, by definition of ki∗−1, we obtain: ki∗−1 + pπi∗−1(ki∗−1)Si∗−1 ki∗−1+1 < ˆrS < rS ≤ ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 ki∗−1+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, ˆki∗−1 = ki∗−1 and ˆτi∗−1 = τi∗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, we obtain that ˆi∗ = i∗ since ˆτi∗−1 = τi∗−1 < ˆrH < rH ≤ νi∗ = ˆνi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, by definition of ℓi∗, we obtain: i∗ � j=1 cj + ℓi∗ � j=1 cπi∗(j) + pi∗ci∗Si∗ ℓi∗+1 < ˆrH < rH ≤ i∗ � j=1 cj + ℓi∗+1 � j=1 cπi∗(j) + pi∗ci∗Si∗ ℓi∗+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, ˆℓi∗ = ℓi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since rH < �i∗ j=1 cj + �ℓi∗+1 j=1 cπi∗(j) + pi∗ci∗Si∗ ℓi∗+2 and rS < ki∗−1 + 1 + pπi∗−1(ki∗−1+1)Si∗−1 ki∗−1+2, then Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='1 implies that all pure NE of the game �Γ with the parameters (n,p,c, ˆrS, ˆrH) for arbitrarily small ε > 0 satisfy the corresponding equilibrium condi- tions (9)-(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore (n,p,c, ˆrS, ˆrH) is arbitrarily close to (n,p,c,rS,rH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proofs of Section 4 Before proving Theorem 2, we show that Algorithm 1 is well defined and terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We denote as κ∗ ∈ Z≥0 ∪ {+∞} the number of iterations of the while loop (3-16) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' ec16 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Each iteration of Algorithm 1 is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, ∀k ∈ �1,κ∗ + 1�, ¯ρk ∈ [0,1]n and n � i=1 ¯ρk i ≤ ¯r, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) ∀k ∈ �1,κ∗�, qk ∈ �1,n� and δk ∈ [0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='33) Proof of Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We show (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) and (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='33) by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We first consider k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By construction, ¯ρ1 = ρ − ⌊ρ⌋ ∈ [0,1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, by definition of � A(b,r), we obtain: ¯r = r − n � i=1 ⌊ρi⌋ = r − n � i=1 ρi + n � i=1 ¯ρ1 i ≥ n � i=1 ¯ρ1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='34) Next, q1 is constructed when the algorithm initiates the while loop (3-16), that is, when ¯ρ1 /∈ {0,1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ¯ρ1 ≥ 0n and ¯r ∈ Z, then 1 ≤ |{i ∈ �1,n� : ¯ρ1 i > 0}| ≤ n and 1 ≤ ⌈�n i=1 ¯ρ1 i ⌉ (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='34) ≤ ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, 1 ≤ q1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, δ1 is well defined since q1 ∈ �1,n�, and δ1 ∈ [0,1] as a consequence of ¯ρ1 ∈ [0,1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We next show by contradiction that δ1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, if δ1 = 1, then we first deduce that for every j ∈ �1,q1�, 1 ≥ ¯ρ1 θ1(j) ≥ ¯ρ1 θ1(q1) ≥ δ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If q1 = n, then this contradicts ¯ρ1 /∈ {0,1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If q1 < n, then, we derive the following inequalities: q1 = q1 � j=1 ¯ρ1 θ1(j) ≤ n � i=1 ¯ρ1 i (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='34) ≤ ¯r, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='35) q1 ≤ ��� i ∈ �1,n� : ¯ρ1 i = 1 ��� ≤ ��� i ∈ �1,n� : ¯ρ1 i > 0 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='36) If q1 = ¯r, (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' q1 = |{i ∈ �1,n� : ¯ρ1 i > 0}|) then (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='35) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='36)) implies that ¯ρ1 θ1(j) = 0 for every j ∈ �q1 + 1,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This also contradicts ¯ρ1 /∈ {0,1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, δ1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we assume that (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) and (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='33) hold for k ∈ �1,κ∗�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since δk < 1, then we obtain: ∀j ∈ �1,qk�, 0 ≤ ¯ρk θk(j) − ¯ρk θk(qk) 1 − δk ≤ ¯ρk θk(j) − δk 1 − δk = ¯ρk+1 θk(j) ≤ 1 − δk 1 − δk = 1, and if qk < n, then ∀j ∈ �qk + 1,n�, 0 ≤ ¯ρk θk(j) 1 − δk = ¯ρk+1 θk(j) ≤ ¯ρk θk(j) ¯ρk θk(qk+1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, for every i ∈ �1,n�, ¯ρk+1 i ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that �n i=1 ¯ρk+1 i ≤ ¯r: n � i=1 ¯ρk+1 i = qk � j=1 ¯ρk θk(j) − δk 1 − δk + n � j=qk+1 ¯ρk θk(j) 1 − δk = 1 1 − δk � n � i=1 ¯ρk i − qkδk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If qk = ¯r, then: n � i=1 ¯ρk+1 i ≤ 1 1 − δk � ¯r − ¯rδk� = ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec17 If on the other hand qk = ��� i ∈ �1,n� : ¯ρk i > 0 ���, then ¯ρk ∈ [0,1]n implies that: n � i=1 ¯ρk+1 i = 1 1 − δk � n � i=1 ¯ρk i − ���i ∈ �1,n� : ¯ρk i > 0���δk � ≤ 1 1 − δk \uf8eb \uf8ed n � i=1 ¯ρk i − δk � {i∈�1,n� : ¯ρk i >0} ¯ρk i \uf8f6 \uf8f8 = n � i=1 ¯ρk i ≤ ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, �n i=1 ¯ρk+1 i ≤ ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since ¯ρk+1 ∈ [0,1]n, then the same argument as the one derived for k = 1 can be applied to conclude that if k < κ∗ and ¯ρk+1 /∈ {0,1}n, then qk+1 ∈ �1,n� and δk+1 ∈ [0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In conclusion, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) and (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='33) hold by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Algorithm 1 terminates after κ∗ ≤ n iterations of the while loop (3-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In particular, for every k ∈ �1,κ∗�, δk > 0, and ��� i ∈ �1,n� : ¯ρk+1 i ∈ {0,1} ��� > ��� i ∈ �1,n� : ¯ρk i ∈ {0,1} ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let k ∈ �1,κ∗�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' First, we show that δk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since qk ≤ ��� i ∈ �1,n� : ¯ρk i > 0 ���, then ¯ρk θk(qk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show by contradiction that if qk < n, then ¯ρk θk(qk+1) < 1: If instead qk < n and ¯ρk θk(qk+1) = 1, then we first deduce that qk < ��� i ∈ �1,n� : ¯ρk i > 0 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, ¯r (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) ≥ n � i=1 ¯ρk i ≥ qk+1 � j=1 ¯ρk θk(j) = qk + 1 > qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This contradicts the definition of qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore if qk < n, then ¯ρk θk(qk+1) < 1, which in turn implies that δk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We now show that ��� i ∈ �1,n� : ¯ρk+1 i ∈ {0,1} ��� > ��� i ∈ �1,n� : ¯ρk i ∈ {0,1} ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Let j′ ∈ �1,n� be such that ¯ρk θk(j′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Necessarily, j′ > ��� i ∈ �1,n� : ¯ρk i > 0 ��� ≥ qk, which implies that ¯ρk+1 θk(j′) = ¯ρk θk(j′)/(1 − δk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we consider j′ ∈ �1,n� such that ¯ρk θk(j′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, j′ ≤ qk, as implied by the following inequalities: j′ = j′ � j=1 ¯ρk θk(j) ≤ n � i=1 ¯ρk i (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) ≤ ¯r, j′ ≤ ��� i ∈ �1,n� : ¯ρk i = 1 ��� ≤ ��� i ∈ �1,n� : ¯ρk i > 0 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, j′ ≤ qk and ¯ρk+1 θk(j′) = (¯ρk θk(j′) −δk)/(1−δk) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This shows that ���i ∈ �1,n� : ¯ρk+1 i ∈ {0,1}��� ≥ ���i ∈ �1,n� : ¯ρk i ∈ {0,1}���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To ensure a strict inequality, we must show that one fractional component of ¯ρk becomes 0 or 1 in ¯ρk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' ec18 e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection We know that 0 < δk < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If δk = ¯ρk θk(qk), then ¯ρk+1 θk(qk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' If qk < n and δk = 1 − ¯ρk θk(qk+1), then ¯ρk+1 θk(qk+1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In both cases, a fractional component of ¯ρk becomes 0 or 1 in ¯ρk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In conclusion ��� i ∈ �1,n� : ¯ρk+1 i ∈ {0,1} ��� > ��� i ∈ �1,n� : ¯ρk i ∈ {0,1} ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since for every k ∈ �1,κ∗ + 1�, ��� i ∈ �1,n� : ¯ρk i ∈ {0,1} ��� ≤ n, then the algorithm must terminate after at most n iterations of the while loop (3-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore κ∗ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □ Now that we proved that Algorithm 1 is well defined and terminates, we can show Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Consider a vector of capacities b ∈ Zn >0, a budget of resources r ∈ Z>0, and a vector ρ ∈ � A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' For convenience, we denote eκ∗+1 := ¯ρκ∗+1 ∈ {0,1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We first show that for every k ∈ �1,κ∗ + 1�, ⌊ρ⌋ + ek ∈ A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' In Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='3, we showed that for every k ∈ �1,κ∗�, if a component i ∈ �1,n� satisfies ¯ρk i = 0, then ¯ρk+1 i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, for every k ∈ �1,κ∗ + 1�, ¯ρk i > 0 only if ¯ρ1 i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' By definition of � A(b,r) and since bi ∈ Z, we deduce that if ¯ρ1 i > 0, then bi ≥ ⌈ρi⌉ = ⌊ρi⌋ + ⌈¯ρ1 i⌉ = ⌊ρi⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, ∀k ∈ �1,κ∗�, n � i=1 (⌊ρi⌋ + ek i ) = qk + n � i=1 ⌊ρi⌋ ≤ ¯r + n � i=1 ⌊ρi⌋ = r, and n � i=1 (⌊ρi⌋ + eκ∗+1 i ) = n � i=1 ¯ρκ∗+1 i + n � i=1 ⌊ρi⌋ (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='32) ≤ ¯r + n � i=1 ⌊ρi⌋ = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, for every k ∈ �1,κ∗ + 1�, ⌊ρ⌋ + ek ∈ A(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Next, we show that σ returned by the algorithm is a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We first note that for every k ∈ �1,κ∗ + 1�, γk ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Furthermore, � z∈A(b,r) σz = γκ∗+1 + κ∗ � k=1 γkδk = γκ∗+1 + κ∗ � k=1 (γk − γk+1) = γκ∗+1 + γ1 − γκ∗+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, σ ∈ ∆(b,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' We now show that σ returned by the algorithm is consistent with the vector ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' To this end, we note the following equality: ∀k ∈ �1,κ∗�, ∀i ∈ �1,n�, γk ¯ρk i − γk+1¯ρk+1 i = γk(¯ρk i − (1 − δk)¯ρk+1 i ) = γkδkek i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Then, we obtain: ∀i ∈ �1,n�, Ez∼σ[zi] = γκ∗+1(⌊ρi⌋ + ¯ρκ∗+1 i ) + κ∗ � k=1 γkδk(⌊ρi⌋ + ek i ) = ⌊ρi⌋ � z∈A(b,r) σz + γκ∗+1¯ρκ∗+1 i + γ1¯ρ1 i − γκ∗+1¯ρκ∗+1 i = ⌊ρi⌋ + ¯ρ1 i = ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Thus, σ returned by Algorithm 1 is consistent with the vector ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since κ∗ ≤ n, then the support of σ is of size at most n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Finally, we argue that Algorithm 1 runs in time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, the first iteration of the while loop (3-16) can be implemented in e-companion to Bahamondes and Dahan: Hide-and-Seek Game with Capacitated Locations and Imperfect Detection ec19 time O(nlogn) by using an efficient sorting algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' merge sort) to sort ¯ρ1 and create the permutation θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Fortunately, the subsequent iterations can be implemented in time O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Indeed, we note that for every k ∈ �1,κ∗ −1�, ¯ρk+1 θk(1) ≥ ··· ≥ ¯ρk+1 θk(qk) and ¯ρk+1 θk(qk+1) ≥ ··· ≥ ¯ρk+1 θk(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Therefore, we can sort ¯ρk+1 and create the permutation θk+1 by merging and sorting the lists (¯ρk+1 θk(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', ¯ρk+1 θk(qk)) and (¯ρk+1 θk(qk+1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=', ¯ρk+1 θk(n)) that are already sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' This operation can be carried out in time O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' Since the number of iterations of the while loop (3-16) is upper bounded by n, then the overall running time of Algorithm 1 is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} +page_content=' □' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FJT4oBgHgl3EQfbyya/content/2301.11541v1.pdf'} diff --git a/s9E1T4oBgHgl3EQfjgRh/content/tmp_files/2301.03263v1.pdf.txt b/s9E1T4oBgHgl3EQfjgRh/content/tmp_files/2301.03263v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2843cbb07e4558b79eae5a8269a5ff3e8119c45 --- /dev/null +++ b/s9E1T4oBgHgl3EQfjgRh/content/tmp_files/2301.03263v1.pdf.txt @@ -0,0 +1,1110 @@ +Observation of magnetic state dependent thermoelectricity in superconducting spin valves +César González-Ruano†,1 Diego Caso†,1 Jabir Ali Ouassou,2 Coriolan Tiusan,3, 4 Yuan Lu,4 Jacob Linder,2 and Farkhad G. Aliev5, ∗ +1Departamento Física de la Materia Condensada C-III, +Universidad Autónoma de Madrid, Madrid 28049, Spain +2Center for Quantum Spintronics, Department of Physics, +Norwegian University of Science and Technology, NO-M7T9Q Trondheim, Norway +3Department of Solid State Physics and Advanced Technologies, +Faculty of Physics, Babes-Bolyai University, Cluj Napoca 400114, Romania +4Institut Jean Lamour, Nancy Universitè, 54506 Vandoeuvre-les-Nancy Cedex, France +5Departamento Física de la Materia Condensada C-III, +Instituto Nicolás Cabrera (INC) and Condensed Matter Physics Institute (IFIMAC), +Universidad Autónoma de Madrid, Madrid 28049, Spain +Superconductor-ferromagnet tunnel junctions demonstrate giant thermoelectric effects which are being exploited +to engineer ultra-sensitive terahertz radiation detectors. Here, we experimentally observe the recently predicted +complete magnetic control over thermoelectric effects in a superconducting spin valve, including the dependence +of its sign on the magnetic state of the spin valve. The description of the experimental results is improved by the +introduction of an interfacial domain wall in the spin filter layer interfacing the superconductor. Surprisingly, the +application of high in-plane magnetic fields induces a double sign inversion of the thermoelectric effect, which +exhibits large values even at applied fields twice the superconducting critical field. +INTRODUCTION +The competition between superconducting (S) and ferromag- +netic (F) ground states can under certain conditions result in +a synergy of these otherwise antagonistic states [1]. In recent +years, a variety of exotic phenomena have been demonstrated +in devices that exploit this synergy. Notable examples include +long-ranged spin-triplet supercurrents [2–4], spin-valve Joseph- +son junctions [5], superconducting spin-valves with record-high +magnetoresistance, and the giant thermoelectric (TE) effect. +These effects are considered as potential ingredients in the next +generation of low-dissipation cryogenic devices [6–8]. +In general, there exists considerable interest in identifying +material platforms for improved TE devices. +At low tem- +peratures, TE effects are expected to be vanishingly small in +both normal metals and bulk superconductors. Instead, they +have been investigated mainly in superconductor/normal-metal +hybrids, where they have been used in micro-refrigeration +and thermometry [9]. More recently, fascinating theoretical +predictions [10–15] have opened the door to unexplored spin +dependent TE effects in S/F hybrids. The transport of spin +and charge due to temperature gradients in such systems have +only been investigated experimentally in a few works [16–19]. +Kolenda et al. [16] reported on the experimental observation of +an enhanced Seebeck coefficient (up to 100 µV/K) when a large +magnetic field of about 1 T splits the quasiparticle band struc- +ture of a superconductor. When combined with a spin-filtering +magnetic interface, this spin splitting breaks the electron–hole +symmetry of the junction. The resulting carrier asymmetry +produces the observed “giant TE effect”, which is now being +exploited to develop ultrasensitive radiation detectors [20]. +So far, the experimental tuning of giant TE effects in S/F +hybrids has been carried out either by applying large magnetic +fields [16] or by exchange coupling a superconductor to a +ferromagnetic insulator [20]. Recently, however, a different +method to control the TE effect has been predicted in super- +conductor/ferromagnet/ferromagnetic insulator systems [21]. +This method can turn the superconducting TE effect on and +off in situ, as well as reversing its sign. Here, by interfacing +a superconductor with a spin valve with large spin filtering +capability, we experimentally demonstrate the above mentioned +complete magnetic control of the superconducting TE effect. +This includes evidence of an antisymmetric TE effect, where +a change of the magnetic state of the spin valve inverts the +direction of the TE current. Controlling the sign of the ther- +mopower, analogous to the inversion of TE signals between +𝑝- and 𝑛-doped semiconductors, enables the design of Peltier +elements based on superconducting spin valves. +EXPERIMENTAL RESULTS +Figure 1(a) illustrates the main experimental set-up and +junctions investigated. +We have measured TE effects +in V(40)/MgO(2)/Fe(10)/MgO(2)/Fe(10)/Co(30) single crys- +talline junctions epitaxially grown on MgO(001) substrates, +with the thickness of each layer given in nanometers in paren- +theses. More details about the junctions, experimental set-up +and procedures can be found in the Supplemental Material +sections 1 and 2. Here, V is a BCS superconductor, Fe and +Co are ferromagnetic metals, and MgO is a symmetry filter- +ing insulator. The magnetically hard Fe/Co electrode allows +for a precise detection of the orientation of the magnetically +free Fe layer interfacing the superconductor. Figure 1(b)-(c) +presents a general electron transport characterization of the +junctions in the superconducting state as a function of the +applied bias and external magnetic field. One observes that +the superconducting gap in the V electrode is suppressed by +in-plane (IP) and out-of-plane (OOP) fields of about 1.7 T and +0.4 T, respectively (Fig. Figure 1(c)). As we will discuss fur- +Typeset by REVTEX +arXiv:2301.03263v1 [cond-mat.supr-con] 9 Jan 2023 + +2 +-3 +-2 +-1 +0 +1 +2 +3 +6 +8 +10 +12 +14 +16 +G (µS) +V(mV) + H=0 Oe + H=10 kOe + H=25 kOe +0 +5 +10 +15 +0.0 +0.5 +1.0 +Sgap (norm) +H (kOe) + HIP + HOOP +-1.0 +-0.5 +0.0 +0.5 +1.0 +0 +10 +20 +30 +40 +50 +TMR(%) +H(kOe) +Fe (10 nm) +Fe (10 nm) +Co (20 nm) +V +MgO +LED ++- VLED +V +∇T +(a) +(b) +(c) +Hc(OOP) +Hc(IP) +T=0.3 K, HIP +(d) +1 +2 +3 +4 +5 +6 +7 +8 +T=0.3 K +Heat flux +Y +Z +X +FIG. 1: (a) Sketch of the S/F/F junctions when heated by a LED. +(b) Typical conductance–bias curves measured at 0.07𝑇𝑐 at three +different applied IP magnetic fields. (c) Normalized superconducting +gap depth (Sgap) taken from 𝐺(𝑉) curves vs applied IP and OOP +magnetic fields. +(d) Typical tunneling magnetoresistance curves +measured with IP magnetic field at 𝑉 = 5 mV and 𝑇 = 0.3 K. +The numbers indicate the order of the field sweeping and resistance +changes. +ther below, the magnetoresistance values of 35-55% provide an +estimation of the effective spin polarization of the Fe electrodes +of around 0.75–0.85 for the different junctions studied based +on the Julliere model, involving two ferromagnetic and one +nonmagnetic (V) electrodes, in line with previous reports [22]. +To study the TE effect as a function of the spin-valve state, the +soft Fe layer was rotated while the hard Fe/Co layer remained +fixed. We did this by applying a rotating in-plane magnetic +field, with a magnitude between the coercive fields of the +two magnetic layers. For the hard layer the coercive field is +typically larger than 500 Oe (Fig. 1(d)), while for the soft layer +it’s smaller than 50 Oe, determined by the magneto-crystalline +anisotropy. This procedure guarantees a reorientation between +the parallel (P), antiparallel (AP) and perpendicular in-plane +(PIP) configurations of the spin valve. For each 3◦ rotation of the +applied field, the temperature gradient ∇𝑇 was re-established +via the LED heater, and the resulting TE response Δ𝑉 was +measured. +Fig. 2(a) shows the variation of the TE voltage generated +during a magnetization rotation of the free layer under an +in-plane applied magnetic field of 70 Oe. While the transition +between the P and PIP states hardly affects the values of the +TE response, a strong reduction of the TE voltage of more than +a factor of 2 is observed when the free layer becomes close to +AP to the fixed Fe/Co layer. Note that there is a slight latency +of about 10–15◦ between the angle of the applied magnetic +field and the real average magnetization orientation of the +50.60 +50.65 +50.70 +-40 +-20 +0 +20 +∆V/∆T (µV/K) +RAP(kΩ) +33.89 +33.90 +33.91 +RP(kΩ) +0 +90 +180 +270 +2 +4 + ∆V (µV) +0 +20 +40 +60 +80 +100 +-2.0 +-1.0 +0.0 +1.0 +∆V (µV) +∆T (mK) +0 +20 +40 +60 +80 +100 +-1 +0 +1 +2 +3 +4 +∆V (µV) +∆T (mK) +T=0.3 K, P State + H=70 Oe + H=100 Oe + H=150 Oe + H=200 Oe + H=300 Oe +(a) +(b) +(c) +(d) +T=0.3 K, AP State + H=-70 Oe + H=-100 Oe + H=-150 Oe + H=-200 Oe + H=-250 Oe + H=-300 Oe +AP +P +15 +30 +45 + TMR (%) +H = 70 Oe + T=0.3 K + T=9 K +TMR +FIG. 2: Thermoelectric response of a S/F/F junction measured under +a rotation and fixed values of the applied magnetic field at 𝑇 = 0.3 K. +(a) TE response at 𝐻 = 70 Oe for in-plane rotations of the magnetic +field at 𝑉LED = 7.3 V (Δ𝑇 ≈ 113 mK), below and above 𝑇𝑐. The +tunnel magnetoresistance of the spin-valve is also displayed against the +rotation angle. (b) Response in the P configuration of the spin-valve +stack. (c) Same experiment for the AP configuration. The TE voltage +changes its sign and intensity depending not directly on the applied +field, but on the saturation of the soft FM layer. The related analysis is +shown in panel (d), where the average value of the TE voltage is plotted +against the measured resistance for the P (blue, upper horizontal axis) +and AP (red, lower horizontal axis) configurations. For the P state, a +lower resistance implies a better polarization, while in the AP state +the polarization is better with a higher resistance. +Fe layer. This is a natural consequence of the experimental +process: the Fe magnetization has to overcome the magneto- +crystalline anisotropy to follow the slowly rotating applied +field (see for example Ref. [23]). The complete 360◦ rotation +takes about two hours, since we stop at each intermediate angle +to measure the TE response. Figures Fig. 2(b) and (c) show +the TE response of the samples vs the induced temperature +gradient for different magnetic configurations and applied fields. +Note that these temperature gradients have been estimated by +first simulating the response to the incoming heat flux and +subsequently recalculating Δ𝑇 from 𝑉LED based on the LED +calibration curves (see Supplemental Material sections 4 and +5). In the P state, changing the applied field by less than a +few hundred Oe does not qualitatively change the TE response +(Fig. 2(b)). However, in the AP state, varying the magnetic field +has a dramatic effect on the TE response and can even change its +sign as seen in Fig. 2(c). We note that no asymmetry of Δ𝑉 or +dependence on𝑉LED was observed above𝑇𝑐 (Fig. 2(a)). Control +experiments on short-circuited junctions also revealed at least +an order of magnitude drop of the TE response regardless of + +3 +0 +50 +100 +-4 +-2 +0 +2 + + +∆V(µV) +∆T(mK) +T=0.3 K, P state + H=50 Oe + H=125 Oe + H=1200 Oe +T=9 K + H=100 Oe +H (kOe) +1.5 +1 +0.5 +0 +-0.2 +-0.5 +-1 +ΔV(μV) +ΔT(mK) +20 +60 +100 +P +AP +1.5 +0 +-3 +(a) +(c) +(b) +0 +50 +100 +-3 +-2 +-1 +0 +T=0.3 K, AP state + H=-60 Oe + H=-200 Oe + H=-600 Oe +T = 9 K + H=-200 Oe +∆V(µV) +∆T(mK) +-0.8 +P +0 +FIG. 3: Thermoelectric voltage of an S/F/F junction demonstrating +sign inversion at high fields in the P state (a) and AP state (b). +(c) Colormap of the recorded TE voltage Δ𝑉 as a function of the +temperature difference Δ𝑇 and the applied in-plane field 𝐻 for the +same junction, indicating the P and AP states. +the magnetic state (see Supplemental Material section 6). +In order to understand the possible reasons for the TE sign +change in the AP state, we analyzed (Fig. 2(d)) the TE response +obtained for a fixed temperature gradient as a function of +the resistance in the P and AP states obtained during each +particular TE experiment at different applied in-plane magnetic +fields (not exceeding 500 Oe). +While the TE response in +the P state is rather robust to the variation of the resistance +(i.e. presence of magnetic inhomogeneities), in the AP state +it changes sign with the reduction of the influence of these +magnetic textures (an increase of the resistance means better +magnetization saturation). +Interestingly, some junctions revealed a TE sign inversion +both in the AP state and also under a sufficiently high applied +in-plane magnetic field in the P state (Fig. 3). The TE response +in the P state is positive and robust at fields below 0.5 kOe +(Fig. 3(a)), and becomes negative for higher magnetic fields. +In contrast, in the AP state the TE response is already negative +for much smaller fields (Fig. 3(b)), right after the spin-valve +switched into the AP state (see steps 1-2 in Fig. 1(d)). Further +increasing the negative magnetic field (path 2-3 in Fig. 1(d)) +reorients the hard layer so the spin-valve is again in the P +configuration, and again the sign of the TE effect follows the +same trend as for positive fields. Figure 3(c) summarizes these +observations with a 3D color plot of the TE voltage signal +against the applied field and evaluated temperature gradient. +We have found that the TE voltage sign inversion in the +P state under high in-plane magnetic fields is a rather robust +effect and is followed by a second TE sign inversion towards +35 +40 +45 +50 +55 +0 +10 +20 +30 +TE inversion field (kOe) +TMRMAX(%) +0.80 +0.82 +0.84 + P +0 +2 +4 +0 +5 +10 +15 +∆V(µV) +H(kOe) +(a) +(b) +0.1 +1 +10 +-10 +0 +10 + S/F/F + S/F +∆V(µV) +H(kOe) +FIG. 4: (a) Thermoelectric voltage of an S/F/F and S/F junction at +𝑉LED = 7.3 V (Δ𝑇 ≈ 113 mK) vs. applied in-plane field at 𝑇 = 0.3 K. +The high-field sign inversion is achieved in both samples, and the TE +effect is maintained with increasing in-plane fields up to 30 kOe. The +inset displays the measured TE voltage in the S/F/F junction under +out-of-plane field. At 𝐻 = 𝐻𝑐, superconductivity and its associated +TE voltage vanish. (b) TE inversion field and polarization against the +maximum TMR value for all the S/F/F junctions under study. +positive values when the magnetic field is further increased +(Fig. 4(a)). Surprisingly, even for maximum applied in-plane +magnetic fields, twice exceeding the second critical magnetic +field (compare Fig. 1(c) and Fig. 4(a)), the TE voltage remains +high and without clear signatures of diminishing. A quali- +tatively similar response has been observed in single barrier +V/MgO/Fe junctions, i.e. without the sensing Fe/Co layer. Fig. +4(b) compares the TE signal inversion field with the TMR and +effective spin polarization values of each corresponding sample. +Apparently, a higher TMR and correspondingly polarization +values shift the TE inversion field range outside our experimen- +tal capabilities (35 kOe). This suggests a possible link between +the observed effect and interfacial domain wall forming in the +Fe electrode. Further experiments are needed to understand +the physical mechanism behind the high field TE effects. +THEORETICAL MODELLING +To better understand the physics behind the experimental +observations, we explored the setup in Fig. 1(a) via numeri- +cal simulations. We employed the Usadel formalism [24–28] +which describes superconductivity in diffusive heterostructures, +together with spin-dependent tunneling boundary conditions +[10, 29–32] valid for arbitrary spin polarizations. To numeri- +cally solve these equations, we used the Ricatti parametrization +[33, 34] to calculate spectral properties and a distribution-trace +parametrization [26–28, 35] to calculate the nonequilibrium +transport properties. The theoretical and numerical approach +is described in more detail in Ref. [21]. +The numerical model used herein is sketched in Fig. 5(a). +The superconductor (V) was treated as a BCS superconducting +reservoir with an effective spin-splitting ℎ = Δ/10, near-zero +temperature 𝑇 = 𝑇𝑐/100, and electrical grounding 𝑉 = 0. The +hard ferromagnet (Fe/Co) was treated as a non-superconducting +metallic reservoir at an elevated temperature 𝑇 = 𝑇𝑐/2 and volt- + +4 +age 𝑉 = Δ𝑉. The interfaces to the soft ferromagnet (V/MgO/Fe +and Fe/MgO/Fe/Co) were treated using spin-polarized tun- +neling boundary conditions with spin polarizations 𝑃1, 𝑃2 +and a low tunneling conductance 𝐺𝑇 = 𝐺𝐷/5 where 𝐺𝐷 is +the Drude conductance of the soft ferromagnet. These pa- +rameters model the high spin filtering capabilities and low +transparencies of the MgO barriers. In the soft ferromagnet +(Fe), we used an exchange splitting ℎ = 30Δ. For each mag- +netic configuration, we (i) solved the Usadel equation for 80 +different 𝑒Δ𝑉/Δ ∈ [−0.04, +0.04]; (ii) used this to calculate +the current 𝐼(Δ𝑉); (iii) interpolated the open-circuit voltage +from 𝐼(Δ𝑉) ≡ 0. This yielded the TE voltage as function of +magnetic configuration. +The magnetic configurations we considered are illustrated +in Fig. 5(a–b). +We take the hard ferromagnet (red) to be +oriented along one in-plane axis (up), and the spin filtering at +the MgO barrier is assumed to be parallel to this orientation +(black). The soft ferromagnet is then rotated by an in-plane +angle 𝜑 relative to the hard ferromagnet (purple), which here is +sketched for the antiparallel case 𝜑 = 𝜋. At the superconductor +interface, we include the possibility for an interfacial domain +wall described by an out-of-plane angle 𝜃. This affects both the +spin filtering at the second MgO barrier (black) and the direction +of spin splitting inside the neighboring superconductor (blue). +Fig. 5(c–e) show the numerical results for the TE voltage Δ𝑉(𝜑) +across the junction as function of the in-plane misalignment +angle between the two ferromagnets. In panel (c), we see +the predicted response for a junction with two identical spin +filters (𝑃1 = 𝑃2) and no interfacial domain wall (𝜃 = 0). We +then predict an asymmetric TE effect, by which we mean that +Δ𝑉(0) is maximal while Δ𝑉(𝜋) → 0. This can intuitively +be understood as follows. If the two spin filters are identical +(𝑃1 = 𝑃2 and 𝜑 = 0), then “filtering the spins twice” does not +significantly change the physics compared to having only one +spin filter. The latter case has in previous work been shown +to produce a giant TE effect in superconductor/ferromagnet +systems [10–19]. On the other hand, the effects of two identical +but oppositely aligned spin filters cancel, so any TE effect +due to spin filtering should vanish for 𝜑 = 𝜋. In panel (d), +we see the case of different spin filters (𝑃1 < 𝑃2). It is now +possible for one ferromagnet to dominate the spin splitting of +the superconducting density of states, while the other dominates +the spin filtering process. Via the mechanism explored in detail +in [21], this leads to an antisymmetric contribution to Δ𝑉(𝜑), +whereby Δ𝑉(0) and Δ𝑉(𝜋) have opposite signs. In the extreme +case of 𝑃1 ≪ 𝑃2 the result is a purely antisymmetric shape for +Δ𝑉(𝜑), whereas for 𝑃1, 𝑃2 of similar magnitude the theory +predicts |Δ𝑉(𝜋)| ≪ |Δ𝑉(0)|. In panel (e), we show the effect +of adding an out-of-plane interfacial domain wall to panel (d), +which clearly suppresses the antisymmetric contribution. +While the simplest model presented in Fig. 5(c) captures the +essential features of Fig. 2(a), including an interfacial domain +wall in the model enhances the agreement with the experiment. +Specifically, in the absence of externally applied fields, such +a domain wall would produce the results in panel Fig. 5(e), +which also agrees well with Fig. 2(a). However, as the in- +180° +90° +(c) +(a) +(b) +φ +θ +(d) +(e) +270° +90° +270° +0° +90° +270° +SC +V = 0 +T ≈ 0 +FM +FM +Usadel +equation +SC +V = 0 +T ≈ 0 +FM +V = ΔV +T ≈ ΔT +FM +FM +SC +FIG. 5: (a) Numerical model, including field directions (shown +here for the AP configuration). (b) Definitions of angles 𝜃 and 𝜑 +with respect to the field directions in the model. (c–e) Numerical +results for the magnetically dependent thermoelectric voltage Δ𝑉(𝜑). +Blue and red correspond to positive and negative voltages, while +the radius in each plot is |Δ𝑉| = 0.04Δ/𝑒 where 𝑒 is the elementary +charge. The three plots correspond to (c) 𝜃 = 0, 𝑃1 = 𝑃2 = 80%; +(d) 𝜃 = 0, 𝑃1 = 60%, 𝑃2 = 80%; (e) 𝜃 = 𝜋/4, 𝑃1 = 60%, 𝑃2 = 80%. +plane applied field is ramped up, the domain wall should be +rotated into the thin-film plane: 𝜃 → 0. In this case, we would +gradually move towards panel (d), where Δ𝑉(𝜑) changes sign. +This qualitatively agrees with the experimental observations in +Fig. 2d, where it is found that Δ𝑉(𝜋) changes sign for increasing +magnetic saturation while Δ𝑉(0) changes only slightly. +DISCUSSION AND CONCLUSIONS +While our numerical modelling qualitatively explains the +experiments at low magnetic fields where the switching between +the P and AP states takes place (Figs. 2, 3 and 5), it does +not account for the unexpected strong variation of the TE +response in the high field limit, where a double sign change +takes place regardless of the presence of the magnetically +hard layer (Fig. 4). This is because the spin-resolved particle- +hole asymmetry in quasiclassical theory is only present in the +superconducting state. A possible factor which may influence +the high field TE response is a transformation of the interfacial +magnetism at the V/MgO interface [36] under an applied +magnetic field. Initially predicted by numerical simulations +[37], spin fluctuations and/or surface atomic layer magnetism +in V have been under debate for decades now [38–42]. In our +experiments, a sufficiently large in-plane magnetic field could +transform the V/MgO interface into an additional, atomically- +thin magnetic layer. The induced surface magnetism might +strongly affect the exchange splitting of the electron bands in V. + +5 +The explanation of the high-field TE response behavior remains +an intriguing open problem. +In conclusion, we report on the experimental control of +the superconducting thermoelectric effect using a spin-valve +device with a spin filter. We demonstrate both experimentally +and by numerical simulations the transition from a strongly +asymmetric to an anti-symmetric response depending on the +saturation of the AP alignment of the spin-valve, which is likely +modulated by an interfacial domain wall. Furthermore, our +results point towards an unexpected thermoelectric response +in superconductor/ferromagnet junctions under high in-plane +magnetic fields. More detailed experimental and theoretical +studies are required to understand this behaviour. +Acknowledgements +Authors thank Michel Hehn for discussions and help with +samples preparation. +The work in Madrid was supported +by Spanish Ministry of Science and Innovation (PID2021- +124585NB-C32 and TED2021-130196B-C22) and Consejería +de Educación e Investigación de la Comunidad de Madrid +(NANOMAGCOST-CM Ref. +P2018/NMT-4321) Grants. +F.G.A. also acknowledges financial support from the Span- +ish Ministry of Science and Innovation through the María de +Maeztu Programme for Units of Excellence in R&D (CEX2018- +000805-M) and “Acción financiada por la Comunidad de +Madrid en el marco del convenio plurianual con la Universidad +Autónoma de Madrid en Línea 3: Excelencia para el Profeso- +rado Universitario”. The work in Trondheim was supported by +the Research Council of Norway through grant 323766, and its +Centres of Excellence funding scheme grant 262633 “QuSpin”. +J. L. and J. A. O. also acknowledge resources provided by +Sigma2—the National Infrastructure for High Performance +Computing and Data Storage in Norway. C. T. acknowledges +the UEFISCDI project “MODESKY”ID PN-III-P4-ID- 880 +PCE-2020-0230-P, grant No. UEFISCDI: PCE 245/02.11.2021. +†C.G.-R and D.C. contributed equally to the manuscript. +SUPPLEMENTAL MATERIAL +1. Samples and experimental set-up +The superconductor–spin-valve multilayer stacks have been +grown by molecular beam epitaxy (MBE) in a chamber with a +base pressure of 5 × 10−11 mbar while the crystalline quality +was controlled by in-situ RHEED measurements, following +the procedure described in Ref. [43]. The resulting layered +structures were then lithographed into squared samples with +lateral sizes ranging from 30 × 30 to 60 × 60 𝜇m2. +The measurements are performed inside a JANIS He3 cryo- +stat (minimum attainable temperature is 0.3 K). The magnetic +field is varied using a 3D vector magnet consisting of one +solenoid (X axis) with 𝐻max = 3.5 T and two Helmholtz coils +(Y and Z axis) with 𝐻max = 1 T. +2. Experimental procedures for the TE response measurements +In order to produce a temperature gradient in the samples, +a commercial Light Emitting Diode (LED) was placed above +the sample (see Fig.1 of the main text). Voltage was supplied +to the LED (model LUXEON 3030 2D) by a Keithley 228A +voltage source. The LED started to emit light at an applied +bias of 5.6 V at room temperature. As no direct visual contact +could be established inside the cryostat, the thermometer and +voltage source were used to check the LED functioning at low +temperatures: First, the thermometer closer to the samples +showed a steady increase in temperature when the applied +voltage was above 6 V. Second, the compliance indicator +present in the Keithley voltage source showed the same signal +at this voltage than it did at room temperature when the LED +was on. +Once this was established, the TE voltage (Δ𝑉) is measured +as follows: first, the resistance and temperature of the sample +are measured, and then voltage measurements are taken with an +applied current of 𝐼 = 0 nA (using the smallest available current +range in the Keithley 220 current source, which has a maximum +compliance current of 𝐼 = 1.9995 nA and a minimum step +of 500 fA with an accuracy of ±2 pA). Then, the LED is +turned on with the desired voltage (𝑉LED), and a second voltage +measurement is taken at zero current after 2 seconds, before the +LED is turned off again. We checked that the voltage readings +were stable for waiting times between 1 and 10 seconds after +turning on the LED. We chose the 2 seconds because it ensures +the gradient temperature without overheating the sample during +the experiments. Each voltage measurement is taken using the +smallest measuring range of the DMM-552 voltmeter card, and +averaging over 100 voltage readings. +We have carried out three types of thermoelectric measure- +ment experiments. First, the magnetic field rotations that have +been explained in the experimental results section. Secondly, +the TE response of the spin-valve (S/F/F) samples was studied +separately in each magnetic configuration. This was done by +setting the desired magnetic orientation (i.e. parallel (P) or +antiparallel (AP) state) at a fixed temperature (usually below +𝑇𝑐 at 𝑇 = 0.3 K or above 𝑇𝑐 at 9 K) and field magnitude, and +measuring the TE voltage Δ𝑉 for increasing values of the LED +heater voltage 𝑉LED. The sample was let to cool down for a +minute after each heating experimental point in order to avoid +overheating. The whole process was repeated 5 to 10 times +and averaged for each 𝑉LED to reduce the uncertainty. This +was made for different magnetic orientations of the sample, +as shown in Figs. 2 and 3 on the main text, where instead of +𝑉LED, the corresponding estimated temperature difference Δ𝑇 +between the Fe and V electrodes is shown (see details on the +temperature gradient evaluation below). +The third type of experiments study the TE response as a +function of the applied magnetic field direction for fixed values + +6 +6.0 +6.5 +7.0 +7.5 +-4 +-3 +-2 +-1 +0 +1 +∆V (µV) +VLED (V) +T=0.3 K + H=1200 Oe +5.5 +6.0 +6.5 +7.0 +7.5 +-3 +-2 +-1 +0 +1 +2 +∆V (µV) +VLED (V) +T=0.3 K + H=1200 Oe +(a) +(b) +FIG. 6: (a) TE voltage observed in a S/F/F junction vs LED voltage +at 𝑇 = 0.3 K (𝐻 = 1200 Oe). The voltage background level has not +been removed yet. (b) Same TE voltage vs 𝑉LED curve as in panel (a) +after the background voltage subtraction. +of 𝑉LED. In all of the experiments, the low bias resistance +(at about 5 mV) was also measured, allowing for a precise +detection of the magnetization configuration of the free Fe layer. +More details on the electric transport measurements, along +with the samples characterization, can be found in Ref. [44] +3. Background voltage removal +Some of the plotted TE voltage curves such as Figs. 2(b–c), +3(a–c) and 4(a) on the main text were constructed assuming +negligible Δ𝑇 up until 𝑉LED = 6 V, considering that below +this voltage practically no temperature gradient was induced +between the superconductor and ferromagnet (see Fig. 8(d) +inset). Figure 6(a) shows a typical TE voltage curve against the +LED voltage, from 5.5 to 7.3 V. As TE effects seen at voltages +below 𝑉LED = 6 V are not related with the heat produced by the +LED, we assume that they are a consequence of the unavoidable +background temperature gradient in the S/F/F junction due to +the thermalization process with the cryostat cooling system +(𝐻𝑒3 pot). Accordingly, the background voltage was removed +on all of the mentioned plots. Fig. 6 displays an example of +background TE signal removal for one of the curves. +4. MODELLING OF THE TEMPERATURE PROFILE +The thermal response of the system was modelled using +COMSOL [45]. The following section explains the modelling +for a fixed voltage of the LED heater of 7 V as a representative +example, since this was one of the most used voltage values +where the TE response was evaluated. +The LED heater worked at 7 V with an applied current of +0.3 mA. The total dissipated power in this situation is therefore +2.1 mW. The heater is located at a distance of ∼ 1 cm from +the sample, so the power per unit area reaching the surface of +the sample holder is estimated as 6.7 W/m2. The measured +Y +Z +X +(a) +(b) +FIG. 7: COMSOL modelling of the S/F/F system. (a) 3D model of the +junction temperature distribution, with a net heat flux of 6.7 × 10−6 W +entering from the top. (b) Temperature profile along the Z direction. +samples surface varies from 30 × 30 to 60 × 60 𝜇m2 depending +on the sample, but the structure is grown as a pillar and located +under a gold contact with a surface of approx 1 × 1 mm2. We +will first assume that all of the energy that falls onto the pillar is +transmitted into the sample’s top layer. Therefore the total heat +flux entering the sample is taken as 6.7 × 10−6 W. The total +temperature gradient had an approximately linear dependence +with this total flux and therefore applied voltage of the LED +heater. +The thermal conductivity and heat capacity of each of the +materials (Co, Fe, MgO and V) at low temperatures was taken +from tabulated values in the literature [46–49]. The model in +COMSOL assumes that the bottom of the V layer has a fixed +temperature of 1 K and the lateral faces are isolated (since the +sample is in a vacuum chamber in the cryostat). With these +parameters, the temperature difference between the Fe and V +layers is in the range of 70 mK (see Fig. 7). With the observed +TE voltage in the order of ∼ 3 𝜇V, this would make for a +Seebeck coefficient of ∼ 50 𝜇V/K, which is in the same order +of magnitude of previous similar studies [50]. +5. RESCALING 𝑉LED INTO Δ𝑇 +Once the temperature gradient was estimated for a fixed bias +of 𝑉 = 7 V with the COMSOL simulations, the whole Δ𝑉 +vs. 𝑉LED curves that were measured for the different studied +samples had to be recalibrated into Δ𝑉 vs. Δ𝑇. For this, first +we took a look at the technical datasheet of the LED used in +the experiments, from which luminosity vs current (fig. 8(a)) +and current vs. voltage (Fig. 8(b)) curves were obtained. +By composing these two curves we obtained a luminosity vs +voltage curve our LED. As the curves were obtained at room +temperature, the applied voltage axis had to be re-scaled in +order to fit the behavior of the LED at low temperatures: for the +minimum working voltage, we looked at the temperature values +measured by the thermometer during 𝑉LED sweeps, observing +that the LED started heating the sample at an applied bias of +∼ 6 V (vs 5.6 V at room 𝑇). As for the maximum luminosity +bias value, we observe that the Keithley 228A power supply + +7 +200 +150 +100 +50 +05.0 +5.5 +6.0 +6.5 +7.0 +VLED(V) +ILED(mA) +Luminosity(a.u.) +1 +00 +50 +100 +150 +200 +0.5 +ILED(mA) +Luminosity(norm) +VLED(V) +6 +6.4 +6.8 +7.2 +1 +0.5 +0 +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +0 +40 +80 +120 +∆T(mK) +VLED(V) +5.0 +6.0 +7.0 +0.0 +0.5 +1.0 +∆T(norm) +VLED(V) +(a) +(b) +(c) +(d) +250 +1.5 +2.0 +FIG. 8: (a) Intensity vs applied bias curve at room temperature for +the commercial LED that was used during the experiments (model +LUXEON 3030 2D). (b) Luminosity vs current curve, for the same +LED. (c) Normalized luminosity vs applied voltage curve, constructed +by composing the two previous ones and with the voltage range already +rescaled to the LED behavior at low temperatures. (d) Δ𝑇 vs 𝑉LED +constructed from the curve in (c) plus the COMSOL simulation results. +The inset shows the normalized increase in temperature measured +experimentally with the thermometer closest to the sample inside the +cryostat during a 𝑉LED sweep. Note that this does not correspond +to the temperature gradient Δ𝑇 established in the junction, as it is +only a local measure near the samples. Nevertheless, the two curves +qualitatively match, which is a good indicator for the reasonable +validity of the estimated Δ𝑇 gradient under the LED heating. +could not exceed 7.35 V during normal operation when the +LED was connected. With this, we can re-scale the luminosity +vs 𝑉LED curve (Fig. 8(c)). The last step is to assume that the +heating will be directly proportional (i.e. linear dependence) to +the emitted luminosity of the LED, and use 2 points to calibrate +the slope: the base point of 𝑉LED = 6 V at which Δ𝑇 is assumed +to be zero, and the simulated point of 𝑉LED = 7 V for which +an estimation of Δ𝑇 = 150 mK was obtained with COMSOL. +Putting all together, we finally have the Δ𝑇 vs. 𝑉LED curve +(Fig. 8(d)), which was used to transform the Δ𝑉 vs. 𝑉LED +experiments into the presented Δ𝑉 vs Δ𝑇 graphs. +6. THERMOELECTRIC RESPONSE IN A +SHORT-CIRCUITED S/F/F JUNCTION +We carried out control experiments of the TE response +in short-circuited junctions. By operating at relatively high +applied biases (between 1 and 2 V), a pinhole was induced in +the MgO barriers of one of the samples where the TE response +was being studied. The outcome was a close to 103 drop in the +resistance, from kOhms to tens of Ohms. The result is shown +in Fig. 9, revealing an absence of TE generated voltage for +0 +50 +100 +-3 +-2 +-1 +0 +1 +2 +∆T(mK) +∆V (µV) +Short-circuited sample + H = 200 Oe, P state + H = -200 Oe, AP state +0.5 +1 +1.5 +2 +-0.5 +2.5 +3 +0 +0 +20 +40 +60 +80 +100 +H(kOe) +ΔT(mK) +2 +0 +-3 +ΔV(μV) +(a) +(b) +FIG. 9: Thermoelectric response of a short-circuited S/F/F sample. +(a) Thermoelectric voltage for 200 Oe and −200 Oe at 𝑇 = 0.3 K. (b) +3D representation of the TE voltage against the in-plane field and Δ𝑇 +at 𝑇 = 0.3 K. +any value of the LED heating and magnetic field. The results +suggest that the electric-channel was short-circuited, which +was accompanied by a strongly enhanced thermal conductivity +and therefore a reduced temperature gradient. +∗ e-mail: farkhad.aliev@uam.es +[1] A. I. Buzdin, Proximity effects in superconductor-ferromagnet +heterostructures, Rev. Mod. Phys. 77, 935 (2005). +[2] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, +G. Xiao, and A. Gupta, A spin triplet supercurrent through the +half-metallic ferromagnet CrO2, Nature 439, 825 (2006). +[3] J. W. A. Robinson, J. D. S. Witt, M. G. Blamire, Controlled +Injection of Spin-Triplet Supercurrents into a Strong Ferromagnet, +Science 329, 59 (2010) +[4] Trupti S. Khaire, Mazin A. Khasawneh, W. P. Pratt, Jr., and +Norman O. Birge, Observation of Spin-Triplet Superconductivity +in Co-Based Josephson Junctions, Phys. Rev. Lett. 104, 137002 +(2010). +[5] B. M. Niedzielski, T. J. Bertus, J. A. Glick, R. Loloee, W. P. +Pratt, Jr. and N. O. Birge, Spin-valve Josephson junctions for +cryogenic memory, Phys. Rev. B 97, 024517 (2018). +[6] G. Yang, C. Ciccarelli, and J. W. A. Robinson, Boosting spin- +tronics with superconductivity, APL Materials 9, 050703 (2021). +[7] A. E. Madden, J. C. Willard, R. Loloee, and N. O. Birge, +Phase controllable Josephson junctions for cryogenic memory, +Supercond. Sci. Technol. 32, 015001 (2018). +[8] L. Ai, E. Zhang et al. Van der Waals ferromagnetic Josephson + +8 +junctions, Nat Commun 12, 6580 (2021). +[9] F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin and J. +P. Pekola, Opportunities for mesoscopics in thermometry and +refrigeration: Physics and applications, Rev. Mod. Phys. 78, 217 +(2009). +[10] P. Machon, M. Eschrig, and W. Belzig, Nonlocal Thermoelectric +Effects and Nonlocal Onsager relations in a Three-Terminal +Proximity-Coupled Superconductor-Ferromagnet Device, Phys. +Rev. Lett. 110, 047002 (2013). +[11] P. Machon, M. Eschrig, and W. Belzig, Giant thermoelectric ef- +fects in a proximity-coupled superconductor–ferromagnet device, +New J. Phys. 16 073002 (2014). +[12] A. Ozaeta, P. Virtanen, F. S. Bergeret, and T. T. Heikkila, +Predicted very large thermoelectric effect in ferromagnet- +superconductor junctions in the presence of a spin-splitting +magnetic field, Phys. Rev. Lett. 112, 057001 (2014). +[13] M. S. Kalenkov and A. D. Zaikin, Electron-hole imbalance +and large thermoelectric effect in superconducting hybrids with +spin-active interfaces, Phys. Rev. B 90, 134502 (2014). +[14] P. Dutta, A. Saha and A. M. Jayannavar, Thermoelectric prop- +erties of a ferromagnet-superconductor hybrid junction: Role +of interfacial Rashba spin-orbit interaction, Phys. Rev. B 96, +115404 (2017). +[15] T. Savander, S. Tamura, C. Flindt, Y. Tanaka and P. Burset, +Thermoelectric detection of Andreev states in unconventional +superconductors, Phys. Rev. Research 2, 043388 (2020). +[16] S. Kolenda, M. J. Wolf and D. Beckmann, Observation of Ther- +moelectric Currents in High-Field Superconductor-Ferromagnet +Tunnel Junctions, Phys. Rev. Lett. 116, 097001 (2016). +[17] S. Kolenda, P. Machon, D. Beckmann and W. Belzig, Nonlinear +thermoelectric effects in high-field superconductor-ferromagnet +tunnel junctions, J. Nanotechnol. 7, 1579–1585 (2016). +[18] S. Kolenda, C. Surgers, G. Fischer, and D. Beckmann, Thermo- +electric effects in superconductor-ferromagnet tunnel junctions +on europium sulfide, Phys. Rev. B 95, 224505 (2017). +[19] J. Heidrich and D. Beckmann, Nonlocal thermoelectric effects in +high-field superconductor-ferromagnet hybrid structures, Phys. +Rev. B 100, 134501 (2019). +[20] T. T. Heikkila, R. Ojajarvi, I. J. Maasilta, E. Strambini, F. Giazotto +and F. S. Bergeret, Thermoelectric Radiation Detector Based on +Superconductor-Ferromagnet Systems, Phys. Rev. Applied 10, +034053 (2018). +[21] J. A. Ouassou, C. González-Ruano, D. Caso, F. G. Aliev, and +J. Linder, Complete magnetic control over the superconducting +thermoelectric effect, Phys. Rev. B 106, 094514 (2022). +[22] I. Martínez, C. Tiusan, M. Hehn, M. Chshiev and F. G. Aliev, +Symmetry broken spin reorientation transition in epitaxial +MgO/Fe/MgO layers with competing anisotropies, Sci Rep 8, +9463 (2018). +[23] M. S. Gabor, T. Petrisor, Jr., C. Tiusan, M. Hehn, and T. Petrisor, +Magnetic and structural anisotropies of Co2FeAl Heusler alloy +epitaxial thin films, Phys. Rev. B 84, 134413 (20211). +[24] K.D. Usadel, Generalized diffusion equation for superconducting +alloys.Phys. Rev. Lett. 25, 507 (1970). +[25] J. Rammer and H. Smith, Quantum field-theoretical methods in +transport theory of metals. Rev. Mod. Phys. 58, 323. +[26] F.S. Bergeret, M. Silaev, P. Virtanen, T.T. Heikkilä, Colloquium: +Nonequilibrium effects in superconductors with a spin-splitting +field.Rev. Mod. Phys. 90, 041001 (2018). +[27] V. Chandrasekhar, Superconductivity, edited by K. Bennemann +and J. Ketterson (Springer, Berlin, 2008). +[28] W. Belzig, F.K. Wilhelm, C. Bruder, G. Schön, A.D. Zaikin, +Quasiclassical Green’s function approach to mesoscopic super- +conductivity. Superlattices and Microstructures 25, 1251 (1999). +[29] J.A. Ouassou, A. Pal, M. Blamire, M. Eschrig, J. Linder, Triplet +Cooper pairs induced in diffusive s-wave superconductors in- +terfaced with strongly spin-polarized magnetic insulators or +half-metallic ferromagnets. Sci Rep 7, 1932 (2017). +[30] M. Eschrig, A. Cottet, W. Belzig, J. Linder, General boundary +conditions for quasiclassical theory of superconductivity in the +diffusive limit: application to strongly spin-polarized systems. +New J. Phys. 17, 083037 (2015). +[31] A. Cottet, D. Huertas-Hernando, W. Belzig, Y.V. Nazarov, Spin- +dependent boundary conditions for isotropic superconducting +Green’s functions. Phys. Rev. B 80, 184511 (2009). +[32] A. Cottet, Spectroscopy and critical temperature of diffusive +superconducting/ferromagnetic hybrid structures with spin-active +interfaces. Phys. Rev. B 76, 224505 (2007). +[33] S.H. Jacobsen, J.A. Ouassou, J. Linder, Critical temperature and +tunneling spectroscopy of superconductor-ferromagnet hybrids +with intrinsic Rashba-Dresselhaus spin-orbit coupling. Phys. Rev. +B 92, 024510 (2015). +[34] N. Schopohl, Transformation of the Eilenberger equations of +superconductivity to a scalar Riccati equation. Unpublished. +[35] J.A. Ouassou, T.D. Vethaak, J. Linder, Voltage-induced thin- +film superconductivity in high magnetic fields. Phys. Rev. B 98, +144509 (2018). +[36] C. Rau, C. Liu, A. Schmalzbauer, and G. Xing, Ferromagnetic +Order at (100) 𝑝(1×1) Surfaces of Bulk Paramagnetic Vanadium, +Phys. Rev. Lett. 57, 2311 (1986). +[37] D. R. Grempel and S. C. Ying, Spin Fluctuations at the Surface +of Vanadium, Phys. Rev. Lett. 45, 1018 (1980). +[38] T. G. Walker and H. Hopster, Induced magnetic order in ultrathin +vanadium films on Fe(100), Phys. Rev. B. 49, 7687 (1994). +[39] P. Fuchs, K. Totland, and M. Landolt, Induced magnetization in +thin epitaxial V films on Fe (100), Phys. Rev. B. 53, 9123 (1996). +[40] T. Bryk, D. M. Bylander, and L. Kleinman, Magnetism of the +V(001) surface in the generalized gradient approximation, Phys. +Rev. B. 61, R3780(R) (2000). +[41] D. Lacina, J. Yang, and J. L. Erskine, Multilayer relaxation +and search for ferromagnetic order at the (100) surface of bulk +paramagnetic vanadium, Phys. Rev. B. 75, 195423 (2007). +[42] A. Rubio-Ponce, D. Olguín, A. Aguayo and R. de Coss, Sur- +face magnetism in vanadium overlayers on W(100), Journal of +Magnetism and Magnetic Materials 514, 15, 167143 (2020). +[43] C. Tiusan, M. Hehn, F. Montaigne, F. Greullet, S. Andrieu and +A. Schuhl, Spin tunneling phenomena in single crystal magnetic +tunnel junction systems, J. Phys.: Condens. Matter 19, 165201 +(2007). +[44] C. González-Ruano, D. Caso, L. G. Johnsen, C. Tiusan, M. Hehn, +N. Banerjee, J. Linder and Farkhad G. Aliev, Superconductivity +assisted change of the perpendicular magnetic anisotropy in +V/MgO/Fe junctions, Scientific Reports 11, 19041 (2021). +[45] COMSOL Multiphysics® www.comsol.com. COMSOL AB, +Stockholm, Sweden. +[46] M. Dixon, F. E. Hoare, T. M. Holden and D. E. Moody, The Low +Temperature Specific Heats of Some Pure Metals (Cu, Ag, Pt, Al, +Ni Fe Co), Proceedings of the Royal Society of London. Series A, +Mathematical and Physical Sciences 285, 1403, 561-580 (1965). +[47] J.W. Gardner and A. C. Anderson, Effect of neutron irradiation +on the low-temperature specific heat and thermal conductivity +of magnesium oxide, Phys. Rev. B 23 415 (1981). +[48] Jung, Wayne Douglas, The thermal conductivity of high purity +vanadium, Retrospective Theses and Dissertations. Paper 5485 +(1975). +[49] R. Radebaugh and P. H. Keesom, Low-Temperature Thermody- +namic Properties of Vanadium. Phys. Rev. 149, 19 (1966). +[50] M. Walter, J. Walowski, V. Zbarsky, M. Munzenberg, M. Schafers, + +9 +D. Ebke, G. Reiss, A. Thomas, P. Peretzki, M. Seibt, J. S. +Moodera, M. Czerner, M. Bachmann and C. Heiliger, Seebeck +effect in magnetic tunnel junctions, Nature Materials 10, 742–746 +(2011). + diff --git a/s9E1T4oBgHgl3EQfjgRh/content/tmp_files/load_file.txt b/s9E1T4oBgHgl3EQfjgRh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b6e1e40514960d024f936ef9e4920784f9990f6 --- /dev/null +++ b/s9E1T4oBgHgl3EQfjgRh/content/tmp_files/load_file.txt @@ -0,0 +1,833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf,len=832 +page_content='Observation of magnetic state dependent thermoelectricity in superconducting spin valves César González-Ruano†,1 Diego Caso†,1 Jabir Ali Ouassou,2 Coriolan Tiusan,3, 4 Yuan Lu,4 Jacob Linder,2 and Farkhad G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Aliev5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' ∗ 1Departamento Física de la Materia Condensada C-III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Universidad Autónoma de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Madrid 28049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Spain 2Center for Quantum Spintronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Norwegian University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' NO-M7T9Q Trondheim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Norway 3Department of Solid State Physics and Advanced Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Babes-Bolyai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Cluj Napoca 400114,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Romania 4Institut Jean Lamour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Nancy Universitè,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 54506 Vandoeuvre-les-Nancy Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' France 5Departamento Física de la Materia Condensada C-III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Instituto Nicolás Cabrera (INC) and Condensed Matter Physics Institute (IFIMAC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Universidad Autónoma de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Madrid 28049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Spain Superconductor-ferromagnet tunnel junctions demonstrate giant thermoelectric effects which are being exploited to engineer ultra-sensitive terahertz radiation detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Here, we experimentally observe the recently predicted complete magnetic control over thermoelectric effects in a superconducting spin valve, including the dependence of its sign on the magnetic state of the spin valve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The description of the experimental results is improved by the introduction of an interfacial domain wall in the spin filter layer interfacing the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Surprisingly, the application of high in-plane magnetic fields induces a double sign inversion of the thermoelectric effect, which exhibits large values even at applied fields twice the superconducting critical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' INTRODUCTION The competition between superconducting (S) and ferromag- netic (F) ground states can under certain conditions result in a synergy of these otherwise antagonistic states [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In recent years, a variety of exotic phenomena have been demonstrated in devices that exploit this synergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Notable examples include long-ranged spin-triplet supercurrents [2–4], spin-valve Joseph- son junctions [5], superconducting spin-valves with record-high magnetoresistance, and the giant thermoelectric (TE) effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' These effects are considered as potential ingredients in the next generation of low-dissipation cryogenic devices [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In general, there exists considerable interest in identifying material platforms for improved TE devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' At low tem- peratures, TE effects are expected to be vanishingly small in both normal metals and bulk superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Instead, they have been investigated mainly in superconductor/normal-metal hybrids, where they have been used in micro-refrigeration and thermometry [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' More recently, fascinating theoretical predictions [10–15] have opened the door to unexplored spin dependent TE effects in S/F hybrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The transport of spin and charge due to temperature gradients in such systems have only been investigated experimentally in a few works [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Kolenda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [16] reported on the experimental observation of an enhanced Seebeck coefficient (up to 100 µV/K) when a large magnetic field of about 1 T splits the quasiparticle band struc- ture of a superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' When combined with a spin-filtering magnetic interface, this spin splitting breaks the electron–hole symmetry of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The resulting carrier asymmetry produces the observed “giant TE effect”, which is now being exploited to develop ultrasensitive radiation detectors [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' So far, the experimental tuning of giant TE effects in S/F hybrids has been carried out either by applying large magnetic fields [16] or by exchange coupling a superconductor to a ferromagnetic insulator [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Recently, however, a different method to control the TE effect has been predicted in super- conductor/ferromagnet/ferromagnetic insulator systems [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This method can turn the superconducting TE effect on and off in situ, as well as reversing its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Here, by interfacing a superconductor with a spin valve with large spin filtering capability, we experimentally demonstrate the above mentioned complete magnetic control of the superconducting TE effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This includes evidence of an antisymmetric TE effect, where a change of the magnetic state of the spin valve inverts the direction of the TE current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Controlling the sign of the ther- mopower, analogous to the inversion of TE signals between 𝑝- and 𝑛-doped semiconductors, enables the design of Peltier elements based on superconducting spin valves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' EXPERIMENTAL RESULTS Figure 1(a) illustrates the main experimental set-up and junctions investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We have measured TE effects in V(40)/MgO(2)/Fe(10)/MgO(2)/Fe(10)/Co(30) single crys- talline junctions epitaxially grown on MgO(001) substrates, with the thickness of each layer given in nanometers in paren- theses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' More details about the junctions, experimental set-up and procedures can be found in the Supplemental Material sections 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Here, V is a BCS superconductor, Fe and Co are ferromagnetic metals, and MgO is a symmetry filter- ing insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The magnetically hard Fe/Co electrode allows for a precise detection of the orientation of the magnetically free Fe layer interfacing the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Figure 1(b)-(c) presents a general electron transport characterization of the junctions in the superconducting state as a function of the applied bias and external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' One observes that the superconducting gap in the V electrode is suppressed by in-plane (IP) and out-of-plane (OOP) fields of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='7 T and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='4 T, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Figure 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' As we will discuss fur- Typeset by REVTEX arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='03263v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='supr-con] 9 Jan 2023 2 3 2 1 0 1 2 3 6 8 10 12 14 16 G (µS) V(mV) H=0 Oe H=10 kOe H=25 kOe 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 Sgap (norm) H (kOe) HIP HOOP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0 10 20 30 40 50 TMR(%) H(kOe) Fe (10 nm) Fe (10 nm) Co (20 nm) V MgO LED +- VLED V ∇T (a) (b) (c) Hc(OOP) Hc(IP) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K, HIP (d) 1 2 3 4 5 6 7 8 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K Heat flux Y Z X FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 1: (a) Sketch of the S/F/F junctions when heated by a LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) Typical conductance–bias curves measured at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='07𝑇𝑐 at three different applied IP magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (c) Normalized superconducting gap depth (Sgap) taken from 𝐺(𝑉) curves vs applied IP and OOP magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (d) Typical tunneling magnetoresistance curves measured with IP magnetic field at 𝑉 = 5 mV and 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The numbers indicate the order of the field sweeping and resistance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' ther below, the magnetoresistance values of 35-55% provide an estimation of the effective spin polarization of the Fe electrodes of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='75–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='85 for the different junctions studied based on the Julliere model, involving two ferromagnetic and one nonmagnetic (V) electrodes, in line with previous reports [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' To study the TE effect as a function of the spin-valve state, the soft Fe layer was rotated while the hard Fe/Co layer remained fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We did this by applying a rotating in-plane magnetic field, with a magnitude between the coercive fields of the two magnetic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' For the hard layer the coercive field is typically larger than 500 Oe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 1(d)), while for the soft layer it’s smaller than 50 Oe, determined by the magneto-crystalline anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This procedure guarantees a reorientation between the parallel (P), antiparallel (AP) and perpendicular in-plane (PIP) configurations of the spin valve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' For each 3◦ rotation of the applied field, the temperature gradient ∇𝑇 was re-established via the LED heater, and the resulting TE response Δ𝑉 was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(a) shows the variation of the TE voltage generated during a magnetization rotation of the free layer under an in-plane applied magnetic field of 70 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' While the transition between the P and PIP states hardly affects the values of the TE response, a strong reduction of the TE voltage of more than a factor of 2 is observed when the free layer becomes close to AP to the fixed Fe/Co layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Note that there is a slight latency of about 10–15◦ between the angle of the applied magnetic field and the real average magnetization orientation of the 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='60 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='65 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='70 40 20 0 20 ∆V/∆T (µV/K) RAP(kΩ) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='89 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='90 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='91 RP(kΩ) 0 90 180 270 2 4 ∆V (µV) 0 20 40 60 80 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 ∆V (µV) ∆T (mK) 0 20 40 60 80 100 1 0 1 2 3 4 ∆V (µV) ∆T (mK) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K, P State H=70 Oe H=100 Oe H=150 Oe H=200 Oe H=300 Oe (a) (b) (c) (d) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K, AP State H=-70 Oe H=-100 Oe H=-150 Oe H=-200 Oe H=-250 Oe H=-300 Oe AP P 15 30 45 TMR (%) H = 70 Oe T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K T=9 K TMR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2: Thermoelectric response of a S/F/F junction measured under a rotation and fixed values of the applied magnetic field at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (a) TE response at 𝐻 = 70 Oe for in-plane rotations of the magnetic field at 𝑉LED = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 V (Δ𝑇 ≈ 113 mK), below and above 𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The tunnel magnetoresistance of the spin-valve is also displayed against the rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) Response in the P configuration of the spin-valve stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (c) Same experiment for the AP configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The TE voltage changes its sign and intensity depending not directly on the applied field, but on the saturation of the soft FM layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The related analysis is shown in panel (d), where the average value of the TE voltage is plotted against the measured resistance for the P (blue, upper horizontal axis) and AP (red, lower horizontal axis) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' For the P state, a lower resistance implies a better polarization, while in the AP state the polarization is better with a higher resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fe layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This is a natural consequence of the experimental process: the Fe magnetization has to overcome the magneto- crystalline anisotropy to follow the slowly rotating applied field (see for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The complete 360◦ rotation takes about two hours, since we stop at each intermediate angle to measure the TE response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Figures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(b) and (c) show the TE response of the samples vs the induced temperature gradient for different magnetic configurations and applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Note that these temperature gradients have been estimated by first simulating the response to the incoming heat flux and subsequently recalculating Δ𝑇 from 𝑉LED based on the LED calibration curves (see Supplemental Material sections 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In the P state, changing the applied field by less than a few hundred Oe does not qualitatively change the TE response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' However, in the AP state, varying the magnetic field has a dramatic effect on the TE response and can even change its sign as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We note that no asymmetry of Δ𝑉 or dependence on𝑉LED was observed above𝑇𝑐 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Control experiments on short-circuited junctions also revealed at least an order of magnitude drop of the TE response regardless of 3 0 50 100 4 2 0 2 ∆V(µV) ∆T(mK) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K, P state H=50 Oe H=125 Oe H=1200 Oe T=9 K H=100 Oe H (kOe) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 1 ΔV(μV) ΔT(mK) 20 60 100 P AP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 0 3 (a) (c) (b) 0 50 100 3 2 1 0 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K, AP state H=-60 Oe H=-200 Oe H=-600 Oe T = 9 K H=-200 Oe ∆V(µV) ∆T(mK) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='8 P 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 3: Thermoelectric voltage of an S/F/F junction demonstrating sign inversion at high fields in the P state (a) and AP state (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (c) Colormap of the recorded TE voltage Δ𝑉 as a function of the temperature difference Δ𝑇 and the applied in-plane field 𝐻 for the same junction, indicating the P and AP states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' the magnetic state (see Supplemental Material section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In order to understand the possible reasons for the TE sign change in the AP state, we analyzed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(d)) the TE response obtained for a fixed temperature gradient as a function of the resistance in the P and AP states obtained during each particular TE experiment at different applied in-plane magnetic fields (not exceeding 500 Oe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' While the TE response in the P state is rather robust to the variation of the resistance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' presence of magnetic inhomogeneities), in the AP state it changes sign with the reduction of the influence of these magnetic textures (an increase of the resistance means better magnetization saturation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Interestingly, some junctions revealed a TE sign inversion both in the AP state and also under a sufficiently high applied in-plane magnetic field in the P state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The TE response in the P state is positive and robust at fields below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 kOe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 3(a)), and becomes negative for higher magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In contrast, in the AP state the TE response is already negative for much smaller fields (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 3(b)), right after the spin-valve switched into the AP state (see steps 1-2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Further increasing the negative magnetic field (path 2-3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 1(d)) reorients the hard layer so the spin-valve is again in the P configuration, and again the sign of the TE effect follows the same trend as for positive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Figure 3(c) summarizes these observations with a 3D color plot of the TE voltage signal against the applied field and evaluated temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We have found that the TE voltage sign inversion in the P state under high in-plane magnetic fields is a rather robust effect and is followed by a second TE sign inversion towards 35 40 45 50 55 0 10 20 30 TE inversion field (kOe) TMRMAX(%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='84 P 0 2 4 0 5 10 15 ∆V(µV) H(kOe) (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='1 1 10 10 0 10 S/F/F S/F ∆V(µV) H(kOe) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 4: (a) Thermoelectric voltage of an S/F/F and S/F junction at 𝑉LED = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 V (Δ𝑇 ≈ 113 mK) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' applied in-plane field at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The high-field sign inversion is achieved in both samples, and the TE effect is maintained with increasing in-plane fields up to 30 kOe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The inset displays the measured TE voltage in the S/F/F junction under out-of-plane field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' At 𝐻 = 𝐻𝑐, superconductivity and its associated TE voltage vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) TE inversion field and polarization against the maximum TMR value for all the S/F/F junctions under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' positive values when the magnetic field is further increased (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Surprisingly, even for maximum applied in-plane magnetic fields, twice exceeding the second critical magnetic field (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 1(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 4(a)), the TE voltage remains high and without clear signatures of diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A quali- tatively similar response has been observed in single barrier V/MgO/Fe junctions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' without the sensing Fe/Co layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 4(b) compares the TE signal inversion field with the TMR and effective spin polarization values of each corresponding sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Apparently, a higher TMR and correspondingly polarization values shift the TE inversion field range outside our experimen- tal capabilities (35 kOe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This suggests a possible link between the observed effect and interfacial domain wall forming in the Fe electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Further experiments are needed to understand the physical mechanism behind the high field TE effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' THEORETICAL MODELLING To better understand the physics behind the experimental observations, we explored the setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 1(a) via numeri- cal simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We employed the Usadel formalism [24–28] which describes superconductivity in diffusive heterostructures, together with spin-dependent tunneling boundary conditions [10, 29–32] valid for arbitrary spin polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' To numeri- cally solve these equations, we used the Ricatti parametrization [33, 34] to calculate spectral properties and a distribution-trace parametrization [26–28, 35] to calculate the nonequilibrium transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The theoretical and numerical approach is described in more detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The numerical model used herein is sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The superconductor (V) was treated as a BCS superconducting reservoir with an effective spin-splitting ℎ = Δ/10, near-zero temperature 𝑇 = 𝑇𝑐/100, and electrical grounding 𝑉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The hard ferromagnet (Fe/Co) was treated as a non-superconducting metallic reservoir at an elevated temperature 𝑇 = 𝑇𝑐/2 and volt- 4 age 𝑉 = Δ𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The interfaces to the soft ferromagnet (V/MgO/Fe and Fe/MgO/Fe/Co) were treated using spin-polarized tun- neling boundary conditions with spin polarizations 𝑃1, 𝑃2 and a low tunneling conductance 𝐺𝑇 = 𝐺𝐷/5 where 𝐺𝐷 is the Drude conductance of the soft ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' These pa- rameters model the high spin filtering capabilities and low transparencies of the MgO barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In the soft ferromagnet (Fe), we used an exchange splitting ℎ = 30Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' For each mag- netic configuration, we (i) solved the Usadel equation for 80 different 𝑒Δ𝑉/Δ ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='04, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='04];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (ii) used this to calculate the current 𝐼(Δ𝑉);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (iii) interpolated the open-circuit voltage from 𝐼(Δ𝑉) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This yielded the TE voltage as function of magnetic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The magnetic configurations we considered are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5(a–b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We take the hard ferromagnet (red) to be oriented along one in-plane axis (up), and the spin filtering at the MgO barrier is assumed to be parallel to this orientation (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The soft ferromagnet is then rotated by an in-plane angle 𝜑 relative to the hard ferromagnet (purple), which here is sketched for the antiparallel case 𝜑 = 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' At the superconductor interface, we include the possibility for an interfacial domain wall described by an out-of-plane angle 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This affects both the spin filtering at the second MgO barrier (black) and the direction of spin splitting inside the neighboring superconductor (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5(c–e) show the numerical results for the TE voltage Δ𝑉(𝜑) across the junction as function of the in-plane misalignment angle between the two ferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In panel (c), we see the predicted response for a junction with two identical spin filters (𝑃1 = 𝑃2) and no interfacial domain wall (𝜃 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We then predict an asymmetric TE effect, by which we mean that Δ𝑉(0) is maximal while Δ𝑉(𝜋) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This can intuitively be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' If the two spin filters are identical (𝑃1 = 𝑃2 and 𝜑 = 0), then “filtering the spins twice” does not significantly change the physics compared to having only one spin filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The latter case has in previous work been shown to produce a giant TE effect in superconductor/ferromagnet systems [10–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' On the other hand, the effects of two identical but oppositely aligned spin filters cancel, so any TE effect due to spin filtering should vanish for 𝜑 = 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In panel (d), we see the case of different spin filters (𝑃1 < 𝑃2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' It is now possible for one ferromagnet to dominate the spin splitting of the superconducting density of states, while the other dominates the spin filtering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Via the mechanism explored in detail in [21], this leads to an antisymmetric contribution to Δ𝑉(𝜑), whereby Δ𝑉(0) and Δ𝑉(𝜋) have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In the extreme case of 𝑃1 ≪ 𝑃2 the result is a purely antisymmetric shape for Δ𝑉(𝜑), whereas for 𝑃1, 𝑃2 of similar magnitude the theory predicts |Δ𝑉(𝜋)| ≪ |Δ𝑉(0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In panel (e), we show the effect of adding an out-of-plane interfacial domain wall to panel (d), which clearly suppresses the antisymmetric contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' While the simplest model presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5(c) captures the essential features of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(a), including an interfacial domain wall in the model enhances the agreement with the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Specifically, in the absence of externally applied fields, such a domain wall would produce the results in panel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5(e), which also agrees well with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' However, as the in- 180° 90° (c) (a) (b) φ θ (d) (e) 270° 90° 270° 0° 90° 270° SC V = 0 T ≈ 0 FM FM Usadel equation SC V = 0 T ≈ 0 FM V = ΔV T ≈ ΔT FM FM SC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5: (a) Numerical model, including field directions (shown here for the AP configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) Definitions of angles 𝜃 and 𝜑 with respect to the field directions in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (c–e) Numerical results for the magnetically dependent thermoelectric voltage Δ𝑉(𝜑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Blue and red correspond to positive and negative voltages, while the radius in each plot is |Δ𝑉| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='04Δ/𝑒 where 𝑒 is the elementary charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The three plots correspond to (c) 𝜃 = 0, 𝑃1 = 𝑃2 = 80%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (d) 𝜃 = 0, 𝑃1 = 60%, 𝑃2 = 80%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (e) 𝜃 = 𝜋/4, 𝑃1 = 60%, 𝑃2 = 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' plane applied field is ramped up, the domain wall should be rotated into the thin-film plane: 𝜃 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In this case, we would gradually move towards panel (d), where Δ𝑉(𝜑) changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This qualitatively agrees with the experimental observations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2d, where it is found that Δ𝑉(𝜋) changes sign for increasing magnetic saturation while Δ𝑉(0) changes only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS While our numerical modelling qualitatively explains the experiments at low magnetic fields where the switching between the P and AP states takes place (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2, 3 and 5), it does not account for the unexpected strong variation of the TE response in the high field limit, where a double sign change takes place regardless of the presence of the magnetically hard layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This is because the spin-resolved particle- hole asymmetry in quasiclassical theory is only present in the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A possible factor which may influence the high field TE response is a transformation of the interfacial magnetism at the V/MgO interface [36] under an applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Initially predicted by numerical simulations [37], spin fluctuations and/or surface atomic layer magnetism in V have been under debate for decades now [38–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In our experiments, a sufficiently large in-plane magnetic field could transform the V/MgO interface into an additional, atomically- thin magnetic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The induced surface magnetism might strongly affect the exchange splitting of the electron bands in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5 The explanation of the high-field TE response behavior remains an intriguing open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In conclusion, we report on the experimental control of the superconducting thermoelectric effect using a spin-valve device with a spin filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We demonstrate both experimentally and by numerical simulations the transition from a strongly asymmetric to an anti-symmetric response depending on the saturation of the AP alignment of the spin-valve, which is likely modulated by an interfacial domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Furthermore, our results point towards an unexpected thermoelectric response in superconductor/ferromagnet junctions under high in-plane magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' More detailed experimental and theoretical studies are required to understand this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Acknowledgements Authors thank Michel Hehn for discussions and help with samples preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The work in Madrid was supported by Spanish Ministry of Science and Innovation (PID2021- 124585NB-C32 and TED2021-130196B-C22) and Consejería de Educación e Investigación de la Comunidad de Madrid (NANOMAGCOST-CM Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' P2018/NMT-4321) Grants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' also acknowledges financial support from the Span- ish Ministry of Science and Innovation through the María de Maeztu Programme for Units of Excellence in R&D (CEX2018- 000805-M) and “Acción financiada por la Comunidad de Madrid en el marco del convenio plurianual con la Universidad Autónoma de Madrid en Línea 3: Excelencia para el Profeso- rado Universitario”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The work in Trondheim was supported by the Research Council of Norway through grant 323766, and its Centres of Excellence funding scheme grant 262633 “QuSpin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' also acknowledge resources provided by Sigma2—the National Infrastructure for High Performance Computing and Data Storage in Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' acknowledges the UEFISCDI project “MODESKY”ID PN-III-P4-ID- 880 PCE-2020-0230-P, grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' UEFISCDI: PCE 245/02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' †C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='-R and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' contributed equally to the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' SUPPLEMENTAL MATERIAL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Samples and experimental set-up The superconductor–spin-valve multilayer stacks have been grown by molecular beam epitaxy (MBE) in a chamber with a base pressure of 5 × 10−11 mbar while the crystalline quality was controlled by in-situ RHEED measurements, following the procedure described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The resulting layered structures were then lithographed into squared samples with lateral sizes ranging from 30 × 30 to 60 × 60 𝜇m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The measurements are performed inside a JANIS He3 cryo- stat (minimum attainable temperature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The magnetic field is varied using a 3D vector magnet consisting of one solenoid (X axis) with 𝐻max = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 T and two Helmholtz coils (Y and Z axis) with 𝐻max = 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Experimental procedures for the TE response measurements In order to produce a temperature gradient in the samples, a commercial Light Emitting Diode (LED) was placed above the sample (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='1 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Voltage was supplied to the LED (model LUXEON 3030 2D) by a Keithley 228A voltage source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The LED started to emit light at an applied bias of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='6 V at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' As no direct visual contact could be established inside the cryostat, the thermometer and voltage source were used to check the LED functioning at low temperatures: First, the thermometer closer to the samples showed a steady increase in temperature when the applied voltage was above 6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Second, the compliance indicator present in the Keithley voltage source showed the same signal at this voltage than it did at room temperature when the LED was on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Once this was established, the TE voltage (Δ𝑉) is measured as follows: first, the resistance and temperature of the sample are measured, and then voltage measurements are taken with an applied current of 𝐼 = 0 nA (using the smallest available current range in the Keithley 220 current source, which has a maximum compliance current of 𝐼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='9995 nA and a minimum step of 500 fA with an accuracy of ±2 pA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Then, the LED is turned on with the desired voltage (𝑉LED), and a second voltage measurement is taken at zero current after 2 seconds, before the LED is turned off again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We checked that the voltage readings were stable for waiting times between 1 and 10 seconds after turning on the LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We chose the 2 seconds because it ensures the gradient temperature without overheating the sample during the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Each voltage measurement is taken using the smallest measuring range of the DMM-552 voltmeter card, and averaging over 100 voltage readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We have carried out three types of thermoelectric measure- ment experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' First, the magnetic field rotations that have been explained in the experimental results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Secondly, the TE response of the spin-valve (S/F/F) samples was studied separately in each magnetic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This was done by setting the desired magnetic orientation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' parallel (P) or antiparallel (AP) state) at a fixed temperature (usually below 𝑇𝑐 at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K or above 𝑇𝑐 at 9 K) and field magnitude, and measuring the TE voltage Δ𝑉 for increasing values of the LED heater voltage 𝑉LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The sample was let to cool down for a minute after each heating experimental point in order to avoid overheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The whole process was repeated 5 to 10 times and averaged for each 𝑉LED to reduce the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' This was made for different magnetic orientations of the sample, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2 and 3 on the main text, where instead of 𝑉LED, the corresponding estimated temperature difference Δ𝑇 between the Fe and V electrodes is shown (see details on the temperature gradient evaluation below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The third type of experiments study the TE response as a function of the applied magnetic field direction for fixed values 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 4 3 2 1 0 1 ∆V (µV) VLED (V) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K H=1200 Oe 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 3 2 1 0 1 2 ∆V (µV) VLED (V) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K H=1200 Oe (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 6: (a) TE voltage observed in a S/F/F junction vs LED voltage at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K (𝐻 = 1200 Oe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The voltage background level has not been removed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) Same TE voltage vs 𝑉LED curve as in panel (a) after the background voltage subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' of 𝑉LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' In all of the experiments, the low bias resistance (at about 5 mV) was also measured, allowing for a precise detection of the magnetization configuration of the free Fe layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' More details on the electric transport measurements, along with the samples characterization, can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [44] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Background voltage removal Some of the plotted TE voltage curves such as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 2(b–c), 3(a–c) and 4(a) on the main text were constructed assuming negligible Δ𝑇 up until 𝑉LED = 6 V, considering that below this voltage practically no temperature gradient was induced between the superconductor and ferromagnet (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 8(d) inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Figure 6(a) shows a typical TE voltage curve against the LED voltage, from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' As TE effects seen at voltages below 𝑉LED = 6 V are not related with the heat produced by the LED, we assume that they are a consequence of the unavoidable background temperature gradient in the S/F/F junction due to the thermalization process with the cryostat cooling system (𝐻𝑒3 pot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Accordingly, the background voltage was removed on all of the mentioned plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 6 displays an example of background TE signal removal for one of the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' MODELLING OF THE TEMPERATURE PROFILE The thermal response of the system was modelled using COMSOL [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The following section explains the modelling for a fixed voltage of the LED heater of 7 V as a representative example, since this was one of the most used voltage values where the TE response was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The LED heater worked at 7 V with an applied current of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The total dissipated power in this situation is therefore 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='1 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The heater is located at a distance of ∼ 1 cm from the sample, so the power per unit area reaching the surface of the sample holder is estimated as 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='7 W/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The measured Y Z X (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 7: COMSOL modelling of the S/F/F system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (a) 3D model of the junction temperature distribution, with a net heat flux of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='7 × 10−6 W entering from the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) Temperature profile along the Z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' samples surface varies from 30 × 30 to 60 × 60 𝜇m2 depending on the sample, but the structure is grown as a pillar and located under a gold contact with a surface of approx 1 × 1 mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' We will first assume that all of the energy that falls onto the pillar is transmitted into the sample’s top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Therefore the total heat flux entering the sample is taken as 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='7 × 10−6 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The total temperature gradient had an approximately linear dependence with this total flux and therefore applied voltage of the LED heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The thermal conductivity and heat capacity of each of the materials (Co, Fe, MgO and V) at low temperatures was taken from tabulated values in the literature [46–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The model in COMSOL assumes that the bottom of the V layer has a fixed temperature of 1 K and the lateral faces are isolated (since the sample is in a vacuum chamber in the cryostat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' With these parameters, the temperature difference between the Fe and V layers is in the range of 70 mK (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' With the observed TE voltage in the order of ∼ 3 𝜇V, this would make for a Seebeck coefficient of ∼ 50 𝜇V/K, which is in the same order of magnitude of previous similar studies [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' RESCALING 𝑉LED INTO Δ𝑇 Once the temperature gradient was estimated for a fixed bias of 𝑉 = 7 V with the COMSOL simulations, the whole Δ𝑉 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 𝑉LED curves that were measured for the different studied samples had to be recalibrated into Δ𝑉 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Δ𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' For this, first we took a look at the technical datasheet of the LED used in the experiments, from which luminosity vs current (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 8(a)) and current vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' voltage (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 8(b)) curves were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' By composing these two curves we obtained a luminosity vs voltage curve our LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' As the curves were obtained at room temperature, the applied voltage axis had to be re-scaled in order to fit the behavior of the LED at low temperatures: for the minimum working voltage, we looked at the temperature values measured by the thermometer during 𝑉LED sweeps, observing that the LED started heating the sample at an applied bias of ∼ 6 V (vs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='6 V at room 𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' As for the maximum luminosity bias value, we observe that the Keithley 228A power supply 7 200 150 100 50 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 VLED(V) ILED(mA) Luminosity(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=') 1 00 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 ILED(mA) Luminosity(norm) VLED(V) 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 0 40 80 120 ∆T(mK) VLED(V) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 ∆T(norm) VLED(V) (a) (b) (c) (d) 250 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 8: (a) Intensity vs applied bias curve at room temperature for the commercial LED that was used during the experiments (model LUXEON 3030 2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) Luminosity vs current curve, for the same LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (c) Normalized luminosity vs applied voltage curve, constructed by composing the two previous ones and with the voltage range already rescaled to the LED behavior at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (d) Δ𝑇 vs 𝑉LED constructed from the curve in (c) plus the COMSOL simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The inset shows the normalized increase in temperature measured experimentally with the thermometer closest to the sample inside the cryostat during a 𝑉LED sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Note that this does not correspond to the temperature gradient Δ𝑇 established in the junction, as it is only a local measure near the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Nevertheless, the two curves qualitatively match, which is a good indicator for the reasonable validity of the estimated Δ𝑇 gradient under the LED heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' could not exceed 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='35 V during normal operation when the LED was connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' With this, we can re-scale the luminosity vs 𝑉LED curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 8(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The last step is to assume that the heating will be directly proportional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' linear dependence) to the emitted luminosity of the LED, and use 2 points to calibrate the slope: the base point of 𝑉LED = 6 V at which Δ𝑇 is assumed to be zero, and the simulated point of 𝑉LED = 7 V for which an estimation of Δ𝑇 = 150 mK was obtained with COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Putting all together, we finally have the Δ𝑇 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 𝑉LED curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 8(d)), which was used to transform the Δ𝑉 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 𝑉LED experiments into the presented Δ𝑉 vs Δ𝑇 graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' THERMOELECTRIC RESPONSE IN A SHORT-CIRCUITED S/F/F JUNCTION We carried out control experiments of the TE response in short-circuited junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' By operating at relatively high applied biases (between 1 and 2 V), a pinhole was induced in the MgO barriers of one of the samples where the TE response was being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The outcome was a close to 103 drop in the resistance, from kOhms to tens of Ohms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 9, revealing an absence of TE generated voltage for 0 50 100 3 2 1 0 1 2 ∆T(mK) ∆V (µV) Short-circuited sample H = 200 Oe, P state H = -200 Oe, AP state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='5 3 0 0 20 40 60 80 100 H(kOe) ΔT(mK) 2 0 3 ΔV(μV) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 9: Thermoelectric response of a short-circuited S/F/F sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (a) Thermoelectric voltage for 200 Oe and −200 Oe at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' (b) 3D representation of the TE voltage against the in-plane field and Δ𝑇 at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' any value of the LED heating and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' The results suggest that the electric-channel was short-circuited, which was accompanied by a strongly enhanced thermal conductivity and therefore a reduced temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' ∗ e-mail: farkhad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='aliev@uam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='es [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Buzdin, Proximity effects in superconductor-ferromagnet heterostructures, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 77, 935 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Keizer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Goennenwein, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Klapwijk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Miao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Xiao, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Gupta, A spin triplet supercurrent through the half-metallic ferromagnet CrO2, Nature 439, 825 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Witt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Blamire, Controlled Injection of Spin-Triplet Supercurrents into a Strong Ferromagnet, Science 329, 59 (2010) [4] Trupti S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Khaire, Mazin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Khasawneh, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Pratt, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=', and Norman O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Birge, Observation of Spin-Triplet Superconductivity in Co-Based Josephson Junctions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 104, 137002 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Niedzielski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bertus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Glick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Loloee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Pratt, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Birge, Spin-valve Josephson junctions for cryogenic memory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 97, 024517 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ciccarelli, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Robinson, Boosting spin- tronics with superconductivity, APL Materials 9, 050703 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Madden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Willard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Loloee, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Birge, Phase controllable Josephson junctions for cryogenic memory, Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 32, 015001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ai, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Van der Waals ferromagnetic Josephson 8 junctions, Nat Commun 12, 6580 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Giazotto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Heikkilä, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Luukanen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Savin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Pekola, Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 78, 217 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Machon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Eschrig, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Belzig, Nonlocal Thermoelectric Effects and Nonlocal Onsager relations in a Three-Terminal Proximity-Coupled Superconductor-Ferromagnet Device, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 110, 047002 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Machon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Eschrig, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Belzig, Giant thermoelectric ef- fects in a proximity-coupled superconductor–ferromagnet device, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 16 073002 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ozaeta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Virtanen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bergeret, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Heikkila, Predicted very large thermoelectric effect in ferromagnet- superconductor junctions in the presence of a spin-splitting magnetic field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 112, 057001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Kalenkov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Zaikin, Electron-hole imbalance and large thermoelectric effect in superconducting hybrids with spin-active interfaces, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 90, 134502 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Dutta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Saha and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Jayannavar, Thermoelectric prop- erties of a ferromagnet-superconductor hybrid junction: Role of interfacial Rashba spin-orbit interaction, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 96, 115404 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Savander, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Tamura, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Flindt, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Tanaka and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Burset, Thermoelectric detection of Andreev states in unconventional superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Research 2, 043388 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Kolenda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Wolf and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Beckmann, Observation of Ther- moelectric Currents in High-Field Superconductor-Ferromagnet Tunnel Junctions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 116, 097001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Kolenda, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Machon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Beckmann and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Belzig, Nonlinear thermoelectric effects in high-field superconductor-ferromagnet tunnel junctions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 7, 1579–1585 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Kolenda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Surgers, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fischer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Beckmann, Thermo- electric effects in superconductor-ferromagnet tunnel junctions on europium sulfide, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 95, 224505 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Heidrich and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Beckmann, Nonlocal thermoelectric effects in high-field superconductor-ferromagnet hybrid structures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 100, 134501 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Heikkila, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ojajarvi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Maasilta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Strambini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Giazotto and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bergeret, Thermoelectric Radiation Detector Based on Superconductor-Ferromagnet Systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Applied 10, 034053 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ouassou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' González-Ruano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Caso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Aliev, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Linder, Complete magnetic control over the superconducting thermoelectric effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 106, 094514 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Martínez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Tiusan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Hehn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Chshiev and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Aliev, Symmetry broken spin reorientation transition in epitaxial MgO/Fe/MgO layers with competing anisotropies, Sci Rep 8, 9463 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Gabor, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Petrisor, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Tiusan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Hehn, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Petrisor, Magnetic and structural anisotropies of Co2FeAl Heusler alloy epitaxial thin films, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 84, 134413 (20211).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Usadel, Generalized diffusion equation for superconducting alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 25, 507 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rammer and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Smith, Quantum field-theoretical methods in transport theory of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 58, 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [26] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bergeret, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Silaev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Virtanen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Heikkilä, Colloquium: Nonequilibrium effects in superconductors with a spin-splitting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 90, 041001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [27] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Chandrasekhar, Superconductivity, edited by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bennemann and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ketterson (Springer, Berlin, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Belzig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Wilhelm, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bruder, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Schön, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Zaikin, Quasiclassical Green’s function approach to mesoscopic super- conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Superlattices and Microstructures 25, 1251 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ouassou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Pal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Blamire, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Eschrig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Linder, Triplet Cooper pairs induced in diffusive s-wave superconductors in- terfaced with strongly spin-polarized magnetic insulators or half-metallic ferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Sci Rep 7, 1932 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Eschrig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Cottet, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Belzig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Linder, General boundary conditions for quasiclassical theory of superconductivity in the diffusive limit: application to strongly spin-polarized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 17, 083037 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Cottet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Huertas-Hernando, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Belzig, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Nazarov, Spin- dependent boundary conditions for isotropic superconducting Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 80, 184511 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Cottet, Spectroscopy and critical temperature of diffusive superconducting/ferromagnetic hybrid structures with spin-active interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 76, 224505 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Jacobsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ouassou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Linder, Critical temperature and tunneling spectroscopy of superconductor-ferromagnet hybrids with intrinsic Rashba-Dresselhaus spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 92, 024510 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [34] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Schopohl, Transformation of the Eilenberger equations of superconductivity to a scalar Riccati equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Unpublished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ouassou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Vethaak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Linder, Voltage-induced thin- film superconductivity in high magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 98, 144509 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Schmalzbauer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Xing, Ferromagnetic Order at (100) 𝑝(1×1) Surfaces of Bulk Paramagnetic Vanadium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 57, 2311 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Grempel and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ying, Spin Fluctuations at the Surface of Vanadium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 45, 1018 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [38] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Walker and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Hopster, Induced magnetic order in ultrathin vanadium films on Fe(100), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 49, 7687 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [39] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Fuchs, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Totland, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Landolt, Induced magnetization in thin epitaxial V films on Fe (100), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 53, 9123 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [40] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bryk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bylander, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Kleinman, Magnetism of the V(001) surface in the generalized gradient approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 61, R3780(R) (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [41] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Lacina, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Yang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Erskine, Multilayer relaxation and search for ferromagnetic order at the (100) surface of bulk paramagnetic vanadium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 75, 195423 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rubio-Ponce, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Olguín, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Aguayo and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' de Coss, Sur- face magnetism in vanadium overlayers on W(100), Journal of Magnetism and Magnetic Materials 514, 15, 167143 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [43] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Tiusan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Hehn, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Montaigne, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Greullet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Andrieu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Schuhl, Spin tunneling phenomena in single crystal magnetic tunnel junction systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Matter 19, 165201 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' González-Ruano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Caso, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Johnsen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Tiusan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Hehn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Banerjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Linder and Farkhad G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Aliev, Superconductivity assisted change of the perpendicular magnetic anisotropy in V/MgO/Fe junctions, Scientific Reports 11, 19041 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [45] COMSOL Multiphysics® www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='comsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' COMSOL AB, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Dixon, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Hoare, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Holden and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Moody, The Low Temperature Specific Heats of Some Pure Metals (Cu, Ag, Pt, Al, Ni Fe Co), Proceedings of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Series A, Mathematical and Physical Sciences 285, 1403, 561-580 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [47] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Gardner and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Anderson, Effect of neutron irradiation on the low-temperature specific heat and thermal conductivity of magnesium oxide, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' B 23 415 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [48] Jung, Wayne Douglas, The thermal conductivity of high purity vanadium, Retrospective Theses and Dissertations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Paper 5485 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Radebaugh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Keesom, Low-Temperature Thermody- namic Properties of Vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' 149, 19 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Walter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Walowski, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Zbarsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Munzenberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Schafers, 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Ebke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Reiss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Thomas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Peretzki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Seibt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Moodera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Czerner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Bachmann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} +page_content=' Heiliger, Seebeck effect in magnetic tunnel junctions, Nature Materials 10, 742–746 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQfjgRh/content/2301.03263v1.pdf'} diff --git a/s9FAT4oBgHgl3EQfgx0D/content/tmp_files/2301.08589v1.pdf.txt b/s9FAT4oBgHgl3EQfgx0D/content/tmp_files/2301.08589v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..38043ef74c0a3cea0be8fe6beb1521b3af24c0c6 --- /dev/null +++ b/s9FAT4oBgHgl3EQfgx0D/content/tmp_files/2301.08589v1.pdf.txt @@ -0,0 +1,667 @@ +Title +Deposition rate controls nucleation and growth during amorphous/nanocrystalline +competition in sputtered Zr-Cr thin films. +Authors +Q. Liebgotta,b, A. Borrotoa, Z. Fernández-Gutiérreza, S. Bruyèrea, F. Mücklichb, D. Horwata,* +a Université de Lorraine, CNRS, IJL, F-54000 Nancy, France +b Department of Materials Science and Engineering, Chair of Functional Materials, Saarland +University, Campus D3.3, D-66123 Saarbrücken, Germany +* Corresponding author. E-mail address: david.horwat@univ-lorraine.fr (D. Horwat) +Abstract +Dual-phase Zr-based thin films synthesized by magnetron co-sputtering and showing +competitive growth between amorphous and crystalline phases have been reported recently. +In such films, the amorphous phase grows as columns, while the crystalline phase grows as +separated cone-shaped crystalline regions made of smaller crystallites. In this paper, we +investigate this phenomenon and propose a model for the development of the crystalline +regions during thin film growth. We evidence using X-ray diffraction (XRD), scanning electron +microscopy (SEM) and transmission electron microscopy (TEM), that this competitive self- +separation also exists in co-sputtered Zr-Cr thin films with Cr contents of ~84-86 at.%, +corresponding to the transition between the amorphous and crystalline compositions, and in +the Zr-V system. Then, to assess the sturdiness of this phenomenon, its existence and +geometrical characteristics are evaluated when varying the film composition and the +deposition rate. The variation of geometrical features, such as the crystalline cone angle, the + +size and density of crystallites, is discussed. Is it shown that a variation in the deposition rate +changes the nucleation and growth kinetics of the crystallites. The surface coverage by the +crystalline phase at a given thickness is also calculated for each deposition rate. Moreover, +comparison is made between Zr-Cr, Zr-V, Zr-Mo and Zr-W dual-phase thin films to compare +their nucleation and growth kinetics. +Keywords: thin films, competitive growth, nucleation, growth. +1. Introduction +During the last decades, interest for new functional surfaces has continuously been growing +[1–3]. This is because the functionalities of surface-modified materials are manifold: +antibacterial surfaces [4–6], solar cells [7], wear or corrosion protection [8–10], to name a +few. Many advances in the field of thin films have been made thanks to elaboration +techniques such as magnetron sputtering, allowing to inexpensively synthesize thin films at +low temperatures. Among these thin films, Zr-based thin film metallic glasses (TFMGs) have +been intensively investigated due to their good mechanical properties such as hardness +beyond 10 GPa, improvement of fatigue resistance of 316L stainless steel, and their +corrosion resistance, for instance [11–13]. +The ease to synthesize such TFMGs is characterized by the glass-forming ability (GFA). The +higher the GFA, the easier to sputter-deposit a TFMG. For the GFA to be the highest, the +sputtered film must contain different elements, with a difference greater than 10 % in +atomic radii, have a negative mixing enthalpy, and a high quenching rate [14]. Moreover, in +the case of Zr-X systems (X = Cr, V, Mo, W), intermetallic phases can be found in the +equilibrium phase diagram, which increase the GFA, despite the atomic radii of the different +elements is rather close and the mixing enthalpy is only slightly negative (-4 to -12 kJ/mol) + +[15]. This means that although it is possible to find amorphous films in a wide range of +compositions, it is still possible to sputter-deposit crystalline films outside of this range. Thus, +sputtering films at the transition between the amorphous and crystalline compositions could +lead to new microstructures, such as the dual-phase crystalline and amorphous thin films +that have been reported in the literature. This dual-phase morphology has already been +observed in Zr-W [16–18] and Zr-Mo [19]. Moreover, similar morphologies have been +reported in the literature for different systems (Ti-Al [20], Ti-O [21], Al-N [22]), where it has +only been mentioned without much attention given to its development. These dual-phase +thin films are the result of a competitive growth between the amorphous and crystalline +phases during film growth, leading to unique surface morphologies [16, 19]. A possible origin +for crystallization in a given range of composition is that GFA is favored, meaning that the +liquid is relatively stable compared to the crystalline phase. In other words, when the +composition gets closer to the crystalline composition nucleation becomes easier. Moreover, +Borroto et al. have evidenced that, in the Zr-W system, films undergo evolution from +amorphous, to nucleation at the column boundaries, to random nucleation as the tungsten +content increases and nucleation rate increases [16]. +Even if this dual-phase morphology has been observed in different binary systems, the +underlying mechanisms governing the development of competitive growth of the two +phases are currently not well understood. Also, the sturdiness of the phenomenon regarding +the deposition conditions is still unknown, and it is unknown if this phenomenon is present +regardless of the deposition conditions, such as deposition rate, or not. This is mainly due to +the facts that this phenomenon has only been first observed recently, and the composition +range in which it occurs is narrow, making it difficult to observe experimentally. + +In this study, we show that this phenomenon can be extended to Zr-Cr co-sputtered thin +films. The competitive growth is tested by changing the deposition rate between 3.8 and 74 +nm/min to test if it can exist over a wide range of deposition rates, if the composition range +in which it occurs changes, and to compare the geometrical features of the obtained +crystalline regions. Finally, nucleation and growth evolutions are explored in order to explain +our results and generalize our understanding of the competitive growth process in Zr-based +sputtered thin films. It is shown that the angle of crystalline cones decreases when +increasing the deposition rate, and that it is due to a difference in nucleation and growth +kinetics of the crystallites inside the cones. Then, these results are compared with dual- +phase Zr-V, Zr-Mo and Zr-W thin films in regard to the geometrical features of the crystalline +phase. +2. Materials and methods +2.1 Thin films synthesis +Nanostructured Zr-Cr thin films were deposited onto (100) monocrystalline Si substrates (15 +x 15 mm²) using DC magnetron co-sputtering of Zr and Cr metallic targets (targets +dimensions: 2 inches diameter, 3 mm thickness and purity higher than 99.9 %) in an argon +atmosphere. The two targets were in confocal configuration and the substrate-holder +rotation was set at 15 rpm to ensure homogeneity of the deposited films. The sputtering +chamber was pumped down via a mechanical and a turbo-molecular pumps allowing a base +pressure of 10-8 mbar. The working argon pressure was 2 Pa, and the targets to substrate +distance was set at 9 cm for the two targets. The deposition rate was controlled by the +discharge current applied to the Zr target (0.04 A, 0.075 A, 0.15 A or 0.3 A). To each +discharge current value corresponds one deposition rate reference (respectively 5, 11, 30 + +and 63 nm/min). For each deposition rate reference, the discharge current applied to the Cr +target varied from 0.09 A to 0.72 A. This allowed controlling the chemical composition of the +sputtered thin film, as the ratio of atoms sputtered from each target was changed with +minor modification of the deposition rate around the reference rate (for more information, +see Fig. S1 in Supplementary Information). +This allowed changing the composition between 82 and 88 at.%Cr for deposition rates +between 3.8 and 74 nm/min. The deposition time for these films varied from 20 to 240 min, +the first being for the highest deposition rate and the latter for the lowest deposition rate. +This allowed to synthesize thin films with a thickness of 1-1.2 µm. For Zr-V thin films, the +chamber parameters were the same as for Zr-Cr thin films. The discharge currents applied to +Zr and V targets (the V target has the same dimensions as Zr and Cr targets, with a purity +higher than 99.9%) were 0.1 A and 0.42 A, respectively. The deposition time was 105 min. +This resulted in 2 µm thick thin films containing 86 at.%V. +For comparison purposes, a Zr-Mo and a Zr-W thin film have also been deposited. For Zr-Mo, +a 3.7 µm thick thin film has been deposited in 150 min (24 nm/min deposition rate) with 0.3 +A and 0.28 A applied to the Zr and Mo targets, respectively. The chamber parameters and +target geometry are the same. For Zr-W, a 3.1 µm thick thin film has been deposited in 60 +min (52 nm/min deposition rate) at a 3 Pa working pressure with 0.3 A and 0.5 A applied to +the Zr and W targets, respectively. The W target geometry was the same as for the other +targets used in this study. +During the experiments, the substrate temperature has been monitored using a +thermocouple and did not exceed 45 °C for all depositions. +2.2 Thin films characterization + +Film thickness was measured with a Bruker Dektak XT contact profilometer. X-ray diffraction +(XRD) measurements were conducted in the Bragg-Brentano configuration with KαCu +radiation (λ = 1.5406 Å) using an AXS Bruker D8 Advance diffractometer. Scanning electron +micrographs were taken with a Zeiss GeminiSEM 500 scanning electron microscope (SEM) to +analyze the surface morphology and the cross-section of the films. Film composition was +obtained using X-ray energy dispersive spectroscopy (EDS) integrated in the SEM. +Transmission electron microscopy (TEM) analysis was conducted using a cold FEG JEOL +ARM200 microscope. Cross-sectional TEM samples of films were prepared using a focused +ion beam (FIB) scanning electron microscope dual beam system (FEI Helios 600). Throughout +the FIB process, the time during the ionic cuts was the shortest possible to avoid any heating +effect. +2.3 Image processing +Surface coverage by the crystalline phase was obtained using a homemade image treatment +MatLab program on at least 3 different cross-section SEM micrographs and analyzing it at a +film thickness of 750 nm (for more information, see Fig. S3 and Fig. S4 in Supplementary +material). We defined the linear coverage by the crystalline phase as the ratio between the +length corresponding to the crystalline phase and the total length of a line drawn at a given +thickness of the film, and we extrapolated it to be equal to the surface coverage by the +crystalline phase. Cone angles and size of crystallites were determined from cross-section +SEM micrographs using ImageJ software using at least 10 cones for each calculation. For the +crystalline geometry results, the deposition rates will be presented as reference deposition +rates, with each reference corresponding to a given applied current on the Zr target, which +means that 4 points representing each applied discharge current on the Zr target will be + +studied. The number of crystallites per cone was obtained by calculating the ratio of the +average crystalline cone volume divided by the average crystallite volume. The average cone +volume was calculated using the formula + + + + , where h is the height of the +cone (set at 500 nm, to avoid miscalculation due to coalescence of the crystalline cones), +and α the crystalline cone angle. The average crystallite volume was obtained by considering +the crystallite as a cylinder; its volume was calculated using the formula + , where R is the radius of the crystallite, and h its length. The obtained ratio is the +number of crystallites per crystalline cone, and was used to extrapolate the nucleation rate +of crystallites inside the cones. +3. Results and discussion +When synthesizing Zr-Cr thin films by magnetron sputtering, a composition-driven transition +from amorphous to crystalline Cr(Zr) solid solutions has been reported at ~89 at.%Cr [23]. +This work aims to study what happens at the transition range, and aims to search for a +competitive growth between an amorphous and crystalline phases, as has been seen for +other Zr-based binary systems in magnetron sputtering [16-19]. For this purpose, different +thin films at compositions around the transition between these two phases were deposited +at a 11 nm/min deposition rate. X-ray diffractograms of three of these thin films are shown +in Fig. 1 (a), with compositions ranging from 83 to 88 at.%Cr, close to the 89 at.%Cr reported +in [23]. The film with 83 at.%Cr is totally X-Ray amorphous, and the film with 88 at.%Cr is +totally crystalline. However, for a composition between these two (at 85 at.%Cr), we can see +a mixture of the two XRD signals, showing that the two phases coexist in the same film. The +shift in diffraction angle for the (110) bcc Cr plane is due to a change in the lattice parameter. +Indeed, increasing the Zr content in the Cr(Zr) solid solution increases the lattice parameter + +as Zr has an atomic radius larger than that of Cr, thus making the peak to shift towards lower +diffraction angles. +The same thin films have been characterized with SEM, and their surface micrographs are +shown in Fig. 1 (b-d). As can be seen, the amorphous and crystalline films present different +surface morphologies. At the composition of the transition, however, two different regions +can be observed, an amorphous and a crystalline, separated by an interface. According to +previous works [16–19], the crystalline region is in the form of disks on the surface, +surrounded by an amorphous matrix. From this micrograph, it is also noticeable that, at +some point, the crystalline regions coalesce. A cross-section SEM micrograph, Fig. 1 (e), +shows that the amorphous phase grows in a columnar microstructure, while the crystalline +phase grows as cones, with a dome on the surface. These cones do not nucleate from the +beginning of deposition, but after a certain thickness, and the origin of each cone is not at +the same thickness. This explains why the area of each cone emerging at the surface is +different. Indeed, the surface of the crystalline region on the surface depends on the +thickness where the cone nucleates, because the larger the cone, the larger its surface area. +The obtained microstructure for Zr-Cr thin films is the same as what has been observed in +the other Zr-based systems. Moreover, the cone angle is nearly the same whatever its +nucleation thickness. + + +Fig. 1: (a) X-ray diffractograms of Zr-Cr thin films with 83 at.%Cr (blue), 85 at.%Cr (black), and +88 at.%Cr (red). (b-d) Surface SEM micrographs showing crystalline (b), dual-phase (c) and +amorphous (d) surface morphologies, corresponding to the X-ray diffractograms on the left. +(e) Cross-section SEM micrographs showing the dual-phased sample, with amorphous +columns and crystalline cones with a dome shape on the surface. +Interestingly, the surface morphology can be tailored by changing the film thickness. Indeed, +as thickness increases, the surface coverage by the crystalline phase increases, until it +eventually covers the whole surface. Moreover, changing the composition inside the +transition zone can also be used to control the surface morphology [15], as the nucleation +density of the crystalline cones increases with an increasing Cr content. Therefore, +combining thickness and composition can help tailoring the surface of the deposited Zr-Cr +thin films. Nonetheless, the range of this competitive growth is less than 5 at.%Cr, thus very + +88 at% Cr +b) +a) +(110) +Diffracted intensity (arb. unit) +Si(400) +400nm +Crystalline +85 at% Cr +c) +Si(200) +bccCr +88 at.% Cr +Crystalline +400nm +85 at.% Cr +83 at% Cr +d) +83 at.% Cr +50 +- +30 +40 +60 +70 +80 +20() +400nm +e +400nmcareful investigations are required to observe what happens during growth and create +models. +To further understand how the competitive growth develops and can be manipulated, the +deposition rate of the Zr-Cr thin films has been varied in the range 5-63 nm/min reference +deposition rates. Fig. 2 shows the surface coverage by the crystalline phase as a function of +the film composition and for various deposition rates. The value is measured at a 750 nm +thickness for all the films. Films with a 0% surface coverage are totally amorphous, while +films with a 100% surface coverage are totally crystalline at the surface, meaning they either +are crystalline from the start of deposition or that crystalline cones overgrew the amorphous +phase and covered the whole surface. The existence of the competitive growth is +approximately in the range of 84 to 86 at.%Cr for all the deposited films. Yet, due to the +narrow composition range to observe this competitive growth, it is difficult to conclude +whether or not the deposition rate affects the transition composition range. + + +100 +Crystalline phase surface coverage (%) +5 nm/min +1 1 nm/min +30 nm/min +63 nm/min +Trendline +50 +82 +84 +86 +88 +at. % CrFig. 2: Values of the surface coverage by the crystalline phase at 750 nm thickness, +extrapolated from cross-section SEM micrographs, for each deposition rate range. Some +error bars are too small to be seen. +However, it is important to note that in this wide range of reference deposition rates, from 5 +to 63 nm/min, the competitive growth has always been observed, which indicates that it is a +process resilient to changes in deposition conditions and is encouraging to find it in other +systems. Yet, it is important to notice that the overall geometry of the structures is affected +by the deposition rate, as shown in Fig. 3 (a-b). As can be seen, the crystalline cone angle for +low deposition rates, in Fig. 3 (a), is larger than for high deposition rates, in Fig. 3 (b). Hence, +a reduction of the crystalline cone angle is observed when increasing the deposition rate. +Furthermore, a difference is observed depending on the alloy deposited, as can be seen +when comparing Zr-Cr thin films with Zr-V thin films in Fig. 3 (c). Note that Fig. 3(c) proves +that the competitive growth also exists in the Zr-V system. + + +a +400 nm +400nm +1umFig. 3: SEM micrographs showing Zr-Cr thin films (~85 at.%Cr) deposited at 5nm/min (a), 63 +nm/min (b), and a Zr-V thin film (86 at.%V) deposited at 18nm/min (c) for comparison. +To ensure that the columns and cones are two different phases, HRTEM and electron +diffraction have been conducted on the sample presented in Fig. 1 (c). A HRTEM micrograph +of a cone and its interface with the amorphous matrix is shown in Fig. 4 (a), and the selected +area diffraction (SAED) patterns of the two phases are shown in Fig. 4 (b, c). These diffraction +patterns confirm the crystallinity of the cones in the thin film, while the columns only +present short range ordering, associated with an amorphous phase, see Fig. S2 (c) of the +supplementary material. The amorphous halo of Fig. 4 (c) presents a radius that is in +agreement with that of (110) bcc Cr shown in Fig. 4 (b). Therefore, we can argue that the +local bcc Cr order present in the amorphous columns is the precursor of the nucleation of +crystalline regions. + + +a) +b) +101/mm +bcc Cr +C +50 nm +10 1/nm +amorphousFig. 4: HRTEM micrograph (a) showing the interface between a cone (in the middle) and a +column (surrounding the cone), and (b) and (c) the selected area diffraction (SAED) patterns +of the zones highlighted in (a). +As these films showing competitive growth contain two distinct phases, their respective +chemical composition has to be assessed in order to evaluate if the nucleation and growth of +the crystalline phase could be triggered by a composition gradient. For this, EDS has been +conducted on this sample inside the transmission electron microscope. The bright field TEM +image in Fig. 5 (a) shows the zone where the EDS measurements have been conducted. The +measurement points are shown on the blue horizontal line, and correspond to the points +where the chemical composition is reported on the graph in Fig. 5 (b). As can be seen, the +chemical composition change between the columns and the cone is not significant, taking +into account the uncertainty associated with the EDS technique. + + +Fig. 5: Bright field TEM image showing the thin film with a crystalline cone surrounded by +amorphous columns (a) and the Pt film on top of it. The points on the cyan line where the + +200 mm +100 +100 +b +90 +80- +-80 +70- +60- +60 +N +at.% +50- +at.%Cr +1Z% +at.% 2 +40- +40 +30- +20- +20 +10- +0- +0 +5 +10 +15 +20 +PointEDS measurements have been conducted, and the composition of the film in Cr and Zr is +given for each measurement point in (b). The vertical dotted lines are the intersection points +between the amorphous/crystalline interface and the EDS measurement line. +For the Zr-Cr thin films, Fig. 6 (a) gives the crystalline cone angle as a function of deposition +rate. When the deposition rate increases, the crystalline cone angle decreases. However, this +decrease is quite small, given the fact that the deposition rate has been increased +approximately 12-fold. The Zr-V thin films were deposited at a single deposition rate of 18 +nm/min, resulting in the cones having an average angle of 48.4°, which is much larger than +for Zr-Cr thin films. Zr-Mo thin film deposited at a 24 nm/min deposition rate resulted in a +39.9° cone angle, while the Zr-W thin film deposited at a 52 nm/min resulted in a 45.4° cone +angle. + + +Fig. 6: Graphs showing the change in crystalline cone angle (a), in crystallite size (b) and in +number of crystallites per cone (c) for Zr-Cr, Zr-V, Zr-Mo and Zr-W thin films depending on +the deposition rate of the thin films. Some error bars are too small to be seen. +To try to explain the difference found depending on the deposition rate, we propose a +model of nucleation and growth of crystallites inside the crystalline cones, where there is +only a small number of crystallites that nucleate at the apex of the cone, and the other +crystallites nucleate at the interface between the growing crystallites and the amorphous +columns upon the development of the crystalline cones. The proposed model is shown in Fig. +7. + +50 +a) +Zr-Cr +Cone angle (°) +45 +Zr-V +Zr-M c +40 +Zr-W +35 +30 +25 +20 +90 +b) +Zr-Cr +Zr-V +80 +oW-Z +50 +Zr-W +40 +30 +120 +r +c) +100 +Zr-Mo +80 +Zr-W +percone +40° +20 +Deposition rate (nm/min) +Fig. 7: Proposed nucleation and growth model for the crystalline cones in Zr-based thin films. +For the nucleation of a crystalline cone to occur, an energy barrier has to be overcome. The +energy needed can mainly be obtained chemically, thermally or mechanically. As suggested +by the EDS results on the TEM sample, the composition regarding Zr and Cr elements is the +same throughout the whole sample, so it is unlikely that a local chemical gradient is at the +origin of the nucleation. Moreover, during the whole deposition, the substrate temperature +did not exceed 45 °C, which makes the hypothesis of thermal energy being the precursor of +the nucleation less plausible. The mechanical energy originates from local changes of the +strain energy due to an uneven distribution of the stress through the film. It has been shown +that these dual-phase thin films exhibit a change of internal mechanical stress at the first +stages of the crystallite nucleation. Thus, it is likely that the origin for the nucleation of the +first crystallite lies in the mechanical energy aquired thanks to mechanical stress [18]. +It can be seen on the SEM micrographs (Fig. 3) that some crystallites grow from the +beginning of the cone, and some other nucleate at the interface between a crystallite and + +Crystallite +Nucleationpoint(amorphous/crystalline) +Nucleationpoint(crystalline/crystalline)the amorphous matrix, or between crystallites. It is easier to observe on the Zr-V thin film, as +the crystalline phase is less dense and the size of the crystallites is larger than for Zr-Cr. On +Fig. 3 (c), on the cone on the right of the micrograph, vertical crystallites can be seen at the +axis of the cone. In contrast, crystallites that nucleated at the interface between the first +crystallites and the amorphous matrix on the right, and on the left some crystallites show a +larger angle, indicating that they nucleated higher in the cone between different crystallites. +Using this model, the change in crystalline cone angle with deposition rate might find its +origin either in a difference in nucleation rate of the crystallites inside the cones, a difference +in their growth kinetics, or both. +The first explanation for the decrease of the crystalline cones angle with increasing +deposition rate is the surface diffusion of adatoms. Indeed, an increased deposition rate +means less time for the adatoms to diffuse, and thus less in-plane growth of the crystalline +cones, reducing the cone angle. Nevertheless, given the high increase in deposition rate, the +in-plane growth rate would need to be 8 times higher if the diffusion was the only parameter +affecting the cone angle, following the equation given by Borroto et al. [16]: + + + + +With the in-plane growth rate, the deposition rate, and the crystalline cone angle, as +shown on Fig. 3. +Hence, changing the deposition rate affects the in-plane growth of the crystallites. This is +confirmed in Fig. 6 (b), where a ~15 % decrease in crystallite size is observed when increasing +the deposition rate from 5 to 63nm/min, due to the fact that the adatoms have less time to + +diffuse on the surface of the film when the deposition rate is higher, and that means less in- +plane growth for the crystallites. +However, this ~15 % decrease in crystallite size alone does not explain the ~30 % decrease in +cone angle. According to our model, it means that there could be another phenomenon +happening, such as a difference in nucleation rate of the crystallites inside the cones. Thus, if +this model of nucleation and growth of the crystalline phase is correct, a decrease of the +nucleation rate should be observed when increasing the deposition rate. +To quantify the difference in nucleation inside the cones, the number of crystallites in a cone +must be known. Yet, two problems arise when counting the number of crystallites per cone. +The first is due to the density of the crystallites in the cones, making it difficult to count them. +Some interfaces can be easily seen, and have been used to calculate the average crystallite +size for each deposition rate range, but it is not the case for the vast majority of the +crystallites, where it becomes difficult to see the separation. The second problem is that +when observing a cross-section SEM micrograph, the cones are not always cut in their center, +and it is of great difficulty to know exactly where a cone has been cut, which means counting +the crystallites from SEM micrographs is misleading. +To overcome these issues, the average number of crystallites per cone will be calculated by +dividing the average cone volume by the average crystallite volume. To calculate the first +one, the volume is calculated by using the cone angle and setting the height to 500 nm, +which means for cones that have not already coalesced. For the latter, the crystallite is +assumed to be a cylinder, of radius half of the crystallite size shown in Fig. 6 (b), and of +length half of the 500 nm height used for the cone, as all crystallites don’t nucleate at the +origin of the cone. Then, the average cone volume is divided by the average crystallite + +volume, giving the plot in Fig. 6 (c). As can be seen, for a deposition rate between 5 and 30 +nm/min, the nucleation density in a cone does not significantly change. However, when +depositing films at 63nm/min, a 40 % reduction of number of crystallites is observed, which +means that the nucleation rate is lower for this deposition rate, contributing to reducing the +crystalline cone angle. +It should be noted that this calculation gives the same results no matter the length of the +crystallites as compared to the cone height. As can be seen in Fig. 6, when increasing the +deposition rate from 5 nm/min up to 30 nm/min, the decrease in crystalline cone angle is +mostly due to the reduction of the crystallite size, and the nucleation density does not +significantly change. Hence, at these deposition rates, the change in crystalline cone angle is +mostly governed by the diffusion of the adatoms on the surface of the film during growth, +whereas increasing the deposition rate from 30 to 63 nm/min did not significantly change +the crystallite size. Instead, the number of crystallites per cone decreased by nearly 40 % as +compared to other deposition rates, which means that the decrease in crystallite cone angle +for very high deposition rates is mostly due to a lower nucleation density of the crystallites in +the cones. +As a comparison, for the Zr-V thin films, as can be seen in Fig. 6 (a), the crystalline cone angle +is much higher than that of Zr-Cr thin films, being 48° and from 21° to 32°, respectively. +Moreover, it can be seen in Fig. 6 (b) that the crystallite size is 87 nm, approximately 2 times +higher than that of Zr-Cr films, and the number of crystallites per cone is nearly the same as +for Zr-Cr thin films as Fig. 6 (c) shows. Thus, the higher cone angle in Zr-V dual phased thin +films as compared to Zr-Cr thin films might find its origin in the higher surface diffusion of V + +adatoms as compared to Cr adatoms, leading to higher in-plane growth of the crystallites in +Zr-V crystalline cones and thus a higher cone angle. +An explanation for the higher surface diffusion of V adatoms as compared to Cr adatoms in +the studied Zr-V and Zr-Cr alloys lies in their different melting temperatures and +solidification intervals. According to the literature, the stable equilibrium peritectic +temperature of our Zr-V thin films is 1573 K [24] and solidification interval is close to 400 K at +the composition range of interest, while for the Zr-Cr thin films the peritectic temperature is +1832 K and the solidification interval is close to 100 K [25]. During sputtering, the surface +diffusion directly depends on the ratio between the deposition temperature and the melting +temperature of the deposited material [26]. At a similar deposition temperature, a lower +melting temperature means a higher adatom surface diffusion. Thus, the adatom diffusion +for the Zr-V system might be higher than for Zr-Cr, allowing a higher in-plane growth of the +crystallites for the Zr-V crystalline cones, explaining the higher cone angle as compared to Zr- +Cr for the same deposition rate. Similarly, higher solidification interval means more time for +the adatoms to diffuse during the solidification process occurring at the film surface. The Zr- +W peritectic temperature is 2483 K, according to its phase diagram [27], and the crystallites +are smaller than in Zr-Cr and Zr-V. However, the Zr-Mo peritectic temperature is 2194 K [28], +but its crystallites are even smaller than for Zr-W. Again, the potential reason for the lower +diffusion length of in Zr-Mo is the smaller solidification interval (approximately 300 K +compared to 700 K for Zr-W), leading to a faster solidification, which allows less time for +diffusion. +4. Conclusion + +To summarize, the competitive growth phenomenon that occurs between amorphous and +crystalline phases that has been observed in the literature for sputtered Zr-Mo, Zr-W and Zr- +V films has been also found in the Zr-Cr system. This study shows that it exists in a wide +range of deposition rate, indicating its resilience against changes of deposition conditions +and giving promises for its experimental observation in other systems. However, changing +the deposition rate also changes the geometrical features of the crystalline phase such as +the cone angle, which can be used as a new parameter to control the growth of the film, +together with the film composition and film thickness. The size and density of crystallites +inside a cone also depend on the deposition rate and the material deposited. Furthermore, +we proposed a model of nucleation and growth for the crystallites inside the crystalline +cones to explain the development of this competitive growth, which is a step towards the +understanding of the mechanisms underlying this process. This allows a fine tuning of the +surface morphology, and hence of the functional properties. We showed that depending on +the deposition rate, the change in the crystalline cone angle can be due to either the +nucleation kinetics of the crystallites, or their lateral growth. +Acknowledgements +The “Université Franco-Allemande” (UFA) and the “Ministère de l’Enseignement Supérieur et +de la Recherche” are deeply acknowledged for the PhD scholarship of Quentin Liebgott +within the PhD-track in Materials Science and Engineering at UFA. +References +[1] +X. Yang, F. Wang, W. Wang, S. Liu, Y. Chen, H. Tang, Comparison of two-step surface +treatment on surface roughness and corrosion resistance of TC4 alloy parts prepared +by SLM and SEBM, J. Alloys Compd. 921 (2022) 165929. + +https://doi.org/10.1016/j.jallcom.2022.165929. +[2] +Y. Ma, L. Li, J. Qian, W. Qu, R. Luo, F. Wu, R. Chen, Materials and structure engineering +by magnetron sputtering for advanced lithium batteries, Energy Storage Mater. 39 +(2021) 203–224. https://doi.org/10.1016/j.ensm.2021.04.012. +[3] +A. Garcia-Giron, J.M. Romano, Y. Liang, B. Dashtbozorg, H. Dong, P. Penchev, S.S. +Dimov, Combined surface hardening and laser patterning approach for functionalising +stainless steel surfaces, Appl. Surf. Sci. 439 (2018) 516–524. +https://doi.org/10.1016/j.apsusc.2018.01.012. +[4] +D.W. Müller, S. Lößlein, E. Terriac, K. Brix, K. Siems, R. Moeller, R. Kautenburger, F. +Mücklich, Increasing Antibacterial Efficiency of Cu Surfaces by targeted Surface +Functionalization via Ultrashort Pulsed Direct Laser Interference Patterning, Adv. +Mater. Interfaces. 8 (2021). https://doi.org/10.1002/admi.202001656. +[5] +H. Ji, M.C. Zhao, B. Xie, Y.C. Zhao, D. Yin, C. Gao, C. Shuai, A. Atrens, Corrosion and +antibacterial performance of novel selective-laser-melted (SLMed) Ti-xCu biomedical +alloys, J. Alloys Compd. 864 (2021) 158415. +https://doi.org/10.1016/j.jallcom.2020.158415. +[6] +C.B. Yi, Y.X. Yuan, L. Zhang, Y.H. Jiang, Z.Y. He, Antibacterial Ti-35Nb-7Zr-xCu alloy with +excellent mechanical properties generated with a spark plasma sintering method for +biological applications, J. Alloys Compd. 879 (2021) 160473. +https://doi.org/10.1016/j.jallcom.2021.160473. +[7] +G. Wu, R. Liang, M. Ge, G. Sun, Y. Zhang, G. Xing, Surface Passivation Using 2D +Perovskites toward Efficient and Stable Perovskite Solar Cells, Adv. Mater. 34 (2022). + +https://doi.org/10.1002/adma.202105635. +[8] +M. Tabish, M.U. Malik, M.A. Khan, G. Yasin, H.M. Asif, M.J. Anjum, W.Q. Khan, S. +Ibraheem, T.A. Nguyen, Y. Slimani, M.T. Nazir, Construction of NiCo/graphene +nanocomposite coating with bulges-like morphology for enhanced mechanical +properties and corrosion resistance performance, J. Alloys Compd. 867 (2021) 159138. +https://doi.org/10.1016/j.jallcom.2021.159138. +[9] +G. Purcek, H. Yanar, M. Demirtas, D. V. Shangina, N.R. Bochvar, S. V. Dobatkin, +Microstructural, mechanical and tribological properties of ultrafine-grained Cu–Cr–Zr +alloy processed by high pressure torsion, J. Alloys Compd. 816 (2020) 152675. +https://doi.org/10.1016/j.jallcom.2019.152675. +[10] +J. Zhu, X. Cheng, L. Zhang, X. Hui, Y. Wu, H. Zheng, Z. Ren, Y. Zhao, W. Wang, S. Zhu, X. +Wang, Microstructures, wear resistance and corrosion resistance of CoCrFeNi high +entropy alloys coating on AZ91 Mg alloy prepared by cold spray, J. Alloys Compd. 925 +(2022) 166698. https://doi.org/10.1016/j.jallcom.2022.166698. +[11] +Y.L. Deng, J.W. Lee, B.S. Lou, J.G. Duh, J.P. Chu, J.S.C. Jang, The fabrication and +property evaluation of Zr-Ti-B-Si thin film metallic glass materials, Surf. Coatings +Technol. 259 (2014) 115–122. https://doi.org/10.1016/j.surfcoat.2014.03.026. +[12] +H. Jia, F. Liu, Z. An, W. Li, G. Wang, J.P. Chu, J.S.C. Jang, Y. Gao, P.K. Liaw, Thin-film +metallic glasses for substrate fatigue-property improvements, Thin Solid Films. 561 +(2014) 2–27. https://doi.org/10.1016/j.tsf.2013.12.024. +[13] +F.X. Liu, F.Q. Yang, Y.F. Gao, W.H. Jiang, Y.F. Guan, P.D. Rack, O. Sergic, P.K. Liaw, +Micro-scratch study of a magnetron-sputtered Zr-based metallic-glass film, Surf. + +Coatings Technol. 203 (2009) 3480–3484. +https://doi.org/10.1016/j.surfcoat.2009.05.017. +[14] +A. Inoue, T. Zhang, T. Masumoto, Glass-forming ability of alloys, J. Non. Cryst. Solids. +156–158 (1993) 473–480. https://doi.org/10.1016/0022-3093(93)90003-G. +[15] +A.I.I. Akira Takeuchi, Classification of Bulk Metallic Glasses by Atomic Size Difference, +Heat of Mixing and Period of Constituent Elements and Its Application to +Characterization of the Main Alloying Element, Mater. Trans. 46 (2005) 2817–2829. +[16] +A. Borroto, A.C. García-Wong, S. Bruyère, S. Migot, D. Pilloud, J.F. Pierson, F. Mücklich, +D. Horwat, Composition-driven transition from amorphous to crystalline films enables +bottom-up design of functional surfaces, Appl. Surf. Sci. 538 (2021) 148133. +https://doi.org/10.1016/j.apsusc.2020.148133. +[17] +P. Zeman, S. Haviar, M. Červená, Self-formation of dual glassy-crystalline structure in +magnetron sputtered W–Zr films, Vacuum. 187 (2021). +https://doi.org/10.1016/j.vacuum.2021.110099. +[18] +A. Borroto, S. Bruyère, S. Migot, J.F. Pierson, F. Mücklich, D. Horwat, Growth kinetics +and origin of residual stress of two-phase crystalline-amorphous nanostructured films, +J. Appl. Phys. 129 (2021). https://doi.org/10.1063/5.0044029. +[19] +A. Borroto, S. Bruyère, S. Migot, J.F. Pierson, T. Gries, F. Mücklich, D. Horwat, +Controlling surface morphology by nanocrystalline/amorphous competitive self-phase +separation in thin films: Thickness-modulated reflectance and interference +phenomena, Acta Mater. 181 (2019) 78–86. +https://doi.org/10.1016/j.actamat.2019.09.036. + +[20] +V. Kolkovsky, J.U. Schmidt, S. Döring, Electrical and structural properties of direct +current magnetron sputtered amorphous TiAl thin films, Thin Solid Films. 669 (2019) +169–173. https://doi.org/10.1016/j.tsf.2018.10.033. +[21] +R. Gago, M. Vinnichenko, A. Redondo-Cubero, Z. Czigány, L. Vázquez, Surface +morphology of heterogeneous nanocrystalline rutile/amorphous anatase TiO2 films +grown by reactive pulsed magnetron sputtering, Plasma Process. Polym. 7 (2010) 813– +823. https://doi.org/10.1002/ppap.200900182. +[22] + . rien, P. Miska, . olle, P. Pigeat, olumnar growth of AL by r.f. magnetron +spu ering Role of the } planes, J. Cryst. Growth. 307 (2007) 245–252. +https://doi.org/10.1016/j.jcrysgro.2007.06.013. +[23] +J.H. Kim, E. Akiyama, H. Habazaki, A. Kawashima, K. Asami, K. Hashimoto, The +corrosion behavior of sputter-deposited amorphous Chromium-Zirconium alloys in +6M HCl solution, Corros. Sci. 34 (1993) 1817–1827. +[24] +J. Štrof, J. Pavlů, U.D. Wdowik, J. uršík, M. Šob, J. řešťál, Laves phases in the -Zr +system below room temperature: Stability analysis using ab initio results and phase +diagram, Calphad Comput. Coupling Phase Diagrams Thermochem. 44 (2014) 62–69. +https://doi.org/10.1016/j.calphad.2013.08.003. +[25] +H.J. Lu, W.B. Wang, N. Zou, J.Y. Shen, X.G. Lu, Y.L. He, Thermodynamic modeling of Cr- +Nb and Zr-Cr with extension to the ternary Zr-Nb-Cr system, Calphad Comput. +Coupling Phase Diagrams Thermochem. 50 (2015) 134–143. +https://doi.org/10.1016/j.calphad.2015.06.002. +[26] +J.A. Thornton, Influence of Apparatus Geometry and Deposition Conditions on the + +Structure and Topography of Thick Sputtered Coatings., J Vac Sci Technol. 11 (1974) +666–670. https://doi.org/10.1116/1.1312732. +[27] +H. Ren, X. Liu, J. Ning, Microstructure and mechanical properties of W-Zr reactive +materials, Mater. Sci. Eng. A. 660 (2016) 205–212. +https://doi.org/10.1016/j.msea.2016.02.009. +[28] +R.J. Pérez, B. Sundman, Thermodynamic assessment of the Mo-Zr binary phase +diagram, Calphad Comput. Coupling Phase Diagrams Thermochem. 27 (2003) 253–262. +https://doi.org/10.1016/j.calphad.2003.09.003. + + diff --git a/s9FAT4oBgHgl3EQfgx0D/content/tmp_files/load_file.txt b/s9FAT4oBgHgl3EQfgx0D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8f8b7dd46da0381d8a84a3683c6bbae28cd5e3f --- /dev/null +++ b/s9FAT4oBgHgl3EQfgx0D/content/tmp_files/load_file.txt @@ -0,0 +1,760 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf,len=759 +page_content='Title Deposition rate controls nucleation and growth during amorphous/nanocrystalline competition in sputtered Zr-Cr thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Authors Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liebgotta,b, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Borrotoa, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Fernández-Gutiérreza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Bruyèrea, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mücklichb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Horwata,* a Université de Lorraine, CNRS, IJL, F-54000 Nancy, France b Department of Materials Science and Engineering, Chair of Functional Materials, Saarland University, Campus D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='3, D-66123 Saarbrücken, Germany Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' E-mail address: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='horwat@univ-lorraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='fr (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Horwat) Abstract Dual-phase Zr-based thin films synthesized by magnetron co-sputtering and showing competitive growth between amorphous and crystalline phases have been reported recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' In such films, the amorphous phase grows as columns, while the crystalline phase grows as separated cone-shaped crystalline regions made of smaller crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' In this paper, we investigate this phenomenon and propose a model for the development of the crystalline regions during thin film growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' We evidence using X-ray diffraction (XRD), scanning electron microscopy (SEM) and transmission electron microscopy (TEM), that this competitive self- separation also exists in co-sputtered Zr-Cr thin films with Cr contents of ~84-86 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%, corresponding to the transition between the amorphous and crystalline compositions, and in the Zr-V system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Then, to assess the sturdiness of this phenomenon, its existence and geometrical characteristics are evaluated when varying the film composition and the deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The variation of geometrical features, such as the crystalline cone angle, the size and density of crystallites, is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Is it shown that a variation in the deposition rate changes the nucleation and growth kinetics of the crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The surface coverage by the crystalline phase at a given thickness is also calculated for each deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, comparison is made between Zr-Cr, Zr-V, Zr-Mo and Zr-W dual-phase thin films to compare their nucleation and growth kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Keywords: thin films, competitive growth, nucleation, growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Introduction During the last decades, interest for new functional surfaces has continuously been growing [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This is because the functionalities of surface-modified materials are manifold: antibacterial surfaces [4–6], solar cells [7], wear or corrosion protection [8–10], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Many advances in the field of thin films have been made thanks to elaboration techniques such as magnetron sputtering, allowing to inexpensively synthesize thin films at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Among these thin films, Zr-based thin film metallic glasses (TFMGs) have been intensively investigated due to their good mechanical properties such as hardness beyond 10 GPa, improvement of fatigue resistance of 316L stainless steel, and their corrosion resistance, for instance [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The ease to synthesize such TFMGs is characterized by the glass-forming ability (GFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The higher the GFA, the easier to sputter-deposit a TFMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For the GFA to be the highest, the sputtered film must contain different elements, with a difference greater than 10 % in atomic radii, have a negative mixing enthalpy, and a high quenching rate [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, in the case of Zr-X systems (X = Cr, V, Mo, W), intermetallic phases can be found in the equilibrium phase diagram, which increase the GFA, despite the atomic radii of the different elements is rather close and the mixing enthalpy is only slightly negative (-4 to -12 kJ/mol) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This means that although it is possible to find amorphous films in a wide range of compositions, it is still possible to sputter-deposit crystalline films outside of this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Thus, sputtering films at the transition between the amorphous and crystalline compositions could lead to new microstructures, such as the dual-phase crystalline and amorphous thin films that have been reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This dual-phase morphology has already been observed in Zr-W [16–18] and Zr-Mo [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, similar morphologies have been reported in the literature for different systems (Ti-Al [20], Ti-O [21], Al-N [22]), where it has only been mentioned without much attention given to its development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' These dual-phase thin films are the result of a competitive growth between the amorphous and crystalline phases during film growth, leading to unique surface morphologies [16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' A possible origin for crystallization in a given range of composition is that GFA is favored, meaning that the liquid is relatively stable compared to the crystalline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' In other words, when the composition gets closer to the crystalline composition nucleation becomes easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, Borroto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' have evidenced that, in the Zr-W system, films undergo evolution from amorphous, to nucleation at the column boundaries, to random nucleation as the tungsten content increases and nucleation rate increases [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Even if this dual-phase morphology has been observed in different binary systems, the underlying mechanisms governing the development of competitive growth of the two phases are currently not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Also, the sturdiness of the phenomenon regarding the deposition conditions is still unknown, and it is unknown if this phenomenon is present regardless of the deposition conditions, such as deposition rate, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This is mainly due to the facts that this phenomenon has only been first observed recently, and the composition range in which it occurs is narrow, making it difficult to observe experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' In this study, we show that this phenomenon can be extended to Zr-Cr co-sputtered thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The competitive growth is tested by changing the deposition rate between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='8 and 74 nm/min to test if it can exist over a wide range of deposition rates, if the composition range in which it occurs changes, and to compare the geometrical features of the obtained crystalline regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Finally, nucleation and growth evolutions are explored in order to explain our results and generalize our understanding of the competitive growth process in Zr-based sputtered thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' It is shown that the angle of crystalline cones decreases when increasing the deposition rate, and that it is due to a difference in nucleation and growth kinetics of the crystallites inside the cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Then, these results are compared with dual- phase Zr-V, Zr-Mo and Zr-W thin films in regard to the geometrical features of the crystalline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Materials and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1 Thin films synthesis Nanostructured Zr-Cr thin films were deposited onto (100) monocrystalline Si substrates (15 x 15 mm²) using DC magnetron co-sputtering of Zr and Cr metallic targets (targets dimensions: 2 inches diameter, 3 mm thickness and purity higher than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='9 %) in an argon atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The two targets were in confocal configuration and the substrate-holder rotation was set at 15 rpm to ensure homogeneity of the deposited films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The sputtering chamber was pumped down via a mechanical and a turbo-molecular pumps allowing a base pressure of 10-8 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The working argon pressure was 2 Pa, and the targets to substrate distance was set at 9 cm for the two targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The deposition rate was controlled by the discharge current applied to the Zr target (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='04 A, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='075 A, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='15 A or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='3 A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To each discharge current value corresponds one deposition rate reference (respectively 5, 11, 30 and 63 nm/min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For each deposition rate reference, the discharge current applied to the Cr target varied from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='09 A to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='72 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This allowed controlling the chemical composition of the sputtered thin film, as the ratio of atoms sputtered from each target was changed with minor modification of the deposition rate around the reference rate (for more information, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' S1 in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This allowed changing the composition between 82 and 88 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr for deposition rates between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='8 and 74 nm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The deposition time for these films varied from 20 to 240 min, the first being for the highest deposition rate and the latter for the lowest deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This allowed to synthesize thin films with a thickness of 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For Zr-V thin films, the chamber parameters were the same as for Zr-Cr thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The discharge currents applied to Zr and V targets (the V target has the same dimensions as Zr and Cr targets, with a purity higher than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='9%) were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1 A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='42 A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The deposition time was 105 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This resulted in 2 µm thick thin films containing 86 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For comparison purposes, a Zr-Mo and a Zr-W thin film have also been deposited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For Zr-Mo, a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='7 µm thick thin film has been deposited in 150 min (24 nm/min deposition rate) with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='3 A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='28 A applied to the Zr and Mo targets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The chamber parameters and target geometry are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For Zr-W, a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1 µm thick thin film has been deposited in 60 min (52 nm/min deposition rate) at a 3 Pa working pressure with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='3 A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='5 A applied to the Zr and W targets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The W target geometry was the same as for the other targets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' During the experiments, the substrate temperature has been monitored using a thermocouple and did not exceed 45 °C for all depositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2 Thin films characterization Film thickness was measured with a Bruker Dektak XT contact profilometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' X-ray diffraction (XRD) measurements were conducted in the Bragg-Brentano configuration with KαCu radiation (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='5406 Å) using an AXS Bruker D8 Advance diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Scanning electron micrographs were taken with a Zeiss GeminiSEM 500 scanning electron microscope (SEM) to analyze the surface morphology and the cross-section of the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Film composition was obtained using X-ray energy dispersive spectroscopy (EDS) integrated in the SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Transmission electron microscopy (TEM) analysis was conducted using a cold FEG JEOL ARM200 microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Cross-sectional TEM samples of films were prepared using a focused ion beam (FIB) scanning electron microscope dual beam system (FEI Helios 600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Throughout the FIB process, the time during the ionic cuts was the shortest possible to avoid any heating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='3 Image processing Surface coverage by the crystalline phase was obtained using a homemade image treatment MatLab program on at least 3 different cross-section SEM micrographs and analyzing it at a film thickness of 750 nm (for more information, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' S3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' S4 in Supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' We defined the linear coverage by the crystalline phase as the ratio between the length corresponding to the crystalline phase and the total length of a line drawn at a given thickness of the film, and we extrapolated it to be equal to the surface coverage by the crystalline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Cone angles and size of crystallites were determined from cross-section SEM micrographs using ImageJ software using at least 10 cones for each calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For the crystalline geometry results, the deposition rates will be presented as reference deposition rates, with each reference corresponding to a given applied current on the Zr target, which means that 4 points representing each applied discharge current on the Zr target will be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The number of crystallites per cone was obtained by calculating the ratio of the average crystalline cone volume divided by the average crystallite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The average cone volume was calculated using the formula , where h is the height of the cone (set at 500 nm, to avoid miscalculation due to coalescence of the crystalline cones), and α the crystalline cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The average crystallite volume was obtained by considering the crystallite as a cylinder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' its volume was calculated using the formula , where R is the radius of the crystallite, and h its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The obtained ratio is the number of crystallites per crystalline cone, and was used to extrapolate the nucleation rate of crystallites inside the cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Results and discussion When synthesizing Zr-Cr thin films by magnetron sputtering, a composition-driven transition from amorphous to crystalline Cr(Zr) solid solutions has been reported at ~89 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This work aims to study what happens at the transition range, and aims to search for a competitive growth between an amorphous and crystalline phases, as has been seen for other Zr-based binary systems in magnetron sputtering [16-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For this purpose, different thin films at compositions around the transition between these two phases were deposited at a 11 nm/min deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' X-ray diffractograms of three of these thin films are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 1 (a), with compositions ranging from 83 to 88 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr, close to the 89 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr reported in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The film with 83 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr is totally X-Ray amorphous, and the film with 88 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr is totally crystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, for a composition between these two (at 85 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr), we can see a mixture of the two XRD signals, showing that the two phases coexist in the same film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The shift in diffraction angle for the (110) bcc Cr plane is due to a change in the lattice parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Indeed, increasing the Zr content in the Cr(Zr) solid solution increases the lattice parameter as Zr has an atomic radius larger than that of Cr, thus making the peak to shift towards lower diffraction angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The same thin films have been characterized with SEM, and their surface micrographs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 1 (b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As can be seen, the amorphous and crystalline films present different surface morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' At the composition of the transition, however, two different regions can be observed, an amorphous and a crystalline, separated by an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' According to previous works [16–19], the crystalline region is in the form of disks on the surface, surrounded by an amorphous matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' From this micrograph, it is also noticeable that, at some point, the crystalline regions coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' A cross-section SEM micrograph, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 1 (e), shows that the amorphous phase grows in a columnar microstructure, while the crystalline phase grows as cones, with a dome on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' These cones do not nucleate from the beginning of deposition, but after a certain thickness, and the origin of each cone is not at the same thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This explains why the area of each cone emerging at the surface is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Indeed, the surface of the crystalline region on the surface depends on the thickness where the cone nucleates, because the larger the cone, the larger its surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The obtained microstructure for Zr-Cr thin films is the same as what has been observed in the other Zr-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, the cone angle is nearly the same whatever its nucleation thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 1: (a) X-ray diffractograms of Zr-Cr thin films with 83 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr (blue), 85 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr (black), and 88 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' (b-d) Surface SEM micrographs showing crystalline (b), dual-phase (c) and amorphous (d) surface morphologies, corresponding to the X-ray diffractograms on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' (e) Cross-section SEM micrographs showing the dual-phased sample, with amorphous columns and crystalline cones with a dome shape on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Interestingly, the surface morphology can be tailored by changing the film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Indeed, as thickness increases, the surface coverage by the crystalline phase increases, until it eventually covers the whole surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, changing the composition inside the transition zone can also be used to control the surface morphology [15], as the nucleation density of the crystalline cones increases with an increasing Cr content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Therefore, combining thickness and composition can help tailoring the surface of the deposited Zr-Cr thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Nonetheless, the range of this competitive growth is less than 5 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr, thus very 88 at% Cr b) a) (110) Diffracted intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' unit) Si(400) 400nm Crystalline 85 at% Cr c) Si(200) bccCr 88 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='% Cr Crystalline 400nm 85 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='% Cr 83 at% Cr d) 83 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='% Cr 50 30 40 60 70 80 20() 400nm e 400nmcareful investigations are required to observe what happens during growth and create models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To further understand how the competitive growth develops and can be manipulated, the deposition rate of the Zr-Cr thin films has been varied in the range 5-63 nm/min reference deposition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 2 shows the surface coverage by the crystalline phase as a function of the film composition and for various deposition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The value is measured at a 750 nm thickness for all the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Films with a 0% surface coverage are totally amorphous, while films with a 100% surface coverage are totally crystalline at the surface, meaning they either are crystalline from the start of deposition or that crystalline cones overgrew the amorphous phase and covered the whole surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The existence of the competitive growth is approximately in the range of 84 to 86 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr for all the deposited films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yet, due to the narrow composition range to observe this competitive growth, it is difficult to conclude whether or not the deposition rate affects the transition composition range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 100 Crystalline phase surface coverage (%) 5 nm/min 1 1 nm/min 30 nm/min 63 nm/min Trendline 50 82 84 86 88 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' % CrFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 2: Values of the surface coverage by the crystalline phase at 750 nm thickness, extrapolated from cross-section SEM micrographs, for each deposition rate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Some error bars are too small to be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, it is important to note that in this wide range of reference deposition rates, from 5 to 63 nm/min, the competitive growth has always been observed, which indicates that it is a process resilient to changes in deposition conditions and is encouraging to find it in other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yet, it is important to notice that the overall geometry of the structures is affected by the deposition rate, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3 (a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As can be seen, the crystalline cone angle for low deposition rates, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3 (a), is larger than for high deposition rates, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Hence, a reduction of the crystalline cone angle is observed when increasing the deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Furthermore, a difference is observed depending on the alloy deposited, as can be seen when comparing Zr-Cr thin films with Zr-V thin films in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3(c) proves that the competitive growth also exists in the Zr-V system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' a 400 nm 400nm 1umFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3: SEM micrographs showing Zr-Cr thin films (~85 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr) deposited at 5nm/min (a), 63 nm/min (b), and a Zr-V thin film (86 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%V) deposited at 18nm/min (c) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To ensure that the columns and cones are two different phases, HRTEM and electron diffraction have been conducted on the sample presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' A HRTEM micrograph of a cone and its interface with the amorphous matrix is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 4 (a), and the selected area diffraction (SAED) patterns of the two phases are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 4 (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' These diffraction patterns confirm the crystallinity of the cones in the thin film, while the columns only present short range ordering, associated with an amorphous phase, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' S2 (c) of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The amorphous halo of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 4 (c) presents a radius that is in agreement with that of (110) bcc Cr shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Therefore, we can argue that the local bcc Cr order present in the amorphous columns is the precursor of the nucleation of crystalline regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' a) b) 101/mm bcc Cr C 50 nm 10 1/nm amorphousFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 4: HRTEM micrograph (a) showing the interface between a cone (in the middle) and a column (surrounding the cone), and (b) and (c) the selected area diffraction (SAED) patterns of the zones highlighted in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As these films showing competitive growth contain two distinct phases, their respective chemical composition has to be assessed in order to evaluate if the nucleation and growth of the crystalline phase could be triggered by a composition gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For this, EDS has been conducted on this sample inside the transmission electron microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The bright field TEM image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 5 (a) shows the zone where the EDS measurements have been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The measurement points are shown on the blue horizontal line, and correspond to the points where the chemical composition is reported on the graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As can be seen, the chemical composition change between the columns and the cone is not significant, taking into account the uncertainty associated with the EDS technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 5: Bright field TEM image showing the thin film with a crystalline cone surrounded by amorphous columns (a) and the Pt film on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The points on the cyan line where the 200 mm 100 100 b 90 80- 80 70- 60- 60 N at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='% 50- at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='%Cr 1Z% at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='% 2 40- 40 30- 20- 20 10- 0- 0 5 10 15 20 PointEDS measurements have been conducted, and the composition of the film in Cr and Zr is given for each measurement point in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The vertical dotted lines are the intersection points between the amorphous/crystalline interface and the EDS measurement line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For the Zr-Cr thin films, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (a) gives the crystalline cone angle as a function of deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' When the deposition rate increases, the crystalline cone angle decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, this decrease is quite small, given the fact that the deposition rate has been increased approximately 12-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The Zr-V thin films were deposited at a single deposition rate of 18 nm/min, resulting in the cones having an average angle of 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='4°, which is much larger than for Zr-Cr thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zr-Mo thin film deposited at a 24 nm/min deposition rate resulted in a 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='9° cone angle, while the Zr-W thin film deposited at a 52 nm/min resulted in a 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='4° cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6: Graphs showing the change in crystalline cone angle (a), in crystallite size (b) and in number of crystallites per cone (c) for Zr-Cr, Zr-V, Zr-Mo and Zr-W thin films depending on the deposition rate of the thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Some error bars are too small to be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To try to explain the difference found depending on the deposition rate, we propose a model of nucleation and growth of crystallites inside the crystalline cones, where there is only a small number of crystallites that nucleate at the apex of the cone, and the other crystallites nucleate at the interface between the growing crystallites and the amorphous columns upon the development of the crystalline cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The proposed model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 50 a) Zr-Cr Cone angle (°) 45 Zr-V Zr-M c 40 Zr-W 35 30 25 20 90 b) Zr-Cr Zr-V 80 oW-Z 50 Zr-W 40 30 120 r c) 100 Zr-Mo 80 Zr-W percone 40° 20 Deposition rate (nm/min) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 7: Proposed nucleation and growth model for the crystalline cones in Zr-based thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For the nucleation of a crystalline cone to occur, an energy barrier has to be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The energy needed can mainly be obtained chemically, thermally or mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As suggested by the EDS results on the TEM sample, the composition regarding Zr and Cr elements is the same throughout the whole sample, so it is unlikely that a local chemical gradient is at the origin of the nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, during the whole deposition, the substrate temperature did not exceed 45 °C, which makes the hypothesis of thermal energy being the precursor of the nucleation less plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The mechanical energy originates from local changes of the strain energy due to an uneven distribution of the stress through the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' It has been shown that these dual-phase thin films exhibit a change of internal mechanical stress at the first stages of the crystallite nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Thus, it is likely that the origin for the nucleation of the first crystallite lies in the mechanical energy aquired thanks to mechanical stress [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' It can be seen on the SEM micrographs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3) that some crystallites grow from the beginning of the cone, and some other nucleate at the interface between a crystallite and Crystallite Nucleationpoint(amorphous/crystalline) Nucleationpoint(crystalline/crystalline)the amorphous matrix, or between crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' It is easier to observe on the Zr-V thin film, as the crystalline phase is less dense and the size of the crystallites is larger than for Zr-Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3 (c), on the cone on the right of the micrograph, vertical crystallites can be seen at the axis of the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' In contrast, crystallites that nucleated at the interface between the first crystallites and the amorphous matrix on the right, and on the left some crystallites show a larger angle, indicating that they nucleated higher in the cone between different crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Using this model, the change in crystalline cone angle with deposition rate might find its origin either in a difference in nucleation rate of the crystallites inside the cones, a difference in their growth kinetics, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The first explanation for the decrease of the crystalline cones angle with increasing deposition rate is the surface diffusion of adatoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Indeed, an increased deposition rate means less time for the adatoms to diffuse, and thus less in-plane growth of the crystalline cones, reducing the cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Nevertheless, given the high increase in deposition rate, the in-plane growth rate would need to be 8 times higher if the diffusion was the only parameter affecting the cone angle, following the equation given by Borroto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [16]: With the in-plane growth rate, the deposition rate, and the crystalline cone angle, as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Hence, changing the deposition rate affects the in-plane growth of the crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This is confirmed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (b), where a ~15 % decrease in crystallite size is observed when increasing the deposition rate from 5 to 63nm/min, due to the fact that the adatoms have less time to diffuse on the surface of the film when the deposition rate is higher, and that means less in- plane growth for the crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, this ~15 % decrease in crystallite size alone does not explain the ~30 % decrease in cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' According to our model, it means that there could be another phenomenon happening, such as a difference in nucleation rate of the crystallites inside the cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Thus, if this model of nucleation and growth of the crystalline phase is correct, a decrease of the nucleation rate should be observed when increasing the deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To quantify the difference in nucleation inside the cones, the number of crystallites in a cone must be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yet, two problems arise when counting the number of crystallites per cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The first is due to the density of the crystallites in the cones, making it difficult to count them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Some interfaces can be easily seen, and have been used to calculate the average crystallite size for each deposition rate range, but it is not the case for the vast majority of the crystallites, where it becomes difficult to see the separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The second problem is that when observing a cross-section SEM micrograph, the cones are not always cut in their center, and it is of great difficulty to know exactly where a cone has been cut, which means counting the crystallites from SEM micrographs is misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To overcome these issues, the average number of crystallites per cone will be calculated by dividing the average cone volume by the average crystallite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' To calculate the first one, the volume is calculated by using the cone angle and setting the height to 500 nm, which means for cones that have not already coalesced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' For the latter, the crystallite is assumed to be a cylinder, of radius half of the crystallite size shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (b), and of length half of the 500 nm height used for the cone, as all crystallites don’t nucleate at the origin of the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Then, the average cone volume is divided by the average crystallite volume, giving the plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As can be seen, for a deposition rate between 5 and 30 nm/min, the nucleation density in a cone does not significantly change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, when depositing films at 63nm/min, a 40 % reduction of number of crystallites is observed, which means that the nucleation rate is lower for this deposition rate, contributing to reducing the crystalline cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' It should be noted that this calculation gives the same results no matter the length of the crystallites as compared to the cone height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6, when increasing the deposition rate from 5 nm/min up to 30 nm/min, the decrease in crystalline cone angle is mostly due to the reduction of the crystallite size, and the nucleation density does not significantly change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Hence, at these deposition rates, the change in crystalline cone angle is mostly governed by the diffusion of the adatoms on the surface of the film during growth, whereas increasing the deposition rate from 30 to 63 nm/min did not significantly change the crystallite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Instead, the number of crystallites per cone decreased by nearly 40 % as compared to other deposition rates, which means that the decrease in crystallite cone angle for very high deposition rates is mostly due to a lower nucleation density of the crystallites in the cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' As a comparison, for the Zr-V thin films, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (a), the crystalline cone angle is much higher than that of Zr-Cr thin films, being 48° and from 21° to 32°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moreover, it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (b) that the crystallite size is 87 nm, approximately 2 times higher than that of Zr-Cr films, and the number of crystallites per cone is nearly the same as for Zr-Cr thin films as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 6 (c) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Thus, the higher cone angle in Zr-V dual phased thin films as compared to Zr-Cr thin films might find its origin in the higher surface diffusion of V adatoms as compared to Cr adatoms, leading to higher in-plane growth of the crystallites in Zr-V crystalline cones and thus a higher cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' An explanation for the higher surface diffusion of V adatoms as compared to Cr adatoms in the studied Zr-V and Zr-Cr alloys lies in their different melting temperatures and solidification intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' According to the literature, the stable equilibrium peritectic temperature of our Zr-V thin films is 1573 K [24] and solidification interval is close to 400 K at the composition range of interest, while for the Zr-Cr thin films the peritectic temperature is 1832 K and the solidification interval is close to 100 K [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' During sputtering, the surface diffusion directly depends on the ratio between the deposition temperature and the melting temperature of the deposited material [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' At a similar deposition temperature, a lower melting temperature means a higher adatom surface diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Thus, the adatom diffusion for the Zr-V system might be higher than for Zr-Cr, allowing a higher in-plane growth of the crystallites for the Zr-V crystalline cones, explaining the higher cone angle as compared to Zr- Cr for the same deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Similarly, higher solidification interval means more time for the adatoms to diffuse during the solidification process occurring at the film surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The Zr- W peritectic temperature is 2483 K, according to its phase diagram [27], and the crystallites are smaller than in Zr-Cr and Zr-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, the Zr-Mo peritectic temperature is 2194 K [28], but its crystallites are even smaller than for Zr-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Again, the potential reason for the lower diffusion length of in Zr-Mo is the smaller solidification interval (approximately 300 K compared to 700 K for Zr-W), leading to a faster solidification, which allows less time for diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Conclusion To summarize, the competitive growth phenomenon that occurs between amorphous and crystalline phases that has been observed in the literature for sputtered Zr-Mo, Zr-W and Zr- V films has been also found in the Zr-Cr system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This study shows that it exists in a wide range of deposition rate, indicating its resilience against changes of deposition conditions and giving promises for its experimental observation in other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' However, changing the deposition rate also changes the geometrical features of the crystalline phase such as the cone angle, which can be used as a new parameter to control the growth of the film, together with the film composition and film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' The size and density of crystallites inside a cone also depend on the deposition rate and the material deposited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Furthermore, we proposed a model of nucleation and growth for the crystallites inside the crystalline cones to explain the development of this competitive growth, which is a step towards the understanding of the mechanisms underlying this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' This allows a fine tuning of the surface morphology, and hence of the functional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' We showed that depending on the deposition rate, the change in the crystalline cone angle can be due to either the nucleation kinetics of the crystallites, or their lateral growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Acknowledgements The “Université Franco-Allemande” (UFA) and the “Ministère de l’Enseignement Supérieur et de la Recherche” are deeply acknowledged for the PhD scholarship of Quentin Liebgott within the PhD-track in Materials Science and Engineering at UFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' References [1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Tang, Comparison of two-step surface treatment on surface roughness and corrosion resistance of TC4 alloy parts prepared by SLM and SEBM, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 921 (2022) 165929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jallcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='165929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Qian, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Qu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Luo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Chen, Materials and structure engineering by magnetron sputtering for advanced lithium batteries, Energy Storage Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 39 (2021) 203–224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='ensm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Garcia-Giron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Romano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Dashtbozorg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Dong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Penchev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Dimov, Combined surface hardening and laser patterning approach for functionalising stainless steel surfaces, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 439 (2018) 516–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='apsusc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Müller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Lößlein, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Terriac, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Brix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Siems, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Moeller, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Kautenburger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mücklich, Increasing Antibacterial Efficiency of Cu Surfaces by targeted Surface Functionalization via Ultrashort Pulsed Direct Laser Interference Patterning, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 8 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1002/admi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='202001656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ji, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Shuai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Atrens, Corrosion and antibacterial performance of novel selective-laser-melted (SLMed) Ti-xCu biomedical alloys, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 864 (2021) 158415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jallcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='158415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yuan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' He, Antibacterial Ti-35Nb-7Zr-xCu alloy with excellent mechanical properties generated with a spark plasma sintering method for biological applications, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 879 (2021) 160473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jallcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='160473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Xing, Surface Passivation Using 2D Perovskites toward Efficient and Stable Perovskite Solar Cells, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 34 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1002/adma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='202105635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Tabish, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Malik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Khan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yasin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Asif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Anjum, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Khan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ibraheem, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Nguyen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Slimani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Nazir, Construction of NiCo/graphene nanocomposite coating with bulges-like morphology for enhanced mechanical properties and corrosion resistance performance, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 867 (2021) 159138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jallcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='159138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Purcek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yanar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Demirtas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Shangina, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Bochvar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Dobatkin, Microstructural, mechanical and tribological properties of ultrafine-grained Cu–Cr–Zr alloy processed by high pressure torsion, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 816 (2020) 152675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jallcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='152675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Cheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Hui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ren, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wang, Microstructures, wear resistance and corrosion resistance of CoCrFeNi high entropy alloys coating on AZ91 Mg alloy prepared by cold spray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 925 (2022) 166698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jallcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='166698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Lee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Lou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Duh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Chu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Jang, The fabrication and property evaluation of Zr-Ti-B-Si thin film metallic glass materials, Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Coatings Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 259 (2014) 115–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='surfcoat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Jia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' An, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Chu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Jang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liaw, Thin-film metallic glasses for substrate fatigue-property improvements, Thin Solid Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 561 (2014) 2–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='tsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Gao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Guan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Rack, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sergic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liaw, Micro-scratch study of a magnetron-sputtered Zr-based metallic-glass film, Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Coatings Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 203 (2009) 3480–3484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='surfcoat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Inoue, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Masumoto, Glass-forming ability of alloys, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Non.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 156–158 (1993) 473–480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/0022-3093(93)90003-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Akira Takeuchi, Classification of Bulk Metallic Glasses by Atomic Size Difference, Heat of Mixing and Period of Constituent Elements and Its Application to Characterization of the Main Alloying Element, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 46 (2005) 2817–2829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Borroto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' García-Wong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Bruyère, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Migot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pilloud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pierson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mücklich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Horwat, Composition-driven transition from amorphous to crystalline films enables bottom-up design of functional surfaces, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 538 (2021) 148133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='apsusc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='148133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zeman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Haviar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Červená, Self-formation of dual glassy-crystalline structure in magnetron sputtered W–Zr films, Vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 187 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='110099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Borroto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Bruyère, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Migot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pierson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mücklich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Horwat, Growth kinetics and origin of residual stress of two-phase crystalline-amorphous nanostructured films, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 129 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='0044029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Borroto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Bruyère, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Migot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pierson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Gries, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Mücklich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Horwat, Controlling surface morphology by nanocrystalline/amorphous competitive self-phase separation in thin films: Thickness-modulated reflectance and interference phenomena, Acta Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 181 (2019) 78–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='actamat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Kolkovsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Schmidt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Döring, Electrical and structural properties of direct current magnetron sputtered amorphous TiAl thin films, Thin Solid Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 669 (2019) 169–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='tsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Gago, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Vinnichenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Redondo-Cubero, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Czigány, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Vázquez, Surface morphology of heterogeneous nanocrystalline rutile/amorphous anatase TiO2 films grown by reactive pulsed magnetron sputtering, Plasma Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Polym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 7 (2010) 813– 823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1002/ppap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='200900182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [22] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' rien, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Miska, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' olle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pigeat, olumnar growth of AL by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' magnetron spu ering Role of the } planes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 307 (2007) 245–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='jcrysgro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Kim, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Akiyama, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Habazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Kawashima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Asami, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Hashimoto, The corrosion behavior of sputter-deposited amorphous Chromium-Zirconium alloys in 6M HCl solution, Corros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 34 (1993) 1817–1827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Štrof, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pavlů, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wdowik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' uršík, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Šob, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' řešťál, Laves phases in the -Zr system below room temperature: Stability analysis using ab initio results and phase diagram, Calphad Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Coupling Phase Diagrams Thermochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 44 (2014) 62–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='calphad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Zou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Shen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' He, Thermodynamic modeling of Cr- Nb and Zr-Cr with extension to the ternary Zr-Nb-Cr system, Calphad Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Coupling Phase Diagrams Thermochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 50 (2015) 134–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='calphad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Thornton, Influence of Apparatus Geometry and Deposition Conditions on the Structure and Topography of Thick Sputtered Coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=', J Vac Sci Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 11 (1974) 666–670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1116/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1312732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Ning, Microstructure and mechanical properties of W-Zr reactive materials, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 660 (2016) 205–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='msea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Pérez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Sundman, Thermodynamic assessment of the Mo-Zr binary phase diagram, Calphad Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' Coupling Phase Diagrams Thermochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' 27 (2003) 253–262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='calphad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FAT4oBgHgl3EQfgx0D/content/2301.08589v1.pdf'} diff --git a/sNE2T4oBgHgl3EQffAfS/content/tmp_files/2301.03923v1.pdf.txt b/sNE2T4oBgHgl3EQffAfS/content/tmp_files/2301.03923v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a31dc82de1b46adec0a189ba6a4fc9515d9b532 --- /dev/null +++ b/sNE2T4oBgHgl3EQffAfS/content/tmp_files/2301.03923v1.pdf.txt @@ -0,0 +1,555 @@ +ΛNN content of Λ-nucleus potential +Eliahu Friedman1,⋆ and Avraham Gal1,⋆⋆ +1Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel +Abstract. A minimally constructed Λ-nucleus density-dependent optical potential is used +to calculate binding energies of observed 1sΛ, 1pΛ states across the periodic table, lead- +ing to a repulsive ΛNN contribution D(3) +Λ ≈14 MeV to the phenomenological Λ-nucleus +potential depth DΛ ≈ −30 MeV. This value is significant in connection with the so-called +’hyperon puzzle.’ +1 Introduction +The Λ-nucleus potential depth provides an important constraint in ongoing attempts to resolve the ‘hy- +peron puzzle’, i.e., whether or not dense neutron-star matter contains hyperons, primarily Λs besides +nucleons [1]. Figure 1 presents compilation of most of the known Λ hypernuclear binding energies +(BΛ) across the periodic table, fitted by a three-parameter Woods-Saxon (WS) attractive potential. As +A → ∞, a limiting value of BΛ(A) → 30 MeV is obtained. Interestingly, studies of density dependent +Λ-nuclear optical potentials VΛ(ρ) in Ref. [2], with ρ the nuclear density normalized to the number +of nucleons A, conclude that a ρ2 term motivated by three-body ΛNN interactions provides a large +repulsive (positive) contribution to the Λ-nuclear potential depth DΛ at nuclear-matter density ρ0: +D(3) +Λ ≈ 30 MeV. This repulsive component of DΛ is more than just compensated at ρ0 by a roughly +twice larger attractive depth value D(2) +Λ ≈ −60 MeV, motivated by a two-body ΛN interaction. Note +that DΛ is defined as VΛ(ρ0) in the limit A → ∞ at a given nuclear-matter density ρ0, with a value +0.17 fm−3 assumed here. +Most hyperon-nucleon potential models overbind Λ hypernuclei, yielding values of D(2) +Λ deeper +than −30 MeV. Whereas such overbinding amounts to only few MeV in the often used Nijmegen +soft-core model versions NSC97e,f [4] it is considerably stronger, by more than 10 MeV, in the recent +Nijmegen extended soft-core model ESC16 [5]. A similar overbinding arises at leading order in chiral +effective field theory (χEFT) [6]. The situation at next-to leading order (NLO) is less clear owing to a +strong dependence of D(2) +Λ on the momentum cutoff scale λ [7]. At λ=500 MeV/c, however, it is found +in Ref. [8] that both versions NLO13 [9] and NLO19 [10] overbind by a few MeV. Finally, recent +Quantum Monte Carlo (QMC) calculations [11, 12], using a ΛN + ΛNN interaction model designed +to bind correctly 5 +ΛHe, result in a strongly attractive D(2) +Λ of order −100 MeV and a correspondingly +large repulsive (positive) D(3) +Λ , reproducing the overall potential depth DΛ ≈ −30 MeV. +Our aim in the present phenomenological study is to check to what extent properly chosen Λ +hypernuclear binding energy data, with minimal extra assumptions, imply positive values of D(3) +Λ , and +⋆Eliahu.Friedman@mail.huji.ac.il +⋆⋆avragal@savion.huji.ac.il +arXiv:2301.03923v1 [nucl-th] 10 Jan 2023 + +Update: +Millener, Dover, Gal PRC 38, 2700 (1988) +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +10 +20 +30 +Binding Energy (MeV) +(pi,K) +(e,e’K) +Emulsion +(K,pi) +Λ Single Particle States +A−2/3 +sΛ +pΛ +dΛ +fΛ +gΛ +208 +139 +89 +51 +4032 +28 +16 +131211 10 +8 +7 +Woods-Saxon +V = 30.05 MeV, +r = 1.165 fm, +a = 0.6 fm +Figure 1. Compilation of Λ binding energies in 7 +ΛLi to 208 +ΛPb from various sources, and as calculated using a +three-parameter WS potential [2]. Figure adapted from Ref. [3] +how large it is [13]. Repulsive three-body ΛNN interactions go beyond just providing solution of the +overbinding problem: as nuclear density is increased beyond nuclear matter density ρ0, the balance +between attractive D(2) +Λ and repulsive D(3) +Λ tilts towards the latter. This results in nearly total expulsion +of Λ hyperons from neutron-star matter, suggesting an equation of state (EoS) sufficiently stiff to +support two solar-mass neutron stars, thereby providing a possible solution to the ‘hyperon puzzle’. +The larger D(3) +Λ is, the more likely it is a solution [14, 15]. However, there is no guarantee that three- +body ΛNN interactions are universally repulsive. For a recent discussion of this problem within an +SU(3) ‘decuplet dominance’ approach practised in modern χEFT studies at NLO, see Ref. [8]. +In this Contribution we adopt the optical potential approach as applied by Dover-Hüfner-Lemmer +to pions in nuclear matter [16]. For the Λ-nucleus system, it provides expansion in powers of the +nuclear density ρ(r), consisting of a linear term induced by a two-body ΛN interaction plus two higher- +power density terms: (i) a long-range Pauli correlations term starting at ρ4/3, and (ii) a short-range +ΛNN interaction term dominated in the present context by three-body ΛNN interactions, starting at ρ2. +As demonstrated below, the contribution of the Pauli correlations term is non negligible, propagating +to higher powers of density terms than just ρ4/3, such as the ρ2 ΛNN interaction term. This explains +why the value derived here, D(3) +Λ = (13.9 ± 1.4) MeV, differs from any of those suggested earlier in +Ref. [2] and in Skyrme Hartree Fock studies [17] where Pauli correlations are usually disregarded. +Our value of D(3) +Λ strongly disagrees with the much larger value inferred in QMC calculations [12]. +We comment on these discrepancies below. +2 Nuclear densities +In optical model applications aimed at establishing relations between components with different pow- +ers of density ρ = ρp + ρn, it is crucial to ensure that the radial extent of the densities, e.g., their r.m.s. + +radii, follows closely values derived from experiment. For proton densities we used charge densities, +with proton finite-size and recoil effects included. Harmonic-oscillator type densities [18] were used +for the lightest elements, assuming the same radial parameters for protons and neutrons. A variation +of 1% in the r.m.s. neutron radius was found to affect calculated Λ binding energies considerably less +than given by most of the experimental uncertainties listed in Table 1 below. For a detailed discussion +in the analogous case of light Ξ− hypernuclei, see Ref. [19]. For species beyond the nuclear 1p shell +we used two-parameter Fermi distributions normalized to Z for protons and N = A − Z for neutrons, +derived from assembled nuclear charge distributions [20]. For medium-weight and heavy nuclei, the +r.m.s. radii of our neutron density distributions assume larger values than those for proton density +distributions, as practiced in analyses of exotic atoms [21]. Furthermore, once neutron orbits extend +beyond proton orbits, it is useful to represent the nuclear density ρ(r) as +ρ(r) = ρcore(r) + ρexcess(r), +(1) +where ρcore refers to the Z protons plus the charge symmetric Z neutrons occupying the same nuclear +‘core’ orbits, and ρexcess refers to the (N − Z) ‘excess’ neutrons associated with the nuclear periphery. +3 Optical potential +The optical potential employed in this work, Vopt +Λ (ρ) = V(2) +Λ (ρ)+V(3) +Λ (ρ), consists of terms representing +two-body ΛN and three-body ΛNN interactions, respectively: +V(2) +Λ (ρ) = − 4π +2µΛ +fA CPauli(ρ) b0ρ, +(2) +V(3) +Λ (ρ) = + 4π +2µΛ +fA B0 +ρ2 +ρ0 +, +(3) +with b0 and B0 strength parameters in units of fm (ℏ = c = 1). In these expressions, ρ(r) is a nuclear +density distribution normalized to the number of nucleons A, ρ0 = 0.17 fm−3 stands for nuclear-matter +density, µΛ is the Λ-nucleus reduced mass and fA is a kinematical factor transforming b0 from the ΛN +c.m. system to the Λ-nucleus c.m. system: +fA = 1 + A − 1 +A +µΛ +mN +. +(4) +This form of fA coincides with the way it is used for V(2) +Λ in atomic/nuclear hadron-nucleus bound- +state problems [21] and its A dependence provides good approximation for V(3) +Λ . Next is the density +dependent factor CPauli(ρ) in Eq. (2), standing for a Pauli correlation function: +CPauli(ρ) = (1 + αP +3kF +2π fAb0)−1, +(5) +with Fermi momentum kF = (3π2ρ/2)1/3. The parameter αP in Eq. (5) switches off (αP=0) or on +(αP=1) Pauli correlations in a form suggested in Ref. [22] and practised in K− atoms studies [23]. To +estimate 1/A correction terms, we also approximated CPauli(ρ) by [19]: +CPauli(ρ) ≈ (1 + αP +3kF +2π blab +0 )−1, +blab +0 += (1 + mΛ +mN +) b0. +(6) +As shown below, including CPauli(ρ) in V(2) +Λ affects strongly the balance between the derived potential +depths D(2) +Λ and D(3) +Λ . However, introducing it also in V(3) +Λ is found to make little difference, which is +why it is skipped in Eq. (3). Finally we note that the low-density limit of Vopt +Λ +requires according to +Ref. [16] that b0 is identified with the c.m. ΛN spin-averaged scattering length (positive here). + +4 Data +Table 1. Binding energies in MeV, including uncertainties, considered here; taken from Table IV of Ref. [3]. +hypernucleus +1sΛ +± +1pΛ +± +12 +ΛB +11.52 +0.02 +0.54 +0.04 +13 +ΛC +12.0 +0.2 +1.1 +0.2 +16 +ΛN +13.76 +0.16 +2.84 +0.18 +28 +ΛSi +17.2 +0.2 +7.6 +0.2 +32 +ΛS +17.5 +0.5 +8.2 +0.5 +51 +ΛV +21.5 +0.6 +13.4 +0.6 +89 +ΛY +23.6 +0.5 +17.7 +0.6 +139 +ΛLa +25.1 +1.2 +21.0 +0.6 +208 +ΛPb +26.9 +0.8 +22.5 +0.6 +The present work does not attempt to reproduce the full range of BΛ data shown in Fig. 1. It is +limited to 1sΛ and 1pΛ states listed in Table 1. We fit to such states in one of the nuclear 1p-shell hy- +pernuclei listed in the table, where the 1sΛ state is bound by over 10 MeV while the 1pΛ state has just +become bound. This helps resolve the density dependence of Vopt +Λ by setting a good balance between +its two components, V(2) +Λ (ρ) and V(3) +Λ (ρ), following it all the way to 208 +ΛPb the heaviest hypernucleus +marked in Fig. 1. We chose to fit the 16 +ΛN precise Bexp +Λ (1s, 1p) values derived, respectively, from the +first and third peaks to the left in Fig. 2. The extremely simple 1p proton hole structure of the 15N +nuclear core in this case removes most of the uncertainty arising from spin-dependent residual ΛN +interactions [25]. The fitted optical-potential parameters b0, Eq. (2), and B0, Eq. (3), are then used to +calculate the B1s,1p +Λ +values of the other eight species listed in Table 1. +-20 +-15 +-10 +-5 +0 +5 +10 +15 +- Binding Energy (MeV) +0 +1 +2 +3 +4 +dσ/(dΩedΩKdEedEb) [nb/(sr +2GeV MeV)] +fit +SLA +BS3 +16 +ΛN +Figure 2. 16O(e, e′K+) spectrum of 16 +ΛN from JLab Hall A measurements. Figure adapted from Ref. [24]. + +5 Results +The two strength parameters b0, B0 of the optical potential terms Eqs. (2,3) were obtained by fitting +to the 16 +ΛN Bexp +Λ (1s, 1p) values listed in Table 1. Suppressing Pauli correlations by setting αP = 0 in +Eqs. (5,6), the resulting Λ potential depth DΛ = −27.4 MeV reflects a sizable cancellation between a +strongly attractive two-body potential depth D(2) +Λ and a strongly repulsive three-body potential depth +D(3) +Λ . The overall agreement between calculations and experiment is acceptable, but some under- +binding appears to develop for increasing mass numbers A, noticed clearly in the three heaviest 1sΛ +and two heaviest 1pΛ states. The resulting b0 is about half of the known Λp scattering length of +(1.7 ± 0.1) fm [26, 27]. +When the full potential Eqs. (2-6) is used (marked here as model X, including Pauli correlations +through αP = 1) the overall picture remains unchanged regarding underbinding for the heavier ele- +ments, see Fig. 3. However, the fit parameter b0=1.85 fm agrees now with the Λp scattering length. +The other parameter, B0 = 0.170 fm, is about twice smaller than for αP = 0. +10 +100 +A +0 +5 +10 +15 +20 +25 +30 +binding energy (MeV) +Λ−A binding energies +1s +1p +model X +Figure 3. B1s,1p +Λ +(A) values from model X compared with data. Continuous lines connect calculated values. +The phenomenon of underbinding associated with the optical potential Eqs. (2-6) is likely to be +a result of the use of ρ2 in nuclei where excess neutrons occupy shell-model orbits higher than those +occupied by protons. This situation occurs in Fig. 3 for the four hypernuclei with A ≳ 50. Expecting +that direct three-body ΛNN contributions involving one ‘core’ nucleon and one ‘excess’ nucleon +vanish upon summing on the T=0 ‘core’ closed-shell nucleons, we modify ρ2 = (ρcore + ρexcess)2 by +discarding the bilinear term ρcore ρexcess, thereby replacing ρ2 in V(3) +Λ , Eq. (3), by +ρ2 +core + ρ2 +excess += (2ρp)2 + (ρn − ρp)2 +(7) +in terms of the input densities ρp and ρn. This ansatz is consistent with an overall isospin factor +τ1 · τ2 in two-pion exchange ΛNN forces, as first realized back in 1958 [28]. Results of applying this +ansatz are shown in the lower part of Fig. 4 as model Y, where the underbinding of calculated 1sΛ and + +10 +100 +A +0 +5 +10 +15 +20 +25 +30 +binding energy (MeV) +10 +100 +0 +5 +10 +15 +20 +25 +30 +binding energy (MeV) +1s +1p +Λ−A binding energies +Λ−A binding energies +1s +1p +model Y0 +model Y +Figure 4. B1s,1p +Λ +(A) values from models Y0 and Y compared with data, see text. Continuous lines connect +calculated values. +1pΛ binding energies noticed in model X is no longer observed. The fit parameters, nevertheless, are +the same as for model X above. In the upper part of Fig. 4, model Y0 shows similar results where +the Pauli-correlations correction in model Y, Eq. (6), is replaced by Eq. (5). This provides a rough +estimate of the impact of 1/A corrections typical for our optical-potential methodology. Potential +depth values in model Y are D(2) +Λ = −41.6 MeV, D(3) +Λ = 13.9 MeV. +To estimate uncertainties, we act as follows: (i) decreasing the input value of B1s +Λ (16 +ΛN) fitted to by +0.2 MeV, thereby getting halfway to the central value of B1s +Λ (16 +ΛO)=(13.4±0.4) MeV for 16 +ΛO [29] the +charge-symmetric partner of 16 +ΛN, results in approximately 10% larger value of D(3) +Λ , and (ii) applying +Pauli correlations to V(3) +Λ too reduces D(3) +Λ roughly by 10%. In both cases D(2) +Λ increases moderately +by ≲1 MeV. On the other hand, D(2) +Λ decreases by 1.7 MeV if Eq. (5) is used for CPauli(ρ) instead of +Eq. (6). Considering these uncertainties, our final values are (in MeV) +D(2) +Λ = −40.6 ± 1.0 +D(3) +Λ = 13.9 ± 1.4 +DΛ = −26.7 ± 1.7 +(8) +6 Discussion +The D(2) +Λ and D(3) +Λ values in Eq. (8) are considerably smaller than those deduced in QMC calcula- +tions [11, 12]. Note that the QMC nuclear densities ρQMC(r) are much too compact with respect to +our realistic densities, with nuclear r.m.s. radii rN(QMC) about 0.8 of the known r.m.s. charge radii +in 16O and 40Ca [30]. Since ρ scales as r−3 +N , applying it to the density dependence of our Vopt +Λ +would +transform D(2) +Λ and D(3) +Λ of Eq. (8) to as large depth values as D(2) +Λ (QMC)=(−79.3±2.0) MeV and + +D(3) +Λ (QMC)=(53.0±5.3) MeV, their sum DΛ(QMC)=(−26.3±5.7) MeV agreeing within uncertainties +with ours. +Table 2. Λ-nuclear potential depths (in MeV) from two SHF calculations fitting BΛ data points, and from our +own Vopt +Λ (αP = 0) two-parameter (b0, B0) fit to the two B1s,1p +Λ +(16 +ΛN) values listed in Table 1. +Method +Data Points +D(2) +Λ +D(3) +Λ +DΛ +SHF [2] +3 +−57.8 +31.4 +−26.4 +SHF [17] +35 +−55.4 +20.4 +−35.0 +Vopt +Λ (αP = 0) [13] +2 +−57.6 +30.2 +−27.4 +Smaller-size but still inflated values of D(2) +Λ and D(3) +Λ are obtained by applying the Skyrme Hartree +Fock (SHF) methodology [2, 17]. Apart from small nonlocal potential terms and effective mass +corrections, the SHF Λ-nuclear mean-field potential VΛ(ρ) consists of two terms: V(2) +Λ (ρ) ∝ ρ and +V(3) +Λ (ρ) ∝ ρ2. A large-scale SHF fit [17] of the corresponding Λ potential depths to 35 BΛ data points +is listed in the middle row of Table 2. We note that the overall DΛ = −35 MeV value becomes +−31 MeV upon including a Λ effective-mass correction, a bit closer to the other DΛ values listed in +the table. Similar results, particularly for D(2) +Λ , can be obtained in fact by choosing a considerably +smaller number of fitted data points, as shown by the fits listed in the other two rows of the table. The +11 MeV difference between the D(3) +Λ values derived in these two SHF calculations arises mostly from +nonlocal lower-power density terms, like ρ5/3, present in [17] but absent in [2]. Interestingly, the last +row lists a fit to the two B1s,1p +Λ +(16 +ΛN) values used here when Pauli correlations are suppressed, αP = 0 +in Eq. (5). The sizable difference between D(2) +Λ and D(3) +Λ values listed in Table 2, all of which disregard +Pauli correlations, and the Vopt +Λ +values listed in Eq. (8) demonstrates the importance of including in +Vopt +Λ a Pauli-correlations term (αP = 1) starting as ρ4/3. +7 Summary +In summary, we have presented a straightforward optical-potential analysis of 1sΛ and 1pΛ binding +energies across the periodic table, 12 ≤ A ≤ 208, based on nuclear densities constrained by charge +r.m.s. radii. The potential is parameterized by constants b0 and B0 in front of two-body ΛN and three- +body ΛNN interaction terms. These parameters were fitted to precise Bexp +Λ (1s, 1p) values in 16 +ΛN [31] +and then used to evaluate B1s,1p +Λ +values in the other hypernuclei considered here. Pauli correlations +were found essential to establish a correct balance between b0 and B0, as judged by b0 coming out in +the final model Y analysis close to the value of the ΛN spin-averaged s-wave scattering length. Good +agreement was reached in this model between the calculated B1s,1p +Λ +values and their corresponding +Bexp +Λ values, see Fig. 4. +The potential depth D(3) +Λ derived here, Eq. (8), suggests that in symmetric nuclear matter the Λ- +nucleus potential becomes repulsive near three times ρ0. Our derived depth D(3) +Λ is larger by a few +MeV than the one yielding µ(Λ) > µ(n) for Λ and neutron chemical potentials in purely neutron +matter, respectively, under a ‘decuplet dominance’ construction for the underlying ΛNN interaction +terms within a χEFT(NLO) model [8]. This suggests that the strength of the corresponding repulsive +V(3) +Λ optical potential component, as constrained in the present work by data, is sufficient to prevent +Λ hyperons from playing active role in neutron-star matter, thereby enabling a stiff EoS that supports +two solar-mass neutron stars. + +Acknowledgments +One of us (A.G.) thanks Jiˇrí Mareš and other members of the HYP2022 organizing team for their generous +hospitality during the Conference. The present work is part of a project funded by the European Union’s Horizon +2020 research & innovation programme, grant agreement 824093. +References +[1] L. Tolos, L. Fabbietti, Prog. Part. Nucl. Phys. 112, 103770 (2020), +and past neutron-star work cited therein +[2] D.J. Millener, C.B. Dover, A. Gal, Phys. Rev. C 38, 2700 (1988) +[3] A. Gal, E.V. Hungerford, D.J. Millener, Rev. Mod. Phys. 88, 035004 (2016) +[4] Th.A. Rijken, V.G.J. Stoks, Y. Yamamoto, Phys. Rev. C 59, 21 (1999) +[5] M.M. Nagels, Th.A. Rijken, Y. Yamamoto, Phys. Rev. C 99, 044003 (2019) +[6] H. Polinder, J. Haidenbauer, U.-G. Meißner, Nucl. Phys. A 779, 244 (2006) +[7] J. Haidenbauer, I. Vidaña, Eur. Phys. J. A 56, 55 (2020) +[8] D. Gerstung, N. Kaiser, W. Weise, Eur. Phys. J. A 56, 175 (2020), +and references cited therein to earlier works on ΛNN interactions in χEFT. +[9] J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. Meißner, A. Nogga, W. Weise, +Nucl. Phys. A 915, 24 (2013) +[10] J. Haidenbauer, U.-G. Meißner, A. Nogga, Eur. Phys. J. A 56, 91 (2020) +[11] D. Lonardoni, S. Gandolfi, F. Pederiva, Phys. Rev. C 87, 041303(R) (2013) +[12] D. Lonardoni, F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014) +[13] E. Friedman, A. Gal, Phys. Lett. B 837, 137669 (2023) +[14] D. Lonardoni, A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015) +[15] D. Logoteta, I. Vidaña, I. Bombaci, Eur. Phys. J. A 55, 207 (2019) +[16] C.B. Dover, J. Hüfner, R.H. Lemmer, Ann. Phys. (NY) 66, 248 (1971) +[17] H.-J. Schulze, E. Hiyama, Phys. Rev. C 90, 047301 (2014), and past SHF work cited therein +[18] L.R.B. Elton, Nuclear Sizes (Oxford University Press, Oxford, 1961) +[19] E. Friedman, A. Gal, Phys. Lett. B 820, 136555 (2021) +[20] I. Angeli, K.P. Marinova, At. Data Nucl. Data Tables 99, 69 (2013) +[21] E. Friedman, A. Gal, Phys. Rep. 452, 89 (2007) +[22] T. Waas, M. Rho, W. Weise, Nucl. Phys. A 617, 449 (1997) +[23] E. Friedman, A. Gal, Nucl. Phys. A 959, 66 (2017), and past K− atoms work cited therein +[24] F. Garibaldi, et al. (Jefferson Lab Hall A Collaboration), Phys. Rev. C 99, 054309 (2019) +[25] D.J. Millener, Nucl. Phys. A 804, 84 (2008), 881, 298 (2012), 914, 109 (2013) +[26] G. Alexander, et al., Phys. Rev. 173, 1452 (1968) +[27] A. Budzanowski, et al. (HIRES Collaboration), Phys. Lett. B 687, 31 (2010) +[28] R. Spitzer, Phys. Rev. 110, 1190 (1958) +[29] M. Agnello, et al. (FINUDA Collaboration), Phys. Lett. B 698, 219 (2011) +[30] D. Lonardoni, Ph.D. thesis, University of Trento, Italy, arXiv:1311.6672 [nucl-th], +in particular see Table 4.7 +[31] F. Cusanno, et al. (Jefferson Lab Hall A Collaboration), Phys. Rev. Lett. 103, 202501 (2009) + diff --git a/sNE2T4oBgHgl3EQffAfS/content/tmp_files/load_file.txt b/sNE2T4oBgHgl3EQffAfS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99751e6540cc51f80da442c1ab95eb6e408b6f2e --- /dev/null +++ b/sNE2T4oBgHgl3EQffAfS/content/tmp_files/load_file.txt @@ -0,0 +1,431 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf,len=430 +page_content='ΛNN content of Λ-nucleus potential Eliahu Friedman1,⋆ and Avraham Gal1,⋆⋆ 1Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A minimally constructed Λ-nucleus density-dependent optical potential is used to calculate binding energies of observed 1sΛ, 1pΛ states across the periodic table, lead- ing to a repulsive ΛNN contribution D(3) Λ ≈14 MeV to the phenomenological Λ-nucleus potential depth DΛ ≈ −30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This value is significant in connection with the so-called ’hyperon puzzle.’ 1 Introduction The Λ-nucleus potential depth provides an important constraint in ongoing attempts to resolve the ‘hy- peron puzzle’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=', whether or not dense neutron-star matter contains hyperons, primarily Λs besides nucleons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Figure 1 presents compilation of most of the known Λ hypernuclear binding energies (BΛ) across the periodic table, fitted by a three-parameter Woods-Saxon (WS) attractive potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' As A → ∞, a limiting value of BΛ(A) → 30 MeV is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Interestingly, studies of density dependent Λ-nuclear optical potentials VΛ(ρ) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [2], with ρ the nuclear density normalized to the number of nucleons A, conclude that a ρ2 term motivated by three-body ΛNN interactions provides a large repulsive (positive) contribution to the Λ-nuclear potential depth DΛ at nuclear-matter density ρ0: D(3) Λ ≈ 30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This repulsive component of DΛ is more than just compensated at ρ0 by a roughly twice larger attractive depth value D(2) Λ ≈ −60 MeV, motivated by a two-body ΛN interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Note that DΛ is defined as VΛ(ρ0) in the limit A → ∞ at a given nuclear-matter density ρ0, with a value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='17 fm−3 assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Most hyperon-nucleon potential models overbind Λ hypernuclei, yielding values of D(2) Λ deeper than −30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Whereas such overbinding amounts to only few MeV in the often used Nijmegen soft-core model versions NSC97e,f [4] it is considerably stronger, by more than 10 MeV, in the recent Nijmegen extended soft-core model ESC16 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A similar overbinding arises at leading order in chiral effective field theory (χEFT) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The situation at next-to leading order (NLO) is less clear owing to a strong dependence of D(2) Λ on the momentum cutoff scale λ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' At λ=500 MeV/c, however, it is found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [8] that both versions NLO13 [9] and NLO19 [10] overbind by a few MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Finally, recent Quantum Monte Carlo (QMC) calculations [11, 12], using a ΛN + ΛNN interaction model designed to bind correctly 5 ΛHe, result in a strongly attractive D(2) Λ of order −100 MeV and a correspondingly large repulsive (positive) D(3) Λ , reproducing the overall potential depth DΛ ≈ −30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Our aim in the present phenomenological study is to check to what extent properly chosen Λ hypernuclear binding energy data, with minimal extra assumptions, imply positive values of D(3) Λ , and ⋆Eliahu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='Friedman@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='il ⋆⋆avragal@savion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='il arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='03923v1 [nucl-th] 10 Jan 2023 Update: Millener, Dover, Gal PRC 38, 2700 (1988) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='25 0 10 20 30 Binding Energy (MeV) (pi,K) (e,e’K) Emulsion (K,pi) Λ Single Particle States A−2/3 sΛ pΛ dΛ fΛ gΛ 208 139 89 51 4032 28 16 131211 10 8 7 Woods-Saxon V = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='05 MeV, r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='165 fm, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 fm Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Compilation of Λ binding energies in 7 ΛLi to 208 ΛPb from various sources, and as calculated using a three-parameter WS potential [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Figure adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [3] how large it is [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Repulsive three-body ΛNN interactions go beyond just providing solution of the overbinding problem: as nuclear density is increased beyond nuclear matter density ρ0, the balance between attractive D(2) Λ and repulsive D(3) Λ tilts towards the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This results in nearly total expulsion of Λ hyperons from neutron-star matter, suggesting an equation of state (EoS) sufficiently stiff to support two solar-mass neutron stars, thereby providing a possible solution to the ‘hyperon puzzle’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The larger D(3) Λ is, the more likely it is a solution [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' However, there is no guarantee that three- body ΛNN interactions are universally repulsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' For a recent discussion of this problem within an SU(3) ‘decuplet dominance’ approach practised in modern χEFT studies at NLO, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' In this Contribution we adopt the optical potential approach as applied by Dover-Hüfner-Lemmer to pions in nuclear matter [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' For the Λ-nucleus system, it provides expansion in powers of the nuclear density ρ(r), consisting of a linear term induced by a two-body ΛN interaction plus two higher- power density terms: (i) a long-range Pauli correlations term starting at ρ4/3, and (ii) a short-range ΛNN interaction term dominated in the present context by three-body ΛNN interactions, starting at ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' As demonstrated below, the contribution of the Pauli correlations term is non negligible, propagating to higher powers of density terms than just ρ4/3, such as the ρ2 ΛNN interaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This explains why the value derived here, D(3) Λ = (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4) MeV, differs from any of those suggested earlier in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [2] and in Skyrme Hartree Fock studies [17] where Pauli correlations are usually disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Our value of D(3) Λ strongly disagrees with the much larger value inferred in QMC calculations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' We comment on these discrepancies below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 2 Nuclear densities In optical model applications aimed at establishing relations between components with different pow- ers of density ρ = ρp + ρn, it is crucial to ensure that the radial extent of the densities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=', their r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' radii, follows closely values derived from experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' For proton densities we used charge densities, with proton finite-size and recoil effects included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Harmonic-oscillator type densities [18] were used for the lightest elements, assuming the same radial parameters for protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A variation of 1% in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' neutron radius was found to affect calculated Λ binding energies considerably less than given by most of the experimental uncertainties listed in Table 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' For a detailed discussion in the analogous case of light Ξ− hypernuclei, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' For species beyond the nuclear 1p shell we used two-parameter Fermi distributions normalized to Z for protons and N = A − Z for neutrons, derived from assembled nuclear charge distributions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' For medium-weight and heavy nuclei, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' radii of our neutron density distributions assume larger values than those for proton density distributions, as practiced in analyses of exotic atoms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Furthermore, once neutron orbits extend beyond proton orbits, it is useful to represent the nuclear density ρ(r) as ρ(r) = ρcore(r) + ρexcess(r), (1) where ρcore refers to the Z protons plus the charge symmetric Z neutrons occupying the same nuclear ‘core’ orbits, and ρexcess refers to the (N − Z) ‘excess’ neutrons associated with the nuclear periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 3 Optical potential The optical potential employed in this work, Vopt Λ (ρ) = V(2) Λ (ρ)+V(3) Λ (ρ), consists of terms representing two-body ΛN and three-body ΛNN interactions, respectively: V(2) Λ (ρ) = − 4π 2µΛ fA CPauli(ρ) b0ρ, (2) V(3) Λ (ρ) = + 4π 2µΛ fA B0 ρ2 ρ0 , (3) with b0 and B0 strength parameters in units of fm (ℏ = c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' In these expressions, ρ(r) is a nuclear density distribution normalized to the number of nucleons A, ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='17 fm−3 stands for nuclear-matter density, µΛ is the Λ-nucleus reduced mass and fA is a kinematical factor transforming b0 from the ΛN c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' system to the Λ-nucleus c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' system: fA = 1 + A − 1 A µΛ mN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (4) This form of fA coincides with the way it is used for V(2) Λ in atomic/nuclear hadron-nucleus bound- state problems [21] and its A dependence provides good approximation for V(3) Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Next is the density dependent factor CPauli(ρ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (2), standing for a Pauli correlation function: CPauli(ρ) = (1 + αP 3kF 2π fAb0)−1, (5) with Fermi momentum kF = (3π2ρ/2)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The parameter αP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (5) switches off (αP=0) or on (αP=1) Pauli correlations in a form suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [22] and practised in K− atoms studies [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' To estimate 1/A correction terms, we also approximated CPauli(ρ) by [19]: CPauli(ρ) ≈ (1 + αP 3kF 2π blab 0 )−1, blab 0 = (1 + mΛ mN ) b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (6) As shown below, including CPauli(ρ) in V(2) Λ affects strongly the balance between the derived potential depths D(2) Λ and D(3) Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' However, introducing it also in V(3) Λ is found to make little difference, which is why it is skipped in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Finally we note that the low-density limit of Vopt Λ requires according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [16] that b0 is identified with the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' ΛN spin-averaged scattering length (positive here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 4 Data Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Binding energies in MeV, including uncertainties, considered here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' taken from Table IV of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' hypernucleus 1sΛ ± 1pΛ ± 12 ΛB 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='04 13 ΛC 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 16 ΛN 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='18 28 ΛSi 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 32 ΛS 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='5 51 ΛV 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 89 ΛY 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 139 ΛLa 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 208 ΛPb 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 The present work does not attempt to reproduce the full range of BΛ data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' It is limited to 1sΛ and 1pΛ states listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' We fit to such states in one of the nuclear 1p-shell hy- pernuclei listed in the table, where the 1sΛ state is bound by over 10 MeV while the 1pΛ state has just become bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This helps resolve the density dependence of Vopt Λ by setting a good balance between its two components, V(2) Λ (ρ) and V(3) Λ (ρ), following it all the way to 208 ΛPb the heaviest hypernucleus marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' We chose to fit the 16 ΛN precise Bexp Λ (1s, 1p) values derived, respectively, from the first and third peaks to the left in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The extremely simple 1p proton hole structure of the 15N nuclear core in this case removes most of the uncertainty arising from spin-dependent residual ΛN interactions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The fitted optical-potential parameters b0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (2), and B0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (3), are then used to calculate the B1s,1p Λ values of the other eight species listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 20 15 10 5 0 5 10 15 Binding Energy (MeV) 0 1 2 3 4 dσ/(dΩedΩKdEedEb) [nb/(sr 2GeV MeV)] fit SLA BS3 16 ΛN Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 16O(e, e′K+) spectrum of 16 ΛN from JLab Hall A measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Figure adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 5 Results The two strength parameters b0, B0 of the optical potential terms Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (2,3) were obtained by fitting to the 16 ΛN Bexp Λ (1s, 1p) values listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Suppressing Pauli correlations by setting αP = 0 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (5,6), the resulting Λ potential depth DΛ = −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 MeV reflects a sizable cancellation between a strongly attractive two-body potential depth D(2) Λ and a strongly repulsive three-body potential depth D(3) Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The overall agreement between calculations and experiment is acceptable, but some under- binding appears to develop for increasing mass numbers A, noticed clearly in the three heaviest 1sΛ and two heaviest 1pΛ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The resulting b0 is about half of the known Λp scattering length of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='1) fm [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' When the full potential Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (2-6) is used (marked here as model X, including Pauli correlations through αP = 1) the overall picture remains unchanged regarding underbinding for the heavier ele- ments, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' However, the fit parameter b0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='85 fm agrees now with the Λp scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The other parameter, B0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='170 fm, is about twice smaller than for αP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 10 100 A 0 5 10 15 20 25 30 binding energy (MeV) Λ−A binding energies 1s 1p model X Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' B1s,1p Λ (A) values from model X compared with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Continuous lines connect calculated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The phenomenon of underbinding associated with the optical potential Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (2-6) is likely to be a result of the use of ρ2 in nuclei where excess neutrons occupy shell-model orbits higher than those occupied by protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This situation occurs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 3 for the four hypernuclei with A ≳ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Expecting that direct three-body ΛNN contributions involving one ‘core’ nucleon and one ‘excess’ nucleon vanish upon summing on the T=0 ‘core’ closed-shell nucleons, we modify ρ2 = (ρcore + ρexcess)2 by discarding the bilinear term ρcore ρexcess, thereby replacing ρ2 in V(3) Λ , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (3), by ρ2 core + ρ2 excess = (2ρp)2 + (ρn − ρp)2 (7) in terms of the input densities ρp and ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This ansatz is consistent with an overall isospin factor τ1 · τ2 in two-pion exchange ΛNN forces, as first realized back in 1958 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Results of applying this ansatz are shown in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 4 as model Y, where the underbinding of calculated 1sΛ and 10 100 A 0 5 10 15 20 25 30 binding energy (MeV) 10 100 0 5 10 15 20 25 30 binding energy (MeV) 1s 1p Λ−A binding energies Λ−A binding energies 1s 1p model Y0 model Y Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' B1s,1p Λ (A) values from models Y0 and Y compared with data, see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Continuous lines connect calculated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 1pΛ binding energies noticed in model X is no longer observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The fit parameters, nevertheless, are the same as for model X above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' In the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 4, model Y0 shows similar results where the Pauli-correlations correction in model Y, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (6), is replaced by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This provides a rough estimate of the impact of 1/A corrections typical for our optical-potential methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Potential depth values in model Y are D(2) Λ = −41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 MeV, D(3) Λ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='9 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' To estimate uncertainties, we act as follows: (i) decreasing the input value of B1s Λ (16 ΛN) fitted to by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 MeV, thereby getting halfway to the central value of B1s Λ (16 ΛO)=(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4) MeV for 16 ΛO [29] the charge-symmetric partner of 16 ΛN, results in approximately 10% larger value of D(3) Λ , and (ii) applying Pauli correlations to V(3) Λ too reduces D(3) Λ roughly by 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' In both cases D(2) Λ increases moderately by ≲1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' On the other hand, D(2) Λ decreases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7 MeV if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (5) is used for CPauli(ρ) instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Considering these uncertainties, our final values are (in MeV) D(2) Λ = −40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='0 D(3) Λ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 DΛ = −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7 (8) 6 Discussion The D(2) Λ and D(3) Λ values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (8) are considerably smaller than those deduced in QMC calcula- tions [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Note that the QMC nuclear densities ρQMC(r) are much too compact with respect to our realistic densities, with nuclear r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' radii rN(QMC) about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='8 of the known r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' charge radii in 16O and 40Ca [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Since ρ scales as r−3 N , applying it to the density dependence of our Vopt Λ would transform D(2) Λ and D(3) Λ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (8) to as large depth values as D(2) Λ (QMC)=(−79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='0) MeV and D(3) Λ (QMC)=(53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='0±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='3) MeV, their sum DΛ(QMC)=(−26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='3±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7) MeV agreeing within uncertainties with ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Λ-nuclear potential depths (in MeV) from two SHF calculations fitting BΛ data points, and from our own Vopt Λ (αP = 0) two-parameter (b0, B0) fit to the two B1s,1p Λ (16 ΛN) values listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Method Data Points D(2) Λ D(3) Λ DΛ SHF [2] 3 −57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='8 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 SHF [17] 35 −55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='0 Vopt Λ (αP = 0) [13] 2 −57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='2 −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='4 Smaller-size but still inflated values of D(2) Λ and D(3) Λ are obtained by applying the Skyrme Hartree Fock (SHF) methodology [2, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Apart from small nonlocal potential terms and effective mass corrections, the SHF Λ-nuclear mean-field potential VΛ(ρ) consists of two terms: V(2) Λ (ρ) ∝ ρ and V(3) Λ (ρ) ∝ ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A large-scale SHF fit [17] of the corresponding Λ potential depths to 35 BΛ data points is listed in the middle row of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' We note that the overall DΛ = −35 MeV value becomes −31 MeV upon including a Λ effective-mass correction, a bit closer to the other DΛ values listed in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Similar results, particularly for D(2) Λ , can be obtained in fact by choosing a considerably smaller number of fitted data points, as shown by the fits listed in the other two rows of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The 11 MeV difference between the D(3) Λ values derived in these two SHF calculations arises mostly from nonlocal lower-power density terms, like ρ5/3, present in [17] but absent in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Interestingly, the last row lists a fit to the two B1s,1p Λ (16 ΛN) values used here when Pauli correlations are suppressed, αP = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The sizable difference between D(2) Λ and D(3) Λ values listed in Table 2, all of which disregard Pauli correlations, and the Vopt Λ values listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (8) demonstrates the importance of including in Vopt Λ a Pauli-correlations term (αP = 1) starting as ρ4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 7 Summary In summary, we have presented a straightforward optical-potential analysis of 1sΛ and 1pΛ binding energies across the periodic table, 12 ≤ A ≤ 208, based on nuclear densities constrained by charge r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The potential is parameterized by constants b0 and B0 in front of two-body ΛN and three- body ΛNN interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' These parameters were fitted to precise Bexp Λ (1s, 1p) values in 16 ΛN [31] and then used to evaluate B1s,1p Λ values in the other hypernuclei considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Pauli correlations were found essential to establish a correct balance between b0 and B0, as judged by b0 coming out in the final model Y analysis close to the value of the ΛN spin-averaged s-wave scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Good agreement was reached in this model between the calculated B1s,1p Λ values and their corresponding Bexp Λ values, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The potential depth D(3) Λ derived here, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (8), suggests that in symmetric nuclear matter the Λ- nucleus potential becomes repulsive near three times ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Our derived depth D(3) Λ is larger by a few MeV than the one yielding µ(Λ) > µ(n) for Λ and neutron chemical potentials in purely neutron matter, respectively, under a ‘decuplet dominance’ construction for the underlying ΛNN interaction terms within a χEFT(NLO) model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' This suggests that the strength of the corresponding repulsive V(3) Λ optical potential component, as constrained in the present work by data, is sufficient to prevent Λ hyperons from playing active role in neutron-star matter, thereby enabling a stiff EoS that supports two solar-mass neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Acknowledgments One of us (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=') thanks Jiˇrí Mareš and other members of the HYP2022 organizing team for their generous hospitality during the Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' The present work is part of a project funded by the European Union’s Horizon 2020 research & innovation programme, grant agreement 824093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Tolos, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Fabbietti, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 112, 103770 (2020), and past neutron-star work cited therein [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Millener, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Dover, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 38, 2700 (1988) [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Hungerford, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Millener, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 88, 035004 (2016) [4] Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rijken, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Stoks, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Yamamoto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 59, 21 (1999) [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Nagels, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rijken, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Yamamoto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 99, 044003 (2019) [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Polinder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Haidenbauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Meißner, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 779, 244 (2006) [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Haidenbauer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Vidaña, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 56, 55 (2020) [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gerstung, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Kaiser, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Weise, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 56, 175 (2020), and references cited therein to earlier works on ΛNN interactions in χEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Haidenbauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Petschauer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Kaiser, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Meißner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Nogga, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Weise, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 915, 24 (2013) [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Haidenbauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Meißner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Nogga, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 56, 91 (2020) [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lonardoni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gandolfi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Pederiva, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 87, 041303(R) (2013) [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lonardoni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Pederiva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gandolfi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 89, 014314 (2014) [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' B 837, 137669 (2023) [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lonardoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lovato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gandolfi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Pederiva, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 114, 092301 (2015) [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Logoteta, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Vidaña, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Bombaci, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 55, 207 (2019) [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Dover, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Hüfner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lemmer, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (NY) 66, 248 (1971) [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Schulze, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Hiyama, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 90, 047301 (2014), and past SHF work cited therein [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Elton, Nuclear Sizes (Oxford University Press, Oxford, 1961) [19] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' B 820, 136555 (2021) [20] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Angeli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Marinova, At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Data Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Data Tables 99, 69 (2013) [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 452, 89 (2007) [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Waas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rho, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Weise, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 617, 449 (1997) [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Gal, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 959, 66 (2017), and past K− atoms work cited therein [24] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Garibaldi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (Jefferson Lab Hall A Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' C 99, 054309 (2019) [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Millener, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' A 804, 84 (2008), 881, 298 (2012), 914, 109 (2013) [26] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Alexander, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 173, 1452 (1968) [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Budzanowski, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (HIRES Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' B 687, 31 (2010) [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Spitzer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 110, 1190 (1958) [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Agnello, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (FINUDA Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' B 698, 219 (2011) [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lonardoni, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' thesis, University of Trento, Italy, arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='6672 [nucl-th], in particular see Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content='7 [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Cusanno, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' (Jefferson Lab Hall A Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} +page_content=' 103, 202501 (2009)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE2T4oBgHgl3EQffAfS/content/2301.03923v1.pdf'} diff --git a/t9E0T4oBgHgl3EQf9QLr/content/tmp_files/2301.02800v1.pdf.txt b/t9E0T4oBgHgl3EQf9QLr/content/tmp_files/2301.02800v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e692c9414af63a5bb2686f9298a4856f49ab93d0 --- /dev/null +++ b/t9E0T4oBgHgl3EQf9QLr/content/tmp_files/2301.02800v1.pdf.txt @@ -0,0 +1,2561 @@ +arXiv manuscript No. +(will be inserted by the editor) +Simulation schemes for the Heston model with Poisson conditioning +Jaehyuk Choi · Yue Kuen Kwok +October 12, 2022 +Abstract Exact simulation schemes under the Heston stochastic volatility model (e.g., Broadie–Kaya and +Glasserman–Kim) suffer from computationally expensive Bessel function evaluations. We propose a new +exact simulation scheme without the Bessel function, based on the observation that the conditional integrated +variance can be simplified when conditioned by the Poisson variate used for simulating the terminal variance. +Our approach also enhances low-bias and time discretization schemes, which are suitable for derivatives with +frequent monitoring. Extensive numerical tests reveal the good performance of the new simulation schemes +in terms of accuracy, efficiency, and reliability when compared with existing methods. +Keywords Heston model · exact simulation · Poisson conditioning · gamma expansion · time discretization +schemes +Mathematics Subject Classification (2020) 60H35 · 65C05 · 91B70 +JEL Classification C63 · G12 · G13 +J. Choi +Rm 755, Peking University HSBC Business School, University Town, Nanshan, Shenzhen 518055, China +Tel.: +86–755–2603–0568 E-mail: jaehyuk@phbs.pku.edu.cn +Y. K. Kwok +Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China +Tel.: +852–2358–7418 E-mail: maykwok@ust.hk +arXiv:2301.02800v1 [q-fin.MF] 7 Jan 2023 + +2 +J. Choi and Y. K. Kwok +1 Introduction +The Heston [11] stochastic volatility model relaxes the constant volatility assumption in the Black–Scholes +(BS) model by taking the instantaneous variance to follow the square root diffusion process with mean +reversion, commonly called the Cox–Ingersoll–Ross (CIR, [9]) process. It is the most popular stochastic +volatility model for market practitioners because of its analytic tractability in computing the prices of +European options. Owing to the existence of a closed-form formula for the characteristic function of the +log-asset price, model calibration of market-observable European option prices can be performed efficiently +using the Fourier inversion algorithm [15, 18]. For pricing path-dependent options under the Heston model, +Monte Carlo (MC) simulations are often used. However, the standard Euler and Milstein time discretization +simulation schemes suffer from a high bias owing to the square root of the diffusion function in the variance +process. Negative values of variance from the simulation must be heuristically set to zero before taking the +square root of variance. In addition, the square root function violates the Lipschitz condition; therefore, the +convergence properties of the discretization scheme may not be guaranteed. There have been numerous fixes +to these issues to minimize discretization biases. A comprehensive review of these discretization schemes +using various fixes can be found in Lord et al. [19]. +A major breakthrough was made by Broadie and Kaya [5] in the simulation of the Heston model from +its exact distribution. Their exact simulation procedures consist of three steps: (i) sampling of the termi- +nal variance conditional on the initial variance, (ii) sampling the integrated variance conditional on the +initial and terminal variance values, and (iii) sampling the asset price process conditional on the variance +and integrated variance. As the exact simulation approach avoids simulation bias, simulation errors remain +inversely proportional to the square root of the computational time budget. However, the Broadie–Kaya +exact simulation algorithm is not competitive in accuracy-speed comparison because it requires extensive +computational time to sample the conditional integrated variance via the numerical inversion of the Laplace +transform in each simulation path. To improve computational efficiency, one may use a caching technique to +sample the terminal variance and conditional integrated variance via precomputation and interpolation of the +appropriate inverse distribution functions [24, 26]. Despite its limitations, Broadie and Kaya [5]’s pioneering +work triggers the construction of exact simulation schemes for other stochastic volatility models, such as the +stochastic-alpha-beta-rho (SABR) [6, 8], 3/2 [2, 26], Wishart [13], and Ornstein–Uhlenbeck-driven stochastic +volatility model [17, 7]. +Based on Pitman and Yor [22]’s decomposition of Bessel bridges, Glasserman and Kim [10] show that con- +ditional integrated variance can be expressed as gamma expansions (GE). Significant computational speedup +can be achieved by sampling conditional integrated variance via the sums of the mixtures of gamma random +variates (with an approximation of the truncated terms). In a related study, Malham et al. [20] construct +a new series expansion for the conditional integrated variance in terms of double infinite weighted sums of +independent random variables through a measure change and decomposition of squared Bessel bridges. Tse +and Wan [23] propose a low-bias simulation algorithm by approximating the conditional integrated variance +with an Inverse Gaussian (IG) variate with matching mean and variance. With its lower computational +cost per time step, the IG scheme can be used as a multiperiod scheme for pricing path-dependent options. +Andersen [1] constructs a time discretization scheme, where the variance at discrete time points is simulated + +Simulation schemes for the Heston model with Poisson conditioning +3 +via the quadratic-exponential (QE) approximation with martingale correction. However, he only uses the +trapezoidal rule to approximate the conditional integrated variance. Such an approximation is acceptable +when the time step is small. +There is no one-size-fits-all solution among the simulation approaches. Each simulation method has its own +advantages depending on the monitoring frequency of the derivatives to price. Tse and Wan [23] summarize +that the decision among GE, IG, or QE schemes depends on the compromise between computational cost and +bias. The exact GE scheme is the best choice for European-style derivatives. Despite the high computation +cost per step, we need to simulate just one step up to expiry. For path-dependent derivatives with frequent +monitoring, the time-discretized QE scheme may be a better choice owing to its low computation cost per +step. When the number of monitoring instants is moderate, one may choose to use the low-bias IG scheme +as a compromise between exact and time discretization schemes. +Despite the advances in Heston simulation algorithms, the computational efficiency still has room for +improvement. The simulation methods mentioned above, except for the QE scheme, involve computationally +demanding evaluations of the modified Bessel function. This has been criticized as a bottleneck in computa- +tion. In particular, the GE scheme [10] involves the Bessel random variable, the sampling of which takes up +a significant portion of the computation time for the same reason. +The contributions of this study are summarized as follows. We propose enhanced Heston simulation +schemes in all ranges based on the key observation that the conditional integrated variance can be further +simplified when conditioned by the Poisson variate involved in the terminal variance. Consequently, the +computationally trivial Poisson variate replaces the Bessel variate in the GE scheme [10]. By adopting the +IG approximation [23] for series truncation also, the Poisson-conditioned GE scheme significantly enhances +both speed and accuracy. The IG scheme [23] is a special case of the new method, but the enhanced IG +scheme no longer requires the Bessel function for mean and variance calculations. Broadie and Kaya [5]’s +Laplace inversion scheme can also benefit from our approach since Poisson conditioning removes the Bessel +function from the Laplace transform. We also propose a Poisson-conditioned time discretization method +with the corresponding martingale correction method, which is suitable for pricing derivatives with frequent +monitoring. The new time discretization scheme compares favorably with Andersen [1]’s QE scheme. +The remainder of this paper is organized as follows. In Section 2, we introduce the Heston model and +its analytical properties. We then review existing simulation algorithms and discuss intrinsic computational +challenges. In Section 3, we show how Poisson conditioning can enhance existing algorithms. In Section 4, we +present extensive numerical tests that compare the performance of Poisson-conditioned simulation schemes +with existing simulation schemes for pricing European options and discretely monitored variance swaps. +Finally, we conclude this paper in Section 5. +2 Heston stochastic volatility model and existing simulation schemes +The dynamics of the asset price process St and instantaneous variance process Vt of the Heston stochastic +volatility model under a risk-neutral measure Q are governed by the following coupled stochastic differential + +4 +J. Choi and Y. K. Kwok +equations: +dSt +St += (r − q) dt + +� +Vt +� +ρ dZt + +� +1 − ρ2 dWt +� +, +(2.1) +dVt = κ(θ − Vt) dt + ξ +� +Vt dZt, +(2.2) +where Zt and Wt are independent Brownian motions, r is the riskless interest rate, q is the continuous dividend +yield, κ is the speed of mean reversion, θ is the mean reversion level, ξ is the volatility of the variance process, +and ρ ∈ [−1, 1] represents the correlation coefficient between St and Vt. The initial conditions S0 and V0 are +assumed to be strictly positive. The joint process (S, V ) is well known to be a time-homogeneous Markov +process. +2.1 Terminal variance, integrated variance, and asset return +We state several analytic properties of the variance, (conditional) integrated variance, and asset return, +which are necessary for the remainder of this paper. Variance Vt is governed by the CIR [9] process. It is well +known that the terminal variance VT given V0 observes a noncentral chi-square distribution characterized by +VT ∼ e− κT +2 +φT (κ) χ2 � +δ, V0 φT (κ)e− κT +2 +� +for +δ = 4κθ +ξ2 +and +φT (κ) = +2κ/ξ2 +sinh(κT/2), +(2.3) +where χ2(δ, λ) denotes a noncentral chi-square random variable with degrees of freedom δ and a noncentrality +parameter λ. The variance process Vt cannot reach zero for t > 0, provided that δ > 2 (the Feller condition). +However, Heston model parameters calibrated to the option market usually violate this condition. The mean +and variance of VT are given by +E(VT ) = θ + (V0 − θ)e−κT +and +Var(VT ) = ξ2 +κ (1 − e−κT ) +� +V0e−κT + θ +2(1 − e−κT ) +� +. +(2.4) +We define the average variance between times 0 and T as +R0,T = 1 +T +� T +0 +Vt dt. +(2.5) +The mean and variance of R0,T , known as [3] +E(R0,T ) = θ + (V0 − θ)1 − e−κT +κT +, +(2.6) +Var(R0,T ) = +ξ2 +κ2T +� +θ − 2(V0 − θ)e−κT + +� +V0 − 5θ +2 + +� +V0 − θ +2 +� +e−κT +� 1 − e−κT +κT +� +, +(2.7) +provide useful insights into this model. For example, mean E(R0,T ) is the fair strike of the continuously +monitored variance swap (see Section 4.2). In the uncorrelated (ρ = 0) Heston model, +� +E(R0,T ) plays the +role of the BS implied volatility in the option price approximation [3]. + +Simulation schemes for the Heston model with Poisson conditioning +5 +We also define the integrated variance between t = 0 and T, conditional on the initial and final variances, +as +I0,T (V0, VT ) = +�� T +0 +Vt dt +��� V0, VT +� +, +(2.8) +that satisfies I0,T = E(TR0,T |V0, VT ). For notational simplicity, we simply write I0,T , assuming conditional +dependence on V0 and VT implicitly. Any expectation regarding I0,T should be understood as a conditional +expectation, E(f(I0,T )) := E(f(I0,T ) | V0, VT ), for any function f unless stated otherwise. +The exact simulation scheme is constructed, starting with the following analytic representation of the +asset price process St of the Heston model: +ST = S0 exp +� +(r − q)T − 1 +2 +� T +0 +Vt dt + ρ +� T +0 +� +Vt dZt + +� +1 − ρ2 +� T +0 +� +Vt dWt +� +. +(2.9) +By integrating (2.2), conditional on VT , we obtain +� T +0 +� +Vt dZt = 1 +ξ [VT − V0 + κ(I0,T − θT)]. +Conditional on VT and I0,T , we deduce that the log return, ln(ST /S0), can be sampled from a normal +distribution: +ln ST +S0 +∼ (r − q)T − I0,T +2 ++ ρ +ξ [VT − V0 + κ(I0,T − θT)] + Σ0,T Z, +(2.10) +where Z is a standard normal variate and Σ0,T is the standard deviation, as defined by +Σ0,T = +� +(1 − ρ2)I0,T . +Therefore, asset price ST can be simulated as a geometric Brownian motion: +ST = FT exp +� +Σ0,T Z − 1 +2Σ2 +0,T +� +, +(2.11) +where FT is the forward stock price, conditional on S0, V0, VT , and I0,T : +FT = E(ST | S0, V0, VT , I0,T ) += S0e(r−q)T exp +� +−ρ2 +2 I0,T + ρ +ξ [VT − V0 + κ(I0,T − θT)] +� +. +(2.12) +Similar to I0,T , we simply write FT in the later expression, assuming the conditional dependence on S0, V0, +VT , and I0,T to be implicit. +Therefore, the simulation of ST given S0 and V0 reduces to sampling VT and I0,T sequentially in each +simulation path. As VT can be sampled with relative ease from a noncentral chi-square distribution, the +challenge in the Heston model simulation lies in sampling the conditional integrated variance I0,T . + +6 +J. Choi and Y. K. Kwok +2.2 Broadie–Kaya exact simulation +Broadie and Kaya [5] perform simulation of I0,T via the numerical Laplace inversion of the conditional +Laplace transform of I0,T with the following analytic from Pitman and Yor [22]: +E +� +e−uI0,T � += exp +� +− V0+VT +2 +cosh( κuT +2 )φT (κu) +� +exp +� +− V0+VT +2 +cosh( κT +2 )φT (κ) +� φT (κu) +φT (κ) +Iν (zu) +Iν (z) , +(2.13) +where +ν = δ +2 − 1 = 2κθ +ξ2 − 1, z = +� +V0 VT φT (κ), zu = +� +V0 VT φT (κu), κu = +� +κ2 + 2ξ2u, +(2.14) +and Iν(z) is a modified Bessel function of the first kind: +Iν(z) = +∞ +� +k=0 +(z/2)ν+2k +k! Γ(k + ν + 1). +(2.15) +This sampling procedure for I0,T is very time-consuming because the Laplace inversion algorithm requires +cumbersome numerical evaluations of Iν(z) over the grid of z values. Moreover, such evaluations must be +performed for each simulation path because the conditional Laplace transform depends on VT . +2.3 Gamma expansion derived from the Bessel bridge decomposition +A significant improvement in computational efficiency can be achieved if numerical inversion of the Laplace +transform can be avoided. Based on Bessel bridge decomposition [22], Glasserman and Kim [10] express the +conditional integrated variance I0,T as the infinite sum of the gamma random variables. +Let POIS(λ) denote the Poisson random variable with rate λ, and Γ1(α) denote the standard (i.e., unit +scale) gamma random variable with shape parameter α. The probability mass function of POIS(λ) and +probability density function of Γ1(α) are respectively given by +PPOIS(j; λ) = λj e−λ +j! +(j = 0, 1, . . .) +and +fΓ1(x; α) = xα−1e−x +Γ(α) +(x > 0). +Then, I0,T can be expressed as [10, Theorem 2.2] +I0,T ∼ X + Zδ/2 + +η0,T +� +j=1 +Z(j) +2 , +(2.16) +where +X ∼ +∞ +� +k=1 +1 +γk +Γ1(nk) +and +Zα ∼ +∞ +� +k=1 +1 +γk +Γ1(α) +for +nk(V0, VT ) ∼ POIS((V0 + VT )λk) i.i.d., +λk = +16k2π2 +ξ2T(κ2T 2 + 4k2π2) +and +γk = κ2T 2 + 4k2π2 +2ξ2T 2 +, + +Simulation schemes for the Heston model with Poisson conditioning +7 +and Z(j) +α +denotes independent identically distributed copies of Zα. The last term in the infinite series (2.16) +involves the summation up to a Bessel random variable η0,T with dependence on V0 and VT , where +η0,T (V0, VT ) ∼ BES(ν, z), +(2.17) +where ν and z are as defined in (2.14). The Bessel random variable, η ∼ BES(ν, z) with ν > −1 and z > 0, +takes non-negative integer values, and its probability mass function is given by the normalized coefficients +of Iν(z) in (2.15): +PBES(j; ν, z) = +(z/2)2j+ν +Iν(z) j! Γ(j + ν + 1) +(j = 0, 1, . . .). +(2.18) +In the actual numerical implementation of evaluating the infinite gamma series, it is necessary to evaluate +the sum of only finite K terms and properly approximate the truncated terms, +XK ∼ +∞ +� +k=K+1 +1 +γk +Γ1(nk) +and +ZK +α ∼ +∞ +� +k=K+1 +1 +γk +Γ1(α). +Glasserman and Kim [10, Proposition 3.2] propose to approximate the three truncated terms of X, Zδ/2, +and Z2 with gamma random variables with matching mean and variance. The mean and variance of the +truncated terms are available analytically. Glasserman and Kim [10] start with the derivation of the mean +and variance of X and Zα: +E(X) = (V0 + VT ) +∞ +� +k=1 +λk +γk += (V0 + VT ) +∞ +� +k=1 +32π2k2 T +(κ2T 2 + 4k2π2)2 = (V0 + VT )mXT, +Var(X) = (V0 + VT ) +∞ +� +k=1 +2λk +γ2 +k += (V0 + VT ) +∞ +� +k=1 +128π2k2 ξ2T 3 +(κ2T 2 + 4k2π2)3 = (V0 + VT )vXξ2T 3, +E(Zα) = +∞ +� +k=1 +α +γk += +∞ +� +k=1 +2αξ2T 2 +κ2T 2 + 4k2π2 = αmZξ2T 2, +Var(Zα) = +∞ +� +k=1 +α +γ2 +k += +∞ +� +k=1 +4αξ4T 4 +(κ2T 2 + 4k2π2)2 = αvZξ4T 4, +(2.19) +where +mX = c1 − ac2 +2a +, +vX = c1 + ac2 − 2a2c1c2 +8a3 +, +mZ = ac1 − 1 +4a2 +, +vZ = ac1 + a2c2 − 2 +16a4 +a = κT +2 , +c1 = +1 +tanh a, +and +c2 = +1 +sinh2 a. +(2.20) +The mean and variance of XK and ZK +α are then obtained by subtracting the first K terms from those of X +and Zα, respectively. For example, +E(XK) = E(X) − (V0 + VT ) +K +� +k=1 +λk +γk +and +E(ZK +α ) = E(Zα) − +K +� +k=1 +α +γk +. + +8 +J. Choi and Y. K. Kwok +It is obvious that K serves as a parameter controlling the accuracy of the simulation scheme; a higher K +implies a smaller error at the tradeoff of a higher computation cost. Note that the numerical implementation +of the GE scheme (2.16) eventually requires (2+η0,T )(K + 1) gamma random variables per simulation path. +The GE representation of I0,T proposed by Glasserman and Kim [10] avoids the tedious numerical +inversion of the conditional Laplace transform of I0,T as implemented by Broadie and Kaya [5]. However, +substantial computational effort is still required to simulate the Bessel random variable η0,T , as it involves +Iν(z) as well.1 +2.4 Inverse Gaussian approximation based on matching moments +Tse and Wan [23] propose a low-bias simulation scheme, where I0,T is approximated by an IG variate with +matching mean and variance of I0,T . They argue that the IG variable is the best candidate for approximating +I0,T because the two converge in the distribution as T → ∞. Let IG(µ, λ) denote the IG random variable +with the parameters µ and λ. The density function of IG(µ, λ) takes the form, +fIG(x; µ, λ) = +� +λ +2πx3 exp +� +−λ(x − µ)2 +2µ2x +� +for +µ > 0, λ > 0, and x > 0, +(2.21) +and the mean and variance of IG(µ, λ) are given by µ and µ3/λ, respectively. Parameters µ and λ are +determined by matching the mean and variance of I0,T : +µ = E(I0,T ) +and +λ = E(I0,T )3 +Var(I0,T ). +The exact mean and variance of I0,T are as follows [23, Proposition 3.1]: +E(I0,T ) = E(X) + E(Zδ/2) + E(η0,T )E(Z2) +Var(I0,T ) = Var(X) + Var(Zδ/2) + E(η0,T )Var(Z2) + Var(η0,T )E(Z2)2, +(2.22) +where the mean and variance of X and Zα are given in (2.19) and +E(η0,T ) = z Iν+1(z) +2Iν(z) +and +Var(η0,T ) = z2 Iν+2(z) +4Iν(z) ++ E(η0,T ) − E(η0,T )2. +(2.23) +Once µ and λ have been determined, it is trivial to sample the IG(µ, λ) variate from the algorithm of Michael +et al. [21]. +Note that the error-controlling parameter, similar to K in the GE scheme, is not present in the IG +approximation. As Tse and Wan [23] demonstrate, the only way to reduce the simulation bias is to decrease +the time interval (i.e., increasing the number of simulation steps). Therefore, we can classify their method +as a low-bias scheme rather than an exact simulation scheme. +However, to obtain E(I0,T ) and Var(I0,T ), the modified Bessel functions (i.e., Iν(z), Iν+1(z), and Iν+2(z)) +must be evaluated for each path, which causes a computational burden. To avoid this problem, Tse and Wan +1 See Glasserman and Kim [10, Table 4] for the computation time sampling the Bessel random variables. + +Simulation schemes for the Heston model with Poisson conditioning +9 +[23] pre-compute E(η0,T ) and Var(η0,T ) for a grid of equally spaced V0VT values, thanks to the property +that η0,T depends on V0 and VT via z = √V0VT φT (κ). Interpolation within the tabulated values is then +performed. In addition, they propose the interpolation–Poisson–zero (IPZ) scheme for faster sampling of the +noncentral chi-square variate for VT (see more comments in Section 4). +2.5 Time discretization with QE approximation, trapezoidal rule, and martingale correction +For pricing path-dependent derivatives such as Asian options and variance swaps, it is necessary to sample +asset prices at frequent time points. For this purpose, the simulation schemes based on time discretization +(e.g., Euler or Milstein) become competitive in terms of the accuracy-speed tradeoff over the exact (e.g., GE) +or low-bias (e.g., IG) schemes. The time discretization scheme is distinguished from those discussed earlier in +that the conditional variance is approximated as a deterministic value rather than as a random variate from +the distribution of I0,T , be it exact or approximate. Therefore, it must be successively applied to small time +intervals. Despite this limitation, the computational cost per time step is lower than that of other schemes. +Therefore, time discretization is the preferred scheme if such a short time step is required for the derivatives +because of the high monitoring frequency. +The Euler and Milstein schemes are direct time discretizations of the stochastic differential equation. In +the Heston model, however, these two simple schemes are notorious for failing with a large bias owing to +the square root process for variance. Among the numerous studies on time discretization schemes under the +Heston model (e.g., [12, 19]), the QE scheme proposed by Andersen [1] has been widely recognized as the +best scheme [24]. The QE scheme is briefly reviewed below: +Sampling variance: QE approximation. Under time discretization, the derivative life, [0, T], is divided +into N equal time intervals of size h (T = Nh). The monitoring points within the time interval are specified +as ti = ih (i = 0, 1, · · · , N). As it is more concise to use index i than time ti, we change the notation +convention in this section (and in Section 3.5 later) by using i in subscripts. For example, we use +Vi := Vti +and +Ii,i+1 := Iti,ti+1 (I0,N := I0,T ). +In the QE scheme, Vi+1 given Vi is sampled using approximate functional forms. First, using (2.4), we +calculate the ratio, ψ = Var(Vi+1|Vi)/E(Vi+1|Vi)2, as a proxy for the probability that Vi+1 hits the origin. +Then, the simulation is split into two cases depending on ψ: +Vi+1 = +� +� +� +a(b + Z)2 +if +ψ ≤ 1.5 +1 +β ln +� +1−p +1−U +� +1U>p +if +ψ > 1.5 +, +(2.24) +where Z and U are standard normal and uniform random variables, respectively; 1x is the indicator function; +and the coefficients a, b, β, and p are determined to match E(Vi+1 | Vi) and Var(Vi+1 | Vi) in each case. + +10 +J. Choi and Y. K. Kwok +Trapezoidal rule. After simulating Vi+1 from Vi, Andersen [1] adopts a simple trapezoidal rule to approx- +imate the conditional integrated variance Ii,i+1 as follows: +ITZ +i,i+1 = (Vi + Vi+1)h +2 . +(2.25) +Given the simulation path {Vi : i = 0, 1, · · · , N}, I0,N over the entire period is approximated by +ITZ +0,N = +N−1 +� +i=0 +ITZ +i,i+1 = (V0 + 2V1 + · · · + 2VN−1 + VN)h +2 . +(2.26) +As previously noted, ITZ +i,i+1 (conditional on Vi and Vi+1) is a deterministic value. This feature marks an +important difference compared with the GE and IG schemes, where Ii,i+1 is sampled as a random variable. +Martingale correction. Finally, to ensure the martingale condition, Si = e(q−r)hE(Fi+1|Si), Andersen [1] +modifies the conditional forward price in (2.12) by adding a correction term, M QE +i,i+1: +Fi+1 = Sie(r−q)h exp +� +−ρ2 +2 ITZ +i,i+1 + ρ +ξ [Vi+1 − Vi + κ(ITZ +i,i+1 − θh)] + M QE +i,i+1 +� +. +The martingale correction term, M QE +i,i+1, is analytically determined from the approximation function (2.24): +M QE +i,i+1 = ρκθ +ξ h − A2Vi + +� +� +� +− A1b2a +1−2A1a + 1 +2 ln(1 − 2A1a) +if ψ ≤ 1.5 +− ln +� +p + β(1−p) +β−A1 +� +if ψ > 1.5 +and +A1,2 = ρh +4 +�2κ +ξ − ρ +� +± ρ +ξ , +where A1 and A2 takes + and − respectively, and a, b, β, and p are the coefficients used in the QE step in +(2.24). +3 Poisson conditioning and enhanced simulation schemes +In this section, we construct efficient simulation schemes for the Heston model based on the key observations +of Poisson conditioning, which simplifies the model formulation. The primary merit of Poisson conditioning +is that it removes the use of Bessel functions or Bessel random variables, the numerical evaluation of which +is computationally demanding. +3.1 Poisson conditioning +It is well known that the noncentral chi-square random variable χ2(δ, γ) is a Poisson mixture of an ordinary +(γ = 0) chi-square variables, and that an ordinary chi-square distribution is a special case of a gamma +distribution, χ2(δ, 0) ∼ 2Γ1(δ/2): +χ2(δ, γ) ∼ χ2 � +δ + 2POIS +�γ +2 +� +, 0 +� +∼ 2 Γ1 +�δ +2 + POIS +�γ +2 +�� +. + +Simulation schemes for the Heston model with Poisson conditioning +11 +With these properties, the exact simulation scheme of VT in (2.3) can be alternatively recast as the following +Poisson mixture gamma scheme: +µ0 ∼ POIS +� +V0 φT (κ)e− κT +2 +2 +� +, +so that +VT ∼ 2e− κT +2 +φT (κ) Γ1 +�δ +2 + µ0 +� +. +(3.1) +Note that VT depends on V0 via µ0. This property is well known and is often used in the Heston simulation +literature [see 24, 10, 23, for example]. Our new simulation schemes also use this Poisson mixture property +to simulate VT . Unlike other schemes, the intermediate Poisson variable µ0 is also essential for simulating +I0,T , in addition to VT in our proposed scheme, as detailed below. +The choice of (3.1) for simulating VT does not significantly increase the computation time. Our numerical +tests with public numerical library show that (3.1) is marginally slower than simulating the noncentral chi- +square variable directly. The small difference is attributed to the generation of the Poisson variable µ0. +Interestingly, we also observe that (3.1) is comparable to Andersen’s QE procedure for generating VT (see +more comments in Section 3.5). +Our key observation for Poisson conditioning is the link between the Poisson random variable µ0 in (3.1) +and the Bessel random variable η0,T in (2.17). According to Pitman and Yor [22, Eq. (5.j)], the Bessel random +variable η ∼ BES(ν, z) can alternatively be represented as a conditional Poisson random variable: +µ ∼ POIS(λ) +conditional on +Γ1(ν + 1 + µ) = z2 +4λ, +where λ is a positive rate parameter that can be chosen arbitrarily. This can be derived from the observation +that the joint probability of µ and Γ1(ν + 1 + µ) is proportional to the probability mass function of η in +(2.18): +PPOIS(k; λ) fΓ1 +� z2 +4λ; ν + 1 + k +� += λk e−λ +k! +� +z2 +4λ +�ν+k +e− z2 +4λ +Γ(ν + 1 + k) = +� z +2λ +�ν +e−λ− z2 +4λ +(z/2)2k+ν +k! Γ(k + ν + 1). +Therefore, the alternative representation is proven as follows: +Prob +� +µ = j +��� Γ1(ν + 1 + µ) = z2 +4λ +� += +PPOIS(j; λ) fΓ1 +� +z2 +4λ; ν + 1 + j +� +�∞ +k=0 PPOIS(k; λ) fΓ1 +� z2 +4λ; ν + 1 + k +� = PBES(j; ν, z). +Instead of taking λ = 1 as in Glasserman and Kim [10, Remark 2.3], we can achieve a useful representation +that resembles that of µ0 by judiciously choosing λ = V0 φT (κ) +2 +e− κT +2 in the context of the Heston model. By +substituting ν = δ +2 − 1 and z = √V0VT φT (κ), η0,T ∼ BES(ν, z) is equivalent to the conditional Poisson +variable: +µ ∼ POIS +� +V0 φT (κ)e− κT +2 +2 +� +conditional on +Γ1 +�δ +2 + µ +� += z2 +4λ = VT φT (κ) +2e− κT +2 +. + +12 +J. Choi and Y. K. Kwok +Consequently, µ coincides with µ0 in (3.1). Simultaneously, we observe that the required condition for µ is +equivalent to the formula for VT in (3.1): +VT = 2e− κT +2 +φT (κ) Γ1 +�δ +2 + µ +� +. +Therefore, we conclude that η0,T is equivalent to µ0 conditional on the terminal variance VT : +η0,T +∼ +µ0 +��� VT = 2e− κT +2 +φT (κ) Γ1 +�δ +2 + µ0 +� +. +(3.2) +This implies that the joint distribution of (VT , η0,T ) is equivalent to that of (VT , µ0) as long as µ0 and +VT follow the relation in (3.1). Therefore, η0,T can be simply replaced by µ0 when sampling I0,T , and the +cumbersome evaluation of the modified Bessel function is completely avoided. Under Poisson conditioning, +(2.16) of the GE scheme can be further simplified to +I0,T | µ0 ∼ X + Zδ/2 + +µ0 +� +j=1 +Z(j) +2 . +(3.3) +This is the key result of an efficient approach known as Poisson conditioning. In the remainder of this section, +we show how various Heston simulation schemes can be simplified under the Poisson conditioning framework. +3.2 Poisson-conditioned IG approximation +First, we enhance Tse and Wan [23]’s IG approximation using Poisson conditioning. The Poisson-conditioned +IG approximation in this section is a special case of the more general Poisson-conditioned GE scheme +introduced in the next section. Nevertheless, we first discuss this scheme because the mean and variance of +I0,T | µ0 to be derived in (3.4) will be used later. +Recall that the modified Bessel function in (2.23) is the computational bottleneck in the original IG +scheme. Under Poisson conditioning, we observe E(η0,T |µ0) = µ0 and Var(η0,T |µ0) = 0 because η0,T is now +conditioned as µ0. Consequently, the mean and variance of I0,T | µ0 can be simplified from (2.22): +E(I0,T | µ0) = E(X) + E(Zδ/2) + µ0E(Z2) = (V0 + VT )mXT + +�δ +2 + 2µ0 +� +mZξ2T 2, +Var(I0,T | µ0) = Var(X) + Var(Zδ/2) + µ0Var(Z2) = (V0 + VT )vXξ2T 3 + +�δ +2 + 2µ0 +� +vZξ4T 4, +(3.4) +which are expressed in terms of elementary functions, with no reference to the Bessel function. Therefore, +we expect significant speedup in calculating the conditional mean and variance of I0,T for moment matching +with the IG variate. + +Simulation schemes for the Heston model with Poisson conditioning +13 +3.3 Poisson-conditioned GE scheme +We present the main results of this study, the Poisson-conditioned GE scheme. For the simulation of I0,T | µ0 +in (3.3), we apply two additional enhancements. +Aggregating gamma random variables. First, using the additive property of the independent gamma +variables, +Γ1(α1) + Γ1(α2) ∼ Γ1(α1 + α2), +we can combine the (2 + µ0) gamma variables in (3.3) as follows: +I0,T | µ0 ∼ +∞ +� +k=1 +� +� 1 +γk +Γ1(nk) + 1 +γk +Γ1(δ/2) + +µ0 +� +j=1 +1 +γk +Γ1(2) +� +� ∼ +∞ +� +k=1 +1 +γk +Γ1 +� +nk + δ +2 + 2µ0 +� +, +(3.5) +turning the (2 + µ0) infinite series into a single series. This reduces the number of required gamma variates +by more than one-third of the original GE scheme. +Series truncation with the IG approximation. In the second enhancement, we improve the accuracy +of truncating the infinite series (3.5) using the IG approximation instead of the gamma approximation in +Glasserman and Kim [10]. Owing to the aggregated gamma variable, we apply the truncation to the whole +I0,T term rather than each of X, Zδ/2, or Z2 terms, as in Glasserman and Kim [10]. As Tse and Wan [23] +have already shown the effectiveness of the IG variate for approximating the entire I0,T , it is natural to +expect that the IG approximation also works for the remaining terms of I0,T . Our numerical tests confirm +this result. +Let IK +0,T | µ0 denote the remainder of the first K terms of (3.5) under Poisson conditioning. +IK +0,T | µ0 ∼ +∞ +� +k=K+1 +1 +γk +Γ1 +� +nk + δ +2 + 2µ0 +� +. +(3.6) +We approximate IK +0,T | µ0 using an IG random variable with a matched mean and variance. The mean and +variance of IK +0,T | µ0 can be easily obtained by subtracting those of the first K terms from (3.4): +E(IK +0,T | µ0) = (V0 + VT ) +� +mXT − +K +� +k=1 +λk +γk +� ++ +�δ +2 + 2µ0 +� � +mZξ2T 2 − +K +� +k=1 +1 +γk +� +, +Var(IK +0,T | µ0) = (V0 + VT ) +� +vXξ2T 3 − +K +� +k=1 +2λk +γ2 +k +� ++ +�δ +2 + 2µ0 +� � +vZξ4T 4 − +K +� +k=1 +1 +γ2 +k +� +. +(3.7) +As in (3.4), there is no reference to Bessel functions. +Note that the K = 0 case in our new scheme is reduced to the Poisson-conditioned IG scheme in +Section 3.2. However, this is not the case with the original GE and IG schemes, in terms of both the +type and number of random variables. The K = 0 case of the GE scheme consists of (2 + η0,T ) IG variates +approximating X, Zδ/2, and copies of Z2 whereas the IG scheme uses one IG variate. Therefore, the proposed + +14 +J. Choi and Y. K. Kwok +Poisson-conditioned GE scheme can be used flexibly as either an exact scheme (K ≥ 1 with one time step) +or a low-bias scheme (K = 0 with multiple time steps). +Finally, the simulation of VT and I0,T under Poisson conditioning can be succinctly performed using the +following simplified steps: +Simulation steps of the Poisson-conditioned GE scheme. +Step 1 Given V0, draw µ0 as +µ0 ∼ POIS +� +V0 φT (κ)e− κT +2 +2 +� +. +Step 2 Given µ0, draw VT as +VT ∼ 2e− κT +2 +φT (κ) Γ1 +�δ +2 + µ0 +� +. +Step 3 Given V0 and µ0, draw I0,T as +I0,T ∼ +K +� +k=1 +1 +γk +Γ1 +� +nk + δ +2 + 2µ0 +� ++ IG(λK, µK) +(3.8) +where nk ∼ POIS((V0 + VT )λk) and IG(λK, µK) is an IG variate with the mean and variance matched +to (3.7). +Step 4 Given V0, VT , I0,T , and S0, draw ST from (2.10). +Here, both µ0 and nk are sampled from the Poisson distribution, which is computationally trivial. The +simulations of VT and I0,T are indifferent to whether the Feller condition for the CIR variance process is +satisfied. +3.4 Poisson-conditioned Broadie–Kaya Laplace inversion scheme +Although the Broadie and Kaya [5] procedure of numerical inversion of the conditional Laplace transform +of I0,T is not recommended as an efficient numerical procedure, it may be instructive to show that Poisson +conditioning can also simplify their numerical procedure. +From Poisson-conditioning decomposition (3.3), the Laplace transform of I0,T conditional on µ0 is ob- +tained as follows: +E +� +e−uI0,T +��� µ0 +� += E +� +e−uX� +E +� +e−uZδ/2� +E +� +e−uZ2�µ0 += exp +� +− V0+VT +2 +cosh( κuT +2 )φT (κu) +� +exp +� +− V0+VT +2 +cosh( κT +2 )φT (κ) +� +�φT (κu) +φT (κ) +�δ/2+2µ0 +, +(3.9) +where κu is as defined in (2.14). This is a direct consequence of Glasserman and Kim [10, Lemma 2.4], who +observe +E +� +e−uX� += exp +� +− V0+VT +2 +cosh( κuT +2 )φT (κu) +� +exp +� +− V0+VT +2 +cosh( κT +2 )φT (κ) +� +and +E +� +e−uZα� += +�φT (κu) +φT (κ) +�α +. + +Simulation schemes for the Heston model with Poisson conditioning +15 +We expect to gain a significant computational benefit compared to the original Broadie–Kaya algorithm +because it no longer requires numerical evaluation of the modified Bessel functions, as in (2.13). +If we use the Laplace inversion of (3.9) instead of the Poisson-conditioned GE scheme in (3.8), Step 3 +can be replaced by the following alternative procedure: +Step 3’ Given V0 and µ0, draw I0,T from the cumulative distribution function obtained from the Laplace +inversion of (3.9). +However, this alternative Broadie–Kaya scheme is still slower than the Poisson-conditioned GE scheme. We +do not include this scheme in our numerical test. +Remark +The original Broadie–Kaya conditional Laplace transform (2.13) can also be derived in a similar manner from +the original decomposition of I0,T in (2.16). This serves as an alternative proof of (2.16), without resorting to +the Bessel bridge decomposition used in Glasserman and Kim [10]. We outline the derivation below, because +it also explains why (2.13) contains Iν(z) and how η0,T is related to Iν(z). +The Laplace transform of (2.16) is expressed as +E +� +e−uI0,T � += E +� +e−uX� +E +� +e−uZδ/2� +E +� +E +� +e−uZ2�η0,T � +. +It is already known that the Laplace transform of X corresponds to the first term in (2.13). For the second +and third terms, we use the probability-weighted average over η0,T . Based on the probability mass function +of η0,T in (2.18) and series expansion of Iν(z) in (2.15), we deduce the remaining terms in (2.13): +E +� +e−uZδ/2� +E +� +E +� +e−uZ2�η0,T � += +�φT (κu) +φT (κ) +�δ/2 ∞ +� +j=0 +P(η0,T = j) +�φT (κu) +φT (κ) +�2j +=φT (κu) +φT (κ) +∞ +� +j=0 +[√V0VT φT (κ)/2]δ/2−1+2j +Iν(√V0VT φT (κ))j! Γ(j + ν + 1) +�φT (κu) +φT (κ) +�δ/2−1+2j +=φT (κu) +φT (κ) +∞ +� +j=0 +[zu/2]ν+2j +Iν(z)j! Γ(j + ν + 1) = φT (κu) +φT (κ) +Iν(zu) +Iν(z) . +(3.10) +3.5 Poisson-conditioned time discretization scheme +The Poisson conditioning framework also enables the construction of a small time-interval simulation of +the integrated variance. The new Poisson-conditioned time discretization scheme competes favorably with +Andersen [1]’s QE scheme. +Sampling variance. To take advantage of Poisson conditioning, we sample Vi using the Gamma-Poisson +scheme in (3.1). In the time discretization setup, sampling Vi+1 given Vi is modified to +µi ∼ POIS +� +Viφh(κ)e− κh +2 +2 +� +, +then +Vi+1 ∼ 2e− κh +2 +φh(κ) Γ1 +�δ +2 + µi +� +, +(3.11) + +16 +J. Choi and Y. K. Kwok +where we use the subscript i instead of t by following a convention similar to that in Section 2.5. +As (3.11) is an exact sampling of Vi+1, it is more accurate than Andersen [1]’s QE scheme for large +time steps. The concern is computing speed. Surprisingly, our numerical tests (see Section 4) verify that the +execution time is comparable to that of QE approximation. It becomes slower than the QE step when time +step h becomes very small. This is because the Poisson rate of µi grows as 2Vi/(ξ2h) for a small h; therefore, +it takes more time to simulate µi. +Poisson-conditioned quadrature approximation. Next, we determine Ii,i+1 given Vi and Vi+1. Given +the results in (3.4), our natural choice for the deterministic value representing Ii,i+1 is its mean: +IPOIS +i,i+1 = E(Ii,i+1 | µi) = (Vi + Vi+1)mXh + +�δ +2 + 2µi +� +mZξ2h2, +(3.12) +where mX and mZ in (2.20) should be redefined with a = κh/2 under the time discretization. As mX and +mZ are constants independent of the simulation path, it is necessary to calculate them only once. Given the +simulation path, the conditional integrated variance over the entire time period is given by +IPOIS +0,N += (V0 + 2V1 + · · · + 2VN−1 + VN)mXh + +�Nδ +2 + 2(µ0 + µ1 + · · · + µN−1) +� +mZξ2h2. +(3.13) +Note that this approach is practically feasible owing to the computationally simple expression in (3.4) under +Poisson conditioning. The same approach without Poisson conditioning is not feasible because of the modified +Bessel function evaluation required in (2.23) to calculate E(I0,T ) and Var(I0,T ) in (2.22). +The new approximation IPOIS +i,i+1 improves over the naive trapezoidal approximation ITZ +i,i+1 in (2.25) and +forms the building block for our time discretization scheme for pricing path-dependent options. Indeed, IPOIS +i,i+1 +can be shown to be related to ITZ +i,i+1 in the limit of h ↓ 0. More precisely, based on the asymptotic expansion +in the powers of h (see Appendix A), it is interesting to observe +lim +h↓0 E +� +IPOIS +i,i+1 +� += +� +Vi + Vi+1 + +� +ViVi+1 +� h +3 ≈ (Vi + Vi+1)h +2 = ITZ +i,i+1. +(3.14) +It is not surprising that the asymptotic form of IPOIS +i,i+1 involves the arithmetic average, +Vi+Vi+1 +2 +, and the +geometric average, +� +ViVi+1, because these two quantities also appear in the Laplace transform of I0,T (see +(2.13)). Furthermore, the asymptotic expansion of IPOIS +i,i+1 in Appendix A reveals that the leading order of +truncation in the trapezoidal rule is O(h2). This may explain why the trapezoidal rule is also accurate in +approximating Ii,i+1 when the time step h is small. +Martingale correction. We derive the corresponding martingale correction in our scheme, which is different +from that in the QE scheme. The martingale correction starts by recognizing that Var(Ii,i+1 | µi), although +small, has been ignored because IPOIS +i,i+1 is a deterministic value. From (3.4), the missing variance is given by +Var(Ii,i+1 | µi) = (Vi + Vi+1)vXξ2h3 + +�δ +2 + 2µi +� +vZξ4h4. + +Simulation schemes for the Heston model with Poisson conditioning +17 +For a random variable X, the following approximation holds if the variance is small: +E(ea+bX) ≈ ea+bE(X)+ 1 +2 b2Var(X). +Therefore, under Poisson-conditioned time discretization, the conditional forward price (2.12) is corrected to +Fi+1|µi = Sie(r−q)hE +� +exp +� +−ρ2 +2 Ii,i+1 + ρ +ξ [Vi+1 − Vi + κ(Ii,i+1 − θT)] +� ��� µi +� +≈ Sie(r−q)h exp +� +−ρ2 +2 IPOIS +i,i+1 + ρ +ξ [Vi+1 − Vi + κ(IPOIS +i,i+1 − θT)] + M POIS +i,i+1 +� +, +(3.15) +where the martingale correction term M POIS +i,i+1 is given by +M POIS +i,i+1 = ρ2 +2 +�κ +ξ − ρ +2 +�2 +Var(Ii,i+1|µi) = ρ2 +2 +�κ +ξ − ρ +2 +�2 � +(Vi + Vi+1)vX + +�δ +2 + 2µi +� +vZξ2h +� +ξ2h3. +(3.16) +Note a subtle difference in the derivation of the martingale correction term compared to the QE scheme. In +the QE scheme, M QE +i,i+1 is fitted to satisfy the unconditional expectation, Si = e(q−r)hE(Fi+1|Si). In contrast, +in our scheme, we use the conditional expectation (2.12). Consequently, M POIS +i,i+1 is a function of both Vi and +Vi+1 whereas M QE +i,i+1 depends solely on Vi. +This approach provides more flexibility in determining the correction for missing variance in different con- +texts. In pricing variance swaps (see Section 4.2), we need to sample the realized return variance ln2(Si+1/Si) +rather than Si itself. Using the equality, +E +� +(a + bX)2� += (a + bE(X))2 + b2Var(X), +we can correct the realized return variance sampling as +ln2(Si+1/Si) ≈ +� +(r − q)h − IPOIS +i,i+1 +2 ++ ρ +ξ +� +Vi+1 − Vi + κ(IPOIS +i,i+1 − θh) +� ++ Σi,i+1Z +�2 ++ M ′POIS +i,i+1, +where the correction terms are different. +M ′POIS +i,i+1 = +�ρκ +ξ − 1 +2 +�2 +Var(Ii,i+1|µi). +(3.17) +In the QE scheme, it is unclear how to find martingale correction in the context of the realized return variance. +We verify the effectiveness of the new martingale corrections, (3.16) and (3.17), for pricing European options +and variance swaps, respectively, in our numerical tests in Section 4. +4 Numerical performance of the simulation schemes +We perform comprehensive numerical tests to assess the performance of the proposed simulation schemes +with Poisson conditioning in comparison to existing schemes. Specifically, we price European vanilla options + +18 +J. Choi and Y. K. Kwok +in Section 4.1 and variance swaps in Section 4.2 using various schemes. For easier reference, we label the +methods to be tested are as follows. +– GE: Glasserman and Kim [10]’s original GE scheme in Section 2.3 +– IG: Tse and Wan [23]’s IG approximation in Section 2.4. +– QEM: Andersen [1]’s QE scheme with the trapezoidal rule and martingale correction in Section 2.5. +– POIS–GE: Poisson-conditioned GE scheme in Section 3.3. +– POIS–TD: Poisson-conditioned time discretization scheme in Section 3.5 +The methods described above are implemented in Python. We aim to keep the implementation of the +schemes as simple as possible for a clean and fair performance comparison. In this regard, we do not adopt +the IPZ algorithm [23, Algorithms 1 and 2], which is a technique to speed up the sampling of (3.1) using the +tabulated inverse distribution function of Γ1(δ/2) corresponding to µ0 = 0. As the IPZ algorithm would ben- +efit both the existing and proposed methods (except QEM), implementing it is unnecessary when comparing +the performance. Instead, we use standard random number generation routines available in Python.2 We do +not use the tabulation–interpolation of E(η0,T ) and Var(η0,T ) in IG. This would speed up IG at the expense +of implementation complexity. Such a trick is not necessary in the corresponding POIS–GE (K = 0). +Case +V0 +θ +ξ +ρ +κ +T +r (%) +q (%) +X +CH +E(R0,T ) +Var(R0,T ) +I +0.04 +0.04 +1 +-0.9 +0.5 +10 +0 +0 +100 +13.08467014 +0.04 +0.011243 +II +0.04 +0.04 +0.9 +-0.5 +0.3 +15 +0 +0 +100 +16.64922292 +0.04 +0.016118 +III +0.010201 +0.019 +0.61 +-0.7 +6.21 +1 +3.19 +0 +100 +6.80611331 +0.017586 +0.000126 +IV +0.04 +0.25 +1 +-0.5 +4 +1 +1 +2 +120 +9.02491348 +0.198462 +0.007109 +Table 4.1 The four sets of the Heston model parameters used in numerical tests. All cases assume S0 = 100. The exact call +option price (CH) with strike price X, mean and variance of the average variance (E(R0,T ) and Var(R0,T )) are also provided +for reference. +We adopt the four sets of the Heston model parameter values in Table 4.1. The first three parameter sets +have been frequently used in earlier studies. +– Case I: Andersen [1], Van Haastrecht and Pelsser [24], Lord et al. [19], Tse and Wan [23] +– Case II: Andersen [1], Van Haastrecht and Pelsser [24] +– Case III: Broadie and Kaya [5], Tse and Wan [23] +We add Case IV [16] to explore a new case. Unlike the first three cases, Case IV does not violate the Feller +condition (i.e., δ = 4) and the strike price is out-of-the-money. Overall, our test cases are fairly diverse. For +example, Cases I and II are long-dated options, and Cases III and IV exhibit strong mean reversion (κ). +Table 4.1 also reports the exact call option price, and the mean and variance of the average variance +for reference. The exact option prices are obtained from the inverse fast Fourier transform [15] with the +unconditional characteristic function of the log asset price, which is free from the branch cut discontinuity [18]. +The prices obtained in this manner agree well with the high-precision values reported in the literature. +2 We use numpy.random.noncentral chisquare if µ0 is not required (e.g., IG and GE). We use numpy.random.poisson and +numpy.random.standard gamma if µ0 is required (e.g., POIS–GE and POIS–TD). + +Simulation schemes for the Heston model with Poisson conditioning +19 +We present a comparison of the efficiency (CPU time) and accuracy (quantified by bias and standard +error) of the derivative prices obtained with various simulation schemes. For the simulation estimator ˆΘ and +the true price Θ, the bias and standard error (SE) of the estimator are respectively defined by +Bias = E( ˆΘ) − Θ +and +SE = +� +E( ˆΘ2) − E( ˆΘ)2. +In all the simulation experiments, we drew 160, 000 paths to obtain an estimator ˆΘ and repeated the exper- +iment 200 times to obtain E( ˆΘ) and E( ˆΘ2). We ran simulations on a PC running Windows 11 with an Intel +i7–11700 (2.5 GHz) CPU and 8 GB RAM. +Case I +GE +POIS–GE +Time +Option +Spot +Time +Option +Spot +N +K +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +0 +0.095 +2.481 (0.025) +1.892 (0.089) +0.038 +0.153 (0.020) +0.069 (0.078) +1 +1 +0.105 +0.987 (0.022) +0.340 (0.076) +0.047 +0.154 (0.020) +0.057 (0.074) +1 +2 +0.116 +0.409 (0.021) +0.075 (0.075) +0.056 +0.084 (0.019) +0.014 (0.074) +1 +4 +0.130 +0.087 (0.020) +0.001 (0.078) +0.074 +0.023 (0.019) +-0.000 (0.074) +1 +8 +0.163 +0.006 (0.019) +-0.000 (0.076) +0.108 +0.002 (0.019) +-0.003 (0.077) +Case I +IG +POIS–GE (K = 0) +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +10 +0.097 +0.159 (0.019) +0.093 (0.077) +0.038 +0.153 (0.020) +0.069 (0.078) +2 +5 +0.168 +-0.057 (0.020) +-0.205 (0.075) +0.057 +-0.057 (0.020) +-0.160 (0.076) +4 +2.5 +0.308 +-0.136 (0.020) +-0.119 (0.081) +0.095 +-0.105 (0.019) +-0.055 (0.076) +8 +1.25 +0.593 +-0.068 (0.018) +-0.023 (0.077) +0.172 +-0.043 (0.020) +-0.014 (0.075) +Case I +QEM +POIS–TD +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +20 +1/2 +0.381 +0.116 (0.021) +-0.011 (0.082) +0.266 +-0.115 (0.019) +0.003 (0.071) +40 +1/4 +0.733 +0.008 (0.019) +-0.011 (0.084) +0.527 +-0.030 (0.020) +-0.009 (0.078) +80 +1/8 +1.505 +-0.015 (0.019) +-0.015 (0.078) +1.100 +-0.004 (0.020) +0.015 (0.076) +Table 4.2 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case I. The bias and +standard error of the at-the-money (X = S0 = 100) call option in (4.1) and the spot price in (4.2) are reported. See Table 4.1 +for the parameter values and reference option price. For the GE schemes (top), we take N = 1 time step (h = T) while the +number of gamma terms, K, is varied. For the IG approximations (middle) and time discretization schemes (bottom), N and +h are varied, respectively. +4.1 European call options +First, we price the European vanilla option. Although the vanilla option can be priced more efficiently with +Fourier inversion, it serves as a nontrivial test case for simulation methods. For pricing, we use the conditional +MC method [25] instead of simulating ST . Conditional on VT and I0,T , ST follows a geometric Brownian +motion in (2.11), and the option price can be obtained using the BS formula with forward price FT in (2.12) +and volatility σ = Σ0,T / +√ +T. Therefore, the unconditional European call option price under the Heston + +20 +J. Choi and Y. K. Kwok +Case II +GE +POIS–GE +Time +Option +Spot +Time +Option +Spot +N +K +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +0 +0.096 +-1.950 (0.011) +0.686 (0.062) +0.037 +-0.107 (0.011) +-0.003 (0.054) +1 +1 +0.104 +-0.823 (0.013) +0.088 (0.055) +0.044 +-0.121 (0.011) +0.014 (0.051) +1 +2 +0.113 +-0.366 (0.012) +0.014 (0.056) +0.052 +-0.075 (0.010) +0.006 (0.055) +1 +4 +0.128 +-0.089 (0.012) +-0.004 (0.058) +0.071 +-0.026 (0.012) +-0.006 (0.052) +1 +8 +0.164 +-0.006 (0.011) +-0.003 (0.054) +0.104 +-0.003 (0.012) +-0.005 (0.054) +Case II +IG +POIS–GE (K = 0) +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +15 +0.089 +-0.117 (0.011) +0.006 (0.056) +0.037 +-0.107 (0.011) +-0.003 (0.054) +2 +7.5 +0.156 +0.064 (0.011) +-0.103 (0.058) +0.056 +0.065 (0.010) +-0.071 (0.056) +4 +3.75 +0.297 +0.122 (0.010) +-0.034 (0.055) +0.096 +0.096 (0.010) +-0.014 (0.053) +8 +1.875 +0.582 +0.066 (0.009) +0.003 (0.052) +0.174 +0.044 (0.011) +0.000 (0.055) +Case II +QEM +POIS–TD +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +30 +1/2 +0.568 +-0.124 (0.011) +-0.004 (0.057) +0.390 +0.078 (0.009) +-0.001 (0.055) +60 +1/4 +1.119 +-0.010 (0.010) +-0.006 (0.060) +0.755 +0.017 (0.010) +0.000 (0.053) +120 +1/8 +2.178 +0.009 (0.010) +-0.004 (0.055) +1.540 +0.005 (0.010) +0.001 (0.054) +Table 4.3 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case II. The bias and +standard error of the at-the-money (X = S0 = 100) call option in (4.1) and the spot price in (4.2) are reported. See Table 4.1 +for the parameter values and reference option price. For the GE schemes (top), we take N = 1 time step (h = T) while the +number of gamma terms, K, is varied. For the IG approximations (middle) and time discretization schemes (bottom), N and +h are varied, respectively. +model can be estimated by taking the MC average of the BS prices over the simulated values of VT and I0,T : +ˆCH = e−rT EMC{CBS(FT , σ, T, X)}, +(4.1) +where CBS is the undiscounted BS call option price, with forward price FT , volatility σ, maturity T, and +strike price X. As the MC variance from the sampling of ST is suppressed by the BS formula, the conditional +MC reduces the MC variance of ˆCH, thereby increasing the accuracy of the bias of the simulation methods. +This method has been used in Broadie and Kaya [5, Tables 4 and 5] for the Heston model and Cai et al. [6] +for the SABR model. +We also measure how accurately the martingale condition is preserved. In theory, the unconditional +expectation of FT should equal the forward price, e(r−q)T S0. Therefore, we reconstruct the spot price by +taking the discounted MC average of FT : +ˆS0 = e(q−r)T EMC{FT }, +(4.2) +and examines the extent to which ˆS0 differs from S0. This serves as another measure to assess the accuracy +of our proposed simulation schemes, in particular, the effectiveness of martingale correction in POIS–TD. In +the two time discretization schemes, we apply the aggregate martingale correction term to the conditional + +Simulation schemes for the Heston model with Poisson conditioning +21 +Case III +GE +POIS–GE +Time +Option +Spot +Time +Option +Spot +N +K +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +0 +0.097 +0.078 (0.011) +0.006 (0.022) +0.042 +0.005 (0.011) +-0.000 (0.025) +1 +1 +0.107 +0.008 (0.010) +0.001 (0.023) +0.053 +0.001 (0.010) +0.000 (0.022) +1 +2 +0.118 +0.001 (0.011) +-0.000 (0.024) +0.063 +0.001 (0.009) +0.001 (0.022) +1 +4 +0.142 +0.001 (0.010) +0.001 (0.022) +0.084 +-0.000 (0.010) +-0.001 (0.022) +1 +8 +0.193 +0.000 (0.010) +0.000 (0.024) +0.131 +-0.000 (0.011) +-0.001 (0.024) +Case III +IG +POIS–GE (K = 0) +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +1 +0.106 +0.007 (0.010) +0.001 (0.024) +0.042 +0.005 (0.011) +-0.000 (0.025) +2 +1/2 +0.213 +-0.008 (0.011) +0.001 (0.023) +0.064 +-0.006 (0.011) +0.001 (0.023) +4 +1/4 +0.444 +-0.005 (0.009) +-0.000 (0.020) +0.105 +-0.001 (0.011) +0.000 (0.023) +8 +1/8 +1.025 +-0.001 (0.009) +-0.002 (0.020) +0.192 +0.001 (0.009) +0.001 (0.021) +Case III +QEM +POIS–TD +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +2 +1/2 +0.061 +0.097 (0.005) +-0.001 (0.016) +0.046 +-0.467 (0.008) +0.003 (0.018) +4 +1/4 +0.098 +0.013 (0.009) +-0.002 (0.022) +0.075 +-0.164 (0.010) +-0.000 (0.021) +8 +1/8 +0.170 +-0.009 (0.010) +-0.001 (0.022) +0.133 +-0.045 (0.010) +0.000 (0.021) +Table 4.4 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case III. The bias and +standard error of the at-the-money (X = S0 = 100) call option in (4.1) and the spot price in (4.2) are reported. See Table 4.1 +for the parameter values and reference option price. For the GE schemes (top), we take N = 1 time step (h = T) while the +number of gamma terms, K, is varied. For the IG approximations (middle) and time discretization schemes (bottom), N and +h are varied, respectively. +forward FT : +M0,N = +N−1 +� +i=0 +Mi,i+1 +In POIS–TD, M POIS +0,N +is proportional to +N−1 +� +i=0 +Var(Ii,i+1|µi) = +� +(V0 + 2V1 + · · · + 2VN−1 + VN)vX + +�Nδ +2 + 2µ0 + · · · + 2µN−1 +� +vZξ2h +� +ξ2h3, +(4.3) +which is a linear combination of the sums of Vi and µi. This adds little extra computation, because the same +terms are already used in IPOIS +0,N +in (3.13). +Tables 4.2–4.5 show the numerical performance of the four cases. In each table, we compare the GE- +based exact schemes (top), IG-based low-bias schemes (middle), and time discretization schemes (bottom). +We comment on each of these comparisons below. +In the GE versus POIS–GE comparison, POIS–GE shows substantially lower bias with CPU time re- +duction (typically 40%). The IG approximation contributes to the lower bias for the truncated series, and +Poisson conditioning contributes to the faster execution. For POIS–GE with low K values, the biases in +Cases I and II are relatively larger than those in the other cases because of the long maturity. Nevertheless, + +22 +J. Choi and Y. K. Kwok +Case IV +GE +POIS–GE +Time +Option +Spot +Time +Option +Spot +N +K +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +0 +0.084 +0.032 (0.013) +0.015 (0.049) +0.035 +-0.001 (0.013) +0.000 (0.053) +1 +1 +0.092 +0.002 (0.012) +0.000 (0.048) +0.043 +0.000 (0.013) +0.002 (0.052) +1 +2 +0.101 +-0.000 (0.012) +-0.001 (0.050) +0.053 +0.001 (0.013) +0.002 (0.054) +1 +4 +0.124 +-0.001 (0.013) +-0.002 (0.054) +0.073 +-0.000 (0.013) +-0.001 (0.049) +1 +8 +0.170 +0.000 (0.013) +0.002 (0.052) +0.107 +0.000 (0.013) +0.000 (0.053) +Case IV +IG +POIS–GE (K = 0) +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +1 +1 +0.091 +-0.001 (0.012) +-0.002 (0.051) +0.035 +-0.001 (0.013) +0.000 (0.053) +2 +1/2 +0.223 +-0.004 (0.012) +-0.003 (0.049) +0.052 +-0.001 (0.013) +0.005 (0.053) +4 +1/4 +0.609 +-0.001 (0.012) +-0.000 (0.050) +0.092 +-0.002 (0.014) +-0.004 (0.055) +8 +1/8 +1.299 +-0.000 (0.013) +-0.001 (0.051) +0.172 +0.002 (0.013) +0.007 (0.053) +Case IV +QEM +POIS–TD +Time +Option +Spot +Time +Option +Spot +N +h +(sec) +Bias (SE) +Bias (SE) +(sec) +Bias (SE) +Bias (SE) +2 +1/2 +0.047 +-0.599 (0.005) +-0.001 (0.014) +0.039 +-0.096 (0.012) +0.013 (0.049) +4 +1/4 +0.071 +-0.166 (0.005) +-0.001 (0.016) +0.063 +-0.034 (0.013) +-0.003 (0.053) +8 +1/8 +0.129 +-0.045 (0.005) +-0.002 (0.016) +0.113 +-0.007 (0.013) +0.008 (0.052) +Table 4.5 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case IV. The bias and +standard error of the out-of-the-money (X = 120) call option in (4.1) and the spot price in (4.2) are reported. See Table 4.1 +for the parameter values and reference option price. For the GE schemes (top), we take N = 1 time step (h = T) while the +number of gamma terms, K, is varied. For the IG approximations (middle) and time discretization schemes (bottom), N and +h are varied, respectively. +the spot price is accurately preserved, indicating that the distributional error in I0,T due to low K manifests +in option prices rather than spot prices. The option price bias becomes exceedingly small when K increases +to 8. +In IG versus POIS–GE (K = 0), the latter reduces the computation time by several factors, although the +bias improvement is marginal. This is because we avoided evaluating the modified Bessel function present in +the original IG scheme. +In QEM versus POIS–TD, POIS–TD compares favorably with QEM. The use of better discretization +rules leads to improved accuracy. It is also noteworthy that the runtime efficiency of POIS–TD is slightly +better than that of the QEM. This is surprising given that the motivation for introducing the QE step +(2.24) is to avoid directly drawing a noncentral chi-squared variate. As expected in the time discretization, +both schemes require a small h to achieve high accuracy. Despite the short maturity, Cases III and IV show +relatively larger biases than the other cases for a large time step h. However, the spot price bias is close +to zero even with a large h, which verifies that the martingale correction is effective in both schemes. The +advantage of POIS–TD is more pronounced for the pricing variance swap, as discussed in the next subsection. +4.2 Discretely monitored variance swap +In general, a swap product is a contractual agreement to exchange the floating and fixed legs of payment for +future time t = T. In the variance swap, the floating leg is given by the annualized log return variance over + +Simulation schemes for the Heston model with Poisson conditioning +23 +N = T/h monitoring periods, which is proxied by +Rh +0,T = 1 +T +N +� +i=1 +ln2(Si/Si−1). +(4.4) +The monitoring frequency is set by the time step h: h = 1/4 for quarterly, h = 1/12 for monthly, and +h = 1/52 for weekly monitoring. The amount of fixed leg, Kh +swap, is typically determined to ensure that the +swap has zero net present value at t = 0: +Kh +swap = E(Rh +0,T ). +This value is called the fair strike of the variance swap. The level of path dependence of these variance +derivatives goes beyond Asian options because their payoff structures involve the squared sum of the log +return of asset price St at t = ti. +The annualized log return variance in the continuous limit (h ↓ 0) is the average variance in (2.5). +lim +h↓0 Rh +0,T → R0,T = 1 +T +� T +0 +Vt dt, +(4.5) +and fair strike Kswap is given by (2.6): +Kswap = E(R) = θ + (V0 − θ)1 − e−κT +κT +. +The fair strike Kh +swap of the discrete variance swap with the monitoring time step h admits a closed-form +solution [4, 14]. We express Kh +swap as an adjustment ∆h +swap to the continuously monitored fair strike Kswap: +Kh +swap = E(Rh +0,T ) = Kswap + ∆h +swap, +(4.6) +where +∆h +swap = h(θ + 2q − 2r) +4 +� +(θ + 2q − 2r) + 2(V0 − θ)1 − e−κT +κT +� ++ θξ +κ +� ξ +4κ − ρ +� � +1 − 1 − e−κh +κh +� ++ (V0 − θ) ξ +κ +� ξ +2κ − ρ +� 1 − e−κT +κT +� +1 + +κh +1 − eκh +� ++ +� ξ2 +κ2 (θ − 2V0) + 2 +κ(V0 − θ)2 +� 1 − e−2κT +8κT +1 − e−κh +1 + e−κh . +The above analytical solution serves as a benchmark to assess the accuracy of the time discretization schemes. +We compare the fair strike Kh +swap for various monitoring frequencies ranging from semi-annual (h = 1/2) +to weekly (h = 1/52). As the time discretization methods are more efficient in derivatives with higher +monitoring frequency, such as variance swap, we only compare QEM with POIS–TD. In POIS–TD, we apply +the martingale correction M ′POIS +i,i+1 in (3.17), which is tailored to log return variance. Given the absence of +such a correction in the QEM, we simply use M QE +i,i+1 for the log return. + +24 +J. Choi and Y. K. Kwok +Case III +QEM +POIS–TD +Benchmark +Time +Bias (SE) +Time +Bias (SE) +N +h +(×10−2) +(sec) +(×10−2) +(sec) +(×10−2) +2 +1/2 +1.870 +0.042 +0.041 (0.010) +0.033 +0.000 (0.007) +4 +1/4 +1.832 +0.091 +-0.024 (0.007) +0.067 +0.001 (0.007) +12 +1/12 +1.790 +0.241 +-0.011 (0.005) +0.221 +-0.001 (0.004) +52 +1/52 +1.767 +0.954 +-0.000 (0.003) +0.920 +0.000 (0.004) +Table 4.6 Speed-accuracy comparison of the time discretization schemes for pricing variance swap with Case III. The bias +and standard error of the fair strike of discretely monitored variance swap are reported for varying monitoring frequencies. The +benchmark fair strike is from the analytical reference price in (4.6). +Case IV +QEM +POIS–TD +Benchmark +Time +Bias (SE) +Time +Bias (SE) +N +h +(×10−2) +(sec) +(×10−2) +(sec) +(×10−2) +2 +1/2 +21.930 +0.038 +-0.750 (0.083) +0.026 +0.002 (0.085) +4 +1/4 +21.132 +0.062 +-0.325 (0.060) +0.057 +0.004 (0.063) +12 +1/12 +20.356 +0.184 +-0.057 (0.036) +0.184 +-0.003 (0.038) +52 +1/52 +19.973 +0.840 +-0.000 (0.021) +0.962 +0.001 (0.029) +Table 4.7 Speed-accuracy comparison of the time discretization schemes for pricing variance swap with Case IV. The bias +and standard error of the fair strike of discretely monitored variance swap are reported for varying monitoring frequencies. The +benchmark fair strike is from the analytical reference price in (4.6). +Tables 4.6 and 4.7 list the results for Cases III and IV, respectively. The two cases are chosen because +variance swaps traded in the market are typically short-dated, such as one year in these cases, and they +show a relatively larger bias in pricing vanilla options in Section 4.1. The results show that the POIS–TD +bias is much smaller than the QEM bias, although the biases from both schemes quickly converge to zero as +the monitoring becomes more frequent. The numerical results also verify that martingale correction (3.17) +is effective. +5 Conclusion +Simulation under the Heston model has been a widely studied topic, and several approaches are available +for different monitoring frequencies: exact [5, 10], low-bias [23], and time discretization [1] schemes. Exist- +ing simulation schemes, however, suffer from computationally expensive evaluations of the modified Bessel +function or Bessel random variables arising from the square-root variance process. Based on the observation +that the conditional integrated variance can be simplified when conditioned by the Poisson variate used for +simulating the terminal variance, we propose simulation methods that enhance the existing methods in all +spectra. +Poisson-conditioned GE is an exact simulation scheme that enhances Glasserman and Kim [10]’s GE +scheme. It achieves significant speedup by expressing the conditional integrated variance without the Bessel +random variable, which is a computational bottleneck. Adopting the IG approximation [23] for the truncation +approximation improves numerical accuracy. A special case of the Poisson-conditioned GE scheme is naturally +reduced to Tse and Wan [23]’s low-bias scheme but without the Bessel function. As the Laplace transform of +the conditional integrated variance can also be expressed without the Bessel function, Broadie and Kaya [5]’s + +Simulation schemes for the Heston model with Poisson conditioning +25 +scheme can be expedited. The Poisson-conditioned time discretization scheme is proposed as an alternative +to Andersen [1]’s QE scheme. Our comprehensive numerical tests illustrate the strong competitiveness of +our schemes in speed-accuracy comparison among existing schemes for pricing derivatives under the Heston +model. In addition to numerical efficiency, our new Heston simulation schemes are simple and straightforward +for practitioners to implement because they involve only elementary functions and random variables. +A Asymptotic expansion of IPOIS +i,i+1 +Taking the limit of h ↓ 0, the coefficients in (2.20) have the following asymptotic expansions in the powers +of a: +mX = c1 − ac2 +2a += 1 +3 +� +1 − 2 +15a2 + +2 +105a4 · · · +� +, +mZ = ac1 − 1 +4a2 += 1 +12 +� +1 − 1 +15a2 + +2 +315a4 · · · +� +, +where a = κh/2. We can easily see that mX → 1 +3 and mZ → +1 +12 as h ↓ 0, with O(h2) as the leading order of +truncation. Furthermore, we obtain asymptotic expansions of Iν(z) and z: +Iν(z) = +ez +√ +2πz +� +1 − 4ν2 − 1 +8z ++ · · · +� +and +z = φh(κ) +� +ViVi+1 = 2κ/ξ2 +sinh κh +2 +� +ViVi+1 → 4 +� +ViVi+1 +ξ2h +. +We also consider E(ηi,i+1) in the h ↓ 0 limit +E(ηi,i+1) = 2µi = z Iν+1(z) +2Iν(z) +→ z +2 +� +1 − 2ν + 1 +2z +� += z +2 − δ +4 → 2 +� +ViVi+1 +ξ2h +− δ +4, +from which we obtain +δ +2 + 2µi = δ +2 + z Iν+1(z) +Iν(z) +→ δ +2 + 4 +� +ViVi+1 +ξ2h +− δ +2 = 4 +� +ViVi+1 +ξ2h +as +h ↓ 0. +Combining these results, we obtain +E +� +ˆIPOIS +i,i+1 +� += (Vi + Vi+1)mXh + +�δ +2 + 2E(µi | Vi, Vi+1) +� +mZξ2h2 += (Vi + Vi+1)mXh + +�δ +2 + 2E(ηi,i+1) +� +mZξ2h2 +→ (Vi + Vi+1)h +3 + 4 +� +ViVi+1 +ξ2h +ξ2h2 +12 += +� +Vi + Vi+1 + +� +ViVi+1 +� h +3 . +This result is consistent with Tse and Wan [23, Proposition 3.5]. + +26 +J. Choi and Y. K. Kwok +References +1. Andersen L (2008) Simple and efficient simulation of the Heston stochastic volatility model. Journal of +Computational Finance 11(3):1–42, doi:10.21314/JCF.2008.189 +2. Baldeaux J (2012) Exact simulation of the 3/2 model. International Journal of Theoretical and Applied +Finance 15(05):1250032, doi:10.1142/S021902491250032X +3. Ball CA, Roma A (1994) Stochastic volatility option pricing. Journal of Financial and Quantitative +Analysis 29(4):589–607, doi:10.2307/2331111 +4. Bernard C, Cui Z (2014) Prices and Asymptotics for Discrete Variance Swaps. Applied Mathematical +Finance 21(2):140–173, doi:10.1080/1350486X.2013.820524 +5. Broadie M, Kaya ¨O (2006) Exact simulation of stochastic volatility and other affine jump diffusion +processes. Operations Research 54(2):217–231, doi:10.1287/opre.1050.0247 +6. Cai N, Song Y, Chen N (2017) Exact simulation of the SABR model. Operations Research 65(4):931–951, +doi:10.1287/opre.2017.1617 +7. Choi J (2023) A new exact simulation of the Ornstein-Uhlenbeck driven stochastic volatility model using +the Karhunen-Lo`eve expansions. Working Paper, Peking University HSBC Business School +8. Choi J, Liu C, Seo BK (2019) Hyperbolic normal stochastic volatility model. Journal of Futures Markets +39(2):186–204, doi:10.1002/fut.21967 +9. Cox JC, Ingersoll JE Jr, Ross SA (1985) A theory of the term structure of interest rates. Econometrica +53(2):385–407, doi:10.2307/1911242 +10. Glasserman P, Kim KK (2011) Gamma expansion of the Heston stochastic volatility model. Finance and +Stochastics 15(2):267–296, doi:10.1007/s00780-009-0115-y +11. Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond +and currency options. Review of Financial Studies 6(2):327–343, doi:10.1093/rfs/6.2.327 +12. Kahl C, J¨ackel P (2006) Fast strong approximation Monte Carlo schemes for stochastic volatility models. +Quantitative Finance 6(6):513–536, doi:10.1080/14697680600841108 +13. Kang C, Kang W, Lee JM (2017) Exact Simulation of the Wishart Multidimensional Stochastic Volatility +Model. Operations Research 65(5):1190–1206, doi:10.1287/opre.2017.1636 +14. Kwok YK, Zheng W (2022) Pricing Models of Volatility Products and Exotic Variance Derivatives. +Chapman & Hall/CRC Financial Mathematics Series, New York +15. Lewis AL (2000) Option Valuation under Stochastic Volatility: With Mathematica Code. Newport Beach, +CA +16. Lewis AL (2019) Heston Model Reference Prices. URL https://financepress.com/2019/02/15/ +heston-model-reference-prices/ +17. Li C, Wu L (2019) Exact simulation of the Ornstein–Uhlenbeck driven stochastic volatility model. +European Journal of Operational Research 275(2):768–779, doi:10.1016/j.ejor.2018.11.057 +18. Lord R, Kahl C (2010) Complex Logarithms in Heston-Like Models. Mathematical Finance 20(4):671– +694, doi:10.1111/j.1467-9965.2010.00416.x +19. Lord R, Koekkoek R, Dijk DV (2010) A comparison of biased simulation schemes for stochastic volatility +models. Quantitative Finance 10(2):177–194, doi:10.1080/14697680802392496 + +Simulation schemes for the Heston model with Poisson conditioning +27 +20. Malham SJA, Shen J, Wiese A (2021) Series Expansions and Direct Inversion for the Heston Model. +SIAM Journal on Financial Mathematics 12(1):487–549, doi:10.1137/19M126791X +21. Michael JR, Schucany WR, Haas RW (1976) Generating random variates using transformations with +multiple roots. The American Statistician 30(2):88–90, doi:10.1080/00031305.1976.10479147 +22. Pitman J, Yor M (1982) A decomposition of Bessel Bridges. Zeitschrift f¨ur Wahrscheinlichkeitstheorie +und Verwandte Gebiete 59(4):425–457, doi:10.1007/BF00532802 +23. Tse ST, Wan JWL (2013) Low-bias simulation scheme for the Heston model by Inverse Gaussian ap- +proximation. Quantitative Finance 13(6):919–937, doi:10.1080/14697688.2012.696678 +24. Van Haastrecht A, Pelsser A (2010) Efficient, almost exact simulation of the Heston stochas- +tic +volatility +model. +International +Journal +of +Theoretical +and +Applied +Finance +13(01):1–43, +doi:10.1142/S0219024910005668 +25. Willard GA (1997) Calculating prices and sensitivities for path-independent derivatives securities in +multifactor models. The Journal of Derivatives 5(1):45–61, doi:10.3905/jod.1997.407982 +26. Zeng P, Xu Z, Jiang P, Kwok YK (2021) Analytical Solvability and Exact Simulation of Stochastic +Volatility Models with Jumps. Working Paper, Southern University of Science and Technology, China, +doi:10.2139/ssrn.3904498 + diff --git a/t9E0T4oBgHgl3EQf9QLr/content/tmp_files/load_file.txt b/t9E0T4oBgHgl3EQf9QLr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29d5afa9607f3e235d84511727c3a830bc2f85ef --- /dev/null +++ b/t9E0T4oBgHgl3EQf9QLr/content/tmp_files/load_file.txt @@ -0,0 +1,1350 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf,len=1349 +page_content='arXiv manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (will be inserted by the editor) Simulation schemes for the Heston model with Poisson conditioning Jaehyuk Choi · Yue Kuen Kwok October 12, 2022 Abstract Exact simulation schemes under the Heston stochastic volatility model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', Broadie–Kaya and Glasserman–Kim) suffer from computationally expensive Bessel function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We propose a new exact simulation scheme without the Bessel function, based on the observation that the conditional integrated variance can be simplified when conditioned by the Poisson variate used for simulating the terminal variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Our approach also enhances low-bias and time discretization schemes, which are suitable for derivatives with frequent monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Extensive numerical tests reveal the good performance of the new simulation schemes in terms of accuracy, efficiency, and reliability when compared with existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Keywords Heston model · exact simulation · Poisson conditioning · gamma expansion · time discretization schemes Mathematics Subject Classification (2020) 60H35 · 65C05 · 91B70 JEL Classification C63 · G12 · G13 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi Rm 755, Peking University HSBC Business School, University Town, Nanshan, Shenzhen 518055, China Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' : +86–755–2603–0568 E-mail: jaehyuk@phbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='cn Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' : +852–2358–7418 E-mail: maykwok@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='hk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='02800v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='MF] 7 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok 1 Introduction The Heston [11] stochastic volatility model relaxes the constant volatility assumption in the Black–Scholes (BS) model by taking the instantaneous variance to follow the square root diffusion process with mean reversion, commonly called the Cox–Ingersoll–Ross (CIR, [9]) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' It is the most popular stochastic volatility model for market practitioners because of its analytic tractability in computing the prices of European options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Owing to the existence of a closed-form formula for the characteristic function of the log-asset price, model calibration of market-observable European option prices can be performed efficiently using the Fourier inversion algorithm [15, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For pricing path-dependent options under the Heston model, Monte Carlo (MC) simulations are often used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, the standard Euler and Milstein time discretization simulation schemes suffer from a high bias owing to the square root of the diffusion function in the variance process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Negative values of variance from the simulation must be heuristically set to zero before taking the square root of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In addition, the square root function violates the Lipschitz condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' therefore, the convergence properties of the discretization scheme may not be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' There have been numerous fixes to these issues to minimize discretization biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' A comprehensive review of these discretization schemes using various fixes can be found in Lord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' A major breakthrough was made by Broadie and Kaya [5] in the simulation of the Heston model from its exact distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Their exact simulation procedures consist of three steps: (i) sampling of the termi- nal variance conditional on the initial variance, (ii) sampling the integrated variance conditional on the initial and terminal variance values, and (iii) sampling the asset price process conditional on the variance and integrated variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As the exact simulation approach avoids simulation bias, simulation errors remain inversely proportional to the square root of the computational time budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, the Broadie–Kaya exact simulation algorithm is not competitive in accuracy-speed comparison because it requires extensive computational time to sample the conditional integrated variance via the numerical inversion of the Laplace transform in each simulation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' To improve computational efficiency, one may use a caching technique to sample the terminal variance and conditional integrated variance via precomputation and interpolation of the appropriate inverse distribution functions [24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Despite its limitations, Broadie and Kaya [5]’s pioneering work triggers the construction of exact simulation schemes for other stochastic volatility models, such as the stochastic-alpha-beta-rho (SABR) [6, 8], 3/2 [2, 26], Wishart [13], and Ornstein–Uhlenbeck-driven stochastic volatility model [17, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Based on Pitman and Yor [22]’s decomposition of Bessel bridges, Glasserman and Kim [10] show that con- ditional integrated variance can be expressed as gamma expansions (GE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Significant computational speedup can be achieved by sampling conditional integrated variance via the sums of the mixtures of gamma random variates (with an approximation of the truncated terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In a related study, Malham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' [20] construct a new series expansion for the conditional integrated variance in terms of double infinite weighted sums of independent random variables through a measure change and decomposition of squared Bessel bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Tse and Wan [23] propose a low-bias simulation algorithm by approximating the conditional integrated variance with an Inverse Gaussian (IG) variate with matching mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' With its lower computational cost per time step, the IG scheme can be used as a multiperiod scheme for pricing path-dependent options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Andersen [1] constructs a time discretization scheme, where the variance at discrete time points is simulated Simulation schemes for the Heston model with Poisson conditioning 3 via the quadratic-exponential (QE) approximation with martingale correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, he only uses the trapezoidal rule to approximate the conditional integrated variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Such an approximation is acceptable when the time step is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' There is no one-size-fits-all solution among the simulation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Each simulation method has its own advantages depending on the monitoring frequency of the derivatives to price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Tse and Wan [23] summarize that the decision among GE, IG, or QE schemes depends on the compromise between computational cost and bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The exact GE scheme is the best choice for European-style derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Despite the high computation cost per step, we need to simulate just one step up to expiry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For path-dependent derivatives with frequent monitoring, the time-discretized QE scheme may be a better choice owing to its low computation cost per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' When the number of monitoring instants is moderate, one may choose to use the low-bias IG scheme as a compromise between exact and time discretization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Despite the advances in Heston simulation algorithms, the computational efficiency still has room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The simulation methods mentioned above, except for the QE scheme, involve computationally demanding evaluations of the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This has been criticized as a bottleneck in computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In particular, the GE scheme [10] involves the Bessel random variable, the sampling of which takes up a significant portion of the computation time for the same reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The contributions of this study are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We propose enhanced Heston simulation schemes in all ranges based on the key observation that the conditional integrated variance can be further simplified when conditioned by the Poisson variate involved in the terminal variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Consequently, the computationally trivial Poisson variate replaces the Bessel variate in the GE scheme [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' By adopting the IG approximation [23] for series truncation also, the Poisson-conditioned GE scheme significantly enhances both speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The IG scheme [23] is a special case of the new method, but the enhanced IG scheme no longer requires the Bessel function for mean and variance calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Broadie and Kaya [5]’s Laplace inversion scheme can also benefit from our approach since Poisson conditioning removes the Bessel function from the Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We also propose a Poisson-conditioned time discretization method with the corresponding martingale correction method, which is suitable for pricing derivatives with frequent monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The new time discretization scheme compares favorably with Andersen [1]’s QE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In Section 2, we introduce the Heston model and its analytical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We then review existing simulation algorithms and discuss intrinsic computational challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In Section 3, we show how Poisson conditioning can enhance existing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In Section 4, we present extensive numerical tests that compare the performance of Poisson-conditioned simulation schemes with existing simulation schemes for pricing European options and discretely monitored variance swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Finally, we conclude this paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 2 Heston stochastic volatility model and existing simulation schemes The dynamics of the asset price process St and instantaneous variance process Vt of the Heston stochastic volatility model under a risk-neutral measure Q are governed by the following coupled stochastic differential 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok equations: dSt St = (r − q) dt + � Vt � ρ dZt + � 1 − ρ2 dWt � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) dVt = κ(θ − Vt) dt + ξ � Vt dZt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) where Zt and Wt are independent Brownian motions, r is the riskless interest rate, q is the continuous dividend yield, κ is the speed of mean reversion, θ is the mean reversion level, ξ is the volatility of the variance process, and ρ ∈ [−1, 1] represents the correlation coefficient between St and Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The initial conditions S0 and V0 are assumed to be strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The joint process (S, V ) is well known to be a time-homogeneous Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 Terminal variance, integrated variance, and asset return We state several analytic properties of the variance, (conditional) integrated variance, and asset return, which are necessary for the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Variance Vt is governed by the CIR [9] process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' It is well known that the terminal variance VT given V0 observes a noncentral chi-square distribution characterized by VT ∼ e− κT 2 φT (κ) χ2 � δ, V0 φT (κ)e− κT 2 � for δ = 4κθ ξ2 and φT (κ) = 2κ/ξ2 sinh(κT/2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3) where χ2(δ, λ) denotes a noncentral chi-square random variable with degrees of freedom δ and a noncentrality parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The variance process Vt cannot reach zero for t > 0, provided that δ > 2 (the Feller condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, Heston model parameters calibrated to the option market usually violate this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The mean and variance of VT are given by E(VT ) = θ + (V0 − θ)e−κT and Var(VT ) = ξ2 κ (1 − e−κT ) � V0e−κT + θ 2(1 − e−κT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4) We define the average variance between times 0 and T as R0,T = 1 T � T 0 Vt dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5) The mean and variance of R0,T , known as [3] E(R0,T ) = θ + (V0 − θ)1 − e−κT κT , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6) Var(R0,T ) = ξ2 κ2T � θ − 2(V0 − θ)e−κT + � V0 − 5θ 2 + � V0 − θ 2 � e−κT � 1 − e−κT κT � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='7) provide useful insights into this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For example, mean E(R0,T ) is the fair strike of the continuously monitored variance swap (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the uncorrelated (ρ = 0) Heston model, � E(R0,T ) plays the role of the BS implied volatility in the option price approximation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 5 We also define the integrated variance between t = 0 and T, conditional on the initial and final variances, as I0,T (V0, VT ) = �� T 0 Vt dt ��� V0, VT � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='8) that satisfies I0,T = E(TR0,T |V0, VT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For notational simplicity, we simply write I0,T , assuming conditional dependence on V0 and VT implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Any expectation regarding I0,T should be understood as a conditional expectation, E(f(I0,T )) := E(f(I0,T ) | V0, VT ), for any function f unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The exact simulation scheme is constructed, starting with the following analytic representation of the asset price process St of the Heston model: ST = S0 exp � (r − q)T − 1 2 � T 0 Vt dt + ρ � T 0 � Vt dZt + � 1 − ρ2 � T 0 � Vt dWt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='9) By integrating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2), conditional on VT , we obtain � T 0 � Vt dZt = 1 ξ [VT − V0 + κ(I0,T − θT)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Conditional on VT and I0,T , we deduce that the log return, ln(ST /S0), can be sampled from a normal distribution: ln ST S0 ∼ (r − q)T − I0,T 2 + ρ ξ [VT − V0 + κ(I0,T − θT)] + Σ0,T Z, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='10) where Z is a standard normal variate and Σ0,T is the standard deviation, as defined by Σ0,T = � (1 − ρ2)I0,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, asset price ST can be simulated as a geometric Brownian motion: ST = FT exp � Σ0,T Z − 1 2Σ2 0,T � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='11) where FT is the forward stock price, conditional on S0, V0, VT , and I0,T : FT = E(ST | S0, V0, VT , I0,T ) = S0e(r−q)T exp � −ρ2 2 I0,T + ρ ξ [VT − V0 + κ(I0,T − θT)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='12) Similar to I0,T , we simply write FT in the later expression, assuming the conditional dependence on S0, V0, VT , and I0,T to be implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, the simulation of ST given S0 and V0 reduces to sampling VT and I0,T sequentially in each simulation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As VT can be sampled with relative ease from a noncentral chi-square distribution, the challenge in the Heston model simulation lies in sampling the conditional integrated variance I0,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2 Broadie–Kaya exact simulation Broadie and Kaya [5] perform simulation of I0,T via the numerical Laplace inversion of the conditional Laplace transform of I0,T with the following analytic from Pitman and Yor [22]: E � e−uI0,T � = exp � − V0+VT 2 cosh( κuT 2 )φT (κu) � exp � − V0+VT 2 cosh( κT 2 )φT (κ) � φT (κu) φT (κ) Iν (zu) Iν (z) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13) where ν = δ 2 − 1 = 2κθ ξ2 − 1, z = � V0 VT φT (κ), zu = � V0 VT φT (κu), κu = � κ2 + 2ξ2u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='14) and Iν(z) is a modified Bessel function of the first kind: Iν(z) = ∞ � k=0 (z/2)ν+2k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Γ(k + ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='15) This sampling procedure for I0,T is very time-consuming because the Laplace inversion algorithm requires cumbersome numerical evaluations of Iν(z) over the grid of z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Moreover, such evaluations must be performed for each simulation path because the conditional Laplace transform depends on VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3 Gamma expansion derived from the Bessel bridge decomposition A significant improvement in computational efficiency can be achieved if numerical inversion of the Laplace transform can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Based on Bessel bridge decomposition [22], Glasserman and Kim [10] express the conditional integrated variance I0,T as the infinite sum of the gamma random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Let POIS(λ) denote the Poisson random variable with rate λ, and Γ1(α) denote the standard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', unit scale) gamma random variable with shape parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The probability mass function of POIS(λ) and probability density function of Γ1(α) are respectively given by PPOIS(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' λ) = λj e−λ j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=') and fΓ1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' α) = xα−1e−x Γ(α) (x > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Then, I0,T can be expressed as [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2] I0,T ∼ X + Zδ/2 + η0,T � j=1 Z(j) 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) where X ∼ ∞ � k=1 1 γk Γ1(nk) and Zα ∼ ∞ � k=1 1 γk Γ1(α) for nk(V0, VT ) ∼ POIS((V0 + VT )λk) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', λk = 16k2π2 ξ2T(κ2T 2 + 4k2π2) and γk = κ2T 2 + 4k2π2 2ξ2T 2 , Simulation schemes for the Heston model with Poisson conditioning 7 and Z(j) α denotes independent identically distributed copies of Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The last term in the infinite series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) involves the summation up to a Bessel random variable η0,T with dependence on V0 and VT , where η0,T (V0, VT ) ∼ BES(ν, z), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='17) where ν and z are as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The Bessel random variable, η ∼ BES(ν, z) with ν > −1 and z > 0, takes non-negative integer values, and its probability mass function is given by the normalized coefficients of Iν(z) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='15): PBES(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' ν, z) = (z/2)2j+ν Iν(z) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Γ(j + ν + 1) (j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='18) In the actual numerical implementation of evaluating the infinite gamma series, it is necessary to evaluate the sum of only finite K terms and properly approximate the truncated terms, XK ∼ ∞ � k=K+1 1 γk Γ1(nk) and ZK α ∼ ∞ � k=K+1 1 γk Γ1(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Glasserman and Kim [10, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2] propose to approximate the three truncated terms of X, Zδ/2, and Z2 with gamma random variables with matching mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The mean and variance of the truncated terms are available analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Glasserman and Kim [10] start with the derivation of the mean and variance of X and Zα: E(X) = (V0 + VT ) ∞ � k=1 λk γk = (V0 + VT ) ∞ � k=1 32π2k2 T (κ2T 2 + 4k2π2)2 = (V0 + VT )mXT, Var(X) = (V0 + VT ) ∞ � k=1 2λk γ2 k = (V0 + VT ) ∞ � k=1 128π2k2 ξ2T 3 (κ2T 2 + 4k2π2)3 = (V0 + VT )vXξ2T 3, E(Zα) = ∞ � k=1 α γk = ∞ � k=1 2αξ2T 2 κ2T 2 + 4k2π2 = αmZξ2T 2, Var(Zα) = ∞ � k=1 α γ2 k = ∞ � k=1 4αξ4T 4 (κ2T 2 + 4k2π2)2 = αvZξ4T 4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='19) where mX = c1 − ac2 2a , vX = c1 + ac2 − 2a2c1c2 8a3 , mZ = ac1 − 1 4a2 , vZ = ac1 + a2c2 − 2 16a4 a = κT 2 , c1 = 1 tanh a, and c2 = 1 sinh2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='20) The mean and variance of XK and ZK α are then obtained by subtracting the first K terms from those of X and Zα, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For example, E(XK) = E(X) − (V0 + VT ) K � k=1 λk γk and E(ZK α ) = E(Zα) − K � k=1 α γk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok It is obvious that K serves as a parameter controlling the accuracy of the simulation scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' a higher K implies a smaller error at the tradeoff of a higher computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Note that the numerical implementation of the GE scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) eventually requires (2+η0,T )(K + 1) gamma random variables per simulation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The GE representation of I0,T proposed by Glasserman and Kim [10] avoids the tedious numerical inversion of the conditional Laplace transform of I0,T as implemented by Broadie and Kaya [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, substantial computational effort is still required to simulate the Bessel random variable η0,T , as it involves Iν(z) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4 Inverse Gaussian approximation based on matching moments Tse and Wan [23] propose a low-bias simulation scheme, where I0,T is approximated by an IG variate with matching mean and variance of I0,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' They argue that the IG variable is the best candidate for approximating I0,T because the two converge in the distribution as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Let IG(µ, λ) denote the IG random variable with the parameters µ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The density function of IG(µ, λ) takes the form, fIG(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' µ, λ) = � λ 2πx3 exp � −λ(x − µ)2 2µ2x � for µ > 0, λ > 0, and x > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='21) and the mean and variance of IG(µ, λ) are given by µ and µ3/λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Parameters µ and λ are determined by matching the mean and variance of I0,T : µ = E(I0,T ) and λ = E(I0,T )3 Var(I0,T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The exact mean and variance of I0,T are as follows [23, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1]: E(I0,T ) = E(X) + E(Zδ/2) + E(η0,T )E(Z2) Var(I0,T ) = Var(X) + Var(Zδ/2) + E(η0,T )Var(Z2) + Var(η0,T )E(Z2)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='22) where the mean and variance of X and Zα are given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='19) and E(η0,T ) = z Iν+1(z) 2Iν(z) and Var(η0,T ) = z2 Iν+2(z) 4Iν(z) + E(η0,T ) − E(η0,T )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='23) Once µ and λ have been determined, it is trivial to sample the IG(µ, λ) variate from the algorithm of Michael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Note that the error-controlling parameter, similar to K in the GE scheme, is not present in the IG approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As Tse and Wan [23] demonstrate, the only way to reduce the simulation bias is to decrease the time interval (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', increasing the number of simulation steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, we can classify their method as a low-bias scheme rather than an exact simulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, to obtain E(I0,T ) and Var(I0,T ), the modified Bessel functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', Iν(z), Iν+1(z), and Iν+2(z)) must be evaluated for each path, which causes a computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' To avoid this problem, Tse and Wan 1 See Glasserman and Kim [10, Table 4] for the computation time sampling the Bessel random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 9 [23] pre-compute E(η0,T ) and Var(η0,T ) for a grid of equally spaced V0VT values, thanks to the property that η0,T depends on V0 and VT via z = √V0VT φT (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Interpolation within the tabulated values is then performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In addition, they propose the interpolation–Poisson–zero (IPZ) scheme for faster sampling of the noncentral chi-square variate for VT (see more comments in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 Time discretization with QE approximation, trapezoidal rule, and martingale correction For pricing path-dependent derivatives such as Asian options and variance swaps, it is necessary to sample asset prices at frequent time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For this purpose, the simulation schemes based on time discretization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', Euler or Milstein) become competitive in terms of the accuracy-speed tradeoff over the exact (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', GE) or low-bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', IG) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The time discretization scheme is distinguished from those discussed earlier in that the conditional variance is approximated as a deterministic value rather than as a random variate from the distribution of I0,T , be it exact or approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, it must be successively applied to small time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Despite this limitation, the computational cost per time step is lower than that of other schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, time discretization is the preferred scheme if such a short time step is required for the derivatives because of the high monitoring frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The Euler and Milstein schemes are direct time discretizations of the stochastic differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the Heston model, however, these two simple schemes are notorious for failing with a large bias owing to the square root process for variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Among the numerous studies on time discretization schemes under the Heston model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', [12, 19]), the QE scheme proposed by Andersen [1] has been widely recognized as the best scheme [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The QE scheme is briefly reviewed below: Sampling variance: QE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Under time discretization, the derivative life, [0, T], is divided into N equal time intervals of size h (T = Nh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The monitoring points within the time interval are specified as ti = ih (i = 0, 1, · · · , N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As it is more concise to use index i than time ti, we change the notation convention in this section (and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 later) by using i in subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For example, we use Vi := Vti and Ii,i+1 := Iti,ti+1 (I0,N := I0,T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the QE scheme, Vi+1 given Vi is sampled using approximate functional forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' First, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4), we calculate the ratio, ψ = Var(Vi+1|Vi)/E(Vi+1|Vi)2, as a proxy for the probability that Vi+1 hits the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Then, the simulation is split into two cases depending on ψ: Vi+1 = � � � a(b + Z)2 if ψ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 1 β ln � 1−p 1−U � 1U>p if ψ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='24) where Z and U are standard normal and uniform random variables, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 1x is the indicator function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' and the coefficients a, b, β, and p are determined to match E(Vi+1 | Vi) and Var(Vi+1 | Vi) in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Trapezoidal rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' After simulating Vi+1 from Vi, Andersen [1] adopts a simple trapezoidal rule to approx- imate the conditional integrated variance Ii,i+1 as follows: ITZ i,i+1 = (Vi + Vi+1)h 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='25) Given the simulation path {Vi : i = 0, 1, · · · , N}, I0,N over the entire period is approximated by ITZ 0,N = N−1 � i=0 ITZ i,i+1 = (V0 + 2V1 + · · · + 2VN−1 + VN)h 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='26) As previously noted, ITZ i,i+1 (conditional on Vi and Vi+1) is a deterministic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This feature marks an important difference compared with the GE and IG schemes, where Ii,i+1 is sampled as a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Martingale correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Finally, to ensure the martingale condition, Si = e(q−r)hE(Fi+1|Si), Andersen [1] modifies the conditional forward price in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='12) by adding a correction term, M QE i,i+1: Fi+1 = Sie(r−q)h exp � −ρ2 2 ITZ i,i+1 + ρ ξ [Vi+1 − Vi + κ(ITZ i,i+1 − θh)] + M QE i,i+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The martingale correction term, M QE i,i+1, is analytically determined from the approximation function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='24): M QE i,i+1 = ρκθ ξ h − A2Vi + � � � − A1b2a 1−2A1a + 1 2 ln(1 − 2A1a) if ψ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 − ln � p + β(1−p) β−A1 � if ψ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 and A1,2 = ρh 4 �2κ ξ − ρ � ± ρ ξ , where A1 and A2 takes + and − respectively, and a, b, β, and p are the coefficients used in the QE step in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 3 Poisson conditioning and enhanced simulation schemes In this section, we construct efficient simulation schemes for the Heston model based on the key observations of Poisson conditioning, which simplifies the model formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The primary merit of Poisson conditioning is that it removes the use of Bessel functions or Bessel random variables, the numerical evaluation of which is computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 Poisson conditioning It is well known that the noncentral chi-square random variable χ2(δ, γ) is a Poisson mixture of an ordinary (γ = 0) chi-square variables, and that an ordinary chi-square distribution is a special case of a gamma distribution, χ2(δ, 0) ∼ 2Γ1(δ/2): χ2(δ, γ) ∼ χ2 � δ + 2POIS �γ 2 � , 0 � ∼ 2 Γ1 �δ 2 + POIS �γ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 11 With these properties, the exact simulation scheme of VT in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3) can be alternatively recast as the following Poisson mixture gamma scheme: µ0 ∼ POIS � V0 φT (κ)e− κT 2 2 � , so that VT ∼ 2e− κT 2 φT (κ) Γ1 �δ 2 + µ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) Note that VT depends on V0 via µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This property is well known and is often used in the Heston simulation literature [see 24, 10, 23, for example].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Our new simulation schemes also use this Poisson mixture property to simulate VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Unlike other schemes, the intermediate Poisson variable µ0 is also essential for simulating I0,T , in addition to VT in our proposed scheme, as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The choice of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) for simulating VT does not significantly increase the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Our numerical tests with public numerical library show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) is marginally slower than simulating the noncentral chi- square variable directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The small difference is attributed to the generation of the Poisson variable µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Interestingly, we also observe that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) is comparable to Andersen’s QE procedure for generating VT (see more comments in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Our key observation for Poisson conditioning is the link between the Poisson random variable µ0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) and the Bessel random variable η0,T in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' According to Pitman and Yor [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='j)], the Bessel random variable η ∼ BES(ν, z) can alternatively be represented as a conditional Poisson random variable: µ ∼ POIS(λ) conditional on Γ1(ν + 1 + µ) = z2 4λ, where λ is a positive rate parameter that can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This can be derived from the observation that the joint probability of µ and Γ1(ν + 1 + µ) is proportional to the probability mass function of η in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='18): PPOIS(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' λ) fΓ1 � z2 4λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' ν + 1 + k � = λk e−λ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' � z2 4λ �ν+k e− z2 4λ Γ(ν + 1 + k) = � z 2λ �ν e−λ− z2 4λ (z/2)2k+ν k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Γ(k + ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, the alternative representation is proven as follows: Prob � µ = j ��� Γ1(ν + 1 + µ) = z2 4λ � = PPOIS(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' λ) fΓ1 � z2 4λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' ν + 1 + j � �∞ k=0 PPOIS(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' λ) fΓ1 � z2 4λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' ν + 1 + k � = PBES(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' ν, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Instead of taking λ = 1 as in Glasserman and Kim [10, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3], we can achieve a useful representation that resembles that of µ0 by judiciously choosing λ = V0 φT (κ) 2 e− κT 2 in the context of the Heston model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' By substituting ν = δ 2 − 1 and z = √V0VT φT (κ), η0,T ∼ BES(ν, z) is equivalent to the conditional Poisson variable: µ ∼ POIS � V0 φT (κ)e− κT 2 2 � conditional on Γ1 �δ 2 + µ � = z2 4λ = VT φT (κ) 2e− κT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Consequently, µ coincides with µ0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simultaneously, we observe that the required condition for µ is equivalent to the formula for VT in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1): VT = 2e− κT 2 φT (κ) Γ1 �δ 2 + µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, we conclude that η0,T is equivalent to µ0 conditional on the terminal variance VT : η0,T ∼ µ0 ��� VT = 2e− κT 2 φT (κ) Γ1 �δ 2 + µ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) This implies that the joint distribution of (VT , η0,T ) is equivalent to that of (VT , µ0) as long as µ0 and VT follow the relation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, η0,T can be simply replaced by µ0 when sampling I0,T , and the cumbersome evaluation of the modified Bessel function is completely avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Under Poisson conditioning, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) of the GE scheme can be further simplified to I0,T | µ0 ∼ X + Zδ/2 + µ0 � j=1 Z(j) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3) This is the key result of an efficient approach known as Poisson conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the remainder of this section, we show how various Heston simulation schemes can be simplified under the Poisson conditioning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2 Poisson-conditioned IG approximation First, we enhance Tse and Wan [23]’s IG approximation using Poisson conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The Poisson-conditioned IG approximation in this section is a special case of the more general Poisson-conditioned GE scheme introduced in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Nevertheless, we first discuss this scheme because the mean and variance of I0,T | µ0 to be derived in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4) will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Recall that the modified Bessel function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='23) is the computational bottleneck in the original IG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Under Poisson conditioning, we observe E(η0,T |µ0) = µ0 and Var(η0,T |µ0) = 0 because η0,T is now conditioned as µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Consequently, the mean and variance of I0,T | µ0 can be simplified from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='22): E(I0,T | µ0) = E(X) + E(Zδ/2) + µ0E(Z2) = (V0 + VT )mXT + �δ 2 + 2µ0 � mZξ2T 2, Var(I0,T | µ0) = Var(X) + Var(Zδ/2) + µ0Var(Z2) = (V0 + VT )vXξ2T 3 + �δ 2 + 2µ0 � vZξ4T 4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4) which are expressed in terms of elementary functions, with no reference to the Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, we expect significant speedup in calculating the conditional mean and variance of I0,T for moment matching with the IG variate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3 Poisson-conditioned GE scheme We present the main results of this study, the Poisson-conditioned GE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the simulation of I0,T | µ0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3), we apply two additional enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Aggregating gamma random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' First, using the additive property of the independent gamma variables, Γ1(α1) + Γ1(α2) ∼ Γ1(α1 + α2), we can combine the (2 + µ0) gamma variables in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3) as follows: I0,T | µ0 ∼ ∞ � k=1 � � 1 γk Γ1(nk) + 1 γk Γ1(δ/2) + µ0 � j=1 1 γk Γ1(2) � � ∼ ∞ � k=1 1 γk Γ1 � nk + δ 2 + 2µ0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5) turning the (2 + µ0) infinite series into a single series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This reduces the number of required gamma variates by more than one-third of the original GE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Series truncation with the IG approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the second enhancement, we improve the accuracy of truncating the infinite series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5) using the IG approximation instead of the gamma approximation in Glasserman and Kim [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Owing to the aggregated gamma variable, we apply the truncation to the whole I0,T term rather than each of X, Zδ/2, or Z2 terms, as in Glasserman and Kim [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As Tse and Wan [23] have already shown the effectiveness of the IG variate for approximating the entire I0,T , it is natural to expect that the IG approximation also works for the remaining terms of I0,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Our numerical tests confirm this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Let IK 0,T | µ0 denote the remainder of the first K terms of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5) under Poisson conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' IK 0,T | µ0 ∼ ∞ � k=K+1 1 γk Γ1 � nk + δ 2 + 2µ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6) We approximate IK 0,T | µ0 using an IG random variable with a matched mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The mean and variance of IK 0,T | µ0 can be easily obtained by subtracting those of the first K terms from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4): E(IK 0,T | µ0) = (V0 + VT ) � mXT − K � k=1 λk γk � + �δ 2 + 2µ0 � � mZξ2T 2 − K � k=1 1 γk � , Var(IK 0,T | µ0) = (V0 + VT ) � vXξ2T 3 − K � k=1 2λk γ2 k � + �δ 2 + 2µ0 � � vZξ4T 4 − K � k=1 1 γ2 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='7) As in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4), there is no reference to Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Note that the K = 0 case in our new scheme is reduced to the Poisson-conditioned IG scheme in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, this is not the case with the original GE and IG schemes, in terms of both the type and number of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The K = 0 case of the GE scheme consists of (2 + η0,T ) IG variates approximating X, Zδ/2, and copies of Z2 whereas the IG scheme uses one IG variate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, the proposed 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Poisson-conditioned GE scheme can be used flexibly as either an exact scheme (K ≥ 1 with one time step) or a low-bias scheme (K = 0 with multiple time steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Finally, the simulation of VT and I0,T under Poisson conditioning can be succinctly performed using the following simplified steps: Simulation steps of the Poisson-conditioned GE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Step 1 Given V0, draw µ0 as µ0 ∼ POIS � V0 φT (κ)e− κT 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Step 2 Given µ0, draw VT as VT ∼ 2e− κT 2 φT (κ) Γ1 �δ 2 + µ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Step 3 Given V0 and µ0, draw I0,T as I0,T ∼ K � k=1 1 γk Γ1 � nk + δ 2 + 2µ0 � + IG(λK, µK) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='8) where nk ∼ POIS((V0 + VT )λk) and IG(λK, µK) is an IG variate with the mean and variance matched to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Step 4 Given V0, VT , I0,T , and S0, draw ST from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Here, both µ0 and nk are sampled from the Poisson distribution, which is computationally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The simulations of VT and I0,T are indifferent to whether the Feller condition for the CIR variance process is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4 Poisson-conditioned Broadie–Kaya Laplace inversion scheme Although the Broadie and Kaya [5] procedure of numerical inversion of the conditional Laplace transform of I0,T is not recommended as an efficient numerical procedure, it may be instructive to show that Poisson conditioning can also simplify their numerical procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' From Poisson-conditioning decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3), the Laplace transform of I0,T conditional on µ0 is ob- tained as follows: E � e−uI0,T ��� µ0 � = E � e−uX� E � e−uZδ/2� E � e−uZ2�µ0 = exp � − V0+VT 2 cosh( κuT 2 )φT (κu) � exp � − V0+VT 2 cosh( κT 2 )φT (κ) � �φT (κu) φT (κ) �δ/2+2µ0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='9) where κu is as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This is a direct consequence of Glasserman and Kim [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4], who observe E � e−uX� = exp � − V0+VT 2 cosh( κuT 2 )φT (κu) � exp � − V0+VT 2 cosh( κT 2 )φT (κ) � and E � e−uZα� = �φT (κu) φT (κ) �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 15 We expect to gain a significant computational benefit compared to the original Broadie–Kaya algorithm because it no longer requires numerical evaluation of the modified Bessel functions, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' If we use the Laplace inversion of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='9) instead of the Poisson-conditioned GE scheme in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='8), Step 3 can be replaced by the following alternative procedure: Step 3’ Given V0 and µ0, draw I0,T from the cumulative distribution function obtained from the Laplace inversion of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, this alternative Broadie–Kaya scheme is still slower than the Poisson-conditioned GE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We do not include this scheme in our numerical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Remark The original Broadie–Kaya conditional Laplace transform (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13) can also be derived in a similar manner from the original decomposition of I0,T in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This serves as an alternative proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16), without resorting to the Bessel bridge decomposition used in Glasserman and Kim [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We outline the derivation below, because it also explains why (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13) contains Iν(z) and how η0,T is related to Iν(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The Laplace transform of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) is expressed as E � e−uI0,T � = E � e−uX� E � e−uZδ/2� E � E � e−uZ2�η0,T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' It is already known that the Laplace transform of X corresponds to the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the second and third terms, we use the probability-weighted average over η0,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Based on the probability mass function of η0,T in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='18) and series expansion of Iν(z) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='15), we deduce the remaining terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13): E � e−uZδ/2� E � E � e−uZ2�η0,T � = �φT (κu) φT (κ) �δ/2 ∞ � j=0 P(η0,T = j) �φT (κu) φT (κ) �2j =φT (κu) φT (κ) ∞ � j=0 [√V0VT φT (κ)/2]δ/2−1+2j Iν(√V0VT φT (κ))j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Γ(j + ν + 1) �φT (κu) φT (κ) �δ/2−1+2j =φT (κu) φT (κ) ∞ � j=0 [zu/2]ν+2j Iν(z)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Γ(j + ν + 1) = φT (κu) φT (κ) Iν(zu) Iν(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 Poisson-conditioned time discretization scheme The Poisson conditioning framework also enables the construction of a small time-interval simulation of the integrated variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The new Poisson-conditioned time discretization scheme competes favorably with Andersen [1]’s QE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Sampling variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' To take advantage of Poisson conditioning, we sample Vi using the Gamma-Poisson scheme in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the time discretization setup, sampling Vi+1 given Vi is modified to µi ∼ POIS � Viφh(κ)e− κh 2 2 � , then Vi+1 ∼ 2e− κh 2 φh(κ) Γ1 �δ 2 + µi � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='11) 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok where we use the subscript i instead of t by following a convention similar to that in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='11) is an exact sampling of Vi+1, it is more accurate than Andersen [1]’s QE scheme for large time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The concern is computing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Surprisingly, our numerical tests (see Section 4) verify that the execution time is comparable to that of QE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' It becomes slower than the QE step when time step h becomes very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This is because the Poisson rate of µi grows as 2Vi/(ξ2h) for a small h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' therefore, it takes more time to simulate µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Poisson-conditioned quadrature approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Next, we determine Ii,i+1 given Vi and Vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Given the results in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4), our natural choice for the deterministic value representing Ii,i+1 is its mean: IPOIS i,i+1 = E(Ii,i+1 | µi) = (Vi + Vi+1)mXh + �δ 2 + 2µi � mZξ2h2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='12) where mX and mZ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='20) should be redefined with a = κh/2 under the time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As mX and mZ are constants independent of the simulation path, it is necessary to calculate them only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Given the simulation path, the conditional integrated variance over the entire time period is given by IPOIS 0,N = (V0 + 2V1 + · · · + 2VN−1 + VN)mXh + �Nδ 2 + 2(µ0 + µ1 + · · · + µN−1) � mZξ2h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13) Note that this approach is practically feasible owing to the computationally simple expression in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4) under Poisson conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The same approach without Poisson conditioning is not feasible because of the modified Bessel function evaluation required in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='23) to calculate E(I0,T ) and Var(I0,T ) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The new approximation IPOIS i,i+1 improves over the naive trapezoidal approximation ITZ i,i+1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='25) and forms the building block for our time discretization scheme for pricing path-dependent options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Indeed, IPOIS i,i+1 can be shown to be related to ITZ i,i+1 in the limit of h ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' More precisely, based on the asymptotic expansion in the powers of h (see Appendix A), it is interesting to observe lim h↓0 E � IPOIS i,i+1 � = � Vi + Vi+1 + � ViVi+1 � h 3 ≈ (Vi + Vi+1)h 2 = ITZ i,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='14) It is not surprising that the asymptotic form of IPOIS i,i+1 involves the arithmetic average, Vi+Vi+1 2 , and the geometric average, � ViVi+1, because these two quantities also appear in the Laplace transform of I0,T (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Furthermore, the asymptotic expansion of IPOIS i,i+1 in Appendix A reveals that the leading order of truncation in the trapezoidal rule is O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This may explain why the trapezoidal rule is also accurate in approximating Ii,i+1 when the time step h is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Martingale correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We derive the corresponding martingale correction in our scheme, which is different from that in the QE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The martingale correction starts by recognizing that Var(Ii,i+1 | µi), although small, has been ignored because IPOIS i,i+1 is a deterministic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4), the missing variance is given by Var(Ii,i+1 | µi) = (Vi + Vi+1)vXξ2h3 + �δ 2 + 2µi � vZξ4h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 17 For a random variable X, the following approximation holds if the variance is small: E(ea+bX) ≈ ea+bE(X)+ 1 2 b2Var(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, under Poisson-conditioned time discretization, the conditional forward price (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='12) is corrected to Fi+1|µi = Sie(r−q)hE � exp � −ρ2 2 Ii,i+1 + ρ ξ [Vi+1 − Vi + κ(Ii,i+1 − θT)] � ��� µi � ≈ Sie(r−q)h exp � −ρ2 2 IPOIS i,i+1 + ρ ξ [Vi+1 − Vi + κ(IPOIS i,i+1 − θT)] + M POIS i,i+1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='15) where the martingale correction term M POIS i,i+1 is given by M POIS i,i+1 = ρ2 2 �κ ξ − ρ 2 �2 Var(Ii,i+1|µi) = ρ2 2 �κ ξ − ρ 2 �2 � (Vi + Vi+1)vX + �δ 2 + 2µi � vZξ2h � ξ2h3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) Note a subtle difference in the derivation of the martingale correction term compared to the QE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the QE scheme, M QE i,i+1 is fitted to satisfy the unconditional expectation, Si = e(q−r)hE(Fi+1|Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In contrast, in our scheme, we use the conditional expectation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Consequently, M POIS i,i+1 is a function of both Vi and Vi+1 whereas M QE i,i+1 depends solely on Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This approach provides more flexibility in determining the correction for missing variance in different con- texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In pricing variance swaps (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2), we need to sample the realized return variance ln2(Si+1/Si) rather than Si itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Using the equality, E � (a + bX)2� = (a + bE(X))2 + b2Var(X), we can correct the realized return variance sampling as ln2(Si+1/Si) ≈ � (r − q)h − IPOIS i,i+1 2 + ρ ξ � Vi+1 − Vi + κ(IPOIS i,i+1 − θh) � + Σi,i+1Z �2 + M ′POIS i,i+1, where the correction terms are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' M ′POIS i,i+1 = �ρκ ξ − 1 2 �2 Var(Ii,i+1|µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='17) In the QE scheme, it is unclear how to find martingale correction in the context of the realized return variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We verify the effectiveness of the new martingale corrections, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='17), for pricing European options and variance swaps, respectively, in our numerical tests in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 4 Numerical performance of the simulation schemes We perform comprehensive numerical tests to assess the performance of the proposed simulation schemes with Poisson conditioning in comparison to existing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Specifically, we price European vanilla options 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 and variance swaps in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2 using various schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For easier reference, we label the methods to be tested are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' – GE: Glasserman and Kim [10]’s original GE scheme in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3 – IG: Tse and Wan [23]’s IG approximation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' – QEM: Andersen [1]’s QE scheme with the trapezoidal rule and martingale correction in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' – POIS–GE: Poisson-conditioned GE scheme in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' – POIS–TD: Poisson-conditioned time discretization scheme in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 The methods described above are implemented in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We aim to keep the implementation of the schemes as simple as possible for a clean and fair performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In this regard, we do not adopt the IPZ algorithm [23, Algorithms 1 and 2], which is a technique to speed up the sampling of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) using the tabulated inverse distribution function of Γ1(δ/2) corresponding to µ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As the IPZ algorithm would ben- efit both the existing and proposed methods (except QEM), implementing it is unnecessary when comparing the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Instead, we use standard random number generation routines available in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2 We do not use the tabulation–interpolation of E(η0,T ) and Var(η0,T ) in IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This would speed up IG at the expense of implementation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Such a trick is not necessary in the corresponding POIS–GE (K = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Case V0 θ ξ ρ κ T r (%) q (%) X CH E(R0,T ) Var(R0,T ) I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 10 0 0 100 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='08467014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011243 II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3 15 0 0 100 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='64922292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='016118 III 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='21 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='19 0 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='80611331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='017586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000126 IV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='25 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 4 1 1 2 120 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='02491348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='198462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007109 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 The four sets of the Heston model parameters used in numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' All cases assume S0 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The exact call option price (CH) with strike price X, mean and variance of the average variance (E(R0,T ) and Var(R0,T )) are also provided for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We adopt the four sets of the Heston model parameter values in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The first three parameter sets have been frequently used in earlier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' – Case I: Andersen [1], Van Haastrecht and Pelsser [24], Lord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' [19], Tse and Wan [23] – Case II: Andersen [1], Van Haastrecht and Pelsser [24] – Case III: Broadie and Kaya [5], Tse and Wan [23] We add Case IV [16] to explore a new case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Unlike the first three cases, Case IV does not violate the Feller condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', δ = 4) and the strike price is out-of-the-money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Overall, our test cases are fairly diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For example, Cases I and II are long-dated options, and Cases III and IV exhibit strong mean reversion (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 also reports the exact call option price, and the mean and variance of the average variance for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The exact option prices are obtained from the inverse fast Fourier transform [15] with the unconditional characteristic function of the log asset price, which is free from the branch cut discontinuity [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The prices obtained in this manner agree well with the high-precision values reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 2 We use numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='noncentral chisquare if µ0 is not required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', IG and GE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We use numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='poisson and numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='standard gamma if µ0 is required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=', POIS–GE and POIS–TD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Simulation schemes for the Heston model with Poisson conditioning 19 We present a comparison of the efficiency (CPU time) and accuracy (quantified by bias and standard error) of the derivative prices obtained with various simulation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the simulation estimator ˆΘ and the true price Θ, the bias and standard error (SE) of the estimator are respectively defined by Bias = E( ˆΘ) − Θ and SE = � E( ˆΘ2) − E( ˆΘ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In all the simulation experiments, we drew 160, 000 paths to obtain an estimator ˆΘ and repeated the exper- iment 200 times to obtain E( ˆΘ) and E( ˆΘ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We ran simulations on a PC running Windows 11 with an Intel i7–11700 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 GHz) CPU and 8 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Case I GE POIS–GE Time Option Spot Time Option Spot N K (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='095 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='481 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='025) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='892 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='089) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='153 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='069 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='987 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='340 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='076) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='154 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='074) 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='409 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='075 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='075) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='084 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='074) 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='087 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='023 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='074) 1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='076) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='077) Case I IG POIS–GE (K = 0) Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='159 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='093 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='077) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='153 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='069 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078) 2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='205 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='075) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='160 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='076) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='136 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='119 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='081) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='105 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='076) 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='068 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='018) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='023 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='077) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='043 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='075) Case I QEM POIS–TD Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 20 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='116 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='082) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='115 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='071) 40 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='084) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='030 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078) 80 1/8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='076) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The bias and standard error of the at-the-money (X = S0 = 100) call option in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) and the spot price in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' See Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 for the parameter values and reference option price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the GE schemes (top), we take N = 1 time step (h = T) while the number of gamma terms, K, is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the IG approximations (middle) and time discretization schemes (bottom), N and h are varied, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 European call options First, we price the European vanilla option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Although the vanilla option can be priced more efficiently with Fourier inversion, it serves as a nontrivial test case for simulation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For pricing, we use the conditional MC method [25] instead of simulating ST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Conditional on VT and I0,T , ST follows a geometric Brownian motion in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='11), and the option price can be obtained using the BS formula with forward price FT in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='12) and volatility σ = Σ0,T / √ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, the unconditional European call option price under the Heston 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Case II GE POIS–GE Time Option Spot Time Option Spot N K (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='950 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='686 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='062) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='107 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='823 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='088 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='121 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='051) 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='366 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='056) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='075 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='089 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='058) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='026 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052) 1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) Case II IG POIS–GE (K = 0) Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='117 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='056) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='107 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='064 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='103 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='058) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='065 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='071 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='056) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='122 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='034 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='096 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='066 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='044 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) Case II QEM POIS–TD Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 30 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='124 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) 60 1/4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='060) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='017 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) 120 1/8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='540 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The bias and standard error of the at-the-money (X = S0 = 100) call option in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) and the spot price in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' See Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 for the parameter values and reference option price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the GE schemes (top), we take N = 1 time step (h = T) while the number of gamma terms, K, is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the IG approximations (middle) and time discretization schemes (bottom), N and h are varied, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' model can be estimated by taking the MC average of the BS prices over the simulated values of VT and I0,T : ˆCH = e−rT EMC{CBS(FT , σ, T, X)}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) where CBS is the undiscounted BS call option price, with forward price FT , volatility σ, maturity T, and strike price X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As the MC variance from the sampling of ST is suppressed by the BS formula, the conditional MC reduces the MC variance of ˆCH, thereby increasing the accuracy of the bias of the simulation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This method has been used in Broadie and Kaya [5, Tables 4 and 5] for the Heston model and Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' [6] for the SABR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We also measure how accurately the martingale condition is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In theory, the unconditional expectation of FT should equal the forward price, e(r−q)T S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Therefore, we reconstruct the spot price by taking the discounted MC average of FT : ˆS0 = e(q−r)T EMC{FT }, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) and examines the extent to which ˆS0 differs from S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This serves as another measure to assess the accuracy of our proposed simulation schemes, in particular, the effectiveness of martingale correction in POIS–TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the two time discretization schemes, we apply the aggregate martingale correction term to the conditional Simulation schemes for the Heston model with Poisson conditioning 21 Case III GE POIS–GE Time Option Spot Time Option Spot N K (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='078 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='025) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='024) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='024) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='024) Case III IG POIS–GE (K = 0) Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='024) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='025) 2 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='023) 4 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='023) 8 1/8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='021) Case III QEM POIS–TD Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 2 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='097 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='467 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='008) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='018) 4 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='164 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='021) 8 1/8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='009 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='045 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='021) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The bias and standard error of the at-the-money (X = S0 = 100) call option in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) and the spot price in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' See Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 for the parameter values and reference option price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the GE schemes (top), we take N = 1 time step (h = T) while the number of gamma terms, K, is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the IG approximations (middle) and time discretization schemes (bottom), N and h are varied, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' forward FT : M0,N = N−1 � i=0 Mi,i+1 In POIS–TD, M POIS 0,N is proportional to N−1 � i=0 Var(Ii,i+1|µi) = � (V0 + 2V1 + · · · + 2VN−1 + VN)vX + �Nδ 2 + 2µ0 + · · · + 2µN−1 � vZξ2h � ξ2h3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3) which is a linear combination of the sums of Vi and µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This adds little extra computation, because the same terms are already used in IPOIS 0,N in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Tables 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 show the numerical performance of the four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In each table, we compare the GE- based exact schemes (top), IG-based low-bias schemes (middle), and time discretization schemes (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We comment on each of these comparisons below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the GE versus POIS–GE comparison, POIS–GE shows substantially lower bias with CPU time re- duction (typically 40%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The IG approximation contributes to the lower bias for the truncated series, and Poisson conditioning contributes to the faster execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For POIS–GE with low K values, the biases in Cases I and II are relatively larger than those in the other cases because of the long maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Nevertheless, 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Case IV GE POIS–GE Time Option Spot Time Option Spot N K (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='032 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='049) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='048) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052) 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='050) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='054) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='049) 1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) Case IV IG POIS–GE (K = 0) Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='051) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) 2 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='049) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) 4 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='050) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='055) 8 1/8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='051) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) Case IV QEM POIS–TD Time Option Spot Time Option Spot N h (sec) Bias (SE) Bias (SE) (sec) Bias (SE) Bias (SE) 2 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='599 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='014) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='096 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='049) 4 1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='166 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='034 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='053) 8 1/8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='045 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='052) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5 Speed-accuracy comparison of the existing (left) and Poisson-conditioned (right) schemes for Case IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The bias and standard error of the out-of-the-money (X = 120) call option in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1) and the spot price in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' See Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1 for the parameter values and reference option price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the GE schemes (top), we take N = 1 time step (h = T) while the number of gamma terms, K, is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' For the IG approximations (middle) and time discretization schemes (bottom), N and h are varied, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' the spot price is accurately preserved, indicating that the distributional error in I0,T due to low K manifests in option prices rather than spot prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The option price bias becomes exceedingly small when K increases to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In IG versus POIS–GE (K = 0), the latter reduces the computation time by several factors, although the bias improvement is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This is because we avoided evaluating the modified Bessel function present in the original IG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In QEM versus POIS–TD, POIS–TD compares favorably with QEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The use of better discretization rules leads to improved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' It is also noteworthy that the runtime efficiency of POIS–TD is slightly better than that of the QEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This is surprising given that the motivation for introducing the QE step (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='24) is to avoid directly drawing a noncentral chi-squared variate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As expected in the time discretization, both schemes require a small h to achieve high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Despite the short maturity, Cases III and IV show relatively larger biases than the other cases for a large time step h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' However, the spot price bias is close to zero even with a large h, which verifies that the martingale correction is effective in both schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The advantage of POIS–TD is more pronounced for the pricing variance swap, as discussed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2 Discretely monitored variance swap In general, a swap product is a contractual agreement to exchange the floating and fixed legs of payment for future time t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In the variance swap, the floating leg is given by the annualized log return variance over Simulation schemes for the Heston model with Poisson conditioning 23 N = T/h monitoring periods, which is proxied by Rh 0,T = 1 T N � i=1 ln2(Si/Si−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='4) The monitoring frequency is set by the time step h: h = 1/4 for quarterly, h = 1/12 for monthly, and h = 1/52 for weekly monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The amount of fixed leg, Kh swap, is typically determined to ensure that the swap has zero net present value at t = 0: Kh swap = E(Rh 0,T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This value is called the fair strike of the variance swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The level of path dependence of these variance derivatives goes beyond Asian options because their payoff structures involve the squared sum of the log return of asset price St at t = ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The annualized log return variance in the continuous limit (h ↓ 0) is the average variance in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' lim h↓0 Rh 0,T → R0,T = 1 T � T 0 Vt dt, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5) and fair strike Kswap is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6): Kswap = E(R) = θ + (V0 − θ)1 − e−κT κT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The fair strike Kh swap of the discrete variance swap with the monitoring time step h admits a closed-form solution [4, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We express Kh swap as an adjustment ∆h swap to the continuously monitored fair strike Kswap: Kh swap = E(Rh 0,T ) = Kswap + ∆h swap, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6) where ∆h swap = h(θ + 2q − 2r) 4 � (θ + 2q − 2r) + 2(V0 − θ)1 − e−κT κT � + θξ κ � ξ 4κ − ρ � � 1 − 1 − e−κh κh � + (V0 − θ) ξ κ � ξ 2κ − ρ � 1 − e−κT κT � 1 + κh 1 − eκh � + � ξ2 κ2 (θ − 2V0) + 2 κ(V0 − θ)2 � 1 − e−2κT 8κT 1 − e−κh 1 + e−κh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The above analytical solution serves as a benchmark to assess the accuracy of the time discretization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We compare the fair strike Kh swap for various monitoring frequencies ranging from semi-annual (h = 1/2) to weekly (h = 1/52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As the time discretization methods are more efficient in derivatives with higher monitoring frequency, such as variance swap, we only compare QEM with POIS–TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In POIS–TD, we apply the martingale correction M ′POIS i,i+1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='17), which is tailored to log return variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Given the absence of such a correction in the QEM, we simply use M QE i,i+1 for the log return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok Case III QEM POIS–TD Benchmark Time Bias (SE) Time Bias (SE) N h (×10−2) (sec) (×10−2) (sec) (×10−2) 2 1/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='041 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007) 4 1/4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='832 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='024 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='007) 12 1/12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='011 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004) 52 1/52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6 Speed-accuracy comparison of the time discretization schemes for pricing variance swap with Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The bias and standard error of the fair strike of discretely monitored variance swap are reported for varying monitoring frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The benchmark fair strike is from the analytical reference price in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Case IV QEM POIS–TD Benchmark Time Bias (SE) Time Bias (SE) N h (×10−2) (sec) (×10−2) (sec) (×10−2) 2 1/2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='930 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='750 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='083) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='085) 4 1/4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='325 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='060) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='063) 12 1/12 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='036) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='038) 52 1/52 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='029) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='7 Speed-accuracy comparison of the time discretization schemes for pricing variance swap with Case IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The bias and standard error of the fair strike of discretely monitored variance swap are reported for varying monitoring frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The benchmark fair strike is from the analytical reference price in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Tables 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='7 list the results for Cases III and IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The two cases are chosen because variance swaps traded in the market are typically short-dated, such as one year in these cases, and they show a relatively larger bias in pricing vanilla options in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The results show that the POIS–TD bias is much smaller than the QEM bias, although the biases from both schemes quickly converge to zero as the monitoring becomes more frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The numerical results also verify that martingale correction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='17) is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 5 Conclusion Simulation under the Heston model has been a widely studied topic, and several approaches are available for different monitoring frequencies: exact [5, 10], low-bias [23], and time discretization [1] schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Exist- ing simulation schemes, however, suffer from computationally expensive evaluations of the modified Bessel function or Bessel random variables arising from the square-root variance process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Based on the observation that the conditional integrated variance can be simplified when conditioned by the Poisson variate used for simulating the terminal variance, we propose simulation methods that enhance the existing methods in all spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Poisson-conditioned GE is an exact simulation scheme that enhances Glasserman and Kim [10]’s GE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' It achieves significant speedup by expressing the conditional integrated variance without the Bessel random variable, which is a computational bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Adopting the IG approximation [23] for the truncation approximation improves numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' A special case of the Poisson-conditioned GE scheme is naturally reduced to Tse and Wan [23]’s low-bias scheme but without the Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' As the Laplace transform of the conditional integrated variance can also be expressed without the Bessel function, Broadie and Kaya [5]’s Simulation schemes for the Heston model with Poisson conditioning 25 scheme can be expedited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The Poisson-conditioned time discretization scheme is proposed as an alternative to Andersen [1]’s QE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Our comprehensive numerical tests illustrate the strong competitiveness of our schemes in speed-accuracy comparison among existing schemes for pricing derivatives under the Heston model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' In addition to numerical efficiency, our new Heston simulation schemes are simple and straightforward for practitioners to implement because they involve only elementary functions and random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' A Asymptotic expansion of IPOIS i,i+1 Taking the limit of h ↓ 0, the coefficients in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='20) have the following asymptotic expansions in the powers of a: mX = c1 − ac2 2a = 1 3 � 1 − 2 15a2 + 2 105a4 · · · � , mZ = ac1 − 1 4a2 = 1 12 � 1 − 1 15a2 + 2 315a4 · · · � , where a = κh/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We can easily see that mX → 1 3 and mZ → 1 12 as h ↓ 0, with O(h2) as the leading order of truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Furthermore, we obtain asymptotic expansions of Iν(z) and z: Iν(z) = ez √ 2πz � 1 − 4ν2 − 1 8z + · · · � and z = φh(κ) � ViVi+1 = 2κ/ξ2 sinh κh 2 � ViVi+1 → 4 � ViVi+1 ξ2h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' We also consider E(ηi,i+1) in the h ↓ 0 limit E(ηi,i+1) = 2µi = z Iν+1(z) 2Iν(z) → z 2 � 1 − 2ν + 1 2z � = z 2 − δ 4 → 2 � ViVi+1 ξ2h − δ 4, from which we obtain δ 2 + 2µi = δ 2 + z Iν+1(z) Iν(z) → δ 2 + 4 � ViVi+1 ξ2h − δ 2 = 4 � ViVi+1 ξ2h as h ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Combining these results, we obtain E � ˆIPOIS i,i+1 � = (Vi + Vi+1)mXh + �δ 2 + 2E(µi | Vi, Vi+1) � mZξ2h2 = (Vi + Vi+1)mXh + �δ 2 + 2E(ηi,i+1) � mZξ2h2 → (Vi + Vi+1)h 3 + 4 � ViVi+1 ξ2h ξ2h2 12 = � Vi + Vi+1 + � ViVi+1 � h 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' This result is consistent with Tse and Wan [23, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Andersen L (2008) Simple and efficient simulation of the Heston stochastic volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Journal of Computational Finance 11(3):1–42, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='21314/JCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='189 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Baldeaux J (2012) Exact simulation of the 3/2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' International Journal of Theoretical and Applied Finance 15(05):1250032, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1142/S021902491250032X 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Ball CA, Roma A (1994) Stochastic volatility option pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Journal of Financial and Quantitative Analysis 29(4):589–607, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2307/2331111 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Bernard C, Cui Z (2014) Prices and Asymptotics for Discrete Variance Swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Applied Mathematical Finance 21(2):140–173, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1080/1350486X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='820524 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Broadie M, Kaya ¨O (2006) Exact simulation of stochastic volatility and other affine jump diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Operations Research 54(2):217–231, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1287/opre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='0247 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Cai N, Song Y, Chen N (2017) Exact simulation of the SABR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Operations Research 65(4):931–951, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1287/opre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1617 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi J (2023) A new exact simulation of the Ornstein-Uhlenbeck driven stochastic volatility model using the Karhunen-Lo`eve expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Working Paper, Peking University HSBC Business School 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Choi J, Liu C, Seo BK (2019) Hyperbolic normal stochastic volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Journal of Futures Markets 39(2):186–204, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1002/fut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='21967 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Cox JC, Ingersoll JE Jr, Ross SA (1985) A theory of the term structure of interest rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Econometrica 53(2):385–407, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2307/1911242 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Glasserman P, Kim KK (2011) Gamma expansion of the Heston stochastic volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Finance and Stochastics 15(2):267–296, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1007/s00780-009-0115-y 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Review of Financial Studies 6(2):327–343, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1093/rfs/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='327 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kahl C, J¨ackel P (2006) Fast strong approximation Monte Carlo schemes for stochastic volatility models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Quantitative Finance 6(6):513–536, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1080/14697680600841108 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kang C, Kang W, Lee JM (2017) Exact Simulation of the Wishart Multidimensional Stochastic Volatility Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Operations Research 65(5):1190–1206, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1287/opre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1636 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Kwok YK, Zheng W (2022) Pricing Models of Volatility Products and Exotic Variance Derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Chapman & Hall/CRC Financial Mathematics Series, New York 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Lewis AL (2000) Option Valuation under Stochastic Volatility: With Mathematica Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Newport Beach, CA 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Lewis AL (2019) Heston Model Reference Prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' URL https://financepress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='com/2019/02/15/ heston-model-reference-prices/ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Li C, Wu L (2019) Exact simulation of the Ornstein–Uhlenbeck driven stochastic volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' European Journal of Operational Research 275(2):768–779, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='ejor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='057 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Lord R, Kahl C (2010) Complex Logarithms in Heston-Like Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Mathematical Finance 20(4):671– 694, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1467-9965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='00416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='x 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Lord R, Koekkoek R, Dijk DV (2010) A comparison of biased simulation schemes for stochastic volatility models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Quantitative Finance 10(2):177–194, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1080/14697680802392496 Simulation schemes for the Heston model with Poisson conditioning 27 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Malham SJA, Shen J, Wiese A (2021) Series Expansions and Direct Inversion for the Heston Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' SIAM Journal on Financial Mathematics 12(1):487–549, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1137/19M126791X 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Michael JR, Schucany WR, Haas RW (1976) Generating random variates using transformations with multiple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The American Statistician 30(2):88–90, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1080/00031305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='10479147 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Pitman J, Yor M (1982) A decomposition of Bessel Bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Zeitschrift f¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete 59(4):425–457, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1007/BF00532802 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Tse ST, Wan JWL (2013) Low-bias simulation scheme for the Heston model by Inverse Gaussian ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Quantitative Finance 13(6):919–937, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1080/14697688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='696678 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Van Haastrecht A, Pelsser A (2010) Efficient, almost exact simulation of the Heston stochas- tic volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' International Journal of Theoretical and Applied Finance 13(01):1–43, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1142/S0219024910005668 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Willard GA (1997) Calculating prices and sensitivities for path-independent derivatives securities in multifactor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' The Journal of Derivatives 5(1):45–61, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3905/jod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='407982 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Zeng P, Xu Z, Jiang P, Kwok YK (2021) Analytical Solvability and Exact Simulation of Stochastic Volatility Models with Jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content=' Working Paper, Southern University of Science and Technology, China, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} +page_content='3904498' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf'} diff --git a/uNE0T4oBgHgl3EQf9gI3/content/tmp_files/2301.02801v1.pdf.txt b/uNE0T4oBgHgl3EQf9gI3/content/tmp_files/2301.02801v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c6bbcbacd384e804f277bda0fe7c2fca096f6fe --- /dev/null +++ b/uNE0T4oBgHgl3EQf9gI3/content/tmp_files/2301.02801v1.pdf.txt @@ -0,0 +1,1221 @@ +arXiv:2301.02801v1 [math.DS] 7 Jan 2023 +Manuscript submitted to +doi:x +AIMS’ Journals +Volume x, Number 0x, xx xx +pp. x–x +A VARIETY OF GLOBALLY STABLE PERIODIC ORBITS IN +PERMUTATION BINARY NEURAL NETWORKS +Mikito Onuki, Kento Saka and Toshimichi Saito∗ +Department of Electrical and Electronic Engineering +HOSEI University, Japan +Abstract. The permutation binary neural networks are characterized by global +permutation connections and local binary connections. Although the parame- +ter space is not large, the networks exhibit various binary periodic orbits. Since +analysis of all the periodic orbits is not easy, we focus on globally stable binary +periodic orbits such that almost all initial points fall into the orbits. For effi- +cient analysis, we define the standard permutation connection that represents +multiple equivalent permutation connections. Applying the brute force attack +to 7-dimensional networks, we present the main result: a list of standard per- +mutation connections for all the globally stable periodic orbits. These results +will be developed into detailed analysis of the networks and its engineering +applications. +1. Introduction. Discrete-time recurrent neural networks (DT-RNNs) are analog +dynamical systems characterized by real valued connection parameters and non- +linear activation functions (e.g., sigmoid function) [1] [2] [3] [4]. The dynamics is +described by autonomous difference equations of real state variables. Depending +on the parameters, the DT-RNNs exhibit various periodic orbits, chaos [6], and +related bifurcation phenomena. +The real/potential applications include associa- +tive memories [1], combinatorial optimization problems solvers [5], and time-series +approximation/prediction in reservoir computing [7] [8] [9]. +The DT-RNNs are +important systems in both basic study of nonlinear dynamics and engineering ap- +plications. However, analysis of the dynamics is hard because of huge parameter +space and complexity of the nonlinear phenomena. +Stability analysis of various +periodic orbits is not easy. +The three-layer dynamic binary neural networks (DBNNs [10] [11]) are digital +dynamical systems characterized by ternary valued connection parameters and the +signum activation function. The dynamics is described by autonomous difference +equations of binary state variables. Since the state space consists of a finite number +of binary variables, the DBNNs cannot generate chaos [6]. However, depending +on the parameters and initial conditions, the DBNNs can generate various peri- +odic orbits of binary vectors (binary periodic orbits, ab. BPOs). As compared +with the DT-RNNs, the DBNNs bring benefits to hardware implementation. An +FPGA based hardware prototype and its application to hexapod walking robots +Key words and phrases. Recurrent neural networks, binary neural networks, permutation, bi- +nary periodic orbits, stability. +∗ Corresponding author: Toshimichi Saito. +1 + +2 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +can be found in [12]. We have presented a parameter setting method that guar- +antees storage and stability of desired BPOs [10]. However, as period of a BPO +increases, the number of hidden neurons increases: parameter space becomes wider +and analysis becomes harder. In the hardware, power consumption becomes larger. +In order to realize efficient analysis and synthesis, reduction of the parameter space +is inevitable. +Simplifying connection parameters of the DBNNs, the permutation binary neural +networks (PBNNs [13]) are constructed. The PBNNs are characterized by two kinds +of connections. The first one is local binary connection between input and hidden +layers. It is defined by a signum-type neuron from three binary inputs to one binary +output. The second one is global one-to-one connection between hidden and output +layers. It is defined by a permutation operator. The parameter space of the PBNNs +is much smaller than that of DBNNs. Depending on the permutation connections, +the PBNNs generate various BPOs. Co-existence of BPOs is possible and the PBNN +exhibits one of the BPOs depending on initial condition. +Since analysis of multiple BPOs are not easy, this paper focuses on globally sta- +ble binary periodic orbits (GBPOs) such that almost all initial points fall into the +GBPOs. As a fundamental concept, we define the standard permutation connection +that represents multiple equivalent permutation connections. Applying the brute +force attack to all the 7-dimensional PBNNs, we present the main result: a list of +the standard permutation connections for all the GBPOs. These results provide +basic information to realize more detailed analysis of PBNNs and its applications. +Real/potential engineering applications of the GBPOs include time-series approx- +imation/prediction [8] [14] [15], control signals of switching power converters [16] +[17] [18], control signals of walking robots [12] [19], and error correcting codes [20]. +The approximate/control signals can be globally stable and robust. As novelty of +this paper, it should be noted that this is the first paper of the GBPOs and standard +permutation connections. +2. Permutation binary neural networks and binary periodic orbits. This +section introduces the 3-layer dynamic binary neural networks (DBNNs, [10]) and +the permutation binary neural networks (PBNNs, [13]). After overview of BPOs, +we show the objective problem. +2.1. Dynamics binary neural network. The DBNNs are recurrent-type 3-layer +networks characterized by ternary connection parameters and signum activation +function. The dynamics is described by the following autonomous difference equa- +tion of N-dimensional binary state variables: +xt+1 +i += sgn + + +M +� +j=1 +cijyt +j + Si + + , yt +j = sgn +� N +� +i=1 +wjixt +i − Tj +� +sgn(x) = +� +1 +if x ≥ 0, +i ∈ {1, · · · , N} +−1 +if x < 0, +j ∈ {1, · · · , M} +(1) +where xt +i ∈ {−1, +1} ≡ B is the i-th binary state variable at discrete time t and +yt +j ∈ B is the j-th binary hidden variable. As shown in Fig. 1, the binary variables +xt +i, yt +j, and xt+1 +i +are located in input, hidden, and output layers, respectively. The +M hidden neurons transform xt +i into yt +j through hidden ternary connections (wji ∈ +{−1, 0, +1}). The N output neurons transform yt +j into xt+1 +i +through output ternary +connections (cij ∈ {−1, 0, +1}). The threshold parameters Si and Tj are integers. + +GLOBALLY STABLE PERIODIC ORBITS +3 +Figure 1. Dynamic binary neural network (DBNN) Red and blue +branches denote positive and negative connections, respectively. +The output xt+1 +i +is fed back to the input layer and the DBNNs generate various +BPOs. Ref. [10] gives a theoretical result of parameter condition that guarantees +storage and stability of desired BPOs. However, as period of a BPO increases, the +number of hidden neurons increases. For example, p hidden neurons are required +for storage of a BPO with period p. As p increases, the parameter space becomes +larger and analysis/implementation becomes harder. +2.2. Permutation binary neural networks. The PBNNs are described by the +following autonomous difference equation of N-dimensional binary state variables: +xt+1 +i += yt +σ(i), yt +i = sgn +� +waxt +i−1 + wbxt +i + wcxt +i+1 +� +σ = +� +1 +2 +· · · +N +σ(1) +σ(2) +· · · +σ(N) +� +i ∈ {1, · · · , N}, N ≥ 3 +(2) +where xt +0 ≡ xt +N and xt +N+1 ≡ xt +1 for ring-type connection as shown in Fig. +2. +As a binary state vector xt ≡ (xt +1, · · · , xt +N) ∈ BN is input at time t, the xt is +transformed into the binary hidden state vector yt ≡ (yt +1, · · · , yt +N) ∈ BN through +hidden neurons with local binary connections. All the hidden neurons have the +same characteristics: the signum activation function from three binary inputs to one +binary output with local binary connection parameters (wa, wb, wc) ∈ B3. The yt +is transformed into xt+1 through one-to-one global permutation connection defined +by the permutation σ. The output vector xt+1 is fed back to the input and the +PBNNs generate sequences of binary vectors. +In comparison with the DBNNs, +the hidden connections wij are replaced with the local binary connections and the +output connections cij are replaced with the global permutation connections. As +shown in Fig. 3, the local binary connections are identified by connection numbers: +CN0 : wl = (−1, −1, −1) +CN1 : wl = (−1, −1, +1) +CN2 : wl = (−1, +1, −1) +CN3 : wl = (−1, +1, +1) +CN4 : wl = (+1, −1, −1) +CN5 : wl = (+1, −1, +1) +CN6 : wl = (+1, +1, −1) +CN7 : wl = (+1, +1, +1) +where wl ≡ (wa, wb, wc). Since CN1 (respectively, CN3) coincides with CN4 (re- +spectively, CN6) by replacement xi → xN−i+1 for i ∈ {1, · · · , N}, we consider +6 connection numbers without CN4 and CN6 hereafter. The global permutation + ++1 +S2 +S5 +S6 +S7 +S1 +y +y2 +yi +T1 +T4 +Ts +T +6 +7 +X2 +x34 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +connections are identified by +Permutation ID: P(σ(1) · · · σ(N)). +Fig. 2 shows examples of 7-dimensional PBNNs for CN1. For identity permuta- +tion P(123456), the PBNN exhibits a BPO with period 14. Applying permutation +P(2613754), the PBNN exhibits a BPO with longer period 20. In the DBNN, 20 +hidden neurons are necessary for period 20. +2.3. Objective problem. In order to visualize the dynamics, we have introduced +the digital return map (Dmap). The domain BN of the PBNNs is equivalent to a set +of 2N points LN ≡ {C1, · · · , C2N }, i.e., C1 ≡ (−1, · · · , −1), C2 ≡ (+1.−1. · · · , −1), +· · · , C2N ≡ (+1, · · · , +1). The dynamics of a PBNN can be integrated into +Dmap: xt+1 = f(xt), xt ∈ BN ≡ LD +(3) +where an N-dimensional binary vector xt is denoted by a point Ci in the Dmap. +Figure 2. Examples of PBNNs and BPOs for CN1, N = 7. Red +and blue branches denote positive and negative local binary con- +nections, respectively. black branches correspond to global permu- +tation connections. White and black squares in spatiotemporal pat- +terns denote xt +i = +1 and xt +i = −1, respectively. (a) P(1234567). +(b) P(2613754). +Figure 3. 8 local binary connections. + +CNO +CN1 +CN2 +CN3 +Wb +Wc +Wa +Wb +Wc +Wa +Wc +Wa +Wa +Wb +Wb +Im +CN4 +CN5 +CN6 +CNT +Wa +Wb +Wc +Wa +Wa +Wb +Wc +Wb +Wa +Wb +Wc +ma +ot+1 +(b) +.t+1 +rt+1 +.t+1 +rt+1 +t+1 +t+1 +t+1 +.t+1 +.t+1 +t+1 +.t+1 +.t+1 +X4 +x3 +x2 +9x +yt +yt +ys +yt +yt +yi +yt +yt +yt +y +y +x2 +x +x +x4 +xt +x +xt +xt +xi +Period 14 +Period 20 +1 +1 +7 +7 +10 +20 +1 +10 +t +1 +1GLOBALLY STABLE PERIODIC ORBITS +5 +Figure 4. Dmap examples (black points) and BPOs (blue orbits) +for CN1, N = 7. (a) P(1234567) (the PBNN is Fig. 2 (a)), BPO +with period 14. (b) P(2613754) (the PBNN is Fig. 2 (b)), BPO +with period 20. +Definition 2.1. A point zp ∈ LD is said to be a binary periodic point (BPP) with +period p if f p(zp) = zp and f(zp) to f p(zp) are all different where f k is the k-fold +composition of f. A sequence of the BPPs, {f(zp), · · · , f p(zp)}, is said to be a BPO +with period p. A point ze is said to be an eventually periodic point (EPP) if ze is +not a BPP but falls into a BPO, i.e., there exists some positive integer l such that +f l(ze) is a BPP. The BPO in the Dmap is equivalent to the BPO in spatiotemporal +pattern from the PBNN. +Fig. 4 shows BPOs in Dmaps corresponding to BPOs in spatiotemporal patterns +in Fig. 2. As parameters (CN and Permutation ID) vary, the PBNN exhibits a vari- +ety of BPOs. The number of CNs (without CN4 and CN6) is 6 whereas the number +of hidden connection parameters wij is 3N 2. The number of Permutation IDs is N! +whereas the number of output connection parameters cij is 3N 2. In addition, the +DBNNs have 2N integer threshold parameters Si and Tj. It goes without saying +that the PBNNs cannot generate more various BPOs than the DBNNs because the +PBNNs are included in the DBNNs. However, the PBNN parameter space is much +smaller than the DBNN parameter space. The objective problem is relationship +between parameters (Permutation ID and CN) and existence/stability of BPOs. +3. Globally stable binary periodic orbits. Depending on parameters, the PBNNs +exhibit various BPOs and multiple BPOs can co-exist for initial state. Since anal- +ysis of multiple BPOs is hard, we try to analyze representative BPOs: the globally +stable binary periodic orbits (GBPOs). This section defines the GBPOs and related +concepts. First, we note two exceptional endpoints in BN: +x− ≡ (−1, · · · , −1) ∈ BN, x+ ≡ (+1, · · · , +1) ∈ BN +(4) +The two endpoints are either fixed points or a BPO with period 2, becuase +f(x+) = x+, f(x−) = x− if wa + wb + wc ≥ +1 +f(x+) = x−, f(x−) = x+ if wa + wb + wc ≤ −1 +(5) + +(a) +(b) +C128 +28 +t+1 +C64 +64 +C64 +C128 +C128 +646 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +Hereafter we omit the two endpoints. The GBPO is defined by +Definition 3.1. A BPO is said to be a globally stable binary periodic orbit (GBPO) +if the BPO is unique (except for x− and x−) and if all the EPPs fall into the BPO +where we assume existence of the EPPs. The number of EPPs plus elements of the +GBPO is 2N − 2. +Fig. 4(b) shows a GBPO with period 20 in the Dmap. In this 7-dimensional +example, (27 − 20 − 2) EPPs fall into the GBPO. As shown in Section 4, depending +on the parameters (permutation ID and CN), the 7-dimensional PBNNs exhibit a +variety of GBPOs and the number of EPPs is more than 27/2. The EPPs represent +global stability corresponding to error correction [20] of binary signals. +As the +number of EPPs increases, the global stability becomes stronger. In the limit case +of the M-sequences (e.g., in the linear feedback shift register [21]), the period is +2N, no EPP exists and is not stable. Such M-sequences are different category from +the GBPOs in this paper. In fundamental viewpoints, uniqueness of the GBPO +is convenient to consider existence and stability. +Analysis of multiple BPOs is +complex. In application viewpoints, the GBPOs are useful as globally stable signal +to approximate/predict time-series [15] and to control switching circuits [16] [17] +[18]. +For simplicity, we focus on the case where N is a prime number Np. +If an +integer N can be factorized into prime factors, classification of the permutation +connections becomes complex. Here, in order to analyze GBPOs, we define several +basic concepts. +Definition 3.2. Let R be a shift operator such that +R : P0(σ0(1) · · · σ0(Np)) → P1(σ1(1) · · · σ1(Np)) +P1 = R(P0), σ1(i + 1) = σ0(i) + 1 mod Np, i ∈ {1, · · · , Np} +(6) +where σ1(Np + 1) ≡ σ1(1). Since the neurons are ring-type connection, the permu- +tation connections P1 and P0 (P and R(P)) are equivalent even if the permutation +IDs are different. +Definition 3.3. Let S be a set of permutation IDs that give equivalent permutation +connections. +The set S is referred to as an equivalent permutation set (EPS). +An EPS is represented by a standard permutation ID Ps(σs(0) · · · σs(Np)) that +corresponds to the minimum element in the EPS by means of base-Np number: +Ps(σs(1) · · · σs(Np)) < Pk(σk(1) · · · σk(Np)) ∈ S, k ̸= s ( base-Np number ) +Fig. 5 shows an example of standard permutation connection and its equivalent +permutation connections for Np = 7. In this example, the EPS is +S = {Ps(1325476), P(7243651), P(2135476), P(7324651), +P(2143576), P(7325461), P(2143576)} +Definition 3.4. A permutation ID Pb is said to be a basic permutation ID if it is a +fixed point of the shift operator: R(Pb) = Pb. Since R(Pb) = Pb iff σb(i+1) = σi+1 +mod Np, the number of basic permutation IDs is Np. +A basic permutation ID +constructs an EPS with one element and is a standard permutation ID. +Fig. 6 shows basic permutation connections for Np = 7. Then we have +Theorem 3.5. In Np-dimensional PBNNs, the number of standard permutation +IDs (i.e., the number of EPSs) is (Np − 1)! + Np − 1 where Np ≥ 3 is a prime +number. + +GLOBALLY STABLE PERIODIC ORBITS +7 +Figure 5. +Equivalent permutation connection examples for Np = +7. Ps: standard permutation connection. R: shift operator. +Figure 6. +7 basic permutation connections for Np = 7. +Figure 7. +Permutation connection examples consisting of 3 sub- +connections for N = 6. +(Proof) Except for Np basic permutations, one standard permutation ID Ps +represents Np equivalent permutation IDs: +RNp(Ps) = Ps, Rk(Ps) ̸= Ps for 1 ≤ k ≤ Np − 1 +where Rk(P) = R(Rk−1(Ps)) is the k-fold composition of the shift operator R. If +there exists an integer l (2 ≤ l < Np) such that Rl(Ps) = Ps, the ring-type con- +nection of Ps is decomposed into the same sub-connections (e.g., 3 sub-connections +R3l(Ps) = RNp(Ps) = Ps as shown in Fig. 7). However, it is impossible for a prime +number Np. Therefore, except for the basic permutations, the number of standard +permutation IDs is (Np! − Np)/Np. +Adding the Np basic permutation IDs, the +number of standard permutation IDs is (N! − Np)/Np + Np = (Np − 1)! + Np − 1. + +Ps(214365) +P(632541) +Ps(214365) +XXX +XXX +XXX +R +P(632541) +Ps(214365) +P(632541) +XXX +R +RPb(1234567) +Pb(2345671) +Pb(3456712) +Pb(4567123) +IIIIII1 +Pb(5671234) +Pb(6712345) +Pb(7123456)P(7243651) +P(2135476) +P(7324651) +Ps(1325476) +IXXX +XXXIXXXX +P(2143576) +P(7325461) +P(2143576) +Ps(1325476) +XXXRXXXIE +RXXIX8 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +4. Brute force attack to explore GBPOs. Table 1 shows the number of stan- +dard permutation IDs for prime numbers Np together with the number of full bi- +nary connection parameters between hidden and output layers in the DBNNs for +N = M = Np. The number of the permutation connections is much smaller than the +number of the full binary connections. However, analysis of the GBPOs becomes +harder as Np increases. +For convenience, we consider GBPOs in 7-dimensional +PBNNs (Np = 7). In the case NP = 7, the number of all the standard permutation +connections is (Np − 1)! + Np − 1 = 726, the number of initial points is 27, and the +brute force attack is possible. We can clarify the number and period of all the GB- +POs precisely. Analysis of the 7-dimensional GBPOs are fundamental to consider +higher-dimensional GBPOs and their engineering applications. +We explore the 7-dimensional GBPOs as the following. First, as state earlier, +objective connection numbers are CN0, CN2, CN2, CN3, CN5, and CN7 (CN1 ≡ +CN4 and CN3 ≡ CN6). Second, applying the shift operator R, we obtain the 726 +standard permutation IDs. Third, applying the brute force attack to each standard +permutation ID and CN, we obtain BPOs and their EPPs where we use the BPO +calculation algorithm in [22]. If the number of a BPP plus its EPPs is 27 − 2 = 126 +then the BPO is declared as the GBPO. The period of the GBPO is stored together +with its standard permutation ID. +In the exploration, it is confirmed that CN0 and CN7 cannot provide GBPO. The +CN0 and CN7 are omitted hereafter. Fig. 8 shows typical examples of PBNNs for +CN1, CN2, CN3, and CN5 that generate GBPO with period 42, period 14, period +26, and period 14, respectively. +Fig. +9 shows the 4 GBPOs as spatiotemporal +patterns and Fig. 10 shows the 4 GBPOs in Dmaps. +As a criterion of the period, we give +Definition 4.1. For identity permutation (Pb(1234567) for NP = 7), the period +of the BPO is said to be basic period. If the PBNN generates multiple BPOs, the +maximum period is adopted. +For CN1, the basic period is 14 as a BPO in Fig. 2 (a) that is a GBPO. We +have confirmed that the identity permutation Pb(1234567) cannot provide a GBPO. +In Figs. 8 to 10, we can see that, adjusting permutation IDs from the identity +permutation Pb(1234567), the PBNNs can generate a variety of BPOs represented +by the GBPOs with longer period. As the main result, tables 2 to 5 show a list +of standard permutation IDs for all the GBPOs. As stated in Definition 3.3, each +standard permutation ID represents 7 equivalent permutation IDs. +We give an +overview of the list for CN1, CN2, CN3, and CN5: +Table 1. The number of standard permutation connections in +PBNN and full binary connections between hidden and output lay- +ers in DBNN. +Np +# standard permutation IDs +# full binary connections +3 +4 +29 +5 +28 +225 +7 +726 +249 +11 +3628810 +2121 +13 +479001612 +2169 +17 +20922789888016 +2289 + +GLOBALLY STABLE PERIODIC ORBITS +9 +Figure 8. PBNN examples (exhibit GBPOs) for Np = 7. +(a) +Ps(1357246), CN1. (b) Ps(1462753), CN2. (c) Ps(1256473), CN3. +(d) Ps(1463725), CN5. +• CN1: The basic period is 14 for Pb(1234567). The PBNNs generate 27 GB- +POs. The maximum period is 42 for Ps(1357246) as shown in Fig. 9 (a). The +number of EPPs is 126 − 42. +• CN2: The basic period is 2. The PBNNs generate 56 GBPOs. The maximum +period is 14 where the number of EPPs is 126−14, e.g. Ps((1462753) as shown +in Fig. 9 (b). +• CN3: The basic period is 14. The PBNNs generate 28 GBPOs. The maximum +period is 26 where the number of EPPs is 126−26, e.g. Ps(1256473) as shown +in Fig. 9 (c). +• CN5: The basic period is 2. The PBNNs generate 62 GBPOs. The maximum +period is 14 where the number of EPPs is 126−14, e.g. Ps(1463725) as shown +in Fig. 9 (d). +These tables clarify relation between parameters (permutation ID and CN) and +periods of the GBPOs. The number of EPPs is 126 minus the period. As the pa- +rameters vary, the 7-dimensional PBNNs can generate a variety of GBPOs. These +results provide fundamental information to analyze various PBNNs and to synthe- +size PBNNs with desired GBPOs. + +(a) +(b) +t+1 +t+1 +t+1 +t+1 +.t+1 +t+1 +t+1 +.t+1 +t+1 +.t+1 +t+1 +t+1 +t+1 +.t+1 +xi +x3 +x +X2 +x3 +CN1 +CN2 +y4 +yt +yt +yt +y4 +y2 +yt +yt +yt +y2 +y6 +yi +x2 +xt +x4 +xt +x2 +xt +x +x +xt +x +xt +xt +(c) +(d) +t+1 +t+1 +t+1 +t+1 +t+1 +t+1 ++1 +t+1 ++1 +t+1 +.t+1 +t +t +t+1 +xi +x +x +6 +6 +CN3 +CN5 +y2 +yyt +yt +y +y4 +yt +yt +yi +t +t +t +.t +xi +xi10 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +Figure 9. GBPO examples as spatiotemporal patterns for Np = +7. (a) Ps(1357246), CN1, GBPO with period 42. (b) Ps(1462753), +CN2, GBPO with period 14. (c) Ps(1256473), CN3, GBPO with +period 26. (d) Ps(1463725), CN5, GBPO with period 14. + +(a) +CNI +Period 42 +1 +7 +1 +10 +20 +30 +40 +50 +(b) +CN2 +Period 14 +1 +i +1 +10 +20 +30 +40 +50 +(c) +CN3 +Period 26 +1 +1 +10 +20 +30 +40 +50 +(d) +CN5 +Period 14 +7 +t +1 +10 +20 +30 +40 +50GLOBALLY STABLE PERIODIC ORBITS +11 +Figure 10. GBPO examples in Dmaps. (a) Ps(1357246), CN1, +GBPO with period 42. (b) Ps(1462753), CN2, GBPO with period +14. (c) Ps(1256473), CN3, GBPO with period 26. (d) Ps(1463725), +CN5, GBPO with period 14. + +(a) CN1 +CN2 +(b) +C128 +C128 +64 +64 +C1 +t +C1 +C64 +C128 +C64 +CN3 +CN5 +(d) +C +C128 +128 +t+1 +xt+1 +C128 +t +tt +C128 +C64 +6412 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +Table 2. Standard permutation ID and period of GBPO for CN1 +ID +period +ID +period +ID +period +1256374 +26 +1625473 +6 +2517436 +18 +1257436 +18 +1627435 +16 +2576314 +12 +1273654 +14 +1657234 +12 +2613754 +20 +1352476 +34 +1657243 +4 +2615374 +12 +1357246 +42 +1672453 +18 +2675314 +8 +1375426 +26 +1672543 +2 +2751436 +8 +1526374 +42 +1673425 +18 +2763154 +20 +1527643 +14 +2175346 +10 +3416725 +8 +1576324 +24 +2417356 +14 +4671325 +24 +Table 3. Standard permutation ID and period of GBPO for CN2 +ID +period +ID +period +ID +period +1367245 +4 +1653724 +14 +2641735 +2 +1427365 +8 +1674352 +6 +2641753 +2 +1436275 +8 +1675234 +4 +2671354 +2 +1436752 +8 +1732645 +8 +2671453 +2 +1457236 +4 +1742653 +2 +2761345 +4 +1462753 +14 +1762453 +2 +2761354 +2 +1467325 +6 +1762543 +6 +2761453 +2 +1476235 +6 +2156734 +14 +2763154 +2 +1476532 +6 +2167345 +14 +3156724 +4 +1527643 +10 +2356714 +8 +3176254 +4 +1564372 +6 +2361754 +2 +3471256 +4 +1567423 +4 +2365174 +2 +3517426 +2 +1572643 +14 +2367145 +4 +3571246 +2 +1627543 +2 +2367154 +2 +3576214 +4 +1642735 +14 +2461753 +2 +4167253 +2 +1643752 +2 +2467135 +2 +4173256 +2 +1645732 +2 +2467153 +2 +4617325 +4 +1647532 +2 +2517643 +4 +4712356 +8 +1653274 +10 +2571634 +2 + +GLOBALLY STABLE PERIODIC ORBITS +13 +Table 4. Standard permutation ID and period of GBPO for CN3 +ID +period +ID +period +ID +period +1235476 +14 +1567243 +12 +2761345 +2 +1246753 +22 +1576324 +24 +3157426 +12 +1256473 +26 +1652473 +10 +3167425 +8 +1267435 +26 +1657243 +4 +3176245 +10 +1362754 +6 +2156374 +22 +3561724 +10 +1375462 +2 +2417635 +6 +3567214 +24 +1425376 +10 +2463175 +20 +3612745 +8 +1463275 +6 +2516374 +2 +3761425 +12 +1465273 +16 +2516473 +8 +1476235 +2 +2641753 +20 +Table 5. Standard permutation ID and period of GBPO for CN5 +ID +period +ID +period +ID +period +1245673 +10 +1436752 +8 +1673452 +10 +1246735 +2 +1437526 +2 +1675234 +4 +1247635 +6 +1453726 +6 +1726543 +2 +1256734 +10 +1457236 +4 +1732645 +8 +1257346 +2 +1457263 +10 +1745326 +6 +1257436 +6 +1463275 +6 +1745623 +10 +1267345 +10 +1463725 +14 +1756243 +2 +1273456 +10 +1465723 +10 +1765243 +6 +1342765 +6 +1467235 +2 +2356714 +8 +1356724 +10 +1467523 +10 +2367145 +4 +1367245 +4 +1472635 +14 +2517643 +4 +1372645 +6 +1532764 +2 +2761345 +4 +1374265 +2 +1547326 +6 +3156724 +4 +1376254 +6 +1567423 +4 +3176254 +4 +1376452 +6 +1572346 +10 +3461725 +10 +1423765 +6 +1572364 +10 +3471256 +4 +1427365 +8 +1654372 +2 +3576214 +4 +1427635 +6 +1657342 +2 +3617245 +10 +1432675 +6 +1657432 +6 +4617325 +4 +1432756 +6 +1672435 +10 +4712356 +8 +1436275 +8 +1672534 +10 + +14 +MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO +5. Conclusions. Fundamental dynamics of the PBNNs has been studied in this +paper. The PBNNs are characterized by global permutation connections and local +binary connections. Although the parameter space is much smaller than existing +recurrent-type neural networks, the PBNN can exhibit various BPOs. In order to +realize precise analysis, we focus on the GBPOs and define standard permutation +connections. Applying the brute force attack to 7-dimensional PBNNs, we have +presented complete list that clarifies relationship between parameters and periods +of GBPOs. Even in the 7-dimensional cases, the PBNNs exhibit a variety of GBPOs. +It suggests that higher dimensional PBNNs exhibit a huge variety of BPOs/EPPs. +Many problems remain in our future works: +• Mechanism to generate the GBPOs. +• Classification and stability analysis of various BPOs. Besides the GBPOs, the +PBNNs exhibit various BPOs, depending on parameters and initial conditions. +• Effective evolutionary algorithms [23] [24] for analysis of higher dimensional +BPOs where the brute force attack is impossible. +• Effective evolutionary algorithms for synthesis of PBNNs with desired BPOs. +• Efficient hardware implementation for engineering applications including ro- +bust control signals of switching circuits and time-series approximation/prediction. +The PBNNs are well suited for FPGA based hardware implementation that +transforms the BPOs into electric signals in the applications. +Declaration of competing interest. The authors declares that he has no known +competing financial interests or personal relationships that could have appeared to +influence the work reported in this paper. +REFERENCES +[1] J. J. Hopfield, Neural networks and physical systems with emergent collective computation +abilities, Proc. Nat. Acad. Sci. 79 (1982) 2554-2558. +[2] F. Pasemann, Characterization of periodic attractors in neural ring networks, Neural Networks +8(3) (1995) 421-429. +[3] M. Adachi, K. Aihara, Associative dynamics in a chaotic neural network, Neural Networks +10(1) (1997) 83-98. +[4] G. Tanaka, R. Nakane, T. Takeuchi, T. Yamane, D. Nakano, Y. Katayama, A. Hirose, Spa- +tially arranged sparse recurrent neural networks for energy efficient associative memory, IEEE +Trans. Neural Netw. Learn. Syst. 31(1) (2020) 24-38. +[5] J. J. Hopfield, D. W. Tank, ‘Neural’ computation of decisions optimization problems, Biolog- +ical Cybern., 52 (1985) 141-152. +[6] E. Ott, Chaos in Dynamical Systems, Second edition. Cambridge University Press (2002) +[7] L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. +Schrauwen, C. R. Mirasso, I. Fischer, Information processing using a single dynamical node +as complex system, Nat. Commun., 2:468; doi: 10.1038/ncomms1476 (2011) +[8] G. Tanaka, T. Yamane, J. B. H´eroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, +D. Nakano, A. Hirose, Recent advances in physical reservoir computing: a review. Neural +Networks, 115 (2019) 100-123. +[9] J. Dong, E. B¨orve, M. Rafayelyan, M. Unser, Asymptotic stability in reservoir computing, in: +Proc. of IJCNN, 2022, 3774. +[10] S. Koyama, T. Saito, Guaranteed storage and stabilization of desired binary periodic orbits +in three-layer dynamic binary neural networks, Neurocomputing 416 (2020) 12-18. +[11] S. Anzai, T. Suzuki, T. Saito, Dynamic binary neural networks with time-variant parameters +and switching of desired periodic orbits, Neurocomputing, 457 (2021) 357-364. +[12] T. +Suzuki, +T. +Saito, +Synthesis +of +three-layer +dynamic +binary +neural +networks +for +control +of +hexapod +walking +robots, +in: +Proc. +of +the +IEEE/CNNA +(2021) +doi: +10.1109/CNNA49188.2021.9610809 + +GLOBALLY STABLE PERIODIC ORBITS +15 +[13] H. Udagawa, T. Okano, T. Saito, Permutation binary neural networks: analysis of periodic +orbits and its applications, Discrete Contin. Dyn. Syst. Ser. B, 28(1) (2023) 748-764. +[14] W. Wada, J. Kuroiwa, S. Nara, Completely reproducible description of digital sound data +with cellular automata, Phys. Lett. A 306 (2002) 110-115. +[15] H. Uchida, Y. Oishi and T. Saito, A simple digital spiking neural network: synchronization +and spike-train approximation, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1479-1494. +[16] W. Holderbaum, Application of neural network to hybrid systems with binary inputs, IEEE +Trans. Neural Netw. 18(4) (2007) 1254-1261. +[17] B. K. Bose, Neural network applications in power electronics and motor drives - an introduc- +tion and perspective, IEEE Trans. Ind. Electron., 54 (2007) 14-33. +[18] P. W. Wheeler, J. Rodriguez, J. C. Clare, L. Empringham, A. Weinstein, Matrix converters: +a technology review, IEEE Tran. Ind. Electron. 49(2) (2002) 276-288. +[19] M. Lodi, A. L. Shilnikov, M. Storace, Design principles for central pattern generators with +preset rhythms, IEEE Trans. Neural Netw. Learn. Syst. 31(9) (2020) 3658-3669. +[20] D. R. Chowdhury, S. Basu, I. S. Gupta, P. P. Chaudhuri, Design of CAECC - cellular automata +based error correcting code, IEEE Trans. Comput. 43 (6): (1994) 759-764. +[21] A. Poorghanad, A. Sadr, A. Kashanipour, Generating high quality pseudo random number +using evolutionary methods, in: Proc. IEEE/CIS, 9 (2008) 331-335. +[22] N. Horimoto, T. Saito, Analysis of digital spike maps based on bifurcating neurons, NOLTA, +IEICE, 3 (4) (2012) 596–605. +[23] Q. Zhang, L. Hui, MOEA/D: A multiobjective evolutionary algorithm based on decomposi- +tion, IEEE Trans. Evol. Comput., 11(6) (2007) 712-731. +[24] X. Ma, Y. Yu, X. Li, Y. Qi, Z. Zhu, A survey of weight vector adjustment methods for +decomposition-based multiobjective evolutionary algorithms, IEEE Trans. Evol. Comput., +24(4) (2020) 636-649. +E-mail address: tsaito@hosei.ac.jp + diff --git a/uNE0T4oBgHgl3EQf9gI3/content/tmp_files/load_file.txt b/uNE0T4oBgHgl3EQf9gI3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a286c1761a280bd94e6a3b87a71b07df0aa9486c --- /dev/null +++ b/uNE0T4oBgHgl3EQf9gI3/content/tmp_files/load_file.txt @@ -0,0 +1,741 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf,len=740 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='02801v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='DS] 7 Jan 2023 Manuscript submitted to doi:x AIMS’ Journals Volume x, Number 0x, xx xx pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' x–x A VARIETY OF GLOBALLY STABLE PERIODIC ORBITS IN PERMUTATION BINARY NEURAL NETWORKS Mikito Onuki, Kento Saka and Toshimichi Saito∗ Department of Electrical and Electronic Engineering HOSEI University, Japan Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The permutation binary neural networks are characterized by global permutation connections and local binary connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Although the parame- ter space is not large, the networks exhibit various binary periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since analysis of all the periodic orbits is not easy, we focus on globally stable binary periodic orbits such that almost all initial points fall into the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For effi- cient analysis, we define the standard permutation connection that represents multiple equivalent permutation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Applying the brute force attack to 7-dimensional networks, we present the main result: a list of standard per- mutation connections for all the globally stable periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' These results will be developed into detailed analysis of the networks and its engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Discrete-time recurrent neural networks (DT-RNNs) are analog dynamical systems characterized by real valued connection parameters and non- linear activation functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', sigmoid function) [1] [2] [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The dynamics is described by autonomous difference equations of real state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Depending on the parameters, the DT-RNNs exhibit various periodic orbits, chaos [6], and related bifurcation phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The real/potential applications include associa- tive memories [1], combinatorial optimization problems solvers [5], and time-series approximation/prediction in reservoir computing [7] [8] [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The DT-RNNs are important systems in both basic study of nonlinear dynamics and engineering ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, analysis of the dynamics is hard because of huge parameter space and complexity of the nonlinear phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Stability analysis of various periodic orbits is not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The three-layer dynamic binary neural networks (DBNNs [10] [11]) are digital dynamical systems characterized by ternary valued connection parameters and the signum activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The dynamics is described by autonomous difference equations of binary state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since the state space consists of a finite number of binary variables, the DBNNs cannot generate chaos [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, depending on the parameters and initial conditions, the DBNNs can generate various peri- odic orbits of binary vectors (binary periodic orbits, ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' BPOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As compared with the DT-RNNs, the DBNNs bring benefits to hardware implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' An FPGA based hardware prototype and its application to hexapod walking robots Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Recurrent neural networks, binary neural networks, permutation, bi- nary periodic orbits, stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' ∗ Corresponding author: Toshimichi Saito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 1 2 MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' We have presented a parameter setting method that guar- antees storage and stability of desired BPOs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, as period of a BPO increases, the number of hidden neurons increases: parameter space becomes wider and analysis becomes harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In the hardware, power consumption becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In order to realize efficient analysis and synthesis, reduction of the parameter space is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Simplifying connection parameters of the DBNNs, the permutation binary neural networks (PBNNs [13]) are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs are characterized by two kinds of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The first one is local binary connection between input and hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' It is defined by a signum-type neuron from three binary inputs to one binary output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The second one is global one-to-one connection between hidden and output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' It is defined by a permutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The parameter space of the PBNNs is much smaller than that of DBNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Depending on the permutation connections, the PBNNs generate various BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Co-existence of BPOs is possible and the PBNN exhibits one of the BPOs depending on initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since analysis of multiple BPOs are not easy, this paper focuses on globally sta- ble binary periodic orbits (GBPOs) such that almost all initial points fall into the GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As a fundamental concept, we define the standard permutation connection that represents multiple equivalent permutation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Applying the brute force attack to all the 7-dimensional PBNNs, we present the main result: a list of the standard permutation connections for all the GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' These results provide basic information to realize more detailed analysis of PBNNs and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Real/potential engineering applications of the GBPOs include time-series approx- imation/prediction [8] [14] [15], control signals of switching power converters [16] [17] [18], control signals of walking robots [12] [19], and error correcting codes [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The approximate/control signals can be globally stable and robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As novelty of this paper, it should be noted that this is the first paper of the GBPOs and standard permutation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Permutation binary neural networks and binary periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' This section introduces the 3-layer dynamic binary neural networks (DBNNs, [10]) and the permutation binary neural networks (PBNNs, [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' After overview of BPOs, we show the objective problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dynamics binary neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The DBNNs are recurrent-type 3-layer networks characterized by ternary connection parameters and signum activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The dynamics is described by the following autonomous difference equa- tion of N-dimensional binary state variables: xt+1 i = sgn \uf8eb \uf8ed M � j=1 cijyt j + Si \uf8f6 \uf8f8 , yt j = sgn � N � i=1 wjixt i − Tj � sgn(x) = � +1 if x ≥ 0, i ∈ {1, · · · , N} −1 if x < 0, j ∈ {1, · · · , M} (1) where xt i ∈ {−1, +1} ≡ B is the i-th binary state variable at discrete time t and yt j ∈ B is the j-th binary hidden variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 1, the binary variables xt i, yt j, and xt+1 i are located in input, hidden, and output layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The M hidden neurons transform xt i into yt j through hidden ternary connections (wji ∈ {−1, 0, +1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The N output neurons transform yt j into xt+1 i through output ternary connections (cij ∈ {−1, 0, +1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The threshold parameters Si and Tj are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' GLOBALLY STABLE PERIODIC ORBITS 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dynamic binary neural network (DBNN) Red and blue branches denote positive and negative connections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The output xt+1 i is fed back to the input layer and the DBNNs generate various BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [10] gives a theoretical result of parameter condition that guarantees storage and stability of desired BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, as period of a BPO increases, the number of hidden neurons increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For example, p hidden neurons are required for storage of a BPO with period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As p increases, the parameter space becomes larger and analysis/implementation becomes harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Permutation binary neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs are described by the following autonomous difference equation of N-dimensional binary state variables: xt+1 i = yt σ(i), yt i = sgn � waxt i−1 + wbxt i + wcxt i+1 � σ = � 1 2 · · N σ(1) σ(2) · · σ(N) � i ∈ {1, · · · , N}, N ≥ 3 (2) where xt 0 ≡ xt N and xt N+1 ≡ xt 1 for ring-type connection as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As a binary state vector xt ≡ (xt 1, · · · , xt N) ∈ BN is input at time t, the xt is transformed into the binary hidden state vector yt ≡ (yt 1, · · · , yt N) ∈ BN through hidden neurons with local binary connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' All the hidden neurons have the same characteristics: the signum activation function from three binary inputs to one binary output with local binary connection parameters (wa, wb, wc) ∈ B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The yt is transformed into xt+1 through one-to-one global permutation connection defined by the permutation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The output vector xt+1 is fed back to the input and the PBNNs generate sequences of binary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In comparison with the DBNNs, the hidden connections wij are replaced with the local binary connections and the output connections cij are replaced with the global permutation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 3, the local binary connections are identified by connection numbers: CN0 : wl = (−1, −1, −1) CN1 : wl = (−1, −1, +1) CN2 : wl = (−1, +1, −1) CN3 : wl = (−1, +1, +1) CN4 : wl = (+1, −1, −1) CN5 : wl = (+1, −1, +1) CN6 : wl = (+1, +1, −1) CN7 : wl = (+1, +1, +1) where wl ≡ (wa, wb, wc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since CN1 (respectively, CN3) coincides with CN4 (re- spectively, CN6) by replacement xi → xN−i+1 for i ∈ {1, · · · , N}, we consider 6 connection numbers without CN4 and CN6 hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The global permutation +1 S2 S5 S6 S7 S1 y y2 yi T1 T4 Ts T 6 7 X2 x34 MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO connections are identified by Permutation ID: P(σ(1) · · · σ(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2 shows examples of 7-dimensional PBNNs for CN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For identity permuta- tion P(123456), the PBNN exhibits a BPO with period 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Applying permutation P(2613754), the PBNN exhibits a BPO with longer period 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In the DBNN, 20 hidden neurons are necessary for period 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Objective problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In order to visualize the dynamics, we have introduced the digital return map (Dmap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The domain BN of the PBNNs is equivalent to a set of 2N points LN ≡ {C1, · · · , C2N }, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', C1 ≡ (−1, · · · , −1), C2 ≡ (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' · · · , −1), · · , C2N ≡ (+1, · · · , +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The dynamics of a PBNN can be integrated into Dmap: xt+1 = f(xt), xt ∈ BN ≡ LD (3) where an N-dimensional binary vector xt is denoted by a point Ci in the Dmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Examples of PBNNs and BPOs for CN1, N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Red and blue branches denote positive and negative local binary con- nections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' black branches correspond to global permu- tation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' White and black squares in spatiotemporal pat- terns denote xt i = +1 and xt i = −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) P(1234567).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (b) P(2613754).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 8 local binary connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' CNO CN1 CN2 CN3 Wb Wc Wa Wb Wc Wa Wc Wa Wa Wb Wb Im CN4 CN5 CN6 CNT Wa Wb Wc Wa Wa Wb Wc Wb Wa Wb Wc ma ot+1 (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 rt+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 rt+1 t+1 t+1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 X4 x3 x2 9x yt yt ys yt yt yi yt yt yt y y x2 x x x4 xt x xt xt xi Period 14 Period 20 1 1 7 7 10 20 1 10 t 1 1GLOBALLY STABLE PERIODIC ORBITS 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dmap examples (black points) and BPOs (blue orbits) for CN1, N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) P(1234567) (the PBNN is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2 (a)), BPO with period 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (b) P(2613754) (the PBNN is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2 (b)), BPO with period 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A point zp ∈ LD is said to be a binary periodic point (BPP) with period p if f p(zp) = zp and f(zp) to f p(zp) are all different where f k is the k-fold composition of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A sequence of the BPPs, {f(zp), · · · , f p(zp)}, is said to be a BPO with period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A point ze is said to be an eventually periodic point (EPP) if ze is not a BPP but falls into a BPO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', there exists some positive integer l such that f l(ze) is a BPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The BPO in the Dmap is equivalent to the BPO in spatiotemporal pattern from the PBNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 4 shows BPOs in Dmaps corresponding to BPOs in spatiotemporal patterns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As parameters (CN and Permutation ID) vary, the PBNN exhibits a vari- ety of BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of CNs (without CN4 and CN6) is 6 whereas the number of hidden connection parameters wij is 3N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of Permutation IDs is N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' whereas the number of output connection parameters cij is 3N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In addition, the DBNNs have 2N integer threshold parameters Si and Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' It goes without saying that the PBNNs cannot generate more various BPOs than the DBNNs because the PBNNs are included in the DBNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, the PBNN parameter space is much smaller than the DBNN parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The objective problem is relationship between parameters (Permutation ID and CN) and existence/stability of BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Globally stable binary periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Depending on parameters, the PBNNs exhibit various BPOs and multiple BPOs can co-exist for initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since anal- ysis of multiple BPOs is hard, we try to analyze representative BPOs: the globally stable binary periodic orbits (GBPOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' This section defines the GBPOs and related concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' First, we note two exceptional endpoints in BN: x− ≡ (−1, · · · , −1) ∈ BN, x+ ≡ (+1, · · · , +1) ∈ BN (4) The two endpoints are either fixed points or a BPO with period 2, becuase f(x+) = x+, f(x−) = x− if wa + wb + wc ≥ +1 f(x+) = x−, f(x−) = x+ if wa + wb + wc ≤ −1 (5) (a) (b) C128 28 t+1 C64 64 C64 C128 C128 646 MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO Hereafter we omit the two endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The GBPO is defined by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A BPO is said to be a globally stable binary periodic orbit (GBPO) if the BPO is unique (except for x− and x−) and if all the EPPs fall into the BPO where we assume existence of the EPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of EPPs plus elements of the GBPO is 2N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 4(b) shows a GBPO with period 20 in the Dmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In this 7-dimensional example, (27 − 20 − 2) EPPs fall into the GBPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As shown in Section 4, depending on the parameters (permutation ID and CN), the 7-dimensional PBNNs exhibit a variety of GBPOs and the number of EPPs is more than 27/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The EPPs represent global stability corresponding to error correction [20] of binary signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As the number of EPPs increases, the global stability becomes stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In the limit case of the M-sequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', in the linear feedback shift register [21]), the period is 2N, no EPP exists and is not stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Such M-sequences are different category from the GBPOs in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In fundamental viewpoints, uniqueness of the GBPO is convenient to consider existence and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Analysis of multiple BPOs is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In application viewpoints, the GBPOs are useful as globally stable signal to approximate/predict time-series [15] and to control switching circuits [16] [17] [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For simplicity, we focus on the case where N is a prime number Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' If an integer N can be factorized into prime factors, classification of the permutation connections becomes complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Here, in order to analyze GBPOs, we define several basic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Let R be a shift operator such that R : P0(σ0(1) · · · σ0(Np)) → P1(σ1(1) · · · σ1(Np)) P1 = R(P0), σ1(i + 1) = σ0(i) + 1 mod Np, i ∈ {1, · · · , Np} (6) where σ1(Np + 1) ≡ σ1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since the neurons are ring-type connection, the permu- tation connections P1 and P0 (P and R(P)) are equivalent even if the permutation IDs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Let S be a set of permutation IDs that give equivalent permutation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The set S is referred to as an equivalent permutation set (EPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' An EPS is represented by a standard permutation ID Ps(σs(0) · · · σs(Np)) that corresponds to the minimum element in the EPS by means of base-Np number: Ps(σs(1) · · · σs(Np)) < Pk(σk(1) · · · σk(Np)) ∈ S, k ̸= s ( base-Np number ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 5 shows an example of standard permutation connection and its equivalent permutation connections for Np = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In this example, the EPS is S = {Ps(1325476), P(7243651), P(2135476), P(7324651), P(2143576), P(7325461), P(2143576)} Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A permutation ID Pb is said to be a basic permutation ID if it is a fixed point of the shift operator: R(Pb) = Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Since R(Pb) = Pb iff σb(i+1) = σi+1 mod Np, the number of basic permutation IDs is Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A basic permutation ID constructs an EPS with one element and is a standard permutation ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 6 shows basic permutation connections for Np = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Then we have Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In Np-dimensional PBNNs, the number of standard permutation IDs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', the number of EPSs) is (Np − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' + Np − 1 where Np ≥ 3 is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' GLOBALLY STABLE PERIODIC ORBITS 7 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Equivalent permutation connection examples for Np = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ps: standard permutation connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' R: shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 7 basic permutation connections for Np = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Permutation connection examples consisting of 3 sub- connections for N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (Proof) Except for Np basic permutations, one standard permutation ID Ps represents Np equivalent permutation IDs: RNp(Ps) = Ps, Rk(Ps) ̸= Ps for 1 ≤ k ≤ Np − 1 where Rk(P) = R(Rk−1(Ps)) is the k-fold composition of the shift operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' If there exists an integer l (2 ≤ l < Np) such that Rl(Ps) = Ps, the ring-type con- nection of Ps is decomposed into the same sub-connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', 3 sub-connections R3l(Ps) = RNp(Ps) = Ps as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, it is impossible for a prime number Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Therefore, except for the basic permutations, the number of standard permutation IDs is (Np!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' − Np)/Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Adding the Np basic permutation IDs, the number of standard permutation IDs is (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' − Np)/Np + Np = (Np − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' + Np − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ps(214365) P(632541) Ps(214365) XXX XXX XXX R P(632541) Ps(214365) P(632541) XXX R RPb(1234567) Pb(2345671) Pb(3456712) Pb(4567123) IIIIII1 Pb(5671234) Pb(6712345) Pb(7123456)P(7243651) P(2135476) P(7324651) Ps(1325476) IXXX XXXIXXXX P(2143576) P(7325461) P(2143576) Ps(1325476) XXXRXXXIE RXXIX8 MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Brute force attack to explore GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Table 1 shows the number of stan- dard permutation IDs for prime numbers Np together with the number of full bi- nary connection parameters between hidden and output layers in the DBNNs for N = M = Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of the permutation connections is much smaller than the number of the full binary connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' However, analysis of the GBPOs becomes harder as Np increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For convenience, we consider GBPOs in 7-dimensional PBNNs (Np = 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In the case NP = 7, the number of all the standard permutation connections is (Np − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' + Np − 1 = 726, the number of initial points is 27, and the brute force attack is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' We can clarify the number and period of all the GB- POs precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Analysis of the 7-dimensional GBPOs are fundamental to consider higher-dimensional GBPOs and their engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' We explore the 7-dimensional GBPOs as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' First, as state earlier, objective connection numbers are CN0, CN2, CN2, CN3, CN5, and CN7 (CN1 ≡ CN4 and CN3 ≡ CN6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Second, applying the shift operator R, we obtain the 726 standard permutation IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Third, applying the brute force attack to each standard permutation ID and CN, we obtain BPOs and their EPPs where we use the BPO calculation algorithm in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' If the number of a BPP plus its EPPs is 27 − 2 = 126 then the BPO is declared as the GBPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The period of the GBPO is stored together with its standard permutation ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In the exploration, it is confirmed that CN0 and CN7 cannot provide GBPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The CN0 and CN7 are omitted hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 8 shows typical examples of PBNNs for CN1, CN2, CN3, and CN5 that generate GBPO with period 42, period 14, period 26, and period 14, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 9 shows the 4 GBPOs as spatiotemporal patterns and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 10 shows the 4 GBPOs in Dmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As a criterion of the period, we give Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For identity permutation (Pb(1234567) for NP = 7), the period of the BPO is said to be basic period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' If the PBNN generates multiple BPOs, the maximum period is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' For CN1, the basic period is 14 as a BPO in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 2 (a) that is a GBPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' We have confirmed that the identity permutation Pb(1234567) cannot provide a GBPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 8 to 10, we can see that, adjusting permutation IDs from the identity permutation Pb(1234567), the PBNNs can generate a variety of BPOs represented by the GBPOs with longer period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As the main result, tables 2 to 5 show a list of standard permutation IDs for all the GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As stated in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3, each standard permutation ID represents 7 equivalent permutation IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' We give an overview of the list for CN1, CN2, CN3, and CN5: Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of standard permutation connections in PBNN and full binary connections between hidden and output lay- ers in DBNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Np # standard permutation IDs # full binary connections 3 4 29 5 28 225 7 726 249 11 3628810 2121 13 479001612 2169 17 20922789888016 2289 GLOBALLY STABLE PERIODIC ORBITS 9 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' PBNN examples (exhibit GBPOs) for Np = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) Ps(1357246), CN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (b) Ps(1462753), CN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (c) Ps(1256473), CN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (d) Ps(1463725), CN5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' CN1: The basic period is 14 for Pb(1234567).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs generate 27 GB- POs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The maximum period is 42 for Ps(1357246) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 9 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of EPPs is 126 − 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' CN2: The basic period is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs generate 56 GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The maximum period is 14 where the number of EPPs is 126−14, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ps((1462753) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 9 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' CN3: The basic period is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs generate 28 GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The maximum period is 26 where the number of EPPs is 126−26, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ps(1256473) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 9 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' CN5: The basic period is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs generate 62 GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The maximum period is 14 where the number of EPPs is 126−14, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ps(1463725) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 9 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' These tables clarify relation between parameters (permutation ID and CN) and periods of the GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The number of EPPs is 126 minus the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' As the pa- rameters vary, the 7-dimensional PBNNs can generate a variety of GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' These results provide fundamental information to analyze various PBNNs and to synthe- size PBNNs with desired GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) (b) t+1 t+1 t+1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 t+1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 t+1 t+1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 xi x3 x X2 x3 CN1 CN2 y4 yt yt yt y4 y2 yt yt yt y2 y6 yi x2 xt x4 xt x2 xt x x xt x xt xt (c) (d) t+1 t+1 t+1 t+1 t+1 t+1 +1 t+1 +1 t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t+1 t t t+1 xi x x 6 6 CN3 CN5 y2 yyt yt y y4 yt yt yi t t t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='t xi xi10 MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' GBPO examples as spatiotemporal patterns for Np = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) Ps(1357246), CN1, GBPO with period 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (b) Ps(1462753), CN2, GBPO with period 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (c) Ps(1256473), CN3, GBPO with period 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (d) Ps(1463725), CN5, GBPO with period 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) CNI Period 42 1 7 1 10 20 30 40 50 (b) CN2 Period 14 1 i 1 10 20 30 40 50 (c) CN3 Period 26 1 1 10 20 30 40 50 (d) CN5 Period 14 7 t 1 10 20 30 40 50GLOBALLY STABLE PERIODIC ORBITS 11 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' GBPO examples in Dmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) Ps(1357246), CN1, GBPO with period 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (b) Ps(1462753), CN2, GBPO with period 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (c) Ps(1256473), CN3, GBPO with period 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (d) Ps(1463725), CN5, GBPO with period 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' (a) CN1 CN2 (b) C128 C128 64 64 C1 t C1 C64 C128 C64 CN3 CN5 (d) C C128 128 t+1 xt+1 C128 t tt C128 C64 6412 MIKITO ONUKI, KENTO SAKA AND TOSHIMICHI SAITO Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Standard permutation ID and period of GBPO for CN1 ID period ID period ID period 1256374 26 1625473 6 2517436 18 1257436 18 1627435 16 2576314 12 1273654 14 1657234 12 2613754 20 1352476 34 1657243 4 2615374 12 1357246 42 1672453 18 2675314 8 1375426 26 1672543 2 2751436 8 1526374 42 1673425 18 2763154 20 1527643 14 2175346 10 3416725 8 1576324 24 2417356 14 4671325 24 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Standard permutation ID and period of GBPO for CN2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1367245 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1653724 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2641735 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1427365 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1674352 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2641753 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1436275 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1675234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2671354 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1436752 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1732645 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2671453 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1457236 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1742653 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2761345 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1462753 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1762453 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2761354 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1467325 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1762543 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2761453 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1476235 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2156734 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2763154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1476532 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2167345 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3156724 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1527643 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2356714 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3176254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1564372 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2361754 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3471256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1567423 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2365174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3517426 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1572643 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2367145 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3571246 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1627543 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2367154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3576214 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1642735 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2461753 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4167253 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1643752 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2467135 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4173256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1645732 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2467153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4617325 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1647532 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2517643 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4712356 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1653274 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2571634 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='GLOBALLY STABLE PERIODIC ORBITS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Standard permutation ID and period of GBPO for CN3 ID period ID period ID period 1235476 14 1567243 12 2761345 2 1246753 22 1576324 24 3157426 12 1256473 26 1652473 10 3167425 8 1267435 26 1657243 4 3176245 10 1362754 6 2156374 22 3561724 10 1375462 2 2417635 6 3567214 24 1425376 10 2463175 20 3612745 8 1463275 6 2516374 2 3761425 12 1465273 16 2516473 8 1476235 2 2641753 20 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Standard permutation ID and period of GBPO for CN5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1245673 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1436752 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1673452 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1246735 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1437526 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1675234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1247635 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1453726 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1726543 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1256734 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1457236 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1732645 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1257346 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1457263 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1745326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1257436 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1463275 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1745623 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1267345 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1463725 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1756243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1273456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1465723 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1765243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1342765 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1467235 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2356714 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1356724 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1467523 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2367145 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1367245 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1472635 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2517643 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1372645 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1532764 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2761345 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1374265 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1547326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3156724 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1376254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1567423 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3176254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1376452 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1572346 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3461725 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1423765 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1572364 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3471256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1427365 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1654372 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3576214 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1427635 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1657342 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='3617245 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1432675 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1657432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4617325 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1432756 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1672435 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='4712356 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1436275 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1672534 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='MIKITO ONUKI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' KENTO SAKA AND TOSHIMICHI SAITO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fundamental dynamics of the PBNNs has been studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs are characterized by global permutation connections and local binary connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Although the parameter space is much smaller than existing recurrent-type neural networks, the PBNN can exhibit various BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' In order to realize precise analysis, we focus on the GBPOs and define standard permutation connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Applying the brute force attack to 7-dimensional PBNNs, we have presented complete list that clarifies relationship between parameters and periods of GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Even in the 7-dimensional cases, the PBNNs exhibit a variety of GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' It suggests that higher dimensional PBNNs exhibit a huge variety of BPOs/EPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Many problems remain in our future works: Mechanism to generate the GBPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Classification and stability analysis of various BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Besides the GBPOs, the PBNNs exhibit various BPOs, depending on parameters and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Effective evolutionary algorithms [23] [24] for analysis of higher dimensional BPOs where the brute force attack is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Effective evolutionary algorithms for synthesis of PBNNs with desired BPOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Efficient hardware implementation for engineering applications including ro- bust control signals of switching circuits and time-series approximation/prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The PBNNs are well suited for FPGA based hardware implementation that transforms the BPOs into electric signals in the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Declaration of competing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' The authors declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Hopfield, Neural networks and physical systems with emergent collective computation abilities, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 79 (1982) 2554-2558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Pasemann, Characterization of periodic attractors in neural ring networks, Neural Networks 8(3) (1995) 421-429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Adachi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Aihara, Associative dynamics in a chaotic neural network, Neural Networks 10(1) (1997) 83-98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Tanaka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Nakane, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Takeuchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Yamane, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Nakano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Katayama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Hirose, Spa- tially arranged sparse recurrent neural networks for energy efficient associative memory, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Neural Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 31(1) (2020) 24-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Hopfield, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Tank, ‘Neural’ computation of decisions optimization problems, Biolog- ical Cybern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', 52 (1985) 141-152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ott, Chaos in Dynamical Systems, Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Cambridge University Press (2002) [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Appeltant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Soriano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Van der Sande, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Danckaert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Massar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dambre, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Schrauwen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Mirasso, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Fischer, Information processing using a single dynamical node as complex system, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', 2:468;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1038/ncomms1476 (2011) [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Tanaka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Yamane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' H´eroux, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Nakane, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Kanazawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Takeda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Numata, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Nakano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Hirose, Recent advances in physical reservoir computing: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Neural Networks, 115 (2019) 100-123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' B¨orve, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Rafayelyan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Unser, Asymptotic stability in reservoir computing, in: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' of IJCNN, 2022, 3774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Koyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Saito, Guaranteed storage and stabilization of desired binary periodic orbits in three-layer dynamic binary neural networks, Neurocomputing 416 (2020) 12-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Anzai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Suzuki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Saito, Dynamic binary neural networks with time-variant parameters and switching of desired periodic orbits, Neurocomputing, 457 (2021) 357-364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Suzuki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Saito, Synthesis of three-layer dynamic binary neural networks for control of hexapod walking robots, in: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' of the IEEE/CNNA (2021) doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='1109/CNNA49188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='9610809 GLOBALLY STABLE PERIODIC ORBITS 15 [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Udagawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Okano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Saito, Permutation binary neural networks: analysis of periodic orbits and its applications, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' B, 28(1) (2023) 748-764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Wada, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Kuroiwa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Nara, Completely reproducible description of digital sound data with cellular automata, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' A 306 (2002) 110-115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Uchida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Oishi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Saito, A simple digital spiking neural network: synchronization and spike-train approximation, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' S, 14 (2021), 1479-1494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Holderbaum, Application of neural network to hybrid systems with binary inputs, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Neural Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 18(4) (2007) 1254-1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Bose, Neural network applications in power electronics and motor drives - an introduc- tion and perspective, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', 54 (2007) 14-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Wheeler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Rodriguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Clare, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Empringham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Weinstein, Matrix converters: a technology review, IEEE Tran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 49(2) (2002) 276-288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Lodi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Shilnikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Storace, Design principles for central pattern generators with preset rhythms, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Neural Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 31(9) (2020) 3658-3669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Chowdhury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Basu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Gupta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Chaudhuri, Design of CAECC - cellular automata based error correcting code, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' 43 (6): (1994) 759-764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Poorghanad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Sadr, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Kashanipour, Generating high quality pseudo random number using evolutionary methods, in: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' IEEE/CIS, 9 (2008) 331-335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [22] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Horimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Saito, Analysis of digital spike maps based on bifurcating neurons, NOLTA, IEICE, 3 (4) (2012) 596–605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [23] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Hui, MOEA/D: A multiobjective evolutionary algorithm based on decomposi- tion, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', 11(6) (2007) 712-731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' [24] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Qi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Zhu, A survey of weight vector adjustment methods for decomposition-based multiobjective evolutionary algorithms, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=', 24(4) (2020) 636-649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content=' E-mail address: tsaito@hosei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} +page_content='jp' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf'} diff --git a/v9FST4oBgHgl3EQfQjh8/content/2301.13759v1.pdf b/v9FST4oBgHgl3EQfQjh8/content/2301.13759v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a6803f4128b89832d3d8caae1eb4f66bc5b71296 --- /dev/null +++ b/v9FST4oBgHgl3EQfQjh8/content/2301.13759v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ed8045c111610fa510aa5bc603e39534e1b6abd7011fe24a4a2b482fd2afb4f5 +size 205210 diff --git a/v9FST4oBgHgl3EQfQjh8/vector_store/index.faiss b/v9FST4oBgHgl3EQfQjh8/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9e5e0649cf48910e691dccb8eec07e70d67e905a --- /dev/null +++ b/v9FST4oBgHgl3EQfQjh8/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8757cfdf32febaf6636e48cf85f36680ddfdeba757e53d64ed0a2f6345a3eea0 +size 2949165 diff --git a/v9FST4oBgHgl3EQfQjh8/vector_store/index.pkl b/v9FST4oBgHgl3EQfQjh8/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0a733a0ce9fbd637a4d98a828db2a8f9851a7dda --- /dev/null +++ b/v9FST4oBgHgl3EQfQjh8/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c58ec0541c4a32cee61bc2c3afcfa5918200cb5c228044e329b0606b949ce527 +size 106047 diff --git a/vdAyT4oBgHgl3EQfafdn/content/2301.00243v1.pdf b/vdAyT4oBgHgl3EQfafdn/content/2301.00243v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..776b6c726b7efeb0054a843d74641008c8da7fde --- /dev/null +++ b/vdAyT4oBgHgl3EQfafdn/content/2301.00243v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4b3a3636ab44e0c6e8508eb71e1d7f531786aff9b733c9f68a4da85ea87e8138 +size 721215 diff --git a/vdE3T4oBgHgl3EQfkwp-/content/tmp_files/2301.04600v1.pdf.txt b/vdE3T4oBgHgl3EQfkwp-/content/tmp_files/2301.04600v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da18d36fb0b5ea009fa19ef0ecd2807595a150c0 --- /dev/null +++ b/vdE3T4oBgHgl3EQfkwp-/content/tmp_files/2301.04600v1.pdf.txt @@ -0,0 +1,640 @@ +arXiv:2301.04600v1 [math.DG] 11 Jan 2023 +Maximal first Betti number rigidity of noncompact +RCD(0,N) spaces +Zhu Ye∗ +Department of Mathematics, Capital Normal University +Abstract +Let (M, d, m) be a noncompact RCD(0,N) space with N ∈ N+ and suppm = +M. We prove that if the first Betti number of M equals N − 1, then (M, d, m) +is either a flat Riemannian N-manifold with a soul T N−1 or the metric product +[0, ∞) × T N−1, both with the measure a multiple of the Riemannian volume, +where T N−1 is a flat torus. +1 +Introduction +Let K ∈ R and N ∈ [1, ∞) throughtout the paper. The RCD(K, N) spaces or +RCD∗(K, N) spaces ([16],[17],[12],[4],[2]) are both synthetic counterparts of Rieman- +nian manifolds with Ricci curvature bounded below by K and dimension bounded +above by N. Many classical results that hold for manifolds with Ricci curvature +bounded below remain true in the synthetic setting. +Here we mention some of +them that will be used in the present paper. +In [9], Gigli established the split- +ting theorem on RCD(0, N) spaces. In [15], the existence of the universal cover of +an RCD∗(K, N) space is established by Mondino-Wei. This universal cover admits +a natural RCD∗(K, N) structure such that the covering projection is a local metric +measure isometry. Recently in [18], based on the results of [15], Wang proved that +an RCD∗(K, N) space is semi-locally simply connected, and hence its universal cover +is simply connected. In [6], Deng proved that an RCD(K, N) space is nonbranching. +∗Supported partially by National Natural Science Foundation of China [11821101] and Beijing +Natural Science Foundation [Z19003]. +Email address: 2210501006@cnu.edu.cn. +1 + +The main purpose of this paper is to generalize the following result in [20] to +RCD(0, N) spaces: +Theorem 1. If M is an open Riemannian n-manifold with RicM ≥ 0, then b1(M) ≤ +n − 1 and the equality holds if and only if M is flat with a soul T n−1. +See [20] for a background of Theorem 1. +Throughtout this paper, for a CD∗(K, N) space (X, d, m), we will always assume +that suppm = X. This is not a restrictive condition since if (X, d, m) is CD∗(K, N) +then (suppm, d, m) is also CD∗(K, N). +The following is the main result of this paper: +Theorem 2. Let N ≥ 2 and let (X, d, m) be a noncompact RCD(0, N) space. Then +the first Betti number b1(X) ≤ [N] − 1. If N ∈ N+, then the equality holds if and +only if (X, d, m) is either a flat N-manifold with a soul T N−1 or [0, ∞) × T N−1, both +with m a multiple of the Lebesgue measure. +Remark 1. The above result is optimal in the sense that when N /∈ N+, the maximal +first Betti number rigidity in the metric measure sense fails. Indeed, for each h > 0, +the space ([0, ∞), d, VhL1) with Vh(x) = f h(x) and f(x) : [0, ∞) → (0, 2) convex +is RCD(0, h + 1), where d is the Euclidean distance and L1 is the Lebesgue measure +(cf. [7] Proposition 3.21). So the product of ([0, ∞), d, VhL1) and T [N]−1 with the +Euclidean disdance and the Lebesgue measure is an RCD(0, N) space for every 0 < +h < N − [N] (see [4] Theorem 4.1 and [1]Theorem 7.6 for related tensorlization +properties of RCD(0, N) spaces). +The author conjectures that the metric rigidity still holds in this case, i.e. that +(X, d) is isomorphic to a flat N-manifold with a soul T N−1 or [0, ∞) × T N−1 if +(X, d, m) is a noncompact RCD(0, N) space with b1(X) = [N] − 1. +One knows from [15] and [18] that for a compact RCD(0, N) space (M, d, m), the +first Betti number b1(M) ≤ [N], with equality holds if and only if (M, d, m) is a T [N] +with m a multiple of the Lebesgue measure. This generalized a classical theorem of +Bochner. For further results related to the first Betti number of compact RCD spaces, +see [14]. +As in [20], our approach to Theorem 2 is the estimate of the orbit growth of +covering group actions. +Definition 1. ([20]) Denote by #(A) the number of elements in a set A. Let (X, d) +be a metric space and let Isom(X) be its isometry group. Let Γ be a subgroup of +Isom(X). For every x ∈ X, put DΓ(x, r) = {g ∈ Γ : d(x, g(x)) ≤ r}. +2 + +Given p ∈ [0, ∞), we say Γ has polynomial orbit growth related to x of order +≥ p (≤ p), if and only if +lim inf +r→∞ +#(DΓ(x, r)) +rp +> 0 (lim sup +r→∞ +#(DΓ(x, r)) +rp +< ∞). +We say Γ has polynomial orbit growth related to x of order > p (< p), if and only if +lim +r→∞ +#(DΓ(x, r)) +rp += ∞ (= 0). +It is easy to check that the polynomial orbit growth properties defined above do +not depend on the choice of the base point x. +Similar to [20], Theorem 2 follows easily from the following theorem on orbit +growth (cf. Proof of Theorem 1 in [20]): +Theorem 3. Let N ≥ 2 and let (X, d, m) be a noncompact RCD(0, N) space. Let +π : ( ˜X, ˜p) → (X, p) be the universal cover with deck transformation group Γ, where +p ∈ X. +If N is an integer, then Γ has polynomial orbit growth of order ≤ N−1. Moreover, +Γ fails to have polynomial orbit growth of order < N − 1, i.e. there exist ri → ∞ +such that +lim +i→∞ +#(DΓ(˜p, ri)) +rN−1 +i +> 0 +if and only if (X, d, m) is a flat N-manifold with an N − 1 dimensional soul or +[0, ∞) × X1 with X1 is a compact flat (N − 1)-manifold, both with m a multiple of +the Riemannian volumes. +If N is not an integer, then Γ has polynomial orbit growth of order < N − 1. +As in [5] and [20], the key to proving Theorem 3 is to control the volume growth +of the set of points that are close to the cut locus of a base point in some sense (see +Lemma 1 (2)). To obtain Lemma 1 (2), we will use the Brunn-Minkowski inequality +to replace the volume element comparison argument used in [20]. The nonbranching +property of RCD(K, N) spaces is also crucial for our proof. +We also generalize Theorem 4 in [20] (which is the orbit version of Theorem 1.1 +in [3] ) to nonbranching CD∗(K, N) spaces: +Theorem 4. Let (X, d, m) be a nonbranching CD∗(K, N) space. +(1) Let p, q ∈ X, p ̸= q. Put E(p, q) = {x = X | d(p, x) = d(q, x)}. Then +m(E(p, q)) = 0. +3 + +(2)Let π : +� ¯X, ¯d, ¯m, ¯x0 +� +→ (X, d, m, x0) be a normal covering with deck transfor- +mation group G (here +� ¯X, ¯d, ¯m +� +is the lift of (X, d, m), cf. 7.2 of [4] or 2.2 of [15]). +Then for every r > 0 we have: +#(DG(¯x0, 2r)) · m(Br(x0)) ≥ ¯m(Br(¯x0)), +(1.1) +#(DG(¯x0, r)) · m(Br(x0)) ≤ ¯m(B2r(¯x0)). +(1.2) +It has been well known since [19] that an open manifold with nonnegative Ricci +curvature has at least linear volume growth (i.e. lim inf +R→∞ +Vol(BR(p)) +R +> 0). This fact also +holds for RCD(0, N) spaces (cf. [10]). So the following definition makes sense: +Definition 2. Let (X, d, m) be a noncompact RCD(0, N) space. We say X has mini- +mal volume growth if and only if +lim sup +R→∞ +m(BR(p)) +R +< ∞ for some p ∈ X. +Let (X, d, m) be a CD(0, N) space. By the Bishop-Gromov inequality, the limit +lim +r→∞ +m(Br(p)) +rN +always exists and does not rely on the choice of p ∈ X. We say that X +has Euclidean volume growth if lim +r→∞ +m(Br(p)) +rN +> 0. Otherwise, we say that X collapses +at infinity. +The following theorem, which generalized Theorem 2 of [20], is an immediate +consequence of Theorem 3 and Theorem 4 (2) (1.1): +Theorem 5. Let (X, d, m) be a noncompact RCD(0, N) space and let p ∈ X. Let +π : ( ˜X, ˜d, ˜m) → (X, d, m) be the universal cover. If X has minimal volume growth, +then: +if N /∈ N+, then ˜X collapses at infinity; +if N ∈ N+, then ˜X has Euclidean volume growth if and only if (X, d) is a flat N- +manifold with an N − 1 dimensional soul or [0, ∞) × X1, where X1 is a compact flat +N − 1 manifold. +Theorem 5 can be viewed as a generalization of the rigidity result of Theorem 2. +Indeed, by Theorem 4, when N ∈ N+, b1(X) = N − 1 implies that X has minimal +volume growth and that ˜X has Euclidean volume growth. +4 + +2 +Preliminaries +In this section, we list the properties of CD∗(K, N) spaces and RCD∗(K, N) spaces +that will be used in this paper. We will not give the definitions here, since they will +not be used explicitly. +1 +When K = 0, the CD∗(0, N) condition is the same as the CD(0, N) condition. +2 +A CD∗(K, N) space (X, d, m) is a proper geodesic space (cf. [13] Lemma 3.5). +3 +An RCD∗(K, N) space is an infinitesimally Hilbertian CD∗(K, N) space (cf. [15] and +related reference there for the definition of the infinitesimally Hilbertian condition). +Especially, an RCD∗(K, N) space is CD∗(K, N). +4 +For K ∈ R, N ≥ 1, 0 ≤ t ≤ 1, and θ ∈ R+, set +σ(t) +K,N(θ) := + + + + + + + + + + + + + + + +∞ +if Kθ2 ≥ Nπ2, +sin(√ +K +N tθ) +sin(√ +K +N θ) +if K > 0, Kθ2 < Nπ2, +t +if K = 0, +sinh(√ +− K +N tθ) +sinh(√ +− K +N θ) +if K < 0. +We will use the following Brunn-Minkowski inequality to prove Lemma 1: +Theorem 6. ([4]Proposition 6.1) Let K, N ∈ R and let N ≥ 1. +Assume that +(X, d, m) is a CD∗(K, N) space. Then for all Borel sets A, B ⊂ M and t ∈ [0, 1], +m(Zt(A, B)) +1 +N ≥ σ(1−t) +K,N (Θ) · m(A) +1 +N + σ(t) +K,N · m(B) +1 +N , +where Zt(A, B) = {x ∈ X | there exist a ∈ A and b ∈ B such that d(a, x) = +td(a, b) and d(b, x) = (1 − t)d(a, b).} and where +Θ := +� +infx0∈A,x1∈B d (x0, x1) , +K ⩾ 0, +supx0∈A,x1∈B d (x0, x1) , +K < 0. +5 + +5 +We also need the splitting theorem in the nonsmooth setting, which generalized +the celebrated Cheeger-Gromoll Splitting Theorem [5]: +Theorem 7. ([9])Let (X, d, m) be an RCD(0, N) space with 1 ≤ N < ∞. Suppose +that X contains a line. Then (X, d, m) is isomorphic to (X′ × R, d′ × dE, m′ × L1), +where dE is the Euclidean distance, L1 the Lebesgue measure and (X′, d′, m′) is an +RCD(0, N − 1) space if N ≥ 2 and a singleton if N < 2. +6 +We require the existence of a simply connected universal cover: +Theorem 8. ([15],[18]) Let (X, d, m) be an RCD∗(K, N)-space for some K ∈ R, 1 < +N < ∞. Then (X, d, m) admits a simply connected universal cover ( ˜X, ˜d, ˜m) which +is itself an RCD∗(K, N)-space. +7 +We say that a geodesic metric space (X, d) is nonbranching if and only if there +are no 4 different points x, y, z, w in X such that d(x, z) = d(x, y) + d(y, z) and +d(x, w) = d(x, y) + d(y, w). +Theorem 9. ([6]) Let (X, d, m) be an RCD(K, N) space. +Then (X, d, m) is non- +branching. +8 +Finally, we will use the following splitting result for metric isometry groups. It +was used in [5] under the condition that N is a Riemannian manifold. +Proposition 1. Let (N, d) be a metric space which contains no line and let k ∈ N+. +Then any isometry F of the metric product N × Rk splits as F = (f, g), where f is +an isometry on N and g is an isometry on Rk. +We will give a proof of Proposition 1 in the Appendix. +3 +Proofs +In this section, we will first prove Theorem 4 and then prove Theorem 3, since +our proof of Theorem 3 slightly uses Theorem 4. +Let (X, d) be a proper geodesic space and let p ∈ X. Put +Cr(p) ={q ∈ X | ∀z ∈ X\Br(q), d(p, q) + d(q, z) > d(p, z)}. +Then Cr(p) is open in X. The set C(p) := +∞� +i=1 +Ci−1(p) is called the cut locus of p. +6 + +Let I = [0, l] or [0, ∞), where 0 < l < ∞. We say that a geodesic γ : I → X +with γ(0) = p is non-extendable (relative to p) if and only if either I = [0, l] and +γ(l) ∈ C(p) or I = [0, ∞), i.e γ is a ray. The set of all non-extendable geodesics +(relative to p) is denoted by NE(p). +The following measure estimate is the key to the proofs of our Theorems. +Lemma 1. Let N > 1 and let (X, d, m) be a nonbranching CD∗(K, N) space. Let +p ∈ X. +(1) If A ⊂ X is a Borel set such that A ∩ γ contains at most one point for every +γ ∈ NE(p). Then m(A) = 0. +(2)If K = 0, then for every r > 0 we have +lim +R→∞ +m(Cr(p) ∩ BR(p)) +RN−1 += 0. +(3.1) +Proof. (1)Without loss of generality, we may assume K = −1. For every i, R ∈ N+, +put +A(R) = A ∩ (BR(p)\BR−1(p)), +Aij(R) = A ∩ +� +BR−1+ j+1 +i (p)\BR−1+ j +i (p) +� +, j = 0, 1, · · · , Ri − 1. +By using the Brunn-Minkowski inequality for Aij(R), {p} and time tij = 1− +(2R)−1 +R−1+ j+1 +i , +we obtain: +m(Ztij(Aij(R), {p})) ≥ c(N, R)m(Aij(R)) +where c(N, R) > 0. Note that the sets Ztij(Aij(R), {p}) are disjoint by the non- +branching condition and by the nature of A. Note also that the set Ztij(Aij(R), {p}) +is contained in B(2R)−1(p)\B(2R)−1− 1 +i (p) for every j. Now, if every Ztij(Aij(R), {p}) +is a Borel set, we have +m(A(R)) = +Ri−1 +� +j=0 +m(Aij(R)) +≤ (c(N, R))−1 +Ri−1 +� +j=0 +m(Ztij(Aij(R), {p})) +≤ (c(N, R))−1m(B(2R)−1(p)\B(2R)−1− 1 +i (p)). +Let i → ∞, we get m(A(R)) = 0. Since A\{p} = +∞� +R=1 +A(R), we obtain m(A) = 0. +7 + +In general, Ztij(Aij(R), {p}) may not be measurable. In this case, we use the +inner regularity of m (note that m is a Radon measure, cf. Theorem 7.8 in [8] ). For +every ǫ > 0 we can find compact Kij ⊂ Aij(R) such that �Ri−1 +j=0 m(Aij(R)\Kij) < ǫ. +Note that Ztij(Kij, {p}) are also compact. So +m(A(R)) < +Ri−1 +� +j=0 +m(Kij) + ǫ +≤ (c(N, R))−1 +Ri−1 +� +j=0 +m(Ztij(Kij, {p})) + ǫ +≤ (c(N, R))−1m(B(2R)−1(p)\B(2R)−1− 1 +i (p)) + ǫ. +Since i and ǫ are arbitary, we conclude that m(A(R)) = 0. +(2)We use the following sublemma: +Sublemma 1. Let R > 1 and let A ⊂ BR+1(p)\BR(p) be compact. Denote by KA +the union of all geodesics connecting p and a point a ∈ A. Put VA = B1(p) ∩ KA. +then m(A) ≤ N(R + 1)N−1m(VA). +Assume the Sublemma 1 holds. Without loss of generality, we may assume r = 1. +Put Ci = C1(p) ∩ (Bi+1(p)\Bi(p)) for i = 2, 3, · · ·. By the inner regularity of m, +We may choose compact C′ +i ⊂ Ci such that �∞ +i=2 m(Ci\C′ +i) < 1. Note that by the +nonbranching condition and the definition of C1(p), VC′ +i ∩ VC′ +j = {p} for |i − j| ≥ 2. +The compactness of C′ +i guarantee that KC′ +i are compact, hence VC′ +i are Borel. So +8 + +�∞ +i=2 m(VC′ +i) ≤ 2m(B1(p)). By Sublemma 1, we have +m(C1(p) ∩ BR(p)) ≤m(B2(p)) + +[R] +� +i=2 +m(Ci) + 0 such that Bl(g1 · ˜x0) ∩ Bl(g2 · ˜x0) = +∅, ∀g1, g2 ∈ Γ, g1 ̸= g2. Since Y contains no line, by contradiction argument there +exists an h > 0 such that +Isommet(Y ) · Bl(y0) ⊂ Bh(y0) ∪ Ch(y0), +where Isommet(Y ) is the metric isometry group of Y (cf. Lemma 1 of [20]). +Since the metric isometry group of ˜X splits, we have +� +g∈Γ,d ˜ +X(˜x0,g·˜x0)≤R +Bl(g · ˜x0) ⊂ ((Bh(y0) ∪ Ch(y0)) ∩ BR+l(y0)) × BR+l(0k), ∀R > 0. +So +˜m(Bl(˜x0)) · #(DΓ +R(˜x0)) ≤ mY ((Bh(y0) ∪ Ch(y0)) ∩ BR+l(y0)) · Lk(BR+l(0k)) +≤ +� +mY (Bh(y0)) + mY (Ch(y0) ∩ BR+l(y0)) +� +ωk(R + l)k += ωkmY (Bh(y0))(R + l)k + ωkf(R + l)(R + l)N−1. (∗) +where ωk = Lk(B1(0k)) and mY (Ch(y0) ∩ BR+l(y0)) = f(R + l)(R + l)N−k−1 and +we have lim +r→∞ f(r) = 0 by Lemma 1 (2). So Γ has polynomial orbit growth of order +< N − 1 in this case. +Now if N /∈ N, we conclude that Γ has polynomial orbit of order < N − 1. +11 + +If N ∈ N, we conclude that Γ has polynomial orbit of order ≤ N − 1. Now +consider the situation that Γ fails to have polynomial orbit of order < N − 1. Then +k can only equals N or N − 1. If k = N, then (X, d, m) is a flat N-manifold with +an N − 1 dimensional soul and with m a multiple of the Riemannian volume. If +k = N − 1, then (Y, dY ) = [0, ∞) and Rk/Γ is a compact flat (N − 1)-manifold as +analysed in Case 2. Since (Y, dY , mY ) is an RCD(0, 1) space, mY can only be a multiple +of L1 (see the Remark 3 below). This completes the proof. +Remark 3. If (Y, dY ) = [0, ∞) and (Y, dY , mY ) is an RCD(0, 1) space, we show that +mY can only be a multiple of L1 here: For i ∈ N+, let Ij = [ j +i, j+1 +i ], j = 0, 1, 2, · · ·. +Use the Brunn-Minkowski inequality for A = Ij ∪ Ij+1, B = { j +i} and t = +1 +2, we +obtain that mY (Ij) ≥ 1 +2mY (A). Simialrly, mY (Ij+1) ≥ 1 +2mY (A). Since a single point +has measure 0, we have mY (Ij) = mY (Ij+1). Since j is arbitrary, we obtain that +mY ([p, q]) = c(q − p), ∀0 ≤ p ≤ q, p, q ∈ Q, where c = mY ([0, 1]) > 0. We conclude +that mY ([a, b]) = c(b − a), ∀0 ≤ a ≤ b, a, b ∈ R by taking limits. Since every open set +of [0, ∞) is a disjoint countable union of intervals, we conclude that mY = cL1 on +open sets. It follows from a classical result in measure theory that mY = cL1 on all +Borel sets. +4 +Appendix: A proof of Proposition 1 +Fix (p, v) ∈ N × Rk, and set F(p, v) = (q, w). Then F({p} × Rk) = {q} × Rk +since F maps a line to another line and since N contains no line. +Claim 1. F(N × {v}) = N × {w}. +Proof. For n ∈ N, assume that F(n, v) = (m, w′). Note that (p, v) is the unique +point on {p} × Rk that is closest to (n, v). So F(p, v) = (q, w) is the unique point on +{q} ×Rk that is closest to F(n, v). Since (q, w′) is the unique point on {q} ×Rk that +is closest to F(n, v) = (m, w′), we obtain that w = w′. So F(N × {v} ⊂ N × {w}). +The same argument shows that F −1(N × {w}) ⊂ N × {v}. This proves the Claim. +By Claim 1, there are isometry f : N → N and g : Rk → Rk such that F(x, v) = +(f(x), w), F(p, y) = (q, g(y)), ∀x ∈ N, y ∈ Rk. +Now we check that F(x, y) = (f(x), g(y)). To see this, note that (x, v) is the +unique point on N × {v} that is closest to (x, y), and that (p, y) is the unique point +on {p} × Rk that is closest to (x, y). So F(x, v) = (f(x), w) is the unique point on +12 + +N × {w} that is closest to F(x, y), and F(p, y) = (q, g(y)) is the unique point on +{q} × Rk that is closest to F(x, y). That is F(x, y) = (f(x), g(y)). +5 +Acknowledgement +The author thanks his advisor Professor Xiaochun Rong for suggesting this prob- +lem and for helpful discussion. +13 + +References +[1] Luigi Ambrosio, Nicola Gigli, Andrea Mondino, and Tapio Rajala. Riemannian +Ricci curvature lower bounds in metric measure spaces with σ-finite measure. +Trans. Am. Math. Soc., 367(7):4661–4701, 2015. +[2] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Metric measure spaces with +Riemannian Ricci curvature bounded from below. Duke Math. J., 163(7):1405– +1490, 2014. +[3] Michael T. Anderson. On the topology of complete manifolds of non-negative +Ricci curvature. Topology, 29(1):41–55, 1990. +[4] Kathrin Bacher and Karl-Theodor Sturm. Localization and tensorization prop- +erties of the curvature-dimension condition for metric measure spaces. J. Funct. +Anal., 259(1):28–56, 2010. +[5] Jeff Cheeger and Detlef Gromoll. The splitting theorem for manifolds of non- +negative Ricci curvature. J. Differ. Geom., 6:119–128, 1971. +[6] Qin Deng. H¨0lder continuity of tangent cones in RCD(K,N) spaces and appli- +cations to non-branching. arXiv:2009.07956v2, 2020. +[7] Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm. On the equiv- +alence of the entropic curvature-dimension condition and Bochner’s inequality +on metric measure spaces. Invent. Math., 201(3):993–1071, 2015. +[8] Gerald B. Folland. +Real analysis. Modern techniques and their applications. +Pure Appl. Math., Wiley-Intersci. Ser. Texts Monogr. Tracts. New York, NY: +Wiley, 2nd ed. edition, 1999. +[9] Nicola Gigli. The splitting theorem in non-smooth context. arXiv:1302.5555, +2013. +[10] Xian-Tao Huang. Non-compact RCD(0, N) spaces with linear volume growth. +J. Geom. Anal., 28(2):1005–1051, 2018. +[11] Yu Kitabeppu and Sajjad Lakzian. +Characterization of low dimensional +RCD∗(K, N) spaces. Anal. Geom. Metr. Spaces, 4:187–215, 2016. +[12] John Lott and Cedric Villani. Ricci curvature for metric-measure spaces via +optimal transport. Ann. Math. (2), 169(3):903–991, 2009. +14 + +[13] Mattia Magnabosco, Lorenzo Portinale, and Tommaso Rossi. +The strong +Brunn–Minkowski inequality and its equivalence with the CD condition. +arXiv:2210.01494v1, 2022. +[14] Ilaria Mondello, Andrea Mondino, and Raquel Perales. An upper bound on the +revised first Betti number and a torus stability result for RCD spaces. Comment. +Math. Helv., 97(3):555–609, 2022. +[15] Andrea Mondino and Guofang Wei. On the universal cover and the fundamental +group of an RCD∗(K, N)-space. J. Reine Angew. Math., 753:211–237, 2019. +[16] Karl-Theodor Sturm. On the geometry of metric measure spaces. I. Acta Math., +196(1):65–131, 2006. +[17] Karl-Theodor Sturm. On the geometry of metric measure spaces. II. Acta Math., +196(1):133–177, 2006. +[18] Jikang Wang. +RCD∗(K, N) spaces are semi-locally simply connected. +arXiv:2211.07087, 2022. +[19] Shing-Tung Yau. Some function-theoretic properties of complete Riemannian +manifold and their applications to geometry. Indiana Univ. Math. J., 25:659– +670, 1976. +[20] Zhu Ye. Maximal first Betti number rigidity for open manifolds of nonnegative +Ricci curvature. arXiv:2212.05530, 2022. +15 + diff --git a/vdE3T4oBgHgl3EQfkwp-/content/tmp_files/load_file.txt b/vdE3T4oBgHgl3EQfkwp-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2bc87a326dfbc34eadc507f58d0c620dc3b1c40 --- /dev/null +++ b/vdE3T4oBgHgl3EQfkwp-/content/tmp_files/load_file.txt @@ -0,0 +1,385 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf,len=384 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='04600v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='DG] 11 Jan 2023 Maximal first Betti number rigidity of noncompact RCD(0,N) spaces Zhu Ye∗ Department of Mathematics, Capital Normal University Abstract Let (M, d, m) be a noncompact RCD(0,N) space with N ∈ N+ and suppm = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We prove that if the first Betti number of M equals N − 1, then (M, d, m) is either a flat Riemannian N-manifold with a soul T N−1 or the metric product [0, ∞) × T N−1, both with the measure a multiple of the Riemannian volume, where T N−1 is a flat torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 1 Introduction Let K ∈ R and N ∈ [1, ∞) throughtout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The RCD(K, N) spaces or RCD∗(K, N) spaces ([16],[17],[12],[4],[2]) are both synthetic counterparts of Rieman- nian manifolds with Ricci curvature bounded below by K and dimension bounded above by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Many classical results that hold for manifolds with Ricci curvature bounded below remain true in the synthetic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Here we mention some of them that will be used in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' In [9], Gigli established the split- ting theorem on RCD(0, N) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' In [15], the existence of the universal cover of an RCD∗(K, N) space is established by Mondino-Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' This universal cover admits a natural RCD∗(K, N) structure such that the covering projection is a local metric measure isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Recently in [18], based on the results of [15], Wang proved that an RCD∗(K, N) space is semi-locally simply connected, and hence its universal cover is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' In [6], Deng proved that an RCD(K, N) space is nonbranching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' ∗Supported partially by National Natural Science Foundation of China [11821101] and Beijing Natural Science Foundation [Z19003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Email address: 2210501006@cnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 1 The main purpose of this paper is to generalize the following result in [20] to RCD(0, N) spaces: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' If M is an open Riemannian n-manifold with RicM ≥ 0, then b1(M) ≤ n − 1 and the equality holds if and only if M is flat with a soul T n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' See [20] for a background of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Throughtout this paper, for a CD∗(K, N) space (X, d, m), we will always assume that suppm = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' This is not a restrictive condition since if (X, d, m) is CD∗(K, N) then (suppm, d, m) is also CD∗(K, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The following is the main result of this paper: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let N ≥ 2 and let (X, d, m) be a noncompact RCD(0, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then the first Betti number b1(X) ≤ [N] − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' If N ∈ N+, then the equality holds if and only if (X, d, m) is either a flat N-manifold with a soul T N−1 or [0, ∞) × T N−1, both with m a multiple of the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The above result is optimal in the sense that when N /∈ N+, the maximal first Betti number rigidity in the metric measure sense fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Indeed, for each h > 0, the space ([0, ∞), d, VhL1) with Vh(x) = f h(x) and f(x) : [0, ∞) → (0, 2) convex is RCD(0, h + 1), where d is the Euclidean distance and L1 is the Lebesgue measure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' [7] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' So the product of ([0, ∞), d, VhL1) and T [N]−1 with the Euclidean disdance and the Lebesgue measure is an RCD(0, N) space for every 0 < h < N − [N] (see [4] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='1 and [1]Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='6 for related tensorlization properties of RCD(0, N) spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The author conjectures that the metric rigidity still holds in this case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' that (X, d) is isomorphic to a flat N-manifold with a soul T N−1 or [0, ∞) × T N−1 if (X, d, m) is a noncompact RCD(0, N) space with b1(X) = [N] − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' One knows from [15] and [18] that for a compact RCD(0, N) space (M, d, m), the first Betti number b1(M) ≤ [N], with equality holds if and only if (M, d, m) is a T [N] with m a multiple of the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' This generalized a classical theorem of Bochner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' For further results related to the first Betti number of compact RCD spaces, see [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' As in [20], our approach to Theorem 2 is the estimate of the orbit growth of covering group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' ([20]) Denote by #(A) the number of elements in a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (X, d) be a metric space and let Isom(X) be its isometry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let Γ be a subgroup of Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' For every x ∈ X, put DΓ(x, r) = {g ∈ Γ : d(x, g(x)) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 2 Given p ∈ [0, ∞), we say Γ has polynomial orbit growth related to x of order ≥ p (≤ p), if and only if lim inf r→∞ #(DΓ(x, r)) rp > 0 (lim sup r→∞ #(DΓ(x, r)) rp < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We say Γ has polynomial orbit growth related to x of order > p (< p), if and only if lim r→∞ #(DΓ(x, r)) rp = ∞ (= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' It is easy to check that the polynomial orbit growth properties defined above do not depend on the choice of the base point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Similar to [20], Theorem 2 follows easily from the following theorem on orbit growth (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Proof of Theorem 1 in [20]): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let N ≥ 2 and let (X, d, m) be a noncompact RCD(0, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let π : ( ˜X, ˜p) → (X, p) be the universal cover with deck transformation group Γ, where p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' If N is an integer, then Γ has polynomial orbit growth of order ≤ N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Moreover, Γ fails to have polynomial orbit growth of order < N − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' there exist ri → ∞ such that lim i→∞ #(DΓ(˜p, ri)) rN−1 i > 0 if and only if (X, d, m) is a flat N-manifold with an N − 1 dimensional soul or [0, ∞) × X1 with X1 is a compact flat (N − 1)-manifold, both with m a multiple of the Riemannian volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' If N is not an integer, then Γ has polynomial orbit growth of order < N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' As in [5] and [20], the key to proving Theorem 3 is to control the volume growth of the set of points that are close to the cut locus of a base point in some sense (see Lemma 1 (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' To obtain Lemma 1 (2), we will use the Brunn-Minkowski inequality to replace the volume element comparison argument used in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The nonbranching property of RCD(K, N) spaces is also crucial for our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We also generalize Theorem 4 in [20] (which is the orbit version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='1 in [3] ) to nonbranching CD∗(K, N) spaces: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (X, d, m) be a nonbranching CD∗(K, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (1) Let p, q ∈ X, p ̸= q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Put E(p, q) = {x = X | d(p, x) = d(q, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then m(E(p, q)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 3 (2)Let π : � ¯X, ¯d, ¯m, ¯x0 � → (X, d, m, x0) be a normal covering with deck transfor- mation group G (here � ¯X, ¯d, ¯m � is the lift of (X, d, m), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='2 of [4] or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='2 of [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then for every r > 0 we have: #(DG(¯x0, 2r)) · m(Br(x0)) ≥ ¯m(Br(¯x0)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='1) #(DG(¯x0, r)) · m(Br(x0)) ≤ ¯m(B2r(¯x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='2) It has been well known since [19] that an open manifold with nonnegative Ricci curvature has at least linear volume growth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' lim inf R→∞ Vol(BR(p)) R > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' This fact also holds for RCD(0, N) spaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' So the following definition makes sense: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (X, d, m) be a noncompact RCD(0, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We say X has mini- mal volume growth if and only if lim sup R→∞ m(BR(p)) R < ∞ for some p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (X, d, m) be a CD(0, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' By the Bishop-Gromov inequality, the limit lim r→∞ m(Br(p)) rN always exists and does not rely on the choice of p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We say that X has Euclidean volume growth if lim r→∞ m(Br(p)) rN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Otherwise, we say that X collapses at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The following theorem, which generalized Theorem 2 of [20], is an immediate consequence of Theorem 3 and Theorem 4 (2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='1): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (X, d, m) be a noncompact RCD(0, N) space and let p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let π : ( ˜X, ˜d, ˜m) → (X, d, m) be the universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' If X has minimal volume growth, then: if N /∈ N+, then ˜X collapses at infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' if N ∈ N+, then ˜X has Euclidean volume growth if and only if (X, d) is a flat N- manifold with an N − 1 dimensional soul or [0, ∞) × X1, where X1 is a compact flat N − 1 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Theorem 5 can be viewed as a generalization of the rigidity result of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Indeed, by Theorem 4, when N ∈ N+, b1(X) = N − 1 implies that X has minimal volume growth and that ˜X has Euclidean volume growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 4 2 Preliminaries In this section, we list the properties of CD∗(K, N) spaces and RCD∗(K, N) spaces that will be used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We will not give the definitions here, since they will not be used explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 1 When K = 0, the CD∗(0, N) condition is the same as the CD(0, N) condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 2 A CD∗(K, N) space (X, d, m) is a proper geodesic space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' [13] Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 3 An RCD∗(K, N) space is an infinitesimally Hilbertian CD∗(K, N) space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' [15] and related reference there for the definition of the infinitesimally Hilbertian condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Especially, an RCD∗(K, N) space is CD∗(K, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 4 For K ∈ R, N ≥ 1, 0 ≤ t ≤ 1, and θ ∈ R+, set σ(t) K,N(θ) := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∞ if Kθ2 ≥ Nπ2, sin(√ K N tθ) sin(√ K N θ) if K > 0, Kθ2 < Nπ2, t if K = 0, sinh(√ − K N tθ) sinh(√ − K N θ) if K < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We will use the following Brunn-Minkowski inequality to prove Lemma 1: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' ([4]Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='1) Let K, N ∈ R and let N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Assume that (X, d, m) is a CD∗(K, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then for all Borel sets A, B ⊂ M and t ∈ [0, 1], m(Zt(A, B)) 1 N ≥ σ(1−t) K,N (Θ) · m(A) 1 N + σ(t) K,N · m(B) 1 N , where Zt(A, B) = {x ∈ X | there exist a ∈ A and b ∈ B such that d(a, x) = td(a, b) and d(b, x) = (1 − t)d(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='} and where Θ := � infx0∈A,x1∈B d (x0, x1) , K ⩾ 0, supx0∈A,x1∈B d (x0, x1) , K < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 5 5 We also need the splitting theorem in the nonsmooth setting, which generalized the celebrated Cheeger-Gromoll Splitting Theorem [5]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' ([9])Let (X, d, m) be an RCD(0, N) space with 1 ≤ N < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Suppose that X contains a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then (X, d, m) is isomorphic to (X′ × R, d′ × dE, m′ × L1), where dE is the Euclidean distance, L1 the Lebesgue measure and (X′, d′, m′) is an RCD(0, N − 1) space if N ≥ 2 and a singleton if N < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 6 We require the existence of a simply connected universal cover: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' ([15],[18]) Let (X, d, m) be an RCD∗(K, N)-space for some K ∈ R, 1 < N < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then (X, d, m) admits a simply connected universal cover ( ˜X, ˜d, ˜m) which is itself an RCD∗(K, N)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 7 We say that a geodesic metric space (X, d) is nonbranching if and only if there are no 4 different points x, y, z, w in X such that d(x, z) = d(x, y) + d(y, z) and d(x, w) = d(x, y) + d(y, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' ([6]) Let (X, d, m) be an RCD(K, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then (X, d, m) is non- branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 8 Finally, we will use the following splitting result for metric isometry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' It was used in [5] under the condition that N is a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (N, d) be a metric space which contains no line and let k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then any isometry F of the metric product N × Rk splits as F = (f, g), where f is an isometry on N and g is an isometry on Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We will give a proof of Proposition 1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 3 Proofs In this section, we will first prove Theorem 4 and then prove Theorem 3, since our proof of Theorem 3 slightly uses Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let (X, d) be a proper geodesic space and let p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Put Cr(p) ={q ∈ X | ∀z ∈ X\\Br(q), d(p, q) + d(q, z) > d(p, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then Cr(p) is open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The set C(p) := ∞� i=1 Ci−1(p) is called the cut locus of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 6 Let I = [0, l] or [0, ∞), where 0 < l < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' We say that a geodesic γ : I → X with γ(0) = p is non-extendable (relative to p) if and only if either I = [0, l] and γ(l) ∈ C(p) or I = [0, ∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='e γ is a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The set of all non-extendable geodesics (relative to p) is denoted by NE(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The following measure estimate is the key to the proofs of our Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let N > 1 and let (X, d, m) be a nonbranching CD∗(K, N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (1) If A ⊂ X is a Borel set such that A ∩ γ contains at most one point for every γ ∈ NE(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Then m(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (2)If K = 0, then for every r > 0 we have lim R→∞ m(Cr(p) ∩ BR(p)) RN−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (1)Without loss of generality, we may assume K = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' For every i, R ∈ N+, put A(R) = A ∩ (BR(p)\\BR−1(p)), Aij(R) = A ∩ � BR−1+ j+1 i (p)\\BR−1+ j i (p) � , j = 0, 1, · · · , Ri − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' By using the Brunn-Minkowski inequality for Aij(R), {p} and time tij = 1− (2R)−1 R−1+ j+1 i , we obtain: m(Ztij(Aij(R), {p})) ≥ c(N, R)m(Aij(R)) where c(N, R) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Note that the sets Ztij(Aij(R), {p}) are disjoint by the non- branching condition and by the nature of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Note also that the set Ztij(Aij(R), {p}) is contained in B(2R)−1(p)\\B(2R)−1− 1 i (p) for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Now, if every Ztij(Aij(R), {p}) is a Borel set, we have m(A(R)) = Ri−1 � j=0 m(Aij(R)) ≤ (c(N, R))−1 Ri−1 � j=0 m(Ztij(Aij(R), {p})) ≤ (c(N, R))−1m(B(2R)−1(p)\\B(2R)−1− 1 i (p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let i → ∞, we get m(A(R)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Since A\\{p} = ∞� R=1 A(R), we obtain m(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' 7 In general, Ztij(Aij(R), {p}) may not be measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' In this case, we use the inner regularity of m (note that m is a Radon measure, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content='8 in [8] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' For every ǫ > 0 we can find compact Kij ⊂ Aij(R) such that �Ri−1 j=0 m(Aij(R)\\Kij) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Note that Ztij(Kij, {p}) are also compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' So m(A(R)) < Ri−1 � j=0 m(Kij) + ǫ ≤ (c(N, R))−1 Ri−1 � j=0 m(Ztij(Kij, {p})) + ǫ ≤ (c(N, R))−1m(B(2R)−1(p)\\B(2R)−1− 1 i (p)) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Since i and ǫ are arbitary, we conclude that m(A(R)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' (2)We use the following sublemma: Sublemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Let R > 1 and let A ⊂ BR+1(p)\\BR(p) be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Denote by KA the union of all geodesics connecting p and a point a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Put VA = B1(p) ∩ KA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' then m(A) ≤ N(R + 1)N−1m(VA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Assume the Sublemma 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Without loss of generality, we may assume r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Put Ci = C1(p) ∩ (Bi+1(p)\\Bi(p)) for i = 2, 3, · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' By the inner regularity of m, We may choose compact C′ i ⊂ Ci such that �∞ i=2 m(Ci\\C′ i) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' Note that by the nonbranching condition and the definition of C1(p), VC′ i ∩ VC′ j = {p} for |i − j| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' The compactness of C′ i guarantee that KC′ i are compact, hence VC′ i are Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' So 8 �∞ i=2 m(VC′ i) ≤ 2m(B1(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf'} +page_content=' By Sublemma 1, we have m(C1(p) ∩ BR(p)) ≤m(B2(p)) + [R] � i=2 m(Ci) the < object >/< sth >. (Please) < action > it < by > the < part > that/which can < affordance >. +3,4 +1-2 +I < verb > the < object >/< sth >. (Please) < action > it < by > the < part > < purpose > < affordance >. +5,6 +1-3 +I < verb > the < object >/< sth >. (Please) < action > it < by > the < part > so that you can < affordance >. +7,8 +1-4 +I < verb > the < object >/< sth >. (Please) < action > it that/which can < affordance >. +9,10 +1-5 +I < verb > the < object >/< sth >. (Please) < action > it < purpose > < affordance >. +11,12 +1-6 +I < verb > the < object >/< sth >. (Please) < action > it so that you can < affordance >. +13,14 +1-7 +I < verb > the < object >/< sth >. (Please) < action > it < by > the < part >. +15,16 +2-1 +(Please) < action > (to) me the < object >/< sth > < by > the < part > < purpose > < affordance >. +17,18 +2-2 +(Please) < action > (to) me the < object >/< sth > < by > the < part > that/which can < affordance >. +19,20 +2-3 +(Please) < action > (to) me the < object >/< sth > < by > the < part > so that you can < affordance >. +21,22 +2-4 +(Please) < action > (to) me the < object >/< sth > < purpose > < affordance >. +23,24 +2-5 +(Please) < action > (to) me the < object >/< sth > that/which can < affordance >. +25,26 +2-6 +(Please) < action > (to) me the < object >/< sth > so that you can < affordance >. +27,28 +2-7 +(Please) < action > (to) me the < object >/< sth > < by > the < part >. +29,30 +3-1 +I < verb > you to < action > the < object >/< sth > < by > the < part > that/which can < affordance >. +31,32 +3-2 +I < verb > you to < action > (to) me the < object >/< sth > < by > the < part > < purpose > < affordance >. +33,34 +3-3 +I < verb > you to < action > (to) me the < object >/< sth > < by > the < part > so that you can < affordance >. +35,36 +3-4 +I < verb > you to < action > the < object >/< sth > that/which can < affordance >. +37,38 +3-5 +I < verb > you to < action > (to) me the < object >/< sth > < purpose > < affordance >. +39,40 +3-6 +I < verb > you to < action > (to) me the < object >/< sth > so that you can < affordance >. +41,42 +3-7 +I < verb > you to < action > (to) me the < object >/< sth > < by > the < part >. +43,44 +4 +The < part > of the < object >/< sth >. +TABLE II: Four types of templates are introduced. The first three are related to human-robot instructions, and the last one is +an object or part description. ⟨verb⟩ denotes intention verb, ⟨action⟩ denotes action verb, ⟨object⟩ denotes an object, ⟨sth⟩ +denotes the referring word, ⟨by⟩ denotes preposition for the part, ⟨part⟩ denotes the part of referring object, ⟨affordance⟩ +denotes the affordance phrase for the part. For index, odd number sentence uses ⟨object⟩, while mean number sentence uses +⟨sth⟩. It is to differentiate whether the object is known. +object O rendered by pyrender 1 and collect 547, 417 point +cloud observations. We repeat the random placement three +times. In total, each example of one object contains 13 point +cloud including one full-view point cloud observation, and +three random placement partial observations (each placement +includes four view observations.). We adopt part labels in +ShapeNet part dataset [18] and map them to sampled point +cloud by ICP and KD-tree search. +In a random object placement pi, four viewpoints vi point +cloud {Cpj +vi , i = 0, 1, 2, 3; j = 0, 1, 2} can be sampled +with 4, 096 points C ∈ R4×4096 with part label in object +coordinates, as shown in Fig. 1. A full-view point cloud +Cfullv are sampled from object mesh directly with 10, 000 +points C ∈ R4×10000 with part label in object coordinates. +A full point cloud observation sets are denoted as C = +1https://github.com/mmatl/pyrender +{Cfullv, Cpj +vi , i = 0, 1, 2, 3; j = 0, 1, 2}. +B. Grasp Sampling and Labeling +We follow the sampling policy of PointNetGPD [10] +using antipodal sampling based on trimesh. 2 Different from +existing work, which samples uniformly on the mesh of +the object, we introduce part semantic to sampling process +and only sample feasible grasps on the specific part surface +with sanity checking. Each grasp also contains a force- +closure metric Qfc and Grasp Wrench Space (GWS) analysis +metric Qgws consistent with PointNetGPD, which is used +to evaluate the grasp quality. Finally, for each example, we +obtain a grasp set G = {(gi, Qi +fc, Qi +gws), pi ∈ Rpart} with +size 60 elements, where gi is a grasp configuration with +position pi, Qi +fc and Qi +gws are grasping evaluation metrics, +2https://github.com/mikedh/trimesh + +Fig. 2: The overall architecture of PIONEER. black arrow trace refers to 3D part language grounding. The red arrow trace +refers to part-aware grasp pose detection. Multiple object observations with point cloud are collected, ICP and downsampled +before fed into PIONEER. +and Rpart is the specific part surface (e.g. surface of Handle +in a bag.). +C. Language Description +We propose 44 templates to generate our language de- +scriptions Q ∈ Q about the part, object and grasp shown +in TABLE. II. We design the templates from reference, +command, object description, intention, and part affordance +to generate the language references, where four types of +templates are interdicted inspired by [35]. The first three +(1-*,2-*,3-* in Type-Index) contain 7 sentence templates +respectively, and the object is replaced by a referring word +to extend a referring version template (total 14 sentences), +where templates 1−3 describe the affordance of a part, 4−6 +remove the part information only describing part affordance, +and 7 does not contain the affordance information. The first +type 1-* is established considering human intention to do +something. The second type 2-* is about human instruction +to give a action command. The third type 3-* is an integrated +version of the above two kinds. Type 4 is an object or part +description phrase. +For ⟨purpose⟩ and ⟨affordance⟩, we seek a large of +scene knowledge from a commonsense knowledge graph +COMET-ATOMIC-2020 [38] with 1.33M everyday inferen- +tial knowledge tuples about entities and people&events. It +represents a large-scale common sense repository of textual +descriptions that encode both the social and the physical as- +pects of common human everyday experiences. We construct +our corpus based on the tuples and people&events to enhance +the practicability of our language data. +For ⟨part⟩, we do not adopt part label in ShapeNet part +dataset, in which same semantic part in different objects are +given different labels (i.e. handle in the bag and handle in the +mug are given two different labels for classification). Instead, +we merge part-level labels containing the same semantic +(affordance) information (from 50 to 35 categories), and +augment semantic labels by considering synonyms and hy- +ponyms in PartNet [39], WordNet [40], and Wikipedia [41]. +IV. PROBLEM DEFINITION +According to existing work [19] about grasp pose detec- +tion using point cloud, given a point cloud and a description +of the geometry of a robotic hand (hand configuration), grasp +pose detection is to predict the grasp configurations based +on the hand configuration, from which a grasp would be +formed if the fingers are to close. A typical solution is to +sample enough grasp configuration candidates and select the +one with the highest score. We formulate it as a probability +model: +P(gi|R, C, Θ), +(1) +where gi is a sampled grasp configuration, C is a point +cloud observation, R is an interesting region to sample grasp +configuration candidates, and Θ is a hand configuration. +However, most existing work considers the region of interest +(ROI) as prior input by object detection or localization in +a scene (e.g. cluttered scene). These result in object wise +coarse-grained grasping, where part semantic of an object is +ignored. +In this paper, we consider more fine-grained grasping +detection by constraining ROI using affordance and intention. +We consider a part-aware probability model of grasp pose +detection using external knowledge from natural language +and decompose it into two parts, given by: +P(gi, R|C, Θ, Q) = P(gi|R, C, Θ) × P(R|C, Q), +(2) + +Object +art +Affordance +BERT model +768 +I want the mug. Please grasp it +128 +by the handgrip that can hold +Duplicate +the coffee. +2048×3 +MLP +0 +2048×128 +PointNet +0.8 +0.2 +PointNet +MLP +D +0.4 +0.6 +Approachwhere Q is a natural language sentence for object description +and instruction. P(gi|R, C, Θ) is given by a grasp pose +detection model (e.g., GPD). P(R|C, Q) is given by a 3D +part language grounding model. Two assumptions are as +follows: +Assumption 1. Natural language sentences are beneficial to +grasp pose detection during human-robot interaction. +Assumption 2. There is at least one positive grasping +candidate that can be detected within the grounding part of +the object under the observation. +V. METHOD +We propose a novel human-in-the-loop framework to +model Eq. 2, named PIONEER (grasP poInt detectiON +with shapE languagE gRounding). The overall architecture +is shown in Fig. 2. It consists of two modules, where the +first is a part-wise 3D language grounding model, which is +used for P(R|C, Q). The second is a part-aware grasp pose +detection model for P(gi|R, C, Θ). +A. Part-wise 3D Language Grounding +Given a query sentence Q from human and robotic point +cloud observation point cloud C, our 3D language grounding +model is to detect the query-related region R. It can be +formulated as a binary classifier function φ for each point +in point cloud C: (Q, C) → R{0, 1}. To achieve this, our +proposed model consists of four modules: language encoder, +point cloud encoder, multimodal fusion module, and a binary +classifier, shown in Fig. 2 with black trace. +A query sentence Q from a human is fed to a pre-trained +language model encoder (we use BERT [17]3) passing two +fully connected layers to calculate a 128-dimension language +feature Zq ∈ R1×128. For point cloud, we choose more than +one viewpoint cloud to merge a relatively complete point +cloud by iterative closest point (ICP) to camera coordinates +and downsample 2, 048 points4 C ∈ R2048×3. The prepro- +cessed C is input into PointNet [42] to calculate a feature +map Zc ∈ R2048×128. After extract language Zq and point +cloud features Zc, we repeat the Zq 2, 048 times to construct +a feature map Z′ +q ∈ R2048×128. We concatenate Zc and Z′ +q +and pass the new feature map to an MLP to extract fusion +feature Zfused. At last, the fusion feature is input to a binary +classifier to predict which points to be grounded. The whole +pipeline can be formulated as: +Zq = Elang (Q) , +Zc = Epoint (C) , +Zfused = MLP (repeat (Zq) ⊕ Zc) , +R = Classifier (Zfused) , +(3) +where Elang and Epoint are language and point cloud +encoders respectively. ⊕ denotes the concatenation operation. +3We use bert-base-uncased model in our work. +4We simplify to ignore viewpoint V representation since we have trans- +formed all points to the same coordinates by ICP. +B. Part-aware Grasp Pose Detection +To achieve part-aware grasp pose detection, we extend +PointNetGPD [10] in candidate sampling policy and grasp +selection. Under Assumption 2, different from sampling uni- +form randomly on the preprocessed point cloud of the whole +object [10], we introduce high-level cognitive semantic R to +constrain sampling region for candidate grasp set gi ∈ S, +shown in Fig. 2 with red trace. During our sampling process, +we sample potential grasp points within R, while making +collision detection and force closure detection to evaluate +sampling quality still using the whole object point cloud. +C. Training and Inference +We train the 3D language grounding model and part-aware +grasp pose detection model separately. To train 3D language +grounding model, we use (C, Q) in our proposed dataset +Lang-SHAPE in Sec. III. The parameters of pre-trained +BERT are frozen during the training process. We use a +binary-class cross-entropy loss to optimize the network with +Adam optimizer. We train the network for 200 epoches with +batchsize 32 and learning rate 1e−3. To the train the part- +aware grasp pose detection model, we use (C, G) in Lang- +SHAPE dataset, in which the oracle point cloud semantic +region is used to constrain the sampling region. We also use a +binary-class cross-entropy loss to optimize the network with +Adam optimizer. The batchsize is 32, training epoch is 60, +and learning rate is 5e−3. +To infer new input data, the grounding region of an object +from 3D language grounding model output is used to inject +into the part-aware grasp pose detection model. A series of +grasp candidate scores are predicted finally. We select the +optimized grasp to execution considering these scores and +robotic reachability. All models are trained and tested under +PyTorch 1.10. +VI. EXPERIMENTS +We conduct both simulation and real-world robot exper- +iments to investigate six research questions (RQ). The first +one is regarding the usefulness of the new dataset, the middle +four are about the analyses of the proposed models, and the +last is concerning the effectiveness of our method on the real +robot. +RQ1: For the effectiveness of proposed dataset, is our +proposed new dataset useful for fine-grained 3D robotic +grasping tasks, especially for affordance-aware task? +RQ2: How does the pre-trained language model perform +compared with existing baseline methods such as the ran- +domly initialized model or similarity-based method in 3D +language grounding with object parts? +RQ3: How much does the pre-trained language model em- +power the embodied inference ability given different-level +prompt language? +RQ4: For compositional generalization, given the fact that +an object can usually be decomposed into a certain number +of parts, how much does our proposed model perform in part +grounding between different objects with at least one but not +all similar parts? + +Langauge Mode +Definition +full data +all 44 sentences in Table II used in the process. +known all +sentences containing object name, part name and affordance, with index [1,3,5,15,17,19,29,31,33]. +object unknown +sentences not containing object name, with index [2,4,6,8,...,40,42,44]. +part unknown +sentences not containing part name, with index [7,8,9,10,11,12,21,22,23,24,25,26,35,36,37,38,39,40]. +part unknown part known +sentences not containing object name, but containing part name, with index [7,9,11,21,23,25,35,37,39]. +part specific +under human intervention to give an optimal grasp part for each object observation. +TABLE III: Multi-level difficulty language configuration. +RQ5: For human intervention, how does our proposed 3D +part language grounding method with human-in-the-loop +perform in fine-grained grasping detection success rate and +effectiveness? +RQ6: For real-wolrd deployment, does our proposed method +perform well on a real-world robot? +A. Data Organization +To train and test our proposed models, we split our Lang- +SHAPE dataset object-wise and the part-wise, respectively, +named Split Mode. Object-wise, we split all examples in +Lang-SHAPE by the object category (16 categories) with +ratios (8 : 1 : 1) for (training/validation/test) sets. Similarly, +part-wise, we split all examples in Lang-SHAPE by the +part category (35 categories) with ratios (8 : 1 : 1) for +(training/validation/test) sets. We further set up fine-grained +language configurations, named Language Mode, defined in +TABLE III. +We provide two compositional generalization sets in TA- +BLE VI. Two extra split modes are introduced: +• related data has two attributes. First, the object cat- +egories of examples in the training set do not occur in +the test set. Second, at least one but not all parts of each +example in the training set are similar to those in the test set. +Nevertheless, the parts contained in the training set cover the +parts in the test set. The details are shown in TABLE VI with +Compositional Factors. In the first setup, chair, laptop, and +skateboard examples in Lang-SHAPE are collected as the +training set, in which they have at least one part such as +leg or board. Table examples are used as the test set, which +consists of legs and boards. In the second setup, guitar and +pistol examples in Lang-SHAPE are used as the training set, +in which they have at least one part such as body or handle. +Mug examples are adopted as the test set, which consists of +body and handle. +• non related data The data in the training set does not +contain any objects or parts that occur in the test set. +B. Evaluation Metrics +Grounding evaluation and 3D grasping detection evalua- +tion are performed in this paper, following [18], [42]. +For 3D part language grounding, we use four metrics: +• Accuracy: Since we formulate 3D part language ground- +ing as a binary classification problem, we calculate classifi- +cation accuracy on points. +• Part avg IoU: We calculate the IoU of grounded points +in each example [18] and average IoUs for each part category +to calculate mIoUs. Finally, we average each part’s mIoUs +to calculate the Part avg IoU. +• Class avg IoU: We calculate the IoU of grounded points +in each example and average IoUs for each object category +to get mIoUs. Finally, we average each object’s mIoUs to +calculate the Class avg IoU. +• Instance avg IoU: We calculate the IoU of grounded +points in each example and average all IoUs directly. +For 3D grasping detection, we define three metrics: +• Success Rate: The percentage of grasps where both +grasp points grounding is correct and pre-grasp prediction is +successful. +• Part-agnostic Success Rate: The percentage of grasps +that pre-grasp prediction is successful. +• Trial Cost: To get a high-quality grasp candidate for +the grasp score module, how many grasp sampling trials are +needed to perform in a standard antipodal grasping sampler +(GPG) [19]. +C. Models +We design 3 baselines for comparison: +• Baseline 1: For 3D part language grounding, inspired by +[25], we compare our method with a zero-shot classifier us- +ing pre-trained models directly. Instead of finetuning BERT, +we use cosine distance between visual and language features +to predict whether each point is grounded or not. Visual +encoder is from a pre-trained part segmentation model [42], +while language encoder is BERT with frozen parameters. +• Baseline 2: For 3D part language grounding, we replace +BERT in our proposed method in Fig. 2 with a Transformer +encoder 5, and train the whole model from scratch. This is +to verify whether the pre-trained model can provide useful +prior knowledge for our task. +• Baseline 3: For 3D grasp pose detection, we use +PointNetGPD [10] without human 3D language grounding +intervention during the sampling process as baseline to verify +the priority of using language human intervention. +We propose two models to solve 3D part language grounding +and grasp pose detection problem: +• PIONEER is what we propose in Fig.2. +• PIONEER-T5: To empower the bidirectional ability +of human-robot interaction, we introduce an extra gener- +ative pre-trained language model (T5 [43]) to infer 3D +part language grounding based on prompt engineering. The +instruction from human is first fed into T5 finetuned by +5https://github.com/pytorch/examples/tree/master/ +word_language_model (6 encoder layers implemented) + +Model +Split Mode +Accuracy +Part avg IoU +Class avg IoU +Instance avg IoU +Language +Encoder +Baseline 1 +part-wise +0.5179 +0.2355 +0.2659 +0.2461 +BERT +Baseline 2 +part-wise +0.8904 +0.5344 +0.5499 +0.6953 +Transformer +PIONEER (ours) +part-wise +0.9251 +0.6696 +0.6873 +0.7826 +BERT +Baseline 1 +object-wise +0.5373 +0.2076 +0.2271 +0.2153 +BERT +Baseline 2 +object-wise +0.8603 +0.4742 +0.5091 +0.6415 +Transformer +PIONEER (ours) +object-wise +0.9226 +0.6490 +0.6805 +0.7770 +BERT +TABLE IV: Overall results of 3D part language grounding in robustness and generalization. +prompt learning to generate object-part description index +43 in TABLE II. Our prior experiments show that model +using the naive object-part description can achieve very high +performance, and thus we combine a T5 based on prompt +learning and a PIONEER trained on index 43. +D. Simulation Experiments on Lang-SHAPE +Based on our proposed Lang-SHAPE dataset, we give a se- +ries of quantitative evaluations to answer research questions +RQ1-RQ5. The model selection is followed by the maximum +Instance avg IoU in the validation set. +1) 3D Part Language Grounding: To evaluate the overall +performance of our proposed model, we compare our model +with Baseline 1 and Baseline 2 in part wise and object- +wise data split mode, shown in TABLE IV. The language +mode used in model training is full data. For RQ1, RQ2, +our proposed model PIONEER outperforms in all metrics. +Compared with Baseline 1 (0.2355 in Part avg IoU), our +model (0.6696 in Part avg IoU) achieves more than double +improvement relative to the zero-shot method. We attribute +the poor performance by Baseline 1 to two points. The +first is no learning process to adjust parameters from the +prior domain to our Lang-SHAPE domain. The second is +that the visual encoder and the language encoder are not +trained jointly, which lacks shared feature space to fuse +multimodal features. This also shows the usefulness of our +proposed Lang-SHAPE dataset, which can be used for point +cloud-language joint training. Compared with Baseline 2, our +model achieves more than 8% improvement in Instance avg +IoU with part wise. The results indicate the advantage of +pre-trained language model over to the randomly initialized +model (i.e. Transformer) in robustness and generalization. +2) Affordance Inference: To evaluate the inference ability +of the proposed model, we set up different corrupted lan- +guage inputs to train models. Three models are used for +comparisons: Baseline 2, PIONEER, and PIONEER-T5. The +first two are trained following language mode and split mode +in TABLE V. In PIONEER-T5, we introduce a finetuned +T5 [43] with prompt engineering. Since T5 is an unimodal +model and cannot perform effective inference when neither +part nor object is unknown, we set up a more fine-grained +language mode part unknown object known, which is T5 +input concatenated with a prompt question. We design four +prompts familiar with [3]: ‘what part should you grasp?’, +‘which part should you take’, ‘how can you grasp it for +me?’, ‘how can you take it for me?’, one of them is randomly +selected. The object-part description index 43 in TABLE II +is the groundtruth of T5 and input of PIONEER. T5 and +PIONEER are trained respectively in PIONEER-T5, and the +test process uses them as a cascade model. +For RQ2, RQ3, in PIONEER, as we can see known all, +object unknown, and part unknown, with different object +attributes being corrupted, the performance of models de- +creases obviously. From Accuracy and Instance avg IoU, +we can find that when the object name is unknown, the +model still performs better than the part name is unknown. +Comparing PIONEER with Baseline 2 in object unknown +and part unknown, we can find that pre-trained language +model can infer the grounded part via affordance information +effectively. For example, compared with Baseline 2, Instance +avg IoU is increased from 0.6253 to 0.7583 in part unknown, +part-wise in TABLE V. +For RQ3, we provide an explicit prompt-based model +PIONEER-T5. From part unknown object known in TA- +BLE V, PIONEER-T5 achieves the best performance, which +again shows the pre-trained language model with prompt +learning can enhance the inference ability of the model. +3) Compsitional Generalization: It is the ability to gen- +eralize systematically to a new data distribution by combin- +ing known components [44]. To measure the compositional +generalization of our models, we propose two compositional +generalization sets defined in Sec. VI-A, shown in TABLE +VI. For RQ4, in PIONEER, with the same test set, the +model trained using our collected set (related data) achieves +better performance in all metrics compared with the model +trained using non related data. This indicates that our pro- +posed model is effectively generalizable. In comparison with +Baseline 1, results show that our proposed model performs +better than the zero-shot method. +4) Grasping Detection and Cost: To evaluate the effec- +tiveness of our proposed method in fine-grained grasping +detection, we test PIONEER on the whole Lang-SHAPE +dataset (including 3D part language grounding and grasp +data). We train 3D part language grounding and Point- +NetGPD respectively following the part-wise and object- +wise splits respectively. Language mode is full data in most +evaluations. The results are shown in TABLE VII. The +grasp sampling rule is that the sampler ends sampling at +a maximum of 150 sample trials or gets 20 high-quality +candidate grasps. It is noted that in the simulation grasping +experiment, the sampling process is on point cloud data + +Split Mode +Model +Language Mode +Accuracy +Part avg IoU +Class avg IoU +Instance avg IoU +Language Encoder +Part-wise +Baseline 2 +object unknown +0.8616 +0.4858 +0.5056 +0.6405 +Transformer +part unknown +0.8579 +0.4905 +0.5017 +0.6253 +part unknown object known +0.8548 +0.4888 +0.5044 +0.6178 +PIONEER (ours) +known all +0.9381 +0.7009 +0.7200 +0.8089 +BERT +object unknown +0.9210 +0.6316 +0.6530 +0.7761 +part unknown +0.9122 +0.6559 +0.6820 +0.7583 +part unknown object known +0.9158 +0.6740 +0.7011 +0.7704 +PIONEER-T5 (ours) +part unknown object known +0.9208 +0.6557 +0.6884 +0.7818 +Object-wise +Baseline 2 +object unknown +0.8894 +0.5445 +0.5778 +0.7074 +Transformer +part unknown +0.8427 +0.4483 +0.4783 +0.6019 +part unknown object known +0.8486 +0.4761 +0.4974 +0.6060 +PIONEER (ours) +known all +0.9381 +0.6943 +0.7287 +0.8116 +BERT +object unknown +0.9230 +0.6569 +0.6782 +0.7776 +part unknown +0.9118 +0.6467 +0.6754 +0.7641 +part unknown object known +0.9165 +0.6548 +0.6936 +0.7709 +PIONEER-T5 (ours) +part unknown object known +0.9203 +0.6868 +0.7032 +0.7853 +TABLE V: Comparisons of inference performance with different corrupted languages. For Split Mode, the top half is in +part wise. The bottom half is in object wise. For Language Encoder, Baseline 2 uses Transformer while our PIONEER +uses BERT. +Model +Split Mode +Accuracy +Part avg IoU +Part 1 IoU +Part 2 IoU +Instance avg IoU +Compositioal Factors +Part 1 + Part 2 +Baseline 1 +related data +0.5004 +0.2622 +0.1769 +0.3475 +0.2629 +chair (leg) + laptop,skateboard (board) += table(leg,board) +PIONEER (ours) +related data +0.7493 +0.5489 +0.5760 +0.5217 +0.5487 +non related data +0.5084 +0.1609 +0.1030 +0.2189 +0.1614 +objects not composed of leg or board +Baseline 1 +related data +0.5023 +0.2355 +0.4088 +0.0622 +0.2365 +guitar (body) + pistol (handle) += mug(body,handle) +PIONEER (ours) +related data +0.7177 +0.4902 +0.6803 +0.3001 +0.4913 +non related data +0.5709 +0.2990 +0.4486 +0.1495 +0.2999 +objects not composed of body or handle +TABLE VI: Results of compositional generalization with two subsets in SHAPE dataset. +instead of object meshes which are used in the dataset +collection process. The sampling process prefers to real- +world setting although we perform experiments in simulation +using dataset. +For RQ1, RQ5, by comparing PIONEER with Baseline +3 in TABLE VII, we can see that our proposed model can +realize part-grounded grasping with more than 40% success +rate, while Baseline 3 can only get 25% approximately. This +indicates the effectiveness of our proposed method in fine- +grained grasping detection. +We find that our method performs relatively weak in part- +agnostic success rate and trial cost. We attribute the reason +to the fact that our method constrains the sampling region, +and some part region is difficult for grasping, which reduces +the whole part-agnostic success rate and costs more time to +sample until the terminal condition. To verify our suppose, +we propose a new language mode part specific, in which +human specifies one grasping part for each object. From +TABLE VII, we can see that our PIONEER improves broadly +in all metrics with knowledge from human intervention. +E. Physical Robot Experiments +To evaluate the effectiveness of our proposed method +in the real world, we deployed our models on a real +robot system to realize part-aware grasping following human +instruction, which includes affordance and intentions. We +choose a single-arm robot Kinova Jaco 7DOF with three +fingers to perform manipulation. An eye-in-hand camera +Model +Split Mode +Language Mode +Success Rate +Part-agnostic +Success Rate +Trial Cost +Baseline 3 +part-wise +full data +0.2511 +0.6333 +12.1644 +PIONEER (ours) +part-wise +full data +0.4207 +0.4876 +15.5332 +Baseline 3 +object-wise +full data +0.2546 +0.6327 +11.4371 +PIONEER (ours) +object-wise +full data +0.4385 +0.4992 +16.1895 +PIONEER (ours) +object-wise +part specific +0.5258 +0.5843 +8.0396 +TABLE VII: Comparisons of fine-grained grasping detection. +Intel RealSense SR300 is fixed on the wrist of end-effector. +The system is deployed on a PC running Ubuntu 18.04 +and ROS Melodic with one Intel Core i7-8700K and one +NVIDIA Geforce GTX 1080Ti GPU. The intrinsic and +extrinsic parameters of the camera are calibrated. We select +our PIONEER model under the training configuration of part- +wise (Split Mode) and full data (Language Mode). +Since our proposed method is to operate in part wise, it +requires a more fine-grained perception of the target object. +For point cloud collection, we design a multi-view (four) +policy to collect each view point cloud and transform into +robot base frame. All viewpoint point cloud are merged by +ICP to get a relatively complete representation of the target +object. We select three categories of household objects. Two +(mug and table) are seen in our Lang-SHAPE, and another +one (hammer) is unseen. The object is randomly placed on +the table, multi-view point cloud collection is to obtain the +outline of the object, and then the merged point cloud is fed +into our PIONEER. The output of PIONEER is a grasp pose + +Fig. 3: Test objects and demonstration of real robot grasping pipeline with language query. +on robot base frame. For Q6, as shown in Fig. 3, our real +robot experiments indicate the effectiveness of our proposed +method in fine-grained grasping with human instruction +including object, part, and affordance. More experiments are +available in the attached video. +VII. CONCLUSION +We investigated part-level affordance on fine-grained +robotic grasping. The Lang-SHAPE dataset is constructed to +facilitate the investigation, and a 3D part language grounding +and a part-aware grasp pose detection model are proposed to +allow fine-grained robotic grasping. Experiments show that +our proposed method outperforms 3D part grasp grounding +in inference and generalizability, and physical robot experi- +ments show its effectiveness in the real world. These results +show the promise of using the pre-trained language model in +affordance grounding and fine-grained grasping for a robot. +REFERENCES +[1] Y. Zhu, T. Gao, L. Fan, S. Huang, M. Edmonds, H. Liu, F. Gao, +C. Zhang, S. Qi, Y. N. Wu, et al., “Dark, beyond deep: A paradigm +shift to cognitive ai with humanlike common sense,” Engineering, +vol. 6, no. 3, pp. 310–345, 2020. +[2] J. J. Gibson, “The theory of affordances,” Hilldale, USA, vol. 1, no. 2, +pp. 67–82, 1977. +[3] M. Ahn, A. Brohan, N. Brown, Y. Chebotar, O. Cortes, B. David, +C. Finn, K. Gopalakrishnan, K. Hausman, A. Herzog, et al., “Do as i +can, not as i say: Grounding language in robotic affordances,” arXiv +preprint arXiv:2204.01691, 2022. +[4] Y. Jiang, S. Moseson, and A. Saxena, “Efficient grasping from rgbd +images: Learning using a new rectangle representation,” in 2011 IEEE +International conference on robotics and automation, pp. 3304–3311, +IEEE, 2011. +[5] I. Lenz, H. Lee, and A. Saxena, “Deep learning for detecting robotic +grasps,” The International Journal of Robotics Research, vol. 34, no. 4- +5, pp. 705–724, 2015. +[6] D. Morrison, P. Corke, and J. Leitner, “Closing the loop for robotic +grasping: A real-time, generative grasp synthesis approach,” Robotics: +Science and Systems XIV, pp. 1–10, 2018. +[7] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. +Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in +Advances in neural information processing systems, pp. 5998–6008, +2017. +[8] S. Wang, Z. Zhou, and Z. Kan, “When transformer meets robotic +grasping: Exploits context for efficient grasp detection,” arXiv preprint +arXiv:2202.11911, 2022. +[9] J. Mahler, J. Liang, S. Niyaz, M. Laskey, R. Doan, X. Liu, J. A. Ojea, +and K. Goldberg, “Dex-net 2.0: Deep learning to plan robust grasps +with synthetic point clouds and analytic grasp metrics,” 2017. +[10] H. Liang, X. Ma, S. Li, M. G¨orner, S. Tang, B. Fang, F. Sun, and +J. Zhang, “Pointnetgpd: Detecting grasp configurations from point +sets,” in 2019 International Conference on Robotics and Automation +(ICRA), pp. 3629–3635, IEEE, 2019. +[11] H.-S. Fang, C. Wang, M. Gou, and C. Lu, “Graspnet-1billion: A large- +scale benchmark for general object grasping,” in Proceedings of the +IEEE/CVF conference on computer vision and pattern recognition, +pp. 11444–11453, 2020. +[12] M. Sundermeyer, A. Mousavian, R. Triebel, and D. Fox, “Contact- +graspnet: Efficient 6-dof grasp generation in cluttered scenes,” in 2021 +IEEE International Conference on Robotics and Automation (ICRA), +pp. 13438–13444, IEEE, 2021. +[13] F.-J. Chu, R. Xu, L. Seguin, and P. A. Vela, “Toward affordance detec- +tion and ranking on novel objects for real-world robotic manipulation,” +IEEE Robotics and Automation Letters, vol. 4, no. 4, pp. 4070–4077, +2019. +[14] R. Xu, F.-J. Chu, C. Tang, W. Liu, and P. A. Vela, “An affordance +keypoint detection network for robot manipulation,” IEEE Robotics +and Automation Letters, vol. 6, no. 2, pp. 2870–2877, 2021. +[15] A. Tekden, M. P. Deisenroth, and Y. Bekiroglu, “Affordance transfer +based on self-aligning implicit representations of local surfaces,” +[16] W. Chen, H. Liang, Z. Chen, F. Sun, and J. Zhang, “Learning 6- +dof task-oriented grasp detection via implicit estimation and visual +affordance,” arXiv preprint arXiv:2210.08537, 2022. +[17] J. D. M.-W. C. Kenton and L. K. Toutanova, “Bert: Pre-training of deep +bidirectional transformers for language understanding,” in Proceedings +of NAACL-HLT, pp. 4171–4186, 2019. +[18] C. R. Qi, H. Su, K. Mo, and L. J. Guibas, “Pointnet: Deep learning +on point sets for 3d classification and segmentation,” in Proceedings +of the IEEE conference on computer vision and pattern recognition, +pp. 652–660, 2017. +[19] A. ten Pas, M. Gualtieri, K. Saenko, and R. Platt, “Grasp pose +detection in point clouds,” The International Journal of Robotics +Research, vol. 36, no. 13-14, pp. 1455–1473, 2017. +[20] D. Z. Chen, A. X. Chang, and M. Nießner, “Scanrefer: 3d object +localization in rgb-d scans using natural language,” in European +Conference on Computer Vision, pp. 202–221, Springer, 2020. +[21] O. Mees and W. Burgard, “Composing pick-and-place tasks by ground- +ing language,” in International Symposium on Experimental Robotics, +pp. 491–501, Springer, 2020. +[22] Y. Chen, R. Xu, Y. Lin, and P. A. Vela, “A joint network for +grasp detection conditioned on natural language commands,” in 2021 +IEEE International Conference on Robotics and Automation (ICRA), +pp. 4576–4582, IEEE, 2021. +[23] H. Zhang, Y. Lu, C. Yu, D. Hsu, X. La, and N. Zheng, “Invigorate: +Interactive visual grounding and grasping in clutter,” in Robotics: +Science and Systems (RSS), 2021. +[24] Y. Song, P. Sun, P. Fang, L. Yang, Y. Xiao, and Y. Zhang, “Human- +in-the-loop robotic grasping using bert scene representation,” in Pro- +ceedings of the 29th International Conference on Computational +Linguistics, pp. 2992–3006, 2022. +[25] J. Thomason, M. Shridhar, Y. Bisk, C. Paxton, and L. Zettlemoyer, +“Language grounding with 3d objects,” in Conference on Robot +Learning, pp. 1691–1701, PMLR, 2022. +[26] J. J. Gibson, “The theory of affordances. the ecological approach to + +mug +keep from touching hot areavisual perception,” The People, Place and, Space Reader; Routledge: +New York, NY, USA, pp. 56–60, 1979. +[27] N. Vahrenkamp, L. Westkamp, N. Yamanobe, E. E. Aksoy, and +T. Asfour, “Part-based grasp planning for familiar objects,” in 2016 +IEEE-RAS 16th International Conference on Humanoid Robots (Hu- +manoids), pp. 919–925, IEEE, 2016. +[28] F.-J. Chu, R. Xu, and P. A. Vela, “Learning affordance segmenta- +tion for real-world robotic manipulation via synthetic images,” IEEE +Robotics and Automation Letters, vol. 4, no. 2, pp. 1140–1147, 2019. +[29] S. Deng, X. Xu, C. Wu, K. Chen, and K. Jia, “3d affordancenet: A +benchmark for visual object affordance understanding,” in Proceed- +ings of the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pp. 1778–1787, 2021. +[30] P. Ard´on, E. Pairet, R. P. Petrick, S. Ramamoorthy, and K. S. Lohan, +“Learning grasp affordance reasoning through semantic relations,” +IEEE Robotics and Automation Letters, vol. 4, no. 4, pp. 4571–4578, +2019. +[31] A. Asemi, A. Ko, and M. Nowkarizi, “Intelligent libraries: a review +on expert systems, artificial intelligence, and robot,” Library Hi Tech, +2020. +[32] L. Hu, Y. Jiang, F. Wang, K. Hwang, M. S. Hossain, and G. Muham- +mad, “Follow me robot-mind: Cloud brain based personalized robot +service with migration,” Future generation computer systems, vol. 107, +pp. 324–332, 2020. +[33] A. Depierre, E. Dellandr´ea, and L. Chen, “Jacquard: A large scale +dataset for robotic grasp detection,” in 2018 IEEE/RSJ International +Conference on Intelligent Robots and Systems (IROS), pp. 3511–3516, +IEEE, 2018. +[34] H. Zhang, X. Lan, S. Bai, X. Zhou, Z. Tian, and N. Zheng, “Roi- +based robotic grasp detection for object overlapping scenes,” in 2019 +IEEE/RSJ International Conference on Intelligent Robots and Systems +(IROS), pp. 4768–4775, IEEE, 2019. +[35] T. Nguyen, N. Gopalan, R. Patel, M. Corsaro, E. Pavlick, and S. Tellex, +“Robot Object Retrieval with Contextual Natural Language Queries,” +in Proceedings of Robotics: Science and Systems, (Corvalis, Oregon, +USA), July 2020. +[36] C. Eppner, A. Mousavian, and D. Fox, “Acronym: A large-scale grasp +dataset based on simulation,” in 2021 IEEE International Conference +on Robotics and Automation (ICRA), pp. 6222–6227, IEEE, 2021. +[37] A. X. Chang, T. Funkhouser, L. Guibas, P. Hanrahan, Q. Huang, +Z. +Li, +S. +Savarese, +M. +Savva, +S. +Song, +H. +Su, +et +al., +“Shapenet: An information-rich 3d model repository,” arXiv preprint +arXiv:1512.03012, 2015. +[38] J. D. Hwang, C. Bhagavatula, R. Le Bras, J. Da, K. Sakaguchi, +A. Bosselut, and Y. Choi, “(comet-) atomic 2020: on symbolic and +neural commonsense knowledge graphs,” in Proceedings of the AAAI +Conference on Artificial Intelligence, vol. 35, pp. 6384–6392, 2021. +[39] K. Mo, S. Zhu, A. X. Chang, L. Yi, S. Tripathi, L. J. Guibas, and H. Su, +“Partnet: A large-scale benchmark for fine-grained and hierarchical +part-level 3d object understanding,” in Proceedings of the IEEE/CVF +conference on computer vision and pattern recognition, pp. 909–918, +2019. +[40] C. Fellbaum, “Wordnet,” in Theory and applications of ontology: +computer applications, pp. 231–243, Springer, 2010. +[41] Wikipedia, Wikipedia. PediaPress, 2004. +[42] C. R. Qi, L. Yi, H. Su, and L. J. Guibas, “Pointnet++: Deep +hierarchical feature learning on point sets in a metric space,” Advances +in neural information processing systems, vol. 30, 2017. +[43] C. Raffel, N. Shazeer, A. Roberts, K. Lee, S. Narang, M. Matena, +Y. Zhou, W. Li, and P. J. Liu, “Exploring the limits of transfer learning +with a unified text-to-text transformer,” Journal of Machine Learning +Research, vol. 21, pp. 1–67, 2020. +[44] J. Kim, P. Ravikumar, J. Ainslie, and S. Onta˜n´on, “Improving compo- +sitional generalization in classification tasks via structure annotations,” +arXiv preprint arXiv:2106.10434, 2021. + diff --git a/xtFJT4oBgHgl3EQfhCzP/content/tmp_files/load_file.txt b/xtFJT4oBgHgl3EQfhCzP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4d5878ecbee2c24797d2df1629d43cf10a00f6f --- /dev/null +++ b/xtFJT4oBgHgl3EQfhCzP/content/tmp_files/load_file.txt @@ -0,0 +1,899 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf,len=898 +page_content='Learning 6-DoF Fine-grained Grasp Detection Based on Part Affordance Grounding Yaoxian Song12†, Penglei Sun1†, Yi Ren4, Yu Zheng4, Yue Zhang23∗ Abstract— Robotic grasping is a fundamental ability for a robot to interact with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Current methods focus on how to obtain a stable and reliable grasping pose in object wise, while little work has been studied on part (shape)-wise grasping which is related to fine-grained grasping and robotic affordance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Parts can be seen as atomic elements to compose an object, which contains rich semantic knowledge and a strong correlation with affordance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' However, lacking a large part-wise 3D robotic dataset limits the development of part represen- tation learning and downstream application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In this paper, we propose a new large Language-guided SHape grAsPing datasEt (named Lang-SHAPE) to learn 3D part-wise affordance and grasping ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We design a novel two-stage fine-grained robotic grasping network (named PIONEER), including a novel 3D part language grounding model, and a part-aware grasp pose detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To evaluate the effectiveness, we perform multi-level difficulty part language grounding grasping experiments and deploy our proposed model on a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Results show our method achieves satisfactory performance and efficiency in reference identification, affordance inference, and 3D part-aware grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Our dataset and code are available on our project website https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='com/view/ lang-shape I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' INTRODUCTION Fine-grained robotic manipulation can allow a robot to tackle human tasks by mincing human hands in embodied AI [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Different from low-level manipulation in control (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g pick and place), fine-grained robotic manipulation not only has the abilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g graspability) but also considers additional details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g which part to grasp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' why to grasp this part), which reduces to affordance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For example, when a human demonstrates the intention of drinking a cup of coffee to hope a robot brings his mug, the robot is expected to grasp the handle and not pollute the inside of the mug subliminally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Behind that, the affordance of manipulated objects is needed and how to empower low-level control policy with high- level concept and knowledge represented by human symbolic language have received increasing attention in the robotics community [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Thanks to recent progress in artificial intelligent fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' computer vision and natural language processing), vision- † Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1The authors are with the School of Computer Science, Fudan University, Emails: plsun20@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='cn 2The authors are with the School of Engineering, Westlake University, Emails: {songyaoxian,zhangyue}@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3The author is with the Institute of Advanced Technology, Westlake Institute for Advanced Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4The authors are with the Tencent Robotcis X Lab, Tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Emails: {evanyren,petezheng}@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='com, yren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='tum@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1: Dataset generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' One random placement on a table scenario and four viewpoints are given to sample the object point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Affordance-related sentences are generated by the knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Positive (cyan) and negative (red) grasps about Handle and Body are sampled respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' based robotic manipulation methods have achieved great suc- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Many efforts are made by researchers to explore steady, dexterous, reliable grasping policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For planar grasping, [4]– [6] propose 2D grasp detection neural network using image- based grasp datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It typically keeps the camera heading vertically to the tabletop and generate grasp pose in a 2D plane, which is simple and easy for using directly computer vision methods such as transformer [7] in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For spatial grasping, [9]–[12] propose point cloud-based networks to predict 6-DoF grasping pose, which allows a robot arm to grasp objects in 3D space more human-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' However, all of this research focuses on how to get a reliable and steady grasping pose based in object wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A more fine-grained grasping is not available for a specific requirement, such as based on some affordance to interact with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For affordance, Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [13] and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [14] and Tekden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [15] explore affordance in grasping detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' However, they use limited objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' only two in [15]) or affordance with small-scale data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' only seven in [16]) to take a trial on simple setup (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' simulation or small test set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For a real- world setup, a large-scale affordance-related grasping dataset and pragmatic grasping detection model in spatial space are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To overcome the challenges in existing work on grasping detection and affordance, we propose a large language- guided shape grasp dataset with human-in-the-loop method for human-robot interaction in spatial grasping detection systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We combine natural language [17], 3D part- segmentation [18], and 6-DoF robotic grasping [10] together arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='11564v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='RO] 27 Jan 2023 A Please hand me the bag via the grip which can pick it up I need you to bring me the bag via the frame so as to put the phone in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' WORDNET: MOSAIC Object Part Affordancto solve fine-grained grasping with affordance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We propose a new large Language-guided SHape grAsP- ing datasEt (named Lang-SHAPE), which contains point cloud, grasping label, and affordance-related language ref- erence considering 35 parts, shown in TABLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Human instructions are constructed by 44 templates shown in TA- BLE II and more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='85 million sentences are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For modeling, we extend a typical spatial grasping detec- tion method (PointNetGPD) [10] with human language in- tervention (named PIONEER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We make two modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' First, we are to provide a visual servo eye-in-hand object scan policy to capture global and detailed point cloud observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Second, we introduce a 3D part language grounding model to constrain sampling region [19] to realize part-aware grasping detection based on the affordance in human language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The whole pipeline is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We give multi-level difficulty language grounding grasping experiments to evaluate our proposed dataset and model in inference and compositional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A real-robot experiment is performed to verify the effectiveness of our proposed model in the real-world qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Results show our method achieves a huge advantage in reference identi- fication, affordance inference, and 3D part-aware grasping with strong generalization benefiting from using pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To the best of our knowledge, we are the first to consider affordance in part-level for spatial robotic grasping using natural language, building the first 3D large language-guided shape grasping dataset covering affordance and intention, named Lang-SHAPE, and proposing the first grasp point detection model with 3D part language ground- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Language Grounding in Robotic Manipulation The task of visual language grounding is to localize a region described by a given referring expression, the query [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For planar grasping, the localized region is usually bounding box [21]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' This work executes on the tabletop and is sensitive to occlusion because of coarse- grained bounding box instead of pixel-wise segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For spatial grasping, a closely related work by [25] studied to reason visual and non-visual language about 3D objects, which is mainly to observe the object instead of grasping it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Furthermore, from the language aspect, although there are several language grounded methods used in robotic grasping [3], [21]–[24], most of them consider direct com- mand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' abstract action) or scene understanding with spa- tial relationship in object wise and object with affordance in part wise has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Different from the above methods, we are the first to consider part-wise grounding language on point cloud and real spatial grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Affordance in Robotic Grasping Detection The possible action an agent could make to interact with the object in the environment and the functionality is a permanent property of an object independent of the charac- teristics of the user [26], which is the core idea of affordance theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To detect grasp point in pixel-wise, Vahrenkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [27] and Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [28] propose an affordance segmen- tation via synthetic images to realize planar grasping based on part affordance in the conventional mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Furthermore, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [14] introduce an affordance keypoint detection by providing structured predictions regarding the position, direction, and extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' However, the pixel-based affordance is barely used by 6-DoF grasping methods, which leads it only to deploy in scenarios such as tabletop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Recently, with the advent of the components like 3D AffordanceNet [29], point- wise spatial grasping detection with affordance based on point cloud is proposed [16], [30], which are to detect limited parts for affordance and lack in expansibility and general- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Compared to existing work considering affordance in an image or a closed set, we give another perspective to solve affordance-based spatial grasping detection using natural language in open world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Benefiting from various affordance knowledge in the form of text and large pre- trained language models, we design a language-vision task to establish the mapping between affordance and objects in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It characterizes effective and flexible deployment and strong generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To meet autonomous decisions for a robot, the natural language can also be abstracted as an interface of the external module (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='expert system [31] or cloud brain system [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' DATA GENERATION We collect our dataset Lang-SHAPE consisting of in- put point cloud observation C rendering, output grasping G labeling, and natural language Q generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To obtain the semantics of object part, we choose ShapeNet part dataset [18] as a source to build our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The dataset is widely used in 3D object part segmentation and provides object point cloud which contains 16,881 shapes from 16 categories, annotated with 50 parts in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To complete observation rendering and grasping labeling, we retrieve object meshes from ShapeNetCore [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For language data, we construct our corpus based on a large commonsense knowledge graph COMET-ATOMIC-2020 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Comparison with existing grasping dataset is shown in TABLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Formally, following the definition of GPD [19], let W ⊆ R3 denote the robot workspace and Craw ⊂ W the 3D point cloud perceived by the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We extend Craw to C ⊆ R4 with extra part semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Each point in the point cloud is paired with at least one viewpoint with camera pose V ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' An object observation with a point cloud can be defined as a tuple C = (C, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We denote a grasp configuration in 3D space g = (p, R) ∈ SE(3), which specifies the position and orientation of the grasping center point of the gripper local coordinate frame to the robot base frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A full example in our proposed dataset is organized as a 3-tuple (C, G, Q) and detailed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Object Semantic Observation In real-world setting, a robot can obtain point cloud to perceive object 3D information using RGB-D camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In our dataset, we establish a table scenario to randomly place one Dataset Plannar /3D Part aware Obser- vations Labels Grasps Objects (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=') Part (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=') Grasps per part Reference Reference per part Cornell [4] □ � Real � 8k 240 Jacquard [33] □ � Sim � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1M 11k VMRD [34] □ � Real � 100k 15k (31) Dex-Net [9] � � Sim f 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7M 1500 (50) GraspNet [11] � � S+R f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1B 88 Affordance language [35] � � Real � 216 655 ACRONYM [36] � � Sim � 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7M 8872 (262) Lang-SHAPE(ours) � � Sim �, f,� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='47M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6k (16) 42k (35) 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='85M 44 TABLE I: Comparison of publicly available grasp datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The label of grasping and text reference is generated either manually(�), by physical simulation(�), or analytical method(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Index Type-Index Template 1,2 1-1 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it < by > the < part > that/which can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3,4 1-2 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it < by > the < part > < purpose > < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 5,6 1-3 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it < by > the < part > so that you can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 7,8 1-4 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it that/which can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 9,10 1-5 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it < purpose > < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 11,12 1-6 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it so that you can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 13,14 1-7 I < verb > the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' (Please) < action > it < by > the < part >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 15,16 2-1 (Please) < action > (to) me the < object >/< sth > < by > the < part > < purpose > < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 17,18 2-2 (Please) < action > (to) me the < object >/< sth > < by > the < part > that/which can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 19,20 2-3 (Please) < action > (to) me the < object >/< sth > < by > the < part > so that you can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 21,22 2-4 (Please) < action > (to) me the < object >/< sth > < purpose > < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 23,24 2-5 (Please) < action > (to) me the < object >/< sth > that/which can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 25,26 2-6 (Please) < action > (to) me the < object >/< sth > so that you can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 27,28 2-7 (Please) < action > (to) me the < object >/< sth > < by > the < part >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 29,30 3-1 I < verb > you to < action > the < object >/< sth > < by > the < part > that/which can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 31,32 3-2 I < verb > you to < action > (to) me the < object >/< sth > < by > the < part > < purpose > < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 33,34 3-3 I < verb > you to < action > (to) me the < object >/< sth > < by > the < part > so that you can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 35,36 3-4 I < verb > you to < action > the < object >/< sth > that/which can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 37,38 3-5 I < verb > you to < action > (to) me the < object >/< sth > < purpose > < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 39,40 3-6 I < verb > you to < action > (to) me the < object >/< sth > so that you can < affordance >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 41,42 3-7 I < verb > you to < action > (to) me the < object >/< sth > < by > the < part >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 43,44 4 The < part > of the < object >/< sth >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' TABLE II: Four types of templates are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The first three are related to human-robot instructions, and the last one is an object or part description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' ⟨verb⟩ denotes intention verb, ⟨action⟩ denotes action verb, ⟨object⟩ denotes an object, ⟨sth⟩ denotes the referring word, ⟨by⟩ denotes preposition for the part, ⟨part⟩ denotes the part of referring object, ⟨affordance⟩ denotes the affordance phrase for the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For index, odd number sentence uses ⟨object⟩, while mean number sentence uses ⟨sth⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It is to differentiate whether the object is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' object O rendered by pyrender 1 and collect 547, 417 point cloud observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We repeat the random placement three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In total, each example of one object contains 13 point cloud including one full-view point cloud observation, and three random placement partial observations (each placement includes four view observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We adopt part labels in ShapeNet part dataset [18] and map them to sampled point cloud by ICP and KD-tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In a random object placement pi, four viewpoints vi point cloud {Cpj vi , i = 0, 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' j = 0, 1, 2} can be sampled with 4, 096 points C ∈ R4×4096 with part label in object coordinates, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A full-view point cloud Cfullv are sampled from object mesh directly with 10, 000 points C ∈ R4×10000 with part label in object coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A full point cloud observation sets are denoted as C = 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='com/mmatl/pyrender {Cfullv, Cpj vi , i = 0, 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' j = 0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Grasp Sampling and Labeling We follow the sampling policy of PointNetGPD [10] using antipodal sampling based on trimesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2 Different from existing work, which samples uniformly on the mesh of the object, we introduce part semantic to sampling process and only sample feasible grasps on the specific part surface with sanity checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Each grasp also contains a force- closure metric Qfc and Grasp Wrench Space (GWS) analysis metric Qgws consistent with PointNetGPD, which is used to evaluate the grasp quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Finally, for each example, we obtain a grasp set G = {(gi, Qi fc, Qi gws), pi ∈ Rpart} with size 60 elements, where gi is a grasp configuration with position pi, Qi fc and Qi gws are grasping evaluation metrics, 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='com/mikedh/trimesh Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2: The overall architecture of PIONEER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' black arrow trace refers to 3D part language grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The red arrow trace refers to part-aware grasp pose detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Multiple object observations with point cloud are collected, ICP and downsampled before fed into PIONEER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' and Rpart is the specific part surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' surface of Handle in a bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Language Description We propose 44 templates to generate our language de- scriptions Q ∈ Q about the part, object and grasp shown in TABLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We design the templates from reference, command, object description, intention, and part affordance to generate the language references, where four types of templates are interdicted inspired by [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The first three (1-*,2-*,3-* in Type-Index) contain 7 sentence templates respectively, and the object is replaced by a referring word to extend a referring version template (total 14 sentences), where templates 1−3 describe the affordance of a part, 4−6 remove the part information only describing part affordance, and 7 does not contain the affordance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The first type 1-* is established considering human intention to do something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The second type 2-* is about human instruction to give a action command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The third type 3-* is an integrated version of the above two kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Type 4 is an object or part description phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For ⟨purpose⟩ and ⟨affordance⟩, we seek a large of scene knowledge from a commonsense knowledge graph COMET-ATOMIC-2020 [38] with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='33M everyday inferen- tial knowledge tuples about entities and people&events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It represents a large-scale common sense repository of textual descriptions that encode both the social and the physical as- pects of common human everyday experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We construct our corpus based on the tuples and people&events to enhance the practicability of our language data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For ⟨part⟩, we do not adopt part label in ShapeNet part dataset, in which same semantic part in different objects are given different labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' handle in the bag and handle in the mug are given two different labels for classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Instead, we merge part-level labels containing the same semantic (affordance) information (from 50 to 35 categories), and augment semantic labels by considering synonyms and hy- ponyms in PartNet [39], WordNet [40], and Wikipedia [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' PROBLEM DEFINITION According to existing work [19] about grasp pose detec- tion using point cloud, given a point cloud and a description of the geometry of a robotic hand (hand configuration), grasp pose detection is to predict the grasp configurations based on the hand configuration, from which a grasp would be formed if the fingers are to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A typical solution is to sample enough grasp configuration candidates and select the one with the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We formulate it as a probability model: P(gi|R, C, Θ), (1) where gi is a sampled grasp configuration, C is a point cloud observation, R is an interesting region to sample grasp configuration candidates, and Θ is a hand configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' However, most existing work considers the region of interest (ROI) as prior input by object detection or localization in a scene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' cluttered scene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' These result in object wise coarse-grained grasping, where part semantic of an object is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In this paper, we consider more fine-grained grasping detection by constraining ROI using affordance and intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We consider a part-aware probability model of grasp pose detection using external knowledge from natural language and decompose it into two parts, given by: P(gi, R|C, Θ, Q) = P(gi|R, C, Θ) × P(R|C, Q), (2) Object art Affordance BERT model 768 I want the mug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Please grasp it 128 by the handgrip that can hold Duplicate the coffee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2048×3 MLP 0 2048×128 PointNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2 PointNet MLP D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6 Approachwhere Q is a natural language sentence for object description and instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' P(gi|R, C, Θ) is given by a grasp pose detection model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', GPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' P(R|C, Q) is given by a 3D part language grounding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Two assumptions are as follows: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Natural language sentences are beneficial to grasp pose detection during human-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' There is at least one positive grasping candidate that can be detected within the grounding part of the object under the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' METHOD We propose a novel human-in-the-loop framework to model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2, named PIONEER (grasP poInt detectiON with shapE languagE gRounding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The overall architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It consists of two modules, where the first is a part-wise 3D language grounding model, which is used for P(R|C, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The second is a part-aware grasp pose detection model for P(gi|R, C, Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Part-wise 3D Language Grounding Given a query sentence Q from human and robotic point cloud observation point cloud C, our 3D language grounding model is to detect the query-related region R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It can be formulated as a binary classifier function φ for each point in point cloud C: (Q, C) → R{0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To achieve this, our proposed model consists of four modules: language encoder, point cloud encoder, multimodal fusion module, and a binary classifier, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2 with black trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A query sentence Q from a human is fed to a pre-trained language model encoder (we use BERT [17]3) passing two fully connected layers to calculate a 128-dimension language feature Zq ∈ R1×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For point cloud, we choose more than one viewpoint cloud to merge a relatively complete point cloud by iterative closest point (ICP) to camera coordinates and downsample 2, 048 points4 C ∈ R2048×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The prepro- cessed C is input into PointNet [42] to calculate a feature map Zc ∈ R2048×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' After extract language Zq and point cloud features Zc, we repeat the Zq 2, 048 times to construct a feature map Z′ q ∈ R2048×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We concatenate Zc and Z′ q and pass the new feature map to an MLP to extract fusion feature Zfused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' At last, the fusion feature is input to a binary classifier to predict which points to be grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The whole pipeline can be formulated as: Zq = Elang (Q) , Zc = Epoint (C) , Zfused = MLP (repeat (Zq) ⊕ Zc) , R = Classifier (Zfused) , (3) where Elang and Epoint are language and point cloud encoders respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' ⊕ denotes the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3We use bert-base-uncased model in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4We simplify to ignore viewpoint V representation since we have trans- formed all points to the same coordinates by ICP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Part-aware Grasp Pose Detection To achieve part-aware grasp pose detection, we extend PointNetGPD [10] in candidate sampling policy and grasp selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Under Assumption 2, different from sampling uni- form randomly on the preprocessed point cloud of the whole object [10], we introduce high-level cognitive semantic R to constrain sampling region for candidate grasp set gi ∈ S, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2 with red trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' During our sampling process, we sample potential grasp points within R, while making collision detection and force closure detection to evaluate sampling quality still using the whole object point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Training and Inference We train the 3D language grounding model and part-aware grasp pose detection model separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To train 3D language grounding model, we use (C, Q) in our proposed dataset Lang-SHAPE in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The parameters of pre-trained BERT are frozen during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We use a binary-class cross-entropy loss to optimize the network with Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We train the network for 200 epoches with batchsize 32 and learning rate 1e−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To the train the part- aware grasp pose detection model, we use (C, G) in Lang- SHAPE dataset, in which the oracle point cloud semantic region is used to constrain the sampling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We also use a binary-class cross-entropy loss to optimize the network with Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The batchsize is 32, training epoch is 60, and learning rate is 5e−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To infer new input data, the grounding region of an object from 3D language grounding model output is used to inject into the part-aware grasp pose detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A series of grasp candidate scores are predicted finally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We select the optimized grasp to execution considering these scores and robotic reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' All models are trained and tested under PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' EXPERIMENTS We conduct both simulation and real-world robot exper- iments to investigate six research questions (RQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The first one is regarding the usefulness of the new dataset, the middle four are about the analyses of the proposed models, and the last is concerning the effectiveness of our method on the real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RQ1: For the effectiveness of proposed dataset, is our proposed new dataset useful for fine-grained 3D robotic grasping tasks, especially for affordance-aware task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RQ2: How does the pre-trained language model perform compared with existing baseline methods such as the ran- domly initialized model or similarity-based method in 3D language grounding with object parts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RQ3: How much does the pre-trained language model em- power the embodied inference ability given different-level prompt language?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RQ4: For compositional generalization, given the fact that an object can usually be decomposed into a certain number of parts, how much does our proposed model perform in part grounding between different objects with at least one but not all similar parts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Langauge Mode Definition full data all 44 sentences in Table II used in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' known all sentences containing object name, part name and affordance, with index [1,3,5,15,17,19,29,31,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' object unknown sentences not containing object name, with index [2,4,6,8,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=',40,42,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' part unknown sentences not containing part name, with index [7,8,9,10,11,12,21,22,23,24,25,26,35,36,37,38,39,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' part unknown part known sentences not containing object name, but containing part name, with index [7,9,11,21,23,25,35,37,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' part specific under human intervention to give an optimal grasp part for each object observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' TABLE III: Multi-level difficulty language configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RQ5: For human intervention, how does our proposed 3D part language grounding method with human-in-the-loop perform in fine-grained grasping detection success rate and effectiveness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' RQ6: For real-wolrd deployment, does our proposed method perform well on a real-world robot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Data Organization To train and test our proposed models, we split our Lang- SHAPE dataset object-wise and the part-wise, respectively, named Split Mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Object-wise, we split all examples in Lang-SHAPE by the object category (16 categories) with ratios (8 : 1 : 1) for (training/validation/test) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Similarly, part-wise, we split all examples in Lang-SHAPE by the part category (35 categories) with ratios (8 : 1 : 1) for (training/validation/test) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We further set up fine-grained language configurations, named Language Mode, defined in TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We provide two compositional generalization sets in TA- BLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Two extra split modes are introduced: related data has two attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' First, the object cat- egories of examples in the training set do not occur in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Second, at least one but not all parts of each example in the training set are similar to those in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Nevertheless, the parts contained in the training set cover the parts in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The details are shown in TABLE VI with Compositional Factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In the first setup, chair, laptop, and skateboard examples in Lang-SHAPE are collected as the training set, in which they have at least one part such as leg or board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Table examples are used as the test set, which consists of legs and boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In the second setup, guitar and pistol examples in Lang-SHAPE are used as the training set, in which they have at least one part such as body or handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mug examples are adopted as the test set, which consists of body and handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' non related data The data in the training set does not contain any objects or parts that occur in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Evaluation Metrics Grounding evaluation and 3D grasping detection evalua- tion are performed in this paper, following [18], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For 3D part language grounding, we use four metrics: Accuracy: Since we formulate 3D part language ground- ing as a binary classification problem, we calculate classifi- cation accuracy on points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Part avg IoU: We calculate the IoU of grounded points in each example [18] and average IoUs for each part category to calculate mIoUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Finally, we average each part’s mIoUs to calculate the Part avg IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Class avg IoU: We calculate the IoU of grounded points in each example and average IoUs for each object category to get mIoUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Finally, we average each object’s mIoUs to calculate the Class avg IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Instance avg IoU: We calculate the IoU of grounded points in each example and average all IoUs directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For 3D grasping detection, we define three metrics: Success Rate: The percentage of grasps where both grasp points grounding is correct and pre-grasp prediction is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Part-agnostic Success Rate: The percentage of grasps that pre-grasp prediction is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Trial Cost: To get a high-quality grasp candidate for the grasp score module, how many grasp sampling trials are needed to perform in a standard antipodal grasping sampler (GPG) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Models We design 3 baselines for comparison: Baseline 1: For 3D part language grounding, inspired by [25], we compare our method with a zero-shot classifier us- ing pre-trained models directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Instead of finetuning BERT, we use cosine distance between visual and language features to predict whether each point is grounded or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Visual encoder is from a pre-trained part segmentation model [42], while language encoder is BERT with frozen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Baseline 2: For 3D part language grounding, we replace BERT in our proposed method in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2 with a Transformer encoder 5, and train the whole model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' This is to verify whether the pre-trained model can provide useful prior knowledge for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Baseline 3: For 3D grasp pose detection, we use PointNetGPD [10] without human 3D language grounding intervention during the sampling process as baseline to verify the priority of using language human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We propose two models to solve 3D part language grounding and grasp pose detection problem: PIONEER is what we propose in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' PIONEER-T5: To empower the bidirectional ability of human-robot interaction, we introduce an extra gener- ative pre-trained language model (T5 [43]) to infer 3D part language grounding based on prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The instruction from human is first fed into T5 finetuned by 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='com/pytorch/examples/tree/master/ word_language_model (6 encoder layers implemented) Model Split Mode Accuracy Part avg IoU Class avg IoU Instance avg IoU Language Encoder Baseline 1 part-wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2461 BERT Baseline 2 part-wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6953 Transformer PIONEER (ours) part-wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7826 BERT Baseline 1 object-wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2153 BERT Baseline 2 object-wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6415 Transformer PIONEER (ours) object-wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7770 BERT TABLE IV: Overall results of 3D part language grounding in robustness and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' prompt learning to generate object-part description index 43 in TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Our prior experiments show that model using the naive object-part description can achieve very high performance, and thus we combine a T5 based on prompt learning and a PIONEER trained on index 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Simulation Experiments on Lang-SHAPE Based on our proposed Lang-SHAPE dataset, we give a se- ries of quantitative evaluations to answer research questions RQ1-RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The model selection is followed by the maximum Instance avg IoU in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1) 3D Part Language Grounding: To evaluate the overall performance of our proposed model, we compare our model with Baseline 1 and Baseline 2 in part wise and object- wise data split mode, shown in TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The language mode used in model training is full data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For RQ1, RQ2, our proposed model PIONEER outperforms in all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Compared with Baseline 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2355 in Part avg IoU), our model (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6696 in Part avg IoU) achieves more than double improvement relative to the zero-shot method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We attribute the poor performance by Baseline 1 to two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The first is no learning process to adjust parameters from the prior domain to our Lang-SHAPE domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The second is that the visual encoder and the language encoder are not trained jointly, which lacks shared feature space to fuse multimodal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' This also shows the usefulness of our proposed Lang-SHAPE dataset, which can be used for point cloud-language joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Compared with Baseline 2, our model achieves more than 8% improvement in Instance avg IoU with part wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The results indicate the advantage of pre-trained language model over to the randomly initialized model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Transformer) in robustness and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2) Affordance Inference: To evaluate the inference ability of the proposed model, we set up different corrupted lan- guage inputs to train models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Three models are used for comparisons: Baseline 2, PIONEER, and PIONEER-T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The first two are trained following language mode and split mode in TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In PIONEER-T5, we introduce a finetuned T5 [43] with prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Since T5 is an unimodal model and cannot perform effective inference when neither part nor object is unknown, we set up a more fine-grained language mode part unknown object known, which is T5 input concatenated with a prompt question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We design four prompts familiar with [3]: ‘what part should you grasp?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', ‘which part should you take’, ‘how can you grasp it for me?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', ‘how can you take it for me?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', one of them is randomly selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The object-part description index 43 in TABLE II is the groundtruth of T5 and input of PIONEER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' T5 and PIONEER are trained respectively in PIONEER-T5, and the test process uses them as a cascade model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For RQ2, RQ3, in PIONEER, as we can see known all, object unknown, and part unknown, with different object attributes being corrupted, the performance of models de- creases obviously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' From Accuracy and Instance avg IoU, we can find that when the object name is unknown, the model still performs better than the part name is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Comparing PIONEER with Baseline 2 in object unknown and part unknown, we can find that pre-trained language model can infer the grounded part via affordance information effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For example, compared with Baseline 2, Instance avg IoU is increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6253 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7583 in part unknown, part-wise in TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For RQ3, we provide an explicit prompt-based model PIONEER-T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' From part unknown object known in TA- BLE V, PIONEER-T5 achieves the best performance, which again shows the pre-trained language model with prompt learning can enhance the inference ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3) Compsitional Generalization: It is the ability to gen- eralize systematically to a new data distribution by combin- ing known components [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To measure the compositional generalization of our models, we propose two compositional generalization sets defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' VI-A, shown in TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For RQ4, in PIONEER, with the same test set, the model trained using our collected set (related data) achieves better performance in all metrics compared with the model trained using non related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' This indicates that our pro- posed model is effectively generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' In comparison with Baseline 1, results show that our proposed model performs better than the zero-shot method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4) Grasping Detection and Cost: To evaluate the effec- tiveness of our proposed method in fine-grained grasping detection, we test PIONEER on the whole Lang-SHAPE dataset (including 3D part language grounding and grasp data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We train 3D part language grounding and Point- NetGPD respectively following the part-wise and object- wise splits respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Language mode is full data in most evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The results are shown in TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The grasp sampling rule is that the sampler ends sampling at a maximum of 150 sample trials or gets 20 high-quality candidate grasps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' It is noted that in the simulation grasping experiment, the sampling process is on point cloud data Split Mode Model Language Mode Accuracy Part avg IoU Class avg IoU Instance avg IoU Language Encoder Part-wise Baseline 2 object unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6405 Transformer part unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6253 part unknown object known 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6178 PIONEER (ours) known all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8089 BERT object unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7761 part unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7583 part unknown object known 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7704 PIONEER-T5 (ours) part unknown object known 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7818 Object-wise Baseline 2 object unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7074 Transformer part unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6019 part unknown object known 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6060 PIONEER (ours) known all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='8116 BERT object unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7776 part unknown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7641 part unknown object known 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7709 PIONEER-T5 (ours) part unknown object known 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='9203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7853 TABLE V: Comparisons of inference performance with different corrupted languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For Split Mode, the top half is in part wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The bottom half is in object wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For Language Encoder, Baseline 2 uses Transformer while our PIONEER uses BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Model Split Mode Accuracy Part avg IoU Part 1 IoU Part 2 IoU Instance avg IoU Compositioal Factors Part 1 + Part 2 Baseline 1 related data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='3475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2629 chair (leg) + laptop,skateboard (board) = table(leg,board) PIONEER (ours) related data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5487 non related data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1614 objects not composed of leg or board Baseline 1 related data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='0622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2365 guitar (body) + pistol (handle) = mug(body,handle) PIONEER (ours) related data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='7177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='3001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4913 non related data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2999 objects not composed of body or handle TABLE VI: Results of compositional generalization with two subsets in SHAPE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' instead of object meshes which are used in the dataset collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The sampling process prefers to real- world setting although we perform experiments in simulation using dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For RQ1, RQ5, by comparing PIONEER with Baseline 3 in TABLE VII, we can see that our proposed model can realize part-grounded grasping with more than 40% success rate, while Baseline 3 can only get 25% approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' This indicates the effectiveness of our proposed method in fine- grained grasping detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We find that our method performs relatively weak in part- agnostic success rate and trial cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We attribute the reason to the fact that our method constrains the sampling region, and some part region is difficult for grasping, which reduces the whole part-agnostic success rate and costs more time to sample until the terminal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' To verify our suppose, we propose a new language mode part specific, in which human specifies one grasping part for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' From TABLE VII, we can see that our PIONEER improves broadly in all metrics with knowledge from human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Physical Robot Experiments To evaluate the effectiveness of our proposed method in the real world, we deployed our models on a real robot system to realize part-aware grasping following human instruction, which includes affordance and intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We choose a single-arm robot Kinova Jaco 7DOF with three fingers to perform manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' An eye-in-hand camera Model Split Mode Language Mode Success Rate Part-agnostic Success Rate Trial Cost Baseline 3 part-wise full data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6333 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1644 PIONEER (ours) part-wise full data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4876 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5332 Baseline 3 object-wise full data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='2546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='6327 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4371 PIONEER (ours) object-wise full data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='4992 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='1895 PIONEER (ours) object-wise part specific 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='5843 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='0396 TABLE VII: Comparisons of fine-grained grasping detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Intel RealSense SR300 is fixed on the wrist of end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The system is deployed on a PC running Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='04 and ROS Melodic with one Intel Core i7-8700K and one NVIDIA Geforce GTX 1080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The intrinsic and extrinsic parameters of the camera are calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We select our PIONEER model under the training configuration of part- wise (Split Mode) and full data (Language Mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Since our proposed method is to operate in part wise, it requires a more fine-grained perception of the target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For point cloud collection, we design a multi-view (four) policy to collect each view point cloud and transform into robot base frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' All viewpoint point cloud are merged by ICP to get a relatively complete representation of the target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' We select three categories of household objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Two (mug and table) are seen in our Lang-SHAPE, and another one (hammer) is unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The object is randomly placed on the table, multi-view point cloud collection is to obtain the outline of the object, and then the merged point cloud is fed into our PIONEER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The output of PIONEER is a grasp pose Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3: Test objects and demonstration of real robot grasping pipeline with language query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' on robot base frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' For Q6, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3, our real robot experiments indicate the effectiveness of our proposed method in fine-grained grasping with human instruction including object, part, and affordance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' More experiments are available in the attached video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' CONCLUSION We investigated part-level affordance on fine-grained robotic grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' The Lang-SHAPE dataset is constructed to facilitate the investigation, and a 3D part language grounding and a part-aware grasp pose detection model are proposed to allow fine-grained robotic grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Experiments show that our proposed method outperforms 3D part grasp grounding in inference and generalizability, and physical robot experi- ments show its effectiveness in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' These results show the promise of using the pre-trained language model in affordance grounding and fine-grained grasping for a robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Edmonds, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Wu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', “Dark, beyond deep: A paradigm shift to cognitive ai with humanlike common sense,” Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 310–345, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gibson, “The theory of affordances,” Hilldale, USA, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 67–82, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ahn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Brohan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Brown, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chebotar, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Cortes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' David, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Finn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gopalakrishnan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hausman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Herzog, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', “Do as i can, not as i say: Grounding language in robotic affordances,” arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='01691, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Moseson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Saxena, “Efficient grasping from rgbd images: Learning using a new rectangle representation,” in 2011 IEEE International conference on robotics and automation, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3304–3311, IEEE, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lenz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lee, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Saxena, “Deep learning for detecting robotic grasps,” The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4- 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 705–724, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Morrison, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Corke, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Leitner, “Closing the loop for robotic grasping: A real-time, generative grasp synthesis approach,” Robotics: Science and Systems XIV, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1–10, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gomez, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Polosukhin, “Attention is all you need,” in Advances in neural information processing systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 5998–6008, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhou, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Kan, “When transformer meets robotic grasping: Exploits context for efficient grasp detection,” arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='11911, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mahler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Niyaz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Laskey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Doan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ojea, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Goldberg, “Dex-net 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='0: Deep learning to plan robust grasps with synthetic point clouds and analytic grasp metrics,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' G¨orner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Tang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Sun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhang, “Pointnetgpd: Detecting grasp configurations from point sets,” in 2019 International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3629–3635, IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gou, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lu, “Graspnet-1billion: A large- scale benchmark for general object grasping,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 11444–11453, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Sundermeyer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mousavian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Triebel, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fox, “Contact- graspnet: Efficient 6-dof grasp generation in cluttered scenes,” in 2021 IEEE International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 13438–13444, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Seguin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Vela, “Toward affordance detec- tion and ranking on novel objects for real-world robotic manipulation,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4070–4077, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Xu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Tang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Vela, “An affordance keypoint detection network for robot manipulation,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2870–2877, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Tekden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Deisenroth, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Bekiroglu, “Affordance transfer based on self-aligning implicit representations of local surfaces,” [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Sun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhang, “Learning 6- dof task-oriented grasp detection via implicit estimation and visual affordance,” arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='08537, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Kenton and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” in Proceedings of NAACL-HLT, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4171–4186, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Qi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Su, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Guibas, “Pointnet: Deep learning on point sets for 3d classification and segmentation,” in Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 652–660, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' ten Pas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gualtieri, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Saenko, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Platt, “Grasp pose detection in point clouds,” The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 13-14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1455–1473, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Nießner, “Scanrefer: 3d object localization in rgb-d scans using natural language,” in European Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 202–221, Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [21] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mees and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Burgard, “Composing pick-and-place tasks by ground- ing language,” in International Symposium on Experimental Robotics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 491–501, Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Vela, “A joint network for grasp detection conditioned on natural language commands,” in 2021 IEEE International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4576–4582, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hsu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' La, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zheng, “Invigorate: Interactive visual grounding and grasping in clutter,” in Robotics: Science and Systems (RSS), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Song, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Sun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Xiao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhang, “Human- in-the-loop robotic grasping using bert scene representation,” in Pro- ceedings of the 29th International Conference on Computational Linguistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2992–3006, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Thomason, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Shridhar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Bisk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Paxton, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zettlemoyer, “Language grounding with 3d objects,” in Conference on Robot Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1691–1701, PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gibson, “The theory of affordances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' the ecological approach to mug keep from touching hot areavisual perception,” The People, Place and, Space Reader;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Routledge: New York, NY, USA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 56–60, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [27] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Vahrenkamp, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Westkamp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Yamanobe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Aksoy, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Asfour, “Part-based grasp planning for familiar objects,” in 2016 IEEE-RAS 16th International Conference on Humanoid Robots (Hu- manoids), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 919–925, IEEE, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Xu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Vela, “Learning affordance segmenta- tion for real-world robotic manipulation via synthetic images,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1140–1147, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Wu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Jia, “3d affordancenet: A benchmark for visual object affordance understanding,” in Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1778–1787, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ard´on, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Pairet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Petrick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ramamoorthy, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lohan, “Learning grasp affordance reasoning through semantic relations,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4571–4578, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Asemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Nowkarizi, “Intelligent libraries: a review on expert systems, artificial intelligence, and robot,” Library Hi Tech, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [32] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Jiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hwang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hossain, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Muham- mad, “Follow me robot-mind: Cloud brain based personalized robot service with migration,” Future generation computer systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 107, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 324–332, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Depierre, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Dellandr´ea, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chen, “Jacquard: A large scale dataset for robotic grasp detection,” in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 3511–3516, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Tian, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zheng, “Roi- based robotic grasp detection for object overlapping scenes,” in 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 4768–4775, IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Nguyen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Gopalan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Patel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Corsaro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Pavlick, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Tellex, “Robot Object Retrieval with Contextual Natural Language Queries,” in Proceedings of Robotics: Science and Systems, (Corvalis, Oregon, USA), July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Eppner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mousavian, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fox, “Acronym: A large-scale grasp dataset based on simulation,” in 2021 IEEE International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 6222–6227, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Funkhouser, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Guibas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hanrahan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Savarese, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Savva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Su, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=', “Shapenet: An information-rich 3d model repository,” arXiv preprint arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='03012, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Hwang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Bhagavatula, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Le Bras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Da, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Sakaguchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Bosselut, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Choi, “(comet-) atomic 2020: on symbolic and neural commonsense knowledge graphs,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 6384–6392, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Mo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Yi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Tripathi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Guibas, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Su, “Partnet: A large-scale benchmark for fine-grained and hierarchical part-level 3d object understanding,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 909–918, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Fellbaum, “Wordnet,” in Theory and applications of ontology: computer applications, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 231–243, Springer, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [41] Wikipedia, Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' PediaPress, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [42] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Qi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Yi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Su, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Guibas, “Pointnet++: Deep hierarchical feature learning on point sets in a metric space,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [43] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Raffel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Shazeer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Roberts, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Narang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Matena, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Li, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Liu, “Exploring the limits of transfer learning with a unified text-to-text transformer,” Journal of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 21, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' 1–67, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ravikumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Ainslie, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content=' Onta˜n´on, “Improving compo- sitional generalization in classification tasks via structure annotations,” arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} +page_content='10434, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf'} diff --git a/zdFST4oBgHgl3EQfUjgb/content/tmp_files/2301.13773v1.pdf.txt b/zdFST4oBgHgl3EQfUjgb/content/tmp_files/2301.13773v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..28a6da8159d72c5c7d04fa6cba145883c92ec08c --- /dev/null +++ b/zdFST4oBgHgl3EQfUjgb/content/tmp_files/2301.13773v1.pdf.txt @@ -0,0 +1,959 @@ +Revisiting one-loop corrections to the trilinear Higgs +boson self-coupling in the Inert Doublet Model +Jaouad El Falaki1 ∗ +1 LPTHE, Physics Department, Faculty of Sciences, Ibn Zohr University, P.O.B. 8106 Agadir, Morocco. +Abstract +We investigate predictions of the trilinear Higgs self-coupling with radiative corrections +in the context of the Inert Doublet Model. The triple Higgs vertex is computed at the one- +loop level based on the on-shell renormalization scheme. We calculate its possible deviation +from the predictions within the standard model, taking into account all relevant theoretical +and experimental constraints, including dark matter searches and the latest bounds on the +branching fraction of the Higgs boson decaying to invisible particles. By scanning the model’s +parameter space, we find that the deviation in the triple Higgs boson self-coupling from +standard model expectations can be substantial, exceeding 100% in certain regions of the +parameter space. +∗E-mail: jaouad.elfalaki@gmail.com,j.elfalaki@uiz.ac.ma +arXiv:2301.13773v1 [hep-ph] 31 Jan 2023 + +1 +Introduction +A great achievement in the history of high energy physics was made on July 4, 2012, with +the discovery of the Higgs boson by ATLAS and CMS, at the CERN Large Hadron Collider +(LHC) [1,2]. Since then, the spectrum of the Standard Model (SM) of particle physics has been +completed. Following this discovery, more accumulated data has been analyzed during LHC +Run I-II and it has been found that the properties of the Higgs boson are in perfect agreement +with the predictions from the SM, with a level of accuracy of 10 − 20% [3,4]. +To establish experimentally the Brout-Englert-Higgs mechanism of electroweak symmetry break- +ing (EWSB), it is necessary to measure not only the couplings of the Higgs boson with the +fermions and the gauge bosons, but also the self-coupling of the Higgs boson, i.e. the triple +λhhh and the quartic λhhhh Higgs boson couplings, in order to be able to reconstruct the shape +of the Higgs scalar potential. The measurement of the Higgs boson quartic coupling is more +challenging because the cross section for triple Higgs production at the LHC is much smaller +(the cross section for pp → 3h production is about 0.1 fb at √s = 14 TeV ), and it is out of reach +even for the high-luminosity LHC (HL-LHC) [5]. On the other hand, the trilinear self-coupling +of the Higgs boson λhhh can be measured from the production of a pair of Higgs bosons at the +LHC [6]. Recently, a statistical combination by ATLAS has been presented [7], in which the +bound on λhhh has been significantly reduced to −0.4 < κλ < 6.3 at 95% confidence level, where +κλ ≡ λhhh +λSM +hhh is the self-coupling modifier parameter. CMS also has derived a limit on κλ which +is constrained to be within −1.24 < κλ < 6.49 at 95% CL [8]. The measurement accuracy of +λhhh will be improved at future experiments such as the HL-LHC where κλ is constrained to be +between 0.1 and 2.3 at 95% CL with 3 ab−1 data [5]. At future linear collider such as the ILC, +the triple Higgs boson coupling is expected to be measured at the precision level of 27% in the +double Higgs-strahlung process e+e− → Zhh at √s = 500 GeV with an integrated luminosity +of 4 ab−1 [9,10]. A relative precision of 10% on λhhh is also possible at 1 TeV from the di-Higgs +production in WW fusion process e+e− → ν¯νhh with an accumulated 8 ab−1 of integrated +luminosity [9, 10]. The expectation for these precise measurements motivates the study of the +radiative corrections to λhhh. +The SM is unable to explain certain phenomena such as dark matter, the hierarchy problem +and tiny neutrino masses. As a result, new physics beyond the SM (BSM) is needed to provide +answers to these unsolved problems. The Higgs sector of the SM only includes one Higgs dou- +blet, but there is no fundamental reason to believe it must be minimal. Among popular BSM +candidates are models with extended Higgs sectors, such as the Inert Doublet Model (IDM). +Originally proposed by Deshpande and Ma [11] and initially suggested for EWSB studies, this +model is highly intriguing due to its potential to generate tiny neutrino masses [12], provide a +dark matter candidate [13–19], and address the naturalness problem [20]. The IDM consists of +adding a second Higgs doublet that does not acquire a vacuum expectation value (VEV) and has +no coupling with SM fermions. An exact Z2 symmetry is imposed, with the SM Higgs doublet +being even and the additional scalar doublet being odd. The preserved Z2 symmetry ensures +that the extra doublet does not interact with matter and its lightest stable neutral component +can act as a dark matter particle. After EWSB, the IDM has a spectrum of five physical scalars: +a CP-even Higgs boson h (identified with the discovered SM Higgs), and four inert scalars (H, +A, and H±). The rich phenomenology of the IDM has been thoroughly studied in the literature, +both in the context of future Higgs factories like the ILC and CLIC and at the LHC [21–57]. +Additionally, there have been many studies of the model that go beyond the lowest order of the +perturbation, including those at one-loop level [58–73] and two-loop order [74–76]. The one-loop +1 + +contributions to the triple Higgs boson coupling from Standard Model particles have been inves- +tigated in [58,77–80], where it was found that these corrections are dominated by loops involving +the top quark. The radiative corrections to λhhh in some non-supersymmetric Higgs models can +be found in references [77–79,81–91], and for corrections in certain supersymmetric models, see +for example Refs [80, 92–97]. These new physics effects in models with extended Higgs sectors +have been shown to be large and can significantly enhance the λhhh coupling in a wide range +of parameter space. Calculations of Higgs boson couplings that include higher-order corrections +are mandatory to compare theory predictions with future precision data from hadron and lepton +colliders. In Ref [58], the one-loop contributions of the inert scalars to λhhh are discussed only +in the degenerate spectra, i.e., mH = mA = mH±. In addition, one loop corrections to λhhh +within the IDM have also been discussed in some scenarios [60], but under the assumption that +mH = mA. In the present letter, we will compute the radiative corrections to the triple Higgs +self-coupling considering non-degenerate masses for the inert scalars, while taking into account +all current theoretical and experimental constraints on the IDM. +The layout of the letter is as follows: In Sec. 2, we briefly introduce the IDM and outline its +theoretical and experimental constraints. +In Sec. 3, we present the on-shell renormalization +scheme and provide a comprehensive explanation of the triple Higgs coupling at the one-loop +level. In Sec. 4, we present our numerical results for the SM and IDM. Conclusions are given in +the last section. +2 +The Inert Doublet Model +2.1 +The Model +The IDM is a simple extension of the Standard Model of particle physics which consists +of the SM, including its Higgs doublet Φ1, and an additional SU(2) doublet Φ2 that brings +in four new scalar particles. An exact Z2 symmetry is introduced such that the SM doublet +is even Φ1 −→ Φ1 while the added extra doublet (inert doublet) is odd Φ2 −→ −Φ2. This +unbroken Z2 parity guarantees the absence of coupling between fermions and the inert doublet +Φ2, therefore there is no flavor-changing neutral currents. Moreover, it ensures that the lightest +neutral component of Φ2 is a natural dark matter candidate. The decomposition for the two +doublets around the vacuum state is given by: +Φ1 = +� +G± +1 +√ +2(v + h + iG0) +� +, +Φ2 = +� +H± +1 +√ +2(H + iA) +� +(1) +Where only the SM doublet Φ1 is involved in EWSB, G0 and G± correspond to the three Nambu- +Goldstone bosons gauged away by the longitudinal component of Z and W ± respectively, h is +the SM Higgs boson and v is the VEV of the SM Higgs doublet. The second doublet Φ2 does +not participate in EWSB and it contains four new inert scalars H, A and H± where either A or +H may act as potential dark matter candidate, depending on the mass hierarchy of these two +inert scalars. +The most general renormalizable tree-level scalar potential in this model can be written as: +V = µ2 +1|Φ1|2 + µ2 +2|Φ2|2 + λ1|Φ1|4 + λ2|Φ2|4 + λ3|Φ1|2|Φ2|2 + λ4|Φ† +1Φ2|2 ++ λ5 +2 +� +(Φ† +1Φ2)2 + h.c +� +, +(2) +2 + +where µ1 and µ2 are the mass of the Φ1 and Φ2 fields, and all λ1,2,3,4 parameters are real since +the above potential must be hermitian while the phase of λ5 can be absorbed into an appropriate +redefinition of Φ1 and Φ2 fields. +After EWSB the five scalar particles of the model acquire their masses which are given by: +m2 +h = −2µ2 +1 = 2λ1v2 +m2 +H = µ2 +2 + λLv2 +m2 +A = µ2 +2 + λSv2 +m2 +H± = µ2 +2 + 1 +2λ3v2 +(3) +where λL,S are defined as: +λL,S = 1 +2(λ3 + λ4 ± λ5) +(4) +The trilinear self-coupling of the Higgs boson at tree level in the IDM involves only SM param- +eters and is given by: +Γtree +hhh = −3m2 +h +v +(5) +In the IDM, there are eight independent parameters: 5λi, 2µi, and v. After eliminating one +parameter through the minimization condition and determining the VEV using the W boson +mass, we are left with six remaining independent parameters which will be selected as follows: +{µ2 +2, λ2, mh, mH±, mH, mA} +(6) +2.2 +Theoretical and Experimental Constraints +In this study, we explore the same parameter space as in our previous published paper [61]. +The IDM parameter space is obtained by performing an extensive parameter scan in the whole +space with all experimental and theoretical constraints applied. The constraints used are sum- +marized below: +• The theoretical constraints: +– The perturbative unitarity [98,99] +– The vacuum stability [11,100] +– The inert vacuum and charge-breaking minima [101] +• The experimental constraints: +– The Higgs data from the LHC [102,103]. +– The invisible Higgs decay [104]. +– The direct collider searches at LEP [23,105–107] +– The electroweak precision tests [20,108,109]. +– The dark matter searches [40,42,110–120]. +3 + +3 +Calculation of one-loop corrections to the triple Higgs cou- +pling +In this section, we briefly discuss the renormalization of the trilinear self-coupling of the +Higgs boson. Using the ’t Hooft-Feynman gauge, we calculate the radiative corrections to the +tree-level formula in Eq. 5 in both the SM and IDM including contributions from all particles in +the loop. Figure 3.1 illustrates some of the Feynman diagrams contributing to the triple Higgs +boson coupling. Note that the dimensional regularization has been used to evaluate the one-loop +Feynman amplitudes. The calculations are carried out using Mathematica packages FeynArts +and FormCalc [121–123]. The numerical evaluation of the scalar one-loop integrals has been +performed with the LoopTools package [124, 125]. It should be noted that the UV-finiteness +of the virtual contributions has been cross-checked numerically and analytically. Compared to +the general two Higgs doublet model (2HDM) [78, 79, 83], the structure of the counter-terms +for the triple Higgs boson coupling in the IDM is simpler and identical to those in the SM +due to the absence of mixing between the SM Higgs boson and the inert scalars. We study an +off-shell Higgs boson that decays into two real Higgs bosons h∗(q) → h(q1)h(q2) at the one-loop +level, where q, q1 and q2 are the four-momenta of the three Higgs bosons satisfying an off-shell +condition q2 ̸= m2 +h for the decaying Higgs boson and on-shell conditions q2 +1 = q2 +2 = m2 +h for the +two real Higgs bosons. The UV divergences that emerge during the calculation’s intermediate +stages should eventually cancel out in the end. +In order to do that, we adopt the on-shell +renormalization scheme which is widely used in quantum field theory because it is simple to +implement and it allows for a clear physical interpretation of the parameters of the theory. As +in the SM, the tree-level trilinear Higgs self-coupling in Eq. 5 depends only on the VEV and +Higgs boson mass. Hence, the renormalization procedure will be the same as the one adopted +in the SM [126–129]. The SM fields and parameters are redefined as follows: +Figure 3.1: Some one-loop Feynman diagrams that do not exist in the SM contribute to the triple Higgs self- +coupling within the IDM. +4 + +h +h +h +h +H +H± +FH +A +H +H± +h +h +h +h +H +±H +HF +A +h +h +h +h +1 +2 +3 +4 +h +H +A +H +h +h +h +2 +h +h +h +H +Ht +A +h +5 +6 +7 +8 +h +h +h +h +h +HF +h +h +HF +h +h +h +h +9 +10 +11 +12 +h +h +3 +h +13m2 +h → m2 +h + δm2 +h +m2 +V → m2 +V + δm2 +V , +V = Z, W ± +sW → sW + δsW +t → t + δt +e → (1 + δZe)e +Zµ → +� +1 + 1 +2δZZZ +� +Zµ + 1 +2δZZAAµ +Aµ → +� +1 + 1 +2δZAA +� +Aµ + 1 +2δZAZZµ +h → Z1/2 +h +h = +� +1 + 1 +2δZh +� +h +(7) +where sW = sin θW is the Weinberg angle and t = v(µ2 +1 − λ1v2) is the tadpole. At tree level, +the tadpole is zero if the minimization condition is satisfied, but it can receive finite corrections +at the one-loop level. To ensure that the VEV of the Higgs field is consistent across all orders +of perturbation theory, it is necessary to renormalize the Higgs tadpole. This can be done by +adding a counter-term to the tadpole, which cancels out any divergences that appear at higher +orders of perturbation theory. Consequently, we set the following condition: +ˆT = δt + T = 0 =⇒ δt = −T +(8) +where T is the one-loop contribution of 1PI diagrams. +The counter-terms of the masses are fixed by the following on-shell condition: +ReˆΣV V +T +(m2 +V ) = 0 =⇒ δm2 +V = ReΣV V +T +(m2 +V ) +V = W, Z +ReˆΣhh(m2 +h) = 0 =⇒ δm2 +h = ReΣhh(m2 +h) +(9) +where Σhh and ΣV V +T +are the one-loop non-renormalized self energies for the Higgs boson and +gauge bosons respectively. +By fixing the residue of the two-point Green functions to be equal to unity, the wave-function +renormalization constant is determined from the following condition. +Re∂ ˆΣhh(k2) +∂k2 +����� +k2=m2 +h += 0 =⇒ δZh = −Re∂Σhh +∂k2 +����� +k2=m2 +h +. +(10) +In the on-shell renormalization scheme, the electric charge is set by ensuring that there are +no higher-order corrections to the e+e−γ vertex in the Thomson limit. +The electric charge +renormalization constant δZe can be expressed as: +δZe = −1 +2δZAA − sW +cW +1 +2δZZA +(11) +where +δZAA = −∂ �AA +T (k2) +∂k2 +����� +k2=0 +and +δZZA = 2 +�AZ +T (0) +m2 +Z +(12) +5 + +To obtain the counter-term δsW one can use the on-shell definition of the weak mixing angle, +which is defined as the ratio of the weak neutral current and the weak charged current. Thus, +δsW is given by: +δsW = c2 +W +2sW +�δm2 +Z +m2 +Z +− δm2 +W +m2 +W +� +(13) +By inserting the redefinition of the parameters into the Lagrangian, we obtain the following +counter-term for the trilinear Higgs self-coupling. +δΓhhh = −3e2 +2sW +m2 +h +mW +� +δZe − δsW +sW ++ δm2 +h +m2 +h ++ +e +2sW +δt +mW m2 +h +− δm2 +W +2m2 +W ++ 3 +2δZh +� +(14) +To obtain the renormalized amplitudes for the triple Higgs coupling, the full one-loop one par- +ticle irreducible vertex Γ1PI +hhh(q2, q2 +1, q2 +2) is added to the corresponding counter-terms δΓhhh as +follows: +ˆΓhhh(q2, q2 +1, q2 +2) = Γtree +hhh + Γ1PI +hhh(q2, q2 +1, q2 +2) + δΓhhh +(15) +4 +Numerical Results +In this section, we present our numerical analysis for the triple Higgs coupling at the one- +loop level in the SM and IDM. In order to parametrize the size of the radiative corrections and +compare it to the Standard Model’s predictions, we define the following ratio in the IDM: +∆Γhhh = +ˆΓhhh(q2, m2 +h, m2 +h)IDM − ˆΓhhh(q2, m2 +h, m2 +h)SM +ˆΓhhh(q2, m2 +h, m2 +h)SM +(16) +While within the SM, we show our numerical results using the following relative ratio: +∆ΓSM +hhh = +ˆΓhhh(q2, m2 +h, m2 +h)SM − Γtree +hhh +Γtree +hhh +(17) +The following numerical values of the input parameters are adopted [109]: +mh = 125.18 GeV +mW = 80.379 GeV +mt = 173.2 GeV +mµ = 0.106 GeV +mZ = 91.198 GeV +mb = 4.660 GeV +mτ = 1.777 GeV +α = 1/137.036 +mc = 1.275 GeV +We scan the entire parameter space for the other IDM parameters, including physical masses mA, +mH and mH± as well as the µ2 +2 parameter. We consider all relevant theoretical and experimental +constraints and perform a random scan over the IDM parameter space in the following ranges: +100 GeV ≤ mH ≤ 700 GeV +20 GeV ≤ mA ≤ 62.5 GeV +80 GeV ≤ mH± ≤ 700 GeV +0 GeV2 ≤ µ2 +2 ≤ 106 GeV2 +(18) +6 + +It’s worth mentioning that our numerical results are independent of λ2 parameter. We will set +λ2 to a fixed value of 2. +It is noteworthy that in this letter, the inert scalar A is selected as the dark matter candidate. +Furthermore, all points have been passed the upper bound from the invisible Higgs boson decay +Br(h → AA) ≤ 11% [104].1 +We visualize in Figure 4.1 the size of the radiative corrections in the SM as a function of the +four-momentum of the off-shell Higgs boson. The measurement of the triple Higgs coupling in +future experiments will be done through the double Higgs production process. In this case, +one of the Higgs bosons will be off-shell, which implies that the dependence on momentum q is +important. One can see that the total corrections to the trilinear self-coupling of the Higgs boson +start from −1% around q = 250 GeV and can reach a maximum value of 8.23% for q = 470 +GeV. It should be emphasized that the top-quark contribution is the dominant correction for +large values of the momentum q. +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +q[TeV] +−20 +−15 +−10 +−5 +0 +5 +∆ΓSM +hhh[%] +Figure 4.1: The relative correction ∆ΓSM +hhh[%] as a function of q, where qµ is the four-momentum of the off-shell +Higgs boson in h∗ → hh. +In the left panel of Figure 4.2, we depict the corrections to the triple Higgs coupling as a +function of the four-momentum q. The color code indicates the allowed charged scalar mass. As +a reference point, we display by a solid-red line the relative corrections ∆ΓSM +hhh to the trilinear +1It bears mentioning that the mass range between 20 GeV and 55 GeV of the dark matter candidate is ruled +out from relic density constraints. It is also worth pointing out that the allowed values of µ2 +2 range between 3000 +GeV 2 and 4400 GeV 2 after passing all constraints. +7 + +Higgs boson self-coupling within the SM. From this plot, it can be seen that the corrections +are small and consistent with the SM predictions for light charged and neutral inert particles, +whose masses are in the range 80 GeV ≤ mH± ≤ 200 GeV and 100 GeV ≤ mH ≤ 200 GeV. +It can also be observed that for inert scalars masses, 300 GeV ≤ mH±, mH ≤ 440 GeV, the +deviation of the triple Higgs coupling from the SM’s predictions is significant and larger than +10% with an enhancement up to 120% for q = 880 GeV. Furthermore, One can infer that for +heavy inert scalars, the corrections are substantial in a large part of the parameter space with +an enhancement of 472% for q = 1216 GeV and mH = 608 GeV. It is worth mentioning that +in the left panel there are two different threshold peaks which are attributed to the opening of +h∗ → H±H∓ for q = 1204 GeV with mH± = 602 GeV, where the corrections can go up to 470%. +The second spike at q = 1216 GeV which amplifies the radiative corrections corresponds to the +threshold effect in h∗ → HH with mH = 608 GeV. The non-decoupling effect in the radiative +correction to the trilinear self-coupling of the Higgs boson is significant when large masses of +the inert scalars are involved, this behavior can be seen on the right panel of Figure 4.2. It is +noteworthy to highlight that this behavior is also observed in the 2HDM, where large corrections +to the triple Higgs coupling at the one-loop level are found to grow as the quartic power of the +extra heavier Higgs bosons [77,78,83]. +Figure 4.2: Left: ∆Γhhh as a function of the momentum q where the the charged scalar mass mH± is shown in +the right column. Right: ∆Γhhh as a function of mH± and the color code indicates the mass mH of the neutral +scalar H. The red line in the left panel represents the relative ratio ∆ΓSM +hhh in the SM. +5 +Conclusion +We computed the trilinear Higgs boson coupling in the IDM at one-loop level with non- +degenerate inert scalar masses, taking into account all relevant theoretical and experimental +constraints, including dark matter searches and Higgs boson invisible decay. We evaluated the +one-loop Feynman amplitudes using dimensional regularization in the ’t Hooft-Feynman gauge +and employed an on-shell scheme renormalization. Our results showed substantial deviations +from the SM predictions for the triple Higgs coupling, with values exceeding 100% in some regions +8 + +009 +400 +500 +300 +OOT +mH+[GeV] +200 +100 +200 +0 +100 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +qTeV]009 +400 +500 +300 +400 +mH[GeV] +200 +008 +100 +200 +0 +100 +200 +300 +400 +500 +600 +mH± +[GeVof the parameter space and reaching up to 472% enhancement due to non-decoupling effects +of the inert scalars. This substantial non-decoupling correction to the triple Higgs boson self- +coupling is known to be associated with a strongly first order electroweak phase transition, which +is necessary for successful electroweak baryogenesis. Detecting significant deviation from the +expected value in the triple Higgs coupling in future colliders can indirectly provide information +on the mass of inert scalar bosons. +References +[1] ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for the +Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B716 +(2012) 1–29, [arXiv:1207.7214]. +[2] CMS Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125 +GeV with the CMS experiment at the LHC, Phys. Lett. B716 (2012) 30–61, +[arXiv:1207.7235]. +[3] ATLAS Collaboration, G. Aad et al., Combined measurements of Higgs boson +production and decay using up to 80 fb−1 of proton-proton collision data at √s = 13 TeV +collected with the ATLAS experiment, Phys. Rev. D 101 (2020), no. 1 012002, +[arXiv:1909.02845]. +[4] CMS Collaboration, A. M. Sirunyan et al., Combined measurements of Higgs boson +couplings in proton–proton collisions at √s = 13 TeV, Eur. Phys. J. C 79 (2019), no. 5 +421, [arXiv:1809.10733]. +[5] M. Cepeda et al., Report from Working Group 2: Higgs Physics at the HL-LHC and +HE-LHC, CERN Yellow Rep. Monogr. 7 (2019) 221–584, [arXiv:1902.00134]. +[6] H. Abouabid, A. Arhrib, D. Azevedo, J. E. Falaki, P. M. Ferreira, M. M¨uhlleitner, and +R. Santos, Benchmarking di-Higgs production in various extended Higgs sector models, +JHEP 09 (2022) 011, [arXiv:2112.12515]. +[7] ATLAS Collaboration, Constraining the Higgs boson self-coupling from single- and +double-Higgs production with the ATLAS detector using pp collisions at √s = 13 TeV, . +[8] CMS Collaboration, A portrait of the Higgs boson by the CMS experiment ten years +after the discovery, Nature 607 (2022), no. 7917 60–68, [arXiv:2207.00043]. +[9] K. Fujii et al., ILC Study Questions for Snowmass 2021, arXiv:2007.03650. +[10] ILC International Development Team Collaboration, A. Aryshev et al., The +International Linear Collider: Report to Snowmass 2021, arXiv:2203.07622. +[11] N. G. Deshpande and E. Ma, Pattern of Symmetry Breaking with Two Higgs Doublets, +Phys. Rev. D18 (1978) 2574. +[12] E. Ma, Verifiable radiative seesaw mechanism of neutrino mass and dark matter, Phys. +Rev. D73 (2006) 077301, [hep-ph/0601225]. +9 + +[13] M. Gustafsson, E. Lundstrom, L. Bergstrom, and J. Edsjo, Significant Gamma Lines +from Inert Higgs Dark Matter, Phys. Rev. Lett. 99 (2007) 041301, [astro-ph/0703512]. +[14] T. Hambye and M. H. G. Tytgat, Electroweak symmetry breaking induced by dark +matter, Phys. Lett. B659 (2008) 651–655, [arXiv:0707.0633]. +[15] P. Agrawal, E. M. Dolle, and C. A. Krenke, Signals of Inert Doublet Dark Matter in +Neutrino Telescopes, Phys. Rev. D79 (2009) 015015, [arXiv:0811.1798]. +[16] E. M. Dolle and S. Su, The Inert Dark Matter, Phys. Rev. D80 (2009) 055012, +[arXiv:0906.1609]. +[17] Q.-H. Cao, E. Ma, and G. Rajasekaran, Observing the Dark Scalar Doublet and its +Impact on the Standard-Model Higgs Boson at Colliders, Phys. Rev. D 76 (2007) 095011, +[arXiv:0708.2939]. +[18] L. Lopez Honorez, E. Nezri, J. F. Oliver, and M. H. G. Tytgat, The Inert Doublet Model: +An Archetype for Dark Matter, JCAP 02 (2007) 028, [hep-ph/0612275]. +[19] A. Goudelis, B. Herrmann, and O. St˚al, Dark matter in the Inert Doublet Model after the +discovery of a Higgs-like boson at the LHC, JHEP 09 (2013) 106, [arXiv:1303.3010]. +[20] R. Barbieri, L. J. Hall, and V. S. Rychkov, Improved naturalness with a heavy Higgs: An +Alternative road to LHC physics, Phys. Rev. D74 (2006) 015007, [hep-ph/0603188]. +[21] E. Dolle, X. Miao, S. Su, and B. Thomas, Dilepton Signals in the Inert Doublet Model, +Phys. Rev. D81 (2010) 035003, [arXiv:0909.3094]. +[22] M. Aoki, S. Kanemura, and H. Yokoya, Reconstruction of Inert Doublet Scalars at the +International Linear Collider, Phys. Lett. B725 (2013) 302–309, [arXiv:1303.6191]. +[23] A. Arhrib, R. Benbrik, and N. Gaur, H → γγ in Inert Higgs Doublet Model, Phys. Rev. +D85 (2012) 095021, [arXiv:1201.2644]. +[24] M. Krawczyk, D. Sokolowska, P. Swaczyna, and B. Swiezewska, Constraining Inert Dark +Matter by Rγγ and WMAP data, JHEP 09 (2013) 055, [arXiv:1305.6266]. +[25] A. Arhrib, R. Benbrik, and T.-C. Yuan, Associated Production of Higgs at Linear +Collider in the Inert Higgs Doublet Model, Eur. Phys. J. C74 (2014) 2892, +[arXiv:1401.6698]. +[26] A. Datta, N. Ganguly, N. Khan, and S. Rakshit, Exploring collider signatures of the +inert Higgs doublet model, Phys. Rev. D95 (2017), no. 1 015017, [arXiv:1610.00648]. +[27] B. Dutta, G. Palacio, J. D. Ruiz-Alvarez, and D. Restrepo, Vector Boson Fusion in the +Inert Doublet Model, Phys. Rev. D97 (2018), no. 5 055045, [arXiv:1709.09796]. +[28] J. Kalinowski, W. Kotlarski, T. Robens, D. Sokolowska, and A. F. Zarnecki, +Benchmarking the Inert Doublet Model for e+e− colliders, JHEP 12 (2018) 081, +[arXiv:1809.07712]. +[29] J. Kalinowski, W. Kotlarski, T. Robens, D. Sokolowska, and A. F. Zarnecki, Exploring +Inert Scalars at CLIC, JHEP 07 (2019) 053, [arXiv:1811.06952]. +10 + +[30] Y. Guo-He, S. Mao, L. Gang, Z. Yu, and G. Jian-You, Searches for dark matter via +charged Higgs pair production in the Inert Doublet Model at γγ collider, +arXiv:2006.06216. +[31] F.-X. Yang, Z.-L. Han, and Y. Jin, Same-Sign Dilepton Signature in the Inert Doublet +Model, arXiv:2101.06862. +[32] J. Kalinowski, T. Robens, D. Sokolowska, and A. F. Zarnecki, IDM Benchmarks for the +LHC and Future Colliders, Symmetry 13 (2021), no. 6 991, [arXiv:2012.14818]. +[33] A. Melfo, M. Nemevsek, F. Nesti, G. Senjanovic, and Y. Zhang, Inert Doublet Dark +Matter and Mirror/Extra Families after Xenon100, Phys. Rev. D84 (2011) 034009, +[arXiv:1105.4611]. +[34] D. Abercrombie et al., Dark Matter Benchmark Models for Early LHC Run-2 Searches: +Report of the ATLAS/CMS Dark Matter Forum, Phys. Dark Univ. 27 (2020) 100371, +[arXiv:1507.00966]. +[35] A. Ilnicka, M. Krawczyk, and T. Robens, Inert Doublet Model in light of LHC Run I and +astrophysical data, Phys. Rev. D93 (2016), no. 5 055026, [arXiv:1508.01671]. +[36] N. Blinov, J. Kozaczuk, D. E. Morrissey, and A. de la Puente, Compressing the Inert +Doublet Model, Phys. Rev. D 93 (2016), no. 3 035020, [arXiv:1510.08069]. +[37] P. Poulose, S. Sahoo, and K. Sridhar, Exploring the Inert Doublet Model through the dijet +plus missing transverse energy channel at the LHC, Phys. Lett. B765 (2017) 300–306, +[arXiv:1604.03045]. +[38] M. Hashemi and S. Najjari, Observability of Inert Scalars at the LHC, Eur. Phys. J. +C77 (2017), no. 9 592, [arXiv:1611.07827]. +[39] N. Wan, N. Li, B. Zhang, H. Yang, M.-F. Zhao, M. Song, G. Li, and J.-Y. Guo, Searches +for Dark Matter via Mono-W Production in Inert Doublet Model at the LHC, Commun. +Theor. Phys. 69 (2018), no. 5 617. +[40] A. Belyaev, T. R. Fernandez Perez Tomei, P. G. Mercadante, C. S. Moon, S. Moretti, +S. F. Novaes, L. Panizzi, F. Rojas, and M. Thomas, Advancing LHC probes of dark +matter from the inert two-Higgs-doublet model with the monojet signal, Phys. Rev. D99 +(2019), no. 1 015011, [arXiv:1809.00933]. +[41] D. Dercks and T. Robens, Constraining the Inert Doublet Model using Vector Boson +Fusion, Eur. Phys. J. C79 (2019), no. 11 924, [arXiv:1812.07913]. +[42] Y.-L. S. Tsai, V. Q. Tran, and C.-T. Lu, Confronting dark matter co-annihilation of +Inert two Higgs Doublet Model with a compressed mass spectrum, JHEP 06 (2020) 033, +[arXiv:1912.08875]. +[43] G. Gil, P. Chankowski, and M. Krawczyk, Inert Dark Matter and Strong Electroweak +Phase Transition, Phys. Lett. B717 (2012) 396–402, [arXiv:1207.0084]. +[44] B. Swiezewska, Inert scalars and vacuum metastability around the electroweak scale, +JHEP 07 (2015) 118, [arXiv:1503.07078]. +11 + +[45] N. Blinov, S. Profumo, and T. Stefaniak, The Electroweak Phase Transition in the Inert +Doublet Model, JCAP 1507 (2015) 028, [arXiv:1504.05949]. +[46] F. P. Huang and J.-H. Yu, Exploring inert dark matter blind spots with gravitational +wave signatures, Phys. Rev. D98 (2018), no. 9 095022, [arXiv:1704.04201]. +[47] A. Belyaev, G. Cacciapaglia, I. P. Ivanov, F. Rojas-Abatte, and M. Thomas, Anatomy of +the Inert Two Higgs Doublet Model in the light of the LHC and non-LHC Dark Matter +Searches, Phys. Rev. D97 (2018), no. 3 035011, [arXiv:1612.00511]. +[48] A. Arhrib, Y.-L. S. Tsai, Q. Yuan, and T.-C. Yuan, An Updated Analysis of Inert Higgs +Doublet Model in light of the Recent Results from LUX, PLANCK, AMS-02 and LHC, +JCAP 1406 (2014) 030, [arXiv:1310.0358]. +[49] B. Eiteneuer, A. Goudelis, and J. Heisig, The inert doublet model in the light of +Fermi-LAT gamma-ray data: a global fit analysis, Eur. Phys. J. C77 (2017), no. 9 624, +[arXiv:1705.01458]. +[50] A. Ghosh, P. Konar, and S. Seth, On precise probing of inert Higgs doublet model at the +LHC, arXiv:2111.15236. +[51] A. Jueid, J. Kim, S. Lee, S. Y. Shim, and J. Song, Phenomenology of the Inert Doublet +Model with a global U(1) symmetry, Phys. Rev. D102 (2020), no. 7 075011, +[arXiv:2006.10263]. +[52] M. Krawczyk, D. Soko�lowska, P. Swaczyna, and B. ´Swie˙zewska, Higgs → γγ, Zγ in the +Inert Doublet Model, Acta Phys. Polon. B44 (2013), no. 11 2163–2170, +[arXiv:1309.7880]. +[53] R. Enberg, J. Rathsman, and G. Wouda, Higgs properties in a broken Inert Doublet +Model, JHEP 08 (2013) 079, [arXiv:1304.1714]. [Erratum: JHEP01,087(2015)]. +[54] M. Krawczyk, D. Sokolowska, and B. Swiezewska, Inert Doublet Model with a 125 GeV +Higgs, in Proceedings, 1st Toyama International Workshop on Higgs as a Probe of New +Physics 2013 (HPNP2013): Toyama, Japan, February 13-16, 2013, 2013. +arXiv:1304.7757. +[55] S. Kanemura, K. Mawatari, and K. Sakurai, Single Higgs production in association with +a photon at electron-positron colliders in extended Higgs models, Phys. Rev. D 99 (2019), +no. 3 035023, [arXiv:1808.10268]. +[56] M. Hashemi, M. Krawczyk, S. Najjari, and A. F. ˙Zarnecki, Production of Inert Scalars +at the high energy e+e− colliders, arXiv:1512.01175. [JHEP02,187(2016)]. +[57] A. Ahriche, A. Arhrib, A. Jueid, S. Nasri, and A. de La Puente, Mono-Higgs Signature +in the Compressed scotogenic Model, arXiv:1811.00490. +[58] A. Arhrib, R. Benbrik, J. El Falaki, and A. Jueid, Radiative corrections to the Triple +Higgs Coupling in the Inert Higgs Doublet Model, JHEP 12 (2015) 007, +[arXiv:1507.03630]. +12 + +[59] H. Abouabid, A. Arhrib, J. E. Falaki, B. Gong, W. Xie, and Q.-S. Yan, Full one-loop +radiative corrections to e+e− → H+H− in the Inert Higgs Doublet Model, +arXiv:2204.05237. +[60] S. Kanemura, M. Kikuchi, and K. Sakurai, Testing the dark matter scenario in the inert +doublet model by future precision measurements of the Higgs boson couplings, Phys. Rev. +D 94 (2016), no. 11 115011, [arXiv:1605.08520]. +[61] H. Abouabid, A. Arhrib, R. Benbrik, J. El Falaki, B. Gong, W. Xie, and Q.-S. Yan, +One-loop radiative corrections to e+e− → Zh0/H0A0 in the Inert Higgs Doublet Model, +JHEP 05 (2021) 100, [arXiv:2009.03250]. +[62] M. Aiko, S. Kanemura, and K. Mawatari, Next-to-leading-order corrections to the Higgs +strahlung process from electron–positron collisions in extended Higgs models, Eur. Phys. +J. C 81 (2021), no. 11 1000, [arXiv:2109.02884]. +[63] M. J. Ramsey-Musolf, J.-H. Yu, and J. Zhou, Probing extended scalar sectors with +precision e+e− → Zh and Higgs diphoton studies, JHEP 10 (2021) 155, +[arXiv:2104.10709]. +[64] S. Banerjee, F. Boudjema, N. Chakrabarty, G. Chalons, and H. Sun, Relic density of +dark matter in the inert doublet model beyond leading order: The heavy mass case, Phys. +Rev. D100 (2019), no. 9 095024, [arXiv:1906.11269]. +[65] S. Banerjee, F. Boudjema, N. Chakrabarty, and H. Sun, Relic density of dark matter in +the inert doublet model beyond leading order for the low mass region: 1. Renormalisation +and constraints, arXiv:2101.02165. +[66] S. Banerjee, F. Boudjema, N. Chakrabarty, and H. Sun, Relic density of dark matter in +the inert doublet model beyond leading order for the low mass region: 2. Co-annihilation, +arXiv:2101.02166. +[67] S. Banerjee, F. Boudjema, N. Chakrabarty, and H. Sun, Relic density of dark matter in +the inert doublet model beyond leading order for the low mass region: 3. Annihilation in +3-body final state, arXiv:2101.02167. +[68] S. Banerjee, F. Boudjema, N. Chakrabarty, and H. Sun, Relic density of dark matter in +the inert doublet model beyond leading order for the low mass region: 4. The Higgs +resonance region, arXiv:2101.02170. +[69] S. Banerjee and N. Chakrabarty, A revisit to scalar dark matter with radiative +corrections, JHEP 05 (2019) 150, [arXiv:1612.01973]. +[70] S. Kanemura, M. Kikuchi, K. Mawatari, K. Sakurai, and K. Yagyu, Full +next-to-leading-order calculations of Higgs boson decay rates in models with non-minimal +scalar sectors, Nucl. Phys. B 949 (2019) 114791, [arXiv:1906.10070]. +[71] S. Kanemura, M. Kikuchi, K. Mawatari, K. Sakurai, and K. Yagyu, H-COUP Version 2: +a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors, +Comput. Phys. Commun. 257 (2020) 107512, [arXiv:1910.12769]. +13 + +[72] S. Kanemura, M. Kikuchi, K. Sakurai, and K. Yagyu, H-COUP: a program for one-loop +corrected Higgs boson couplings in non-minimal Higgs sectors, Comput. Phys. Commun. +233 (2018) 134–144, [arXiv:1710.04603]. +[73] P. M. Ferreira and B. Swiezewska, One-loop contributions to neutral minima in the inert +doublet model, JHEP 04 (2016) 099, [arXiv:1511.02879]. +[74] J. Braathen and S. Kanemura, Leading two-loop corrections to the Higgs boson +self-couplings in models with extended scalar sectors, Eur. Phys. J. C80 (2020), no. 3 +227, [arXiv:1911.11507]. +[75] J. Braathen and S. Kanemura, On two-loop corrections to the Higgs trilinear coupling in +models with extended scalar sectors, Phys. Lett. B796 (2019) 38–46, [arXiv:1903.05417]. +[76] E. Senaha, Radiative Corrections to Triple Higgs Coupling and Electroweak Phase +Transition: Beyond One-loop Analysis, Phys. Rev. D100 (2019), no. 5 055034, +[arXiv:1811.00336]. +[77] S. Kanemura, Y. Okada, E. Senaha, and C. P. Yuan, Higgs coupling constants as a probe +of new physics, Phys. Rev. D 70 (2004) 115002, [hep-ph/0408364]. +[78] S. Kanemura, S. Kiyoura, Y. Okada, E. Senaha, and C. P. Yuan, New physics effect on +the Higgs selfcoupling, Phys. Lett. B 558 (2003) 157–164, [hep-ph/0211308]. +[79] S. Kanemura, M. Kikuchi, and K. Yagyu, Fingerprinting the extended Higgs sector using +one-loop corrected Higgs boson couplings and future precision measurements, Nucl. Phys. +B 896 (2015) 80–137, [arXiv:1502.07716]. +[80] W. Hollik and S. Penaranda, Yukawa coupling quantum corrections to the selfcouplings of +the lightest MSSM Higgs boson, Eur. Phys. J. C 23 (2002) 163–172, [hep-ph/0108245]. +[81] H. Bahl, J. Braathen, and G. Weiglein, New Constraints on Extended Higgs Sectors from +the Trilinear Higgs Coupling, Phys. Rev. Lett. 129 (2022), no. 23 231802, +[arXiv:2202.03453]. +[82] J. Braathen, S. Kanemura, and M. Shimoda, Two-loop analysis of classically +scale-invariant models with extended Higgs sectors, JHEP 03 (2021) 297, +[arXiv:2011.07580]. +[83] S. Kanemura, M. Kikuchi, K. Sakurai, and K. Yagyu, Gauge invariant one-loop +corrections to Higgs boson couplings in non-minimal Higgs models, Phys. Rev. D 96 +(2017), no. 3 035014, [arXiv:1705.05399]. +[84] S. Kanemura, M. Kikuchi, and K. Yagyu, One-loop corrections to the Higgs self-couplings +in the singlet extension, Nucl. Phys. B 917 (2017) 154–177, [arXiv:1608.01582]. +[85] S.-P. He and S.-h. Zhu, One-loop radiative correction to the triple Higgs coupling in the +Higgs singlet model, Phys. Lett. B 764 (2017) 31–37, [arXiv:1607.04497]. [Erratum: +Phys.Lett.B 797, 134782 (2019)]. +[86] M. Aoki, S. Kanemura, M. Kikuchi, and K. Yagyu, Radiative corrections to the Higgs +boson couplings in the triplet model, Phys. Rev. D 87 (2013), no. 1 015012, +[arXiv:1211.6029]. +14 + +[87] P. Basler, M. M¨uhlleitner, and J. Wittbrodt, The CP-Violating 2HDM in Light of a +Strong First Order Electroweak Phase Transition and Implications for Higgs Pair +Production, JHEP 03 (2018) 061, [arXiv:1711.04097]. +[88] P. Basler, M. M¨uhlleitner, and J. M¨uller, Electroweak Phase Transition in Non-Minimal +Higgs Sectors, JHEP 05 (2020) 016, [arXiv:1912.10477]. +[89] A. Moyotl, S. Chamorro, H. Castilla-Valdez, and M. A. P´erez, New physics effects in the +Higgs trilinear self-coupling through one-loop radiative corrections, arXiv:1610.06299. +[90] M. Krause, R. Lorenz, M. Muhlleitner, R. Santos, and H. Ziesche, Gauge-independent +Renormalization of the 2-Higgs-Doublet Model, JHEP 09 (2016) 143, +[arXiv:1605.04853]. +[91] P. Osland, P. N. Pandita, and L. Selbuz, Trilinear Higgs couplings in the two Higgs +doublet model with CP violation, Phys. Rev. D 78 (2008) 015003, [arXiv:0802.0060]. +[92] V. D. Barger, M. S. Berger, A. L. Stange, and R. J. N. Phillips, Supersymmetric Higgs +boson hadroproduction and decays including radiative corrections, Phys. Rev. D 45 +(1992) 4128–4147. +[93] C. Borschensky, T. N. Dao, M. Gabelmann, M. M¨uhlleitner, and H. Rzehak, The +Trilinear Higgs Self-Couplings at O(α2 +t ) in the CP-Violating NMSSM, +arXiv:2210.02104. +[94] M. Brucherseifer, R. Gavin, and M. Spira, Minimal supersymmetric Higgs boson +self-couplings: Two-loop O(αtαs) corrections, Phys. Rev. D 90 (2014), no. 11 117701, +[arXiv:1309.3140]. +[95] D. T. Nhung, M. Muhlleitner, J. Streicher, and K. Walz, Higher Order Corrections to the +Trilinear Higgs Self-Couplings in the Real NMSSM, JHEP 11 (2013) 181, +[arXiv:1306.3926]. +[96] A. Dobado, M. J. Herrero, W. Hollik, and S. Penaranda, Selfinteractions of the lightest +MSSM Higgs boson in the large pseudoscalar mass limit, Phys. Rev. D 66 (2002) 095016, +[hep-ph/0208014]. +[97] S. Kanemura, T. Shindou, and K. Yagyu, Non-decoupling effects in supersymmetric +Higgs sectors, Phys. Lett. B 699 (2011) 258–263, [arXiv:1009.1836]. +[98] S. Kanemura, T. Kubota, and E. Takasugi, Lee-Quigg-Thacker bounds for Higgs boson +masses in a two doublet model, Phys. Lett. B313 (1993) 155–160, [hep-ph/9303263]. +[99] A. G. Akeroyd, A. Arhrib, and E.-M. Naimi, Note on tree level unitarity in the general +two Higgs doublet model, Phys. Lett. B490 (2000) 119–124, [hep-ph/0006035]. +[100] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Theory +and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1–102, +[arXiv:1106.0034]. +[101] I. F. Ginzburg, K. A. Kanishev, M. Krawczyk, and D. Sokolowska, Evolution of Universe +to the present inert phase, Phys. Rev. D82 (2010) 123533, [arXiv:1009.4593]. +15 + +[102] ATLAS Collaboration, M. Aaboud et al., Measurements of Higgs boson properties in the +diphoton decay channel with 36 fb−1 of pp collision data at √s = 13 TeV with the +ATLAS detector, Phys. Rev. D 98 (2018) 052005, [arXiv:1802.04146]. +[103] CMS Collaboration, A. M. Sirunyan et al., Measurements of Higgs boson properties in +the diphoton decay channel in proton-proton collisions at √s = 13 TeV, JHEP 11 (2018) +185, [arXiv:1804.02716]. +[104] ATLAS Collaboration, Combination of searches for invisible Higgs boson decays with +the ATLAS experiment, ATLAS-CONF-2020-052. +[105] B. Swiezewska and M. Krawczyk, Diphoton rate in the inert doublet model with a 125 +GeV Higgs boson, Phys. Rev. D88 (2013), no. 3 035019, [arXiv:1212.4100]. +[106] G. Belanger, B. Dumont, A. Goudelis, B. Herrmann, S. Kraml, and D. Sengupta, +Dilepton constraints in the Inert Doublet Model from Run 1 of the LHC, Phys. Rev. D91 +(2015), no. 11 115011, [arXiv:1503.07367]. +[107] E. Lundstrom, M. Gustafsson, and J. Edsjo, The Inert Doublet Model and LEP II +Limits, Phys. Rev. D79 (2009) 035013, [arXiv:0810.3924]. +[108] M. E. Peskin and T. Takeuchi, Estimation of oblique electroweak corrections, Phys. Rev. +D46 (1992) 381–409. +[109] Particle Data Group Collaboration, M. Tanabashi et al., Review of Particle Physics, +Phys. Rev. D98 (2018), no. 3 030001. +[110] Particle Data Group Collaboration, P. A. Zyla et al., Review of Particle Physics, +PTEP 2020 (2020), no. 8 083C01. +[111] L. F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. +120B (1983) 133–136. +[112] M. Dine and W. Fischler, The Not So Harmless Axion, Phys. Lett. 120B (1983) 137–141. +[113] J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. +120B (1983) 127–132. +[114] G. Belanger, A. Mjallal, and A. Pukhov, Recasting direct detection limits within +micrOMEGAs and implication for non-standard Dark Matter scenarios, pre-print (3, +2020) [arXiv:2003.08621]. +[115] XENON Collaboration, E. Aprile et al., Dark Matter Search Results from a One +Ton-Year Exposure of XENON1T, Phys. Rev. Lett. 121 (2018), no. 11 111302, +[arXiv:1805.12562]. +[116] PICO Collaboration, C. Amole et al., Dark Matter Search Results from the Complete +Exposure of the PICO-60 C3F8 Bubble Chamber, Phys. Rev. D 100 (2019), no. 2 022001, +[arXiv:1902.04031]. +[117] CRESST Collaboration, A. H. Abdelhameed et al., First results from the CRESST-III +low-mass dark matter program, Phys. Rev. D 100 (2019), no. 10 102002, +[arXiv:1904.00498]. +16 + +[118] DarkSide Collaboration, P. Agnes et al., Low-Mass Dark Matter Search with the +DarkSide-50 Experiment, Phys. Rev. Lett. 121 (2018), no. 8 081307, +[arXiv:1802.06994]. +[119] ATLAS Collaboration, M. Aaboud et al., Search for dark matter and other new +phenomena in events with an energetic jet and large missing transverse momentum using +the ATLAS detector, JHEP 01 (2018) 126, [arXiv:1711.03301]. +[120] CMS Collaboration, A. M. Sirunyan et al., Search for dark matter produced with an +energetic jet or a hadronically decaying W or Z boson at √s = 13 TeV, JHEP 07 (2017) +014, [arXiv:1703.01651]. +[121] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. +Commun. 140 (2001) 418–431, [hep-ph/0012260]. +[122] T. Hahn and M. Perez-Victoria, Automatized one loop calculations in four-dimensions +and D-dimensions, Comput. Phys. Commun. 118 (1999) 153–165, [hep-ph/9807565]. +[123] T. Hahn and M. Rauch, News from FormCalc and LoopTools, Nucl. Phys. Proc. Suppl. +157 (2006) 236–240, [hep-ph/0601248]. [,236(2006)]. +[124] T. Hahn, Loop calculations with FeynArts, FormCalc, and LoopTools, Acta Phys. Polon. +B30 (1999) 3469–3475, [hep-ph/9910227]. +[125] T. Hahn, Feynman Diagram Calculations with FeynArts, FormCalc, and LoopTools, PoS +ACAT2010 (2010) 078, [arXiv:1006.2231]. +[126] M. Bohm, H. Spiesberger, and W. Hollik, On the One Loop Renormalization of the +Electroweak Standard Model and Its Application to Leptonic Processes, Fortsch. Phys. 34 +(1986) 687–751. +[127] W. F. L. Hollik, Radiative Corrections in the Standard Model and their Role for +Precision Tests of the Electroweak Theory, Fortsch. Phys. 38 (1990) 165–260. +[128] A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop +level and results for W physics at LEP-200, Fortsch. Phys. 41 (1993) 307–420, +[arXiv:0709.1075]. +[129] A. Denner and S. Dittmaier, Electroweak Radiative Corrections for Collider Physics, +Phys. Rept. 864 (2020) 1–163, [arXiv:1912.06823]. +17 + diff --git a/zdFST4oBgHgl3EQfUjgb/content/tmp_files/load_file.txt b/zdFST4oBgHgl3EQfUjgb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..397263af0ab37997445b8a6c085a2d5dd3c6ccc9 --- /dev/null +++ b/zdFST4oBgHgl3EQfUjgb/content/tmp_files/load_file.txt @@ -0,0 +1,1129 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf,len=1128 +page_content='Revisiting one-loop corrections to the trilinear Higgs boson self-coupling in the Inert Doublet Model Jaouad El Falaki1 ∗ 1 LPTHE, Physics Department, Faculty of Sciences, Ibn Zohr University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 8106 Agadir, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abstract We investigate predictions of the trilinear Higgs self-coupling with radiative corrections in the context of the Inert Doublet Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The triple Higgs vertex is computed at the one- loop level based on the on-shell renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' We calculate its possible deviation from the predictions within the standard model, taking into account all relevant theoretical and experimental constraints, including dark matter searches and the latest bounds on the branching fraction of the Higgs boson decaying to invisible particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' By scanning the model’s parameter space, we find that the deviation in the triple Higgs boson self-coupling from standard model expectations can be substantial, exceeding 100% in certain regions of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' ∗E-mail: jaouad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='elfalaki@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='com,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='elfalaki@uiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='ma arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='13773v1 [hep-ph] 31 Jan 2023 1 Introduction A great achievement in the history of high energy physics was made on July 4, 2012, with the discovery of the Higgs boson by ATLAS and CMS, at the CERN Large Hadron Collider (LHC) [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Since then, the spectrum of the Standard Model (SM) of particle physics has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Following this discovery, more accumulated data has been analyzed during LHC Run I-II and it has been found that the properties of the Higgs boson are in perfect agreement with the predictions from the SM, with a level of accuracy of 10 − 20% [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' To establish experimentally the Brout-Englert-Higgs mechanism of electroweak symmetry break- ing (EWSB), it is necessary to measure not only the couplings of the Higgs boson with the fermions and the gauge bosons, but also the self-coupling of the Higgs boson, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' the triple λhhh and the quartic λhhhh Higgs boson couplings, in order to be able to reconstruct the shape of the Higgs scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The measurement of the Higgs boson quartic coupling is more challenging because the cross section for triple Higgs production at the LHC is much smaller (the cross section for pp → 3h production is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1 fb at √s = 14 TeV ), and it is out of reach even for the high-luminosity LHC (HL-LHC) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' On the other hand, the trilinear self-coupling of the Higgs boson λhhh can be measured from the production of a pair of Higgs bosons at the LHC [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Recently, a statistical combination by ATLAS has been presented [7], in which the bound on λhhh has been significantly reduced to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='4 < κλ < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3 at 95% confidence level, where κλ ≡ λhhh λSM hhh is the self-coupling modifier parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' CMS also has derived a limit on κλ which is constrained to be within −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='24 < κλ < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='49 at 95% CL [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The measurement accuracy of λhhh will be improved at future experiments such as the HL-LHC where κλ is constrained to be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3 at 95% CL with 3 ab−1 data [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' At future linear collider such as the ILC, the triple Higgs boson coupling is expected to be measured at the precision level of 27% in the double Higgs-strahlung process e+e− → Zhh at √s = 500 GeV with an integrated luminosity of 4 ab−1 [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' A relative precision of 10% on λhhh is also possible at 1 TeV from the di-Higgs production in WW fusion process e+e− → ν¯νhh with an accumulated 8 ab−1 of integrated luminosity [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The expectation for these precise measurements motivates the study of the radiative corrections to λhhh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The SM is unable to explain certain phenomena such as dark matter, the hierarchy problem and tiny neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' As a result, new physics beyond the SM (BSM) is needed to provide answers to these unsolved problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The Higgs sector of the SM only includes one Higgs dou- blet, but there is no fundamental reason to believe it must be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Among popular BSM candidates are models with extended Higgs sectors, such as the Inert Doublet Model (IDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Originally proposed by Deshpande and Ma [11] and initially suggested for EWSB studies, this model is highly intriguing due to its potential to generate tiny neutrino masses [12], provide a dark matter candidate [13–19], and address the naturalness problem [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The IDM consists of adding a second Higgs doublet that does not acquire a vacuum expectation value (VEV) and has no coupling with SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' An exact Z2 symmetry is imposed, with the SM Higgs doublet being even and the additional scalar doublet being odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The preserved Z2 symmetry ensures that the extra doublet does not interact with matter and its lightest stable neutral component can act as a dark matter particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' After EWSB, the IDM has a spectrum of five physical scalars: a CP-even Higgs boson h (identified with the discovered SM Higgs), and four inert scalars (H, A, and H±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The rich phenomenology of the IDM has been thoroughly studied in the literature, both in the context of future Higgs factories like the ILC and CLIC and at the LHC [21–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Additionally, there have been many studies of the model that go beyond the lowest order of the perturbation, including those at one-loop level [58–73] and two-loop order [74–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The one-loop 1 contributions to the triple Higgs boson coupling from Standard Model particles have been inves- tigated in [58,77–80], where it was found that these corrections are dominated by loops involving the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The radiative corrections to λhhh in some non-supersymmetric Higgs models can be found in references [77–79,81–91], and for corrections in certain supersymmetric models, see for example Refs [80, 92–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' These new physics effects in models with extended Higgs sectors have been shown to be large and can significantly enhance the λhhh coupling in a wide range of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Calculations of Higgs boson couplings that include higher-order corrections are mandatory to compare theory predictions with future precision data from hadron and lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In Ref [58], the one-loop contributions of the inert scalars to λhhh are discussed only in the degenerate spectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', mH = mA = mH±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In addition, one loop corrections to λhhh within the IDM have also been discussed in some scenarios [60], but under the assumption that mH = mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In the present letter, we will compute the radiative corrections to the triple Higgs self-coupling considering non-degenerate masses for the inert scalars, while taking into account all current theoretical and experimental constraints on the IDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The layout of the letter is as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 2, we briefly introduce the IDM and outline its theoretical and experimental constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3, we present the on-shell renormalization scheme and provide a comprehensive explanation of the triple Higgs coupling at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 4, we present our numerical results for the SM and IDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Conclusions are given in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 2 The Inert Doublet Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1 The Model The IDM is a simple extension of the Standard Model of particle physics which consists of the SM, including its Higgs doublet Φ1, and an additional SU(2) doublet Φ2 that brings in four new scalar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' An exact Z2 symmetry is introduced such that the SM doublet is even Φ1 −→ Φ1 while the added extra doublet (inert doublet) is odd Φ2 −→ −Φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' This unbroken Z2 parity guarantees the absence of coupling between fermions and the inert doublet Φ2, therefore there is no flavor-changing neutral currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Moreover, it ensures that the lightest neutral component of Φ2 is a natural dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The decomposition for the two doublets around the vacuum state is given by: Φ1 = � G± 1 √ 2(v + h + iG0) � , Φ2 = � H± 1 √ 2(H + iA) � (1) Where only the SM doublet Φ1 is involved in EWSB, G0 and G± correspond to the three Nambu- Goldstone bosons gauged away by the longitudinal component of Z and W ± respectively, h is the SM Higgs boson and v is the VEV of the SM Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The second doublet Φ2 does not participate in EWSB and it contains four new inert scalars H, A and H± where either A or H may act as potential dark matter candidate, depending on the mass hierarchy of these two inert scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The most general renormalizable tree-level scalar potential in this model can be written as: V = µ2 1|Φ1|2 + µ2 2|Φ2|2 + λ1|Φ1|4 + λ2|Φ2|4 + λ3|Φ1|2|Φ2|2 + λ4|Φ† 1Φ2|2 + λ5 2 � (Φ† 1Φ2)2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='c � , (2) 2 where µ1 and µ2 are the mass of the Φ1 and Φ2 fields, and all λ1,2,3,4 parameters are real since the above potential must be hermitian while the phase of λ5 can be absorbed into an appropriate redefinition of Φ1 and Φ2 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' After EWSB the five scalar particles of the model acquire their masses which are given by: m2 h = −2µ2 1 = 2λ1v2 m2 H = µ2 2 + λLv2 m2 A = µ2 2 + λSv2 m2 H± = µ2 2 + 1 2λ3v2 (3) where λL,S are defined as: λL,S = 1 2(λ3 + λ4 ± λ5) (4) The trilinear self-coupling of the Higgs boson at tree level in the IDM involves only SM param- eters and is given by: Γtree hhh = −3m2 h v (5) In the IDM, there are eight independent parameters: 5λi, 2µi, and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' After eliminating one parameter through the minimization condition and determining the VEV using the W boson mass, we are left with six remaining independent parameters which will be selected as follows: {µ2 2, λ2, mh, mH±, mH, mA} (6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2 Theoretical and Experimental Constraints In this study, we explore the same parameter space as in our previous published paper [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The IDM parameter space is obtained by performing an extensive parameter scan in the whole space with all experimental and theoretical constraints applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The constraints used are sum- marized below: The theoretical constraints: – The perturbative unitarity [98,99] – The vacuum stability [11,100] – The inert vacuum and charge-breaking minima [101] The experimental constraints: – The Higgs data from the LHC [102,103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' – The invisible Higgs decay [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' – The direct collider searches at LEP [23,105–107] – The electroweak precision tests [20,108,109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' – The dark matter searches [40,42,110–120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 3 Calculation of one-loop corrections to the triple Higgs cou- pling In this section, we briefly discuss the renormalization of the trilinear self-coupling of the Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Using the ’t Hooft-Feynman gauge, we calculate the radiative corrections to the tree-level formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 in both the SM and IDM including contributions from all particles in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1 illustrates some of the Feynman diagrams contributing to the triple Higgs boson coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Note that the dimensional regularization has been used to evaluate the one-loop Feynman amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The calculations are carried out using Mathematica packages FeynArts and FormCalc [121–123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The numerical evaluation of the scalar one-loop integrals has been performed with the LoopTools package [124, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It should be noted that the UV-finiteness of the virtual contributions has been cross-checked numerically and analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Compared to the general two Higgs doublet model (2HDM) [78, 79, 83], the structure of the counter-terms for the triple Higgs boson coupling in the IDM is simpler and identical to those in the SM due to the absence of mixing between the SM Higgs boson and the inert scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' We study an off-shell Higgs boson that decays into two real Higgs bosons h∗(q) → h(q1)h(q2) at the one-loop level, where q, q1 and q2 are the four-momenta of the three Higgs bosons satisfying an off-shell condition q2 ̸= m2 h for the decaying Higgs boson and on-shell conditions q2 1 = q2 2 = m2 h for the two real Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The UV divergences that emerge during the calculation’s intermediate stages should eventually cancel out in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In order to do that, we adopt the on-shell renormalization scheme which is widely used in quantum field theory because it is simple to implement and it allows for a clear physical interpretation of the parameters of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' As in the SM, the tree-level trilinear Higgs self-coupling in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 depends only on the VEV and Higgs boson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hence, the renormalization procedure will be the same as the one adopted in the SM [126–129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The SM fields and parameters are redefined as follows: Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1: Some one-loop Feynman diagrams that do not exist in the SM contribute to the triple Higgs self- coupling within the IDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 4 h h h h H H± FH A H H± h h h h H ±H HF A h h h h 1 2 3 4 h H A H h h h 2 h h h H Ht A h 5 6 7 8 h h h h h HF h h HF h h h h 9 10 11 12 h h 3 h 13m2 h → m2 h + δm2 h m2 V → m2 V + δm2 V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' V = Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' W ± sW → sW + δsW t → t + δt e → (1 + δZe)e Zµ → � 1 + 1 2δZZZ � Zµ + 1 2δZZAAµ Aµ → � 1 + 1 2δZAA � Aµ + 1 2δZAZZµ h → Z1/2 h h = � 1 + 1 2δZh � h (7) where sW = sin θW is the Weinberg angle and t = v(µ2 1 − λ1v2) is the tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' At tree level, the tadpole is zero if the minimization condition is satisfied, but it can receive finite corrections at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' To ensure that the VEV of the Higgs field is consistent across all orders of perturbation theory, it is necessary to renormalize the Higgs tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' This can be done by adding a counter-term to the tadpole, which cancels out any divergences that appear at higher orders of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Consequently, we set the following condition: ˆT = δt + T = 0 =⇒ δt = −T (8) where T is the one-loop contribution of 1PI diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The counter-terms of the masses are fixed by the following on-shell condition: ReˆΣV V T (m2 V ) = 0 =⇒ δm2 V = ReΣV V T (m2 V ) V = W, Z ReˆΣhh(m2 h) = 0 =⇒ δm2 h = ReΣhh(m2 h) (9) where Σhh and ΣV V T are the one-loop non-renormalized self energies for the Higgs boson and gauge bosons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' By fixing the residue of the two-point Green functions to be equal to unity, the wave-function renormalization constant is determined from the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Re∂ ˆΣhh(k2) ∂k2 ����� k2=m2 h = 0 =⇒ δZh = −Re∂Σhh ∂k2 ����� k2=m2 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' (10) In the on-shell renormalization scheme, the electric charge is set by ensuring that there are no higher-order corrections to the e+e−γ vertex in the Thomson limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The electric charge renormalization constant δZe can be expressed as: δZe = −1 2δZAA − sW cW 1 2δZZA (11) where δZAA = −∂ �AA T (k2) ∂k2 ����� k2=0 and δZZA = 2 �AZ T (0) m2 Z (12) 5 To obtain the counter-term δsW one can use the on-shell definition of the weak mixing angle, which is defined as the ratio of the weak neutral current and the weak charged current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Thus, δsW is given by: δsW = c2 W 2sW �δm2 Z m2 Z − δm2 W m2 W � (13) By inserting the redefinition of the parameters into the Lagrangian, we obtain the following counter-term for the trilinear Higgs self-coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' δΓhhh = −3e2 2sW m2 h mW � δZe − δsW sW + δm2 h m2 h + e 2sW δt mW m2 h − δm2 W 2m2 W + 3 2δZh � (14) To obtain the renormalized amplitudes for the triple Higgs coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' the full one-loop one par- ticle irreducible vertex Γ1PI hhh(q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' q2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' q2 2) is added to the corresponding counter-terms δΓhhh as follows: ˆΓhhh(q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' q2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' q2 2) = Γtree hhh + Γ1PI hhh(q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' q2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' q2 2) + δΓhhh (15) 4 Numerical Results In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' we present our numerical analysis for the triple Higgs coupling at the one- loop level in the SM and IDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In order to parametrize the size of the radiative corrections and compare it to the Standard Model’s predictions, we define the following ratio in the IDM: ∆Γhhh = ˆΓhhh(q2, m2 h, m2 h)IDM − ˆΓhhh(q2, m2 h, m2 h)SM ˆΓhhh(q2, m2 h, m2 h)SM (16) While within the SM, we show our numerical results using the following relative ratio: ∆ΓSM hhh = ˆΓhhh(q2, m2 h, m2 h)SM − Γtree hhh Γtree hhh (17) The following numerical values of the input parameters are adopted [109]: mh = 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='18 GeV mW = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='379 GeV mt = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2 GeV mµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='106 GeV mZ = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='198 GeV mb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='660 GeV mτ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='777 GeV α = 1/137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='036 mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='275 GeV We scan the entire parameter space for the other IDM parameters, including physical masses mA, mH and mH± as well as the µ2 2 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' We consider all relevant theoretical and experimental constraints and perform a random scan over the IDM parameter space in the following ranges: 100 GeV ≤ mH ≤ 700 GeV 20 GeV ≤ mA ≤ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='5 GeV 80 GeV ≤ mH± ≤ 700 GeV 0 GeV2 ≤ µ2 2 ≤ 106 GeV2 (18) 6 It’s worth mentioning that our numerical results are independent of λ2 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' We will set λ2 to a fixed value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It is noteworthy that in this letter, the inert scalar A is selected as the dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Furthermore, all points have been passed the upper bound from the invisible Higgs boson decay Br(h → AA) ≤ 11% [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1 We visualize in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1 the size of the radiative corrections in the SM as a function of the four-momentum of the off-shell Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The measurement of the triple Higgs coupling in future experiments will be done through the double Higgs production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In this case, one of the Higgs bosons will be off-shell, which implies that the dependence on momentum q is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' One can see that the total corrections to the trilinear self-coupling of the Higgs boson start from −1% around q = 250 GeV and can reach a maximum value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='23% for q = 470 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It should be emphasized that the top-quark contribution is the dominant correction for large values of the momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0 q[TeV] −20 −15 −10 −5 0 5 ∆ΓSM hhh[%] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1: The relative correction ∆ΓSM hhh[%] as a function of q, where qµ is the four-momentum of the off-shell Higgs boson in h∗ → hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' In the left panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2, we depict the corrections to the triple Higgs coupling as a function of the four-momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The color code indicates the allowed charged scalar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' As a reference point, we display by a solid-red line the relative corrections ∆ΓSM hhh to the trilinear 1It bears mentioning that the mass range between 20 GeV and 55 GeV of the dark matter candidate is ruled out from relic density constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It is also worth pointing out that the allowed values of µ2 2 range between 3000 GeV 2 and 4400 GeV 2 after passing all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 7 Higgs boson self-coupling within the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' From this plot, it can be seen that the corrections are small and consistent with the SM predictions for light charged and neutral inert particles, whose masses are in the range 80 GeV ≤ mH± ≤ 200 GeV and 100 GeV ≤ mH ≤ 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It can also be observed that for inert scalars masses, 300 GeV ≤ mH±, mH ≤ 440 GeV, the deviation of the triple Higgs coupling from the SM’s predictions is significant and larger than 10% with an enhancement up to 120% for q = 880 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Furthermore, One can infer that for heavy inert scalars, the corrections are substantial in a large part of the parameter space with an enhancement of 472% for q = 1216 GeV and mH = 608 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It is worth mentioning that in the left panel there are two different threshold peaks which are attributed to the opening of h∗ → H±H∓ for q = 1204 GeV with mH± = 602 GeV, where the corrections can go up to 470%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The second spike at q = 1216 GeV which amplifies the radiative corrections corresponds to the threshold effect in h∗ → HH with mH = 608 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The non-decoupling effect in the radiative correction to the trilinear self-coupling of the Higgs boson is significant when large masses of the inert scalars are involved, this behavior can be seen on the right panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' It is noteworthy to highlight that this behavior is also observed in the 2HDM, where large corrections to the triple Higgs coupling at the one-loop level are found to grow as the quartic power of the extra heavier Higgs bosons [77,78,83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2: Left: ∆Γhhh as a function of the momentum q where the the charged scalar mass mH± is shown in the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Right: ∆Γhhh as a function of mH± and the color code indicates the mass mH of the neutral scalar H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The red line in the left panel represents the relative ratio ∆ΓSM hhh in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 Conclusion We computed the trilinear Higgs boson coupling in the IDM at one-loop level with non- degenerate inert scalar masses, taking into account all relevant theoretical and experimental constraints, including dark matter searches and Higgs boson invisible decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' We evaluated the one-loop Feynman amplitudes using dimensional regularization in the ’t Hooft-Feynman gauge and employed an on-shell scheme renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Our results showed substantial deviations from the SM predictions for the triple Higgs coupling, with values exceeding 100% in some regions 8 009 400 500 300 OOT mH+[GeV] 200 100 200 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00 qTeV]009 400 500 300 400 mH[GeV] 200 008 100 200 0 100 200 300 400 500 600 mH± [GeVof the parameter space and reaching up to 472% enhancement due to non-decoupling effects of the inert scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' This substantial non-decoupling correction to the triple Higgs boson self- coupling is known to be associated with a strongly first order electroweak phase transition, which is necessary for successful electroweak baryogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Detecting significant deviation from the expected value in the triple Higgs coupling in future colliders can indirectly provide information on the mass of inert scalar bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' References [1] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B716 (2012) 1–29, [arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='7214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [2] CMS Collaboration, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chatrchyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B716 (2012) 30–61, [arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='7235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [3] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Combined measurements of Higgs boson production and decay using up to 80 fb−1 of proton-proton collision data at √s = 13 TeV collected with the ATLAS experiment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 101 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 1 012002, [arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02845].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [4] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Combined measurements of Higgs boson couplings in proton–proton collisions at √s = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C 79 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 421, [arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='10733].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Cepeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC, CERN Yellow Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Monogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 7 (2019) 221–584, [arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abouabid, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Azevedo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Falaki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M¨uhlleitner, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Santos, Benchmarking di-Higgs production in various extended Higgs sector models, JHEP 09 (2022) 011, [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='12515].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [7] ATLAS Collaboration, Constraining the Higgs boson self-coupling from single- and double-Higgs production with the ATLAS detector using pp collisions at √s = 13 TeV, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [8] CMS Collaboration, A portrait of the Higgs boson by the CMS experiment ten years after the discovery, Nature 607 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 7917 60–68, [arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', ILC Study Questions for Snowmass 2021, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='03650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [10] ILC International Development Team Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aryshev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', The International Linear Collider: Report to Snowmass 2021, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Deshpande and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ma, Pattern of Symmetry Breaking with Two Higgs Doublets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D18 (1978) 2574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ma, Verifiable radiative seesaw mechanism of neutrino mass and dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D73 (2006) 077301, [hep-ph/0601225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 9 [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gustafsson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lundstrom, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Bergstrom, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Edsjo, Significant Gamma Lines from Inert Higgs Dark Matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 99 (2007) 041301, [astro-ph/0703512].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hambye and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Tytgat, Electroweak symmetry breaking induced by dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B659 (2008) 651–655, [arXiv:0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0633].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Agrawal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dolle, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krenke, Signals of Inert Doublet Dark Matter in Neutrino Telescopes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D79 (2009) 015015, [arXiv:0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1798].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dolle and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Su, The Inert Dark Matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D80 (2009) 055012, [arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1609].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [17] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Cao, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ma, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rajasekaran, Observing the Dark Scalar Doublet and its Impact on the Standard-Model Higgs Boson at Colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 76 (2007) 095011, [arXiv:0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2939].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lopez Honorez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Nezri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Oliver, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Tytgat, The Inert Doublet Model: An Archetype for Dark Matter, JCAP 02 (2007) 028, [hep-ph/0612275].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Goudelis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Herrmann, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' St˚al, Dark matter in the Inert Doublet Model after the discovery of a Higgs-like boson at the LHC, JHEP 09 (2013) 106, [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Barbieri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hall, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rychkov, Improved naturalness with a heavy Higgs: An Alternative road to LHC physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D74 (2006) 015007, [hep-ph/0603188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dolle, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Miao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Su, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Thomas, Dilepton Signals in the Inert Doublet Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D81 (2010) 035003, [arXiv:0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3094].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aoki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yokoya, Reconstruction of Inert Doublet Scalars at the International Linear Collider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B725 (2013) 302–309, [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='6191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Benbrik, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gaur, H → γγ in Inert Higgs Doublet Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D85 (2012) 095021, [arXiv:1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2644].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sokolowska, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swaczyna, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swiezewska, Constraining Inert Dark Matter by Rγγ and WMAP data, JHEP 09 (2013) 055, [arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='6266].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Benbrik, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yuan, Associated Production of Higgs at Linear Collider in the Inert Higgs Doublet Model, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C74 (2014) 2892, [arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='6698].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Datta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ganguly, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Khan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rakshit, Exploring collider signatures of the inert Higgs doublet model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D95 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 1 015017, [arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00648].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [27] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dutta, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Palacio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ruiz-Alvarez, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Restrepo, Vector Boson Fusion in the Inert Doublet Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D97 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 055045, [arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='09796].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kalinowski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kotlarski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Robens, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sokolowska, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zarnecki, Benchmarking the Inert Doublet Model for e+e− colliders, JHEP 12 (2018) 081, [arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07712].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kalinowski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kotlarski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Robens, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sokolowska, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zarnecki, Exploring Inert Scalars at CLIC, JHEP 07 (2019) 053, [arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='06952].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 10 [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Guo-He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Jian-You, Searches for dark matter via charged Higgs pair production in the Inert Doublet Model at γγ collider, arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='06216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Han, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Jin, Same-Sign Dilepton Signature in the Inert Doublet Model, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='06862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kalinowski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Robens, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sokolowska, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zarnecki, IDM Benchmarks for the LHC and Future Colliders, Symmetry 13 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 6 991, [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='14818].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Melfo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Nemevsek, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Nesti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Senjanovic, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zhang, Inert Doublet Dark Matter and Mirror/Extra Families after Xenon100, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D84 (2011) 034009, [arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='4611].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abercrombie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Dark Matter Benchmark Models for Early LHC Run-2 Searches: Report of the ATLAS/CMS Dark Matter Forum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 27 (2020) 100371, [arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00966].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ilnicka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Robens, Inert Doublet Model in light of LHC Run I and astrophysical data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D93 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 055026, [arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='01671].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [36] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Blinov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kozaczuk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Morrissey, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' de la Puente, Compressing the Inert Doublet Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 93 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 035020, [arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='08069].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Poulose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sahoo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sridhar, Exploring the Inert Doublet Model through the dijet plus missing transverse energy channel at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B765 (2017) 300–306, [arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='03045].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hashemi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Najjari, Observability of Inert Scalars at the LHC, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C77 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 9 592, [arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07827].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [39] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Wan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zhao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Song, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Guo, Searches for Dark Matter via Mono-W Production in Inert Doublet Model at the LHC, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 69 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Belyaev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Fernandez Perez Tomei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mercadante, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Moon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Moretti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Novaes, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Panizzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rojas, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Thomas, Advancing LHC probes of dark matter from the inert two-Higgs-doublet model with the monojet signal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D99 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 1 015011, [arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00933].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [41] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dercks and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Robens, Constraining the Inert Doublet Model using Vector Boson Fusion, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C79 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 924, [arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07913].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [42] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Tsai, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Tran, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lu, Confronting dark matter co-annihilation of Inert two Higgs Doublet Model with a compressed mass spectrum, JHEP 06 (2020) 033, [arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='08875].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gil, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chankowski, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, Inert Dark Matter and Strong Electroweak Phase Transition, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B717 (2012) 396–402, [arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0084].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [44] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swiezewska, Inert scalars and vacuum metastability around the electroweak scale, JHEP 07 (2015) 118, [arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07078].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 [45] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Blinov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Profumo, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Stefaniak, The Electroweak Phase Transition in the Inert Doublet Model, JCAP 1507 (2015) 028, [arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='05949].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [46] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Huang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yu, Exploring inert dark matter blind spots with gravitational wave signatures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D98 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 9 095022, [arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Belyaev, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Cacciapaglia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ivanov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rojas-Abatte, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Thomas, Anatomy of the Inert Two Higgs Doublet Model in the light of the LHC and non-LHC Dark Matter Searches, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D97 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 035011, [arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00511].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Tsai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yuan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yuan, An Updated Analysis of Inert Higgs Doublet Model in light of the Recent Results from LUX, PLANCK, AMS-02 and LHC, JCAP 1406 (2014) 030, [arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0358].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [49] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Eiteneuer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Goudelis, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Heisig, The inert doublet model in the light of Fermi-LAT gamma-ray data: a global fit analysis, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C77 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 9 624, [arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='01458].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ghosh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Konar, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Seth, On precise probing of inert Higgs doublet model at the LHC, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='15236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Jueid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Shim, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Song, Phenomenology of the Inert Doublet Model with a global U(1) symmetry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D102 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 7 075011, [arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='10263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Soko�lowska, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swaczyna, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' ´Swie˙zewska, Higgs → γγ, Zγ in the Inert Doublet Model, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B44 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 2163–2170, [arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='7880].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [53] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Enberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rathsman, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Wouda, Higgs properties in a broken Inert Doublet Model, JHEP 08 (2013) 079, [arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1714].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [Erratum: JHEP01,087(2015)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [54] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sokolowska, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swiezewska, Inert Doublet Model with a 125 GeV Higgs, in Proceedings, 1st Toyama International Workshop on Higgs as a Probe of New Physics 2013 (HPNP2013): Toyama, Japan, February 13-16, 2013, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='7757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mawatari, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sakurai, Single Higgs production in association with a photon at electron-positron colliders in extended Higgs models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 99 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 035023, [arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='10268].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hashemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Najjari, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' ˙Zarnecki, Production of Inert Scalars at the high energy e+e− colliders, arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='01175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [JHEP02,187(2016)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [57] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ahriche, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Jueid, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Nasri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' de La Puente, Mono-Higgs Signature in the Compressed scotogenic Model, arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [58] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Benbrik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' El Falaki, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Jueid, Radiative corrections to the Triple Higgs Coupling in the Inert Higgs Doublet Model, JHEP 12 (2015) 007, [arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='03630].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 12 [59] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abouabid, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Falaki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Xie, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yan, Full one-loop radiative corrections to e+e− → H+H− in the Inert Higgs Doublet Model, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='05237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sakurai, Testing the dark matter scenario in the inert doublet model by future precision measurements of the Higgs boson couplings, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 94 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 115011, [arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='08520].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [61] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abouabid, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Benbrik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' El Falaki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Xie, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yan, One-loop radiative corrections to e+e− → Zh0/H0A0 in the Inert Higgs Doublet Model, JHEP 05 (2021) 100, [arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='03250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [62] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aiko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mawatari, Next-to-leading-order corrections to the Higgs strahlung process from electron–positron collisions in extended Higgs models, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C 81 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 1000, [arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02884].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [63] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ramsey-Musolf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zhou, Probing extended scalar sectors with precision e+e− → Zh and Higgs diphoton studies, JHEP 10 (2021) 155, [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='10709].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [64] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Banerjee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Boudjema, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chakrabarty, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chalons, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sun, Relic density of dark matter in the inert doublet model beyond leading order: The heavy mass case, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D100 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 9 095024, [arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='11269].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [65] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Banerjee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Boudjema, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chakrabarty, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sun, Relic density of dark matter in the inert doublet model beyond leading order for the low mass region: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Renormalisation and constraints, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [66] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Banerjee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Boudjema, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chakrabarty, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sun, Relic density of dark matter in the inert doublet model beyond leading order for the low mass region: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Co-annihilation, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [67] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Banerjee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Boudjema, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chakrabarty, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sun, Relic density of dark matter in the inert doublet model beyond leading order for the low mass region: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Annihilation in 3-body final state, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Banerjee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Boudjema, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chakrabarty, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sun, Relic density of dark matter in the inert doublet model beyond leading order for the low mass region: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' The Higgs resonance region, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [69] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Banerjee and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chakrabarty, A revisit to scalar dark matter with radiative corrections, JHEP 05 (2019) 150, [arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='01973].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [70] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mawatari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sakurai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, Full next-to-leading-order calculations of Higgs boson decay rates in models with non-minimal scalar sectors, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B 949 (2019) 114791, [arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='10070].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [71] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mawatari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sakurai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, H-COUP Version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 257 (2020) 107512, [arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='12769].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 13 [72] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sakurai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, H-COUP: a program for one-loop corrected Higgs boson couplings in non-minimal Higgs sectors, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 233 (2018) 134–144, [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04603].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [73] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ferreira and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swiezewska, One-loop contributions to neutral minima in the inert doublet model, JHEP 04 (2016) 099, [arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02879].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [74] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Braathen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, Leading two-loop corrections to the Higgs boson self-couplings in models with extended scalar sectors, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C80 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 227, [arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='11507].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [75] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Braathen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, On two-loop corrections to the Higgs trilinear coupling in models with extended scalar sectors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B796 (2019) 38–46, [arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='05417].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [76] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Senaha, Radiative Corrections to Triple Higgs Coupling and Electroweak Phase Transition: Beyond One-loop Analysis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D100 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 5 055034, [arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00336].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [77] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Okada, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Senaha, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yuan, Higgs coupling constants as a probe of new physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 70 (2004) 115002, [hep-ph/0408364].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [78] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kiyoura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Okada, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Senaha, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yuan, New physics effect on the Higgs selfcoupling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B 558 (2003) 157–164, [hep-ph/0211308].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [79] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, Fingerprinting the extended Higgs sector using one-loop corrected Higgs boson couplings and future precision measurements, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B 896 (2015) 80–137, [arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07716].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [80] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hollik and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Penaranda, Yukawa coupling quantum corrections to the selfcouplings of the lightest MSSM Higgs boson, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C 23 (2002) 163–172, [hep-ph/0108245].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [81] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Bahl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Braathen, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Weiglein, New Constraints on Extended Higgs Sectors from the Trilinear Higgs Coupling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 129 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 23 231802, [arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='03453].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [82] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Braathen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Shimoda, Two-loop analysis of classically scale-invariant models with extended Higgs sectors, JHEP 03 (2021) 297, [arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07580].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [83] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sakurai, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, Gauge invariant one-loop corrections to Higgs boson couplings in non-minimal Higgs models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 96 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 035014, [arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='05399].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [84] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, One-loop corrections to the Higgs self-couplings in the singlet extension, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B 917 (2017) 154–177, [arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='01582].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [85] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' He and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zhu, One-loop radiative correction to the triple Higgs coupling in the Higgs singlet model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B 764 (2017) 31–37, [arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04497].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='B 797, 134782 (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [86] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aoki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kikuchi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, Radiative corrections to the Higgs boson couplings in the triplet model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 87 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 1 015012, [arXiv:1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='6029].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 14 [87] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Basler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M¨uhlleitner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Wittbrodt, The CP-Violating 2HDM in Light of a Strong First Order Electroweak Phase Transition and Implications for Higgs Pair Production, JHEP 03 (2018) 061, [arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04097].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [88] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Basler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M¨uhlleitner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M¨uller, Electroweak Phase Transition in Non-Minimal Higgs Sectors, JHEP 05 (2020) 016, [arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='10477].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [89] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Moyotl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Chamorro, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Castilla-Valdez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' P´erez, New physics effects in the Higgs trilinear self-coupling through one-loop radiative corrections, arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='06299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [90] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krause, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lorenz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Muhlleitner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Santos, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ziesche, Gauge-independent Renormalization of the 2-Higgs-Doublet Model, JHEP 09 (2016) 143, [arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04853].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [91] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Osland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Pandita, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Selbuz, Trilinear Higgs couplings in the two Higgs doublet model with CP violation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 78 (2008) 015003, [arXiv:0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0060].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [92] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Barger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Berger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Stange, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phillips, Supersymmetric Higgs boson hadroproduction and decays including radiative corrections, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 45 (1992) 4128–4147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [93] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Borschensky, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gabelmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M¨uhlleitner, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rzehak, The Trilinear Higgs Self-Couplings at O(α2 t ) in the CP-Violating NMSSM, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [94] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Brucherseifer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gavin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Spira, Minimal supersymmetric Higgs boson self-couplings: Two-loop O(αtαs) corrections, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 90 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 117701, [arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [95] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Nhung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Muhlleitner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Streicher, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Walz, Higher Order Corrections to the Trilinear Higgs Self-Couplings in the Real NMSSM, JHEP 11 (2013) 181, [arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3926].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [96] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dobado, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Herrero, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hollik, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Penaranda, Selfinteractions of the lightest MSSM Higgs boson in the large pseudoscalar mass limit, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 66 (2002) 095016, [hep-ph/0208014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [97] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Shindou, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Yagyu, Non-decoupling effects in supersymmetric Higgs sectors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B 699 (2011) 258–263, [arXiv:1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1836].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [98] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanemura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kubota, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Takasugi, Lee-Quigg-Thacker bounds for Higgs boson masses in a two doublet model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B313 (1993) 155–160, [hep-ph/9303263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [99] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Akeroyd, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Arhrib, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Naimi, Note on tree level unitarity in the general two Higgs doublet model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B490 (2000) 119–124, [hep-ph/0006035].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [100] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Branco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ferreira, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lavoura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rebelo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sher, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Silva, Theory and phenomenology of two-Higgs-doublet models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 516 (2012) 1–102, [arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='0034].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [101] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Ginzburg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kanishev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sokolowska, Evolution of Universe to the present inert phase, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D82 (2010) 123533, [arXiv:1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='4593].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 15 [102] ATLAS Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aaboud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Measurements of Higgs boson properties in the diphoton decay channel with 36 fb−1 of pp collision data at √s = 13 TeV with the ATLAS detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 98 (2018) 052005, [arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [103] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Measurements of Higgs boson properties in the diphoton decay channel in proton-proton collisions at √s = 13 TeV, JHEP 11 (2018) 185, [arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='02716].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [104] ATLAS Collaboration, Combination of searches for invisible Higgs boson decays with the ATLAS experiment, ATLAS-CONF-2020-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [105] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Swiezewska and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Krawczyk, Diphoton rate in the inert doublet model with a 125 GeV Higgs boson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D88 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 035019, [arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='4100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [106] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Belanger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dumont, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Goudelis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Herrmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Kraml, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sengupta, Dilepton constraints in the Inert Doublet Model from Run 1 of the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D91 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 115011, [arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='07367].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [107] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lundstrom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Gustafsson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Edsjo, The Inert Doublet Model and LEP II Limits, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D79 (2009) 035013, [arXiv:0810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='3924].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [108] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Peskin and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Takeuchi, Estimation of oblique electroweak corrections, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D46 (1992) 381–409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [109] Particle Data Group Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Tanabashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Review of Particle Physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D98 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 3 030001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [110] Particle Data Group Collaboration, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Review of Particle Physics, PTEP 2020 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 8 083C01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [111] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abbott and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sikivie, A Cosmological Bound on the Invisible Axion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 120B (1983) 133–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [112] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dine and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Fischler, The Not So Harmless Axion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 120B (1983) 137–141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [113] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Preskill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Wise, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Wilczek, Cosmology of the Invisible Axion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 120B (1983) 127–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [114] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Belanger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Mjallal, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Pukhov, Recasting direct detection limits within micrOMEGAs and implication for non-standard Dark Matter scenarios, pre-print (3, 2020) [arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='08621].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [115] XENON Collaboration, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aprile et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Dark Matter Search Results from a One Ton-Year Exposure of XENON1T, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 121 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 11 111302, [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='12562].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [116] PICO Collaboration, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Amole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Dark Matter Search Results from the Complete Exposure of the PICO-60 C3F8 Bubble Chamber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 100 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 2 022001, [arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='04031].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [117] CRESST Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Abdelhameed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', First results from the CRESST-III low-mass dark matter program, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' D 100 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 10 102002, [arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='00498].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 16 [118] DarkSide Collaboration, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Agnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Low-Mass Dark Matter Search with the DarkSide-50 Experiment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 121 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 8 081307, [arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='06994].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [119] ATLAS Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Aaboud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Search for dark matter and other new phenomena in events with an energetic jet and large missing transverse momentum using the ATLAS detector, JHEP 01 (2018) 126, [arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='03301].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [120] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=', Search for dark matter produced with an energetic jet or a hadronically decaying W or Z boson at √s = 13 TeV, JHEP 07 (2017) 014, [arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='01651].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [121] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 140 (2001) 418–431, [hep-ph/0012260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [122] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hahn and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Perez-Victoria, Automatized one loop calculations in four-dimensions and D-dimensions, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 118 (1999) 153–165, [hep-ph/9807565].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [123] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hahn and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rauch, News from FormCalc and LoopTools, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 157 (2006) 236–240, [hep-ph/0601248].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [,236(2006)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [124] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hahn, Loop calculations with FeynArts, FormCalc, and LoopTools, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' B30 (1999) 3469–3475, [hep-ph/9910227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [125] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hahn, Feynman Diagram Calculations with FeynArts, FormCalc, and LoopTools, PoS ACAT2010 (2010) 078, [arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='2231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [126] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Bohm, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Spiesberger, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hollik, On the One Loop Renormalization of the Electroweak Standard Model and Its Application to Leptonic Processes, Fortsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 34 (1986) 687–751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [127] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Hollik, Radiative Corrections in the Standard Model and their Role for Precision Tests of the Electroweak Theory, Fortsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 38 (1990) 165–260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [128] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 41 (1993) 307–420, [arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='1075].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' [129] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Denner and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Dittmaier, Electroweak Radiative Corrections for Collider Physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 864 (2020) 1–163, [arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content='06823].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFST4oBgHgl3EQfUjgb/content/2301.13773v1.pdf'} diff --git a/ztFQT4oBgHgl3EQfCjXs/content/tmp_files/2301.13231v1.pdf.txt b/ztFQT4oBgHgl3EQfCjXs/content/tmp_files/2301.13231v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..12743df1c86310e3d397a9bc1398ae7db0d4e4c9 --- /dev/null +++ b/ztFQT4oBgHgl3EQfCjXs/content/tmp_files/2301.13231v1.pdf.txt @@ -0,0 +1,2364 @@ +Prepared for submission to JHEP +Logarithmic, Fractal and Volume-Law +Entanglement in a Kitaev chain with long-range +hopping and pairing +Andrea Solfanelli,a,b,c,1 Stefano Ruffoa,b,d Sauro Succic,e Nicolò Defenuf +aSISSA, via Bonomea 265, I-34136 Trieste, Italy +bINFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy +cCenter for Life Nano-Neuro Science @ La Sapienza, Italian Institute of Technology, 00161 Roma, Italy +dIstituto dei Sistemi Complessi, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy +ePhysics Department, Harvard University, Oxford Street 17, Cambridge, USA +fInstitut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Str. 27 Zürich, Switzerland +E-mail: asolfane@sissa.it +Abstract: Thanks to their prominent collective character, long-range interactions pro- +mote information spreading and generate forms of entanglement scaling, which cannot be +observed in traditional systems with local interactions. In this work, we study the asymp- +totic behavior of the entanglement entropy for Kitaev chains with long-range hopping and +pairing couplings decaying with a power law of the distance. We provide a fully-fledged +analytical and numerical characterization of the asymptotic growth of the ground state en- +tanglement in the large subsystem size limit, finding that the truly non-local nature of the +model leads to an extremely rich phenomenology. Most significantly, in the strong long- +range regime, we discovered that the system ground state may have a logarithmic, fractal, +or volume-law entanglement scaling, depending on the value of the chemical potential and +on the strength of the power law decay. +1Corresponding author. +arXiv:2301.13231v1 [quant-ph] 30 Jan 2023 + +Contents +1 +Introduction +1 +2 +Kitaev chain with long-range couplings +4 +3 +Entanglement scaling in free fermioninc systems +8 +4 +Weak long-range regime +11 +5 +Strong long-range regime +15 +6 +Conclusion and outlooks +19 +A Derivation of the matrix symbol +22 +B Coefficients of the Fisher-Hartwig expansion +22 +C Dispersion relation around the critical modes +26 +D Discontunities in the strong long-range regime +27 +1 +Introduction +The interest of the quantum community in long-range physics has steadily risen in recent +years since long-range interacting quantum systems, i.e., systems in which the coupling en- +ergy between couples of microscopic constituents Vi,j decays as a power law of their distance +r = |i − j|: Vi,j ∝ r−α, with α > 0 [1, 2] are emerging as promising platforms for quantum +technological applications. This is due to their stability against external perturbations, +which allows keeping the impact of dynamically generated excitations under control, there- +fore mitigating their detrimental effects [2, 3]. An example of the rigidity of long-range +interacting platforms against external drivings and of its utility for quantum technological +applications is the possibility for such systems to host clean discrete Floquet time crystal +phases [4–7]. Another example is the recently introduced advantage in the finite time per- +formance of quantum heat-engines with a working substance hosting long-range couplings +[8]. Moreover, this technological and theoretical interest is also supported from the ex- +perimental side by the possibility to implement long-range interacting systems in typical +quantum simulation platforms, such as atomic molecular and optical (AMO) systems [9– +13]. Interestingly, trapped ions setups allow tuning the power law exponent α, dictating +the decay of the interaction energy with distance, from α ≃ 0 to α ≃ 3 [9]. +The most important feature a system should have to be a good candidate for quantum +technologies is the capability of hosting highly entangled states in its spectrum. Indeed, this +– 1 – + +crucial property is the essential ingredient to perform tasks that are classically impossible +or very inefficient [14]. More precisely, entanglement is the property that makes quantum +computation overtake classical one providing the computational speed-up in quantum algo- +rithms as compared to algorithms based on the processes of classical physics [15]. Moreover, +it is crucial for many quantum technological applications such as quantum teleportation [16], +quantum cryptography [17] or quantum metrology [18]. +A set of key quantities entering the characterization of entanglement is provided by +the entanglement Rényi entropies. For their definition, one takes a partition of a given +system in two subsystems A and B (the complement of A), determines the reduced density +matrix of a subsystem (say, of A) ρA by tracing out the degrees of freedom of B, and then +computes its Rényi entropies: Sν = ln Tr[ρν +A]/(1 − ν) [19]. One of the most fundamental +properties of entanglement Rényi entropies is their behavior with the size of the subsystem +considered. The celebrated area law [20, 21] refers to the fact that typically entanglement +grows as the boundary of the subsystem considered, i.e., for a system in d dimensions and a +subsystem of size L having volume ∼ Ld and area ∼ Ld−1, then the entanglement entropy +of the subsystem scales as ∼ Ld−1. +In particular, the area law has been proven to be +satisfied in the ground state of one-dimensional systems with mass gap and short-range +couplings when the size of the subsystem is much larger than the correlation length [22]. +At a quantum critical point, where the correlation length diverges, the area law is known +to be violated by a logarithmic term proportional to the central charge of the conformal +field theory (CFT) that describes the low-energy spectrum of the model [23–28]. These +facts motivated initially the study of this quantity due to its similarity to the black hole +entropy [20, 29], and have eventually revealed the important role that entanglement plays +in high-energy physics [30–33] as well as in the investigation of condensed matter systems +[34–36]. +The previous discussion changes and becomes more involved for systems with long-range +couplings [2, 37, 38]. Indeed the prominent collective character of such non-local systems +promotes entanglement spreading and leads to novel forms of equilibrium and dynamical +scaling, which cannot be observed in traditional systems with local interactions [39–41]. In +particular, the anomalous scaling of entanglement in the presence of long-range couplings +has recently attracted great interest in the context of the so-called measurement-induced +transitions [42–48]. +In this case, the dynamical generation of entanglement is weakend +by the presence of local measures applied randomly during the system evolution. More +precisely, if the measurement rate is high enough, the steady state entanglement saturates +to an area law value independent of the considered subsystem size, if only nearest neighbor +interactions are present [3]. On the other hand, in the presence of long-range couplings, +subvolume law scalings [3, 49–52], also referred to as fractal entanglement phases [53, 54], +appear. +These interesting dynamical phenomena have no clear equilibrium counterpart showing +that their origin is directly related to the presence of long-range interactions. The entangle- +ment properties of the ground state of a fermionic chain with long-range pairing couplings +and nearest neighbors hopping amplitudes were fully characterized in Refs. [55–59] which +reported standard logarithmic violations of the area law in the weak long-range regime. +– 2 – + +Moreover, an anomalous logarithmic growth was found even if the mass gap is not zero, +associated to the divergence of unnormalized couplings, in the strong long-range regime +characterized by a power law decay exponent smaller than the system dimension. On the +other hand, the authors of Refs. [60, 61] considered a model of fermions with strong long- +range hopping amplitudes and no pairing discovering a volume law entanglement scaling. +Moreover, the entanglement properties of the Sachdev-Ye-Kitaev (SYK) model [62, 63], +i.e. a fully connected fermionic model with random interactions, have been extensively +studied [64]. Also in this case, the eigenstates of the SYK Hamiltonian display a volume +law entanglement scaling whose coefficient has been computed numerically using exact di- +agonalization techniques [65, 66] and analytically assuming the eigenstate thermalization +hypothesis [67] or using a path-integral approach which becomes exact in the large-N limit +[68, 69]. Finally, also in long-range bosonic [70] and in fully connected spin systems [71–75] +only logarithmic violations of the area law were reported. +Despite the extensive amount of literature on the topic summarized above, none of the +considered long-range models display a fractal entanglement scaling at equilibrium unless +additional ingredients are added such as modifications of the couplings which violate time +translational symmetry or the presence of a fractal Fermi surface [60]. Here, we are going +to show that the subvolume law observed in measurement induced transitions [3, 49–54] is +directly caused by long-range interactions and also appears at equilibrium, provided certain +conditions are met. +To prove our claim, we study the ground state entanglement scaling in a prototypical +model of fermions with power-law decaying hopping and pairing amplitudes, also known +as the long-range Kitaev chain [2, 76]. This model is sufficiently simple to allow us to +perform analytic calculations but at the same time it turns out to host an extremely rich +phenomenology. Using the well-known Fisher-Hartwig expansion [77, 78], we were able to +analytically determine the leading order dependence of the ground state entanglement on +the subsystem size L in the scaling limit of an infinite chain of N → ∞ sites and infinite +subsystem L → ∞ with fixed l = L/N, for different values of the available parameters. +In particular, we can distinguish two main regimes: the weak long-range regime in which +the coupling’s power law decaying exponents are larger than the system dimension and +the strong long-range regime in which they are smaller. In the former case, the system +shows standard logarithmic deviations from the entanglement area law in correspondence +with the quantum critical points, however, in the most interesting case of equal long- +range hopping and pairing the coefficients in front of these logarithmic divergences show a +nontrivial dependence on the power law decay exponent α which is not compatible with the +standard scaling predicted by critical conformal field theory [23, 24]. On the other hand, in +the strong long-range case, the system becomes genuinely non-additive, therefore showing a +logarithmic deviation from the area law even away from criticality. Most significantly, when +the system chemical potential is zero, no local terms are present in the Hamiltonian (as +we will see this simple fact strongly affects the nature of the ground state which becomes +highly degenerate) thus resulting into a subvolume law entanglement scaling, S ∼ L1−2α. +Summarizing, our work correctly reproduces previously known results in different limits, +thus bringing several disparate results present in the literature into a coherent picture. +– 3 – + +Moreover, we are able to detect a fractal entanglement scaling phase which is entirely +due to the non-additive nature of the model and does not need the dynamical setting of +measurement induced transitions to be observed. +The paper is organized as follows. In Section 2 we introduce the long-range Kitaev +model and we describe its phase diagram. In Section 3 we briefly review the techniques +which allow us to study the entanglement scaling of generic quadratic fermionic models +(the expert reader may safely skip this part). Finally, Section 4 and 5 are devoted to the +detailed characterization of the ground state entanglement scaling of the model in the weak +and strong long-range regimes, respectively. +2 +Kitaev chain with long-range couplings +We consider a generic model of spinless fermions hopping across the N sites of a one- +dimensional chain in the presence of pairing interactions, and with a chemical potential h. +Assuming periodic boundary conditions, the system Hamiltonian reads +H = − +N +� +j=1 +N/2−1 +� +r=1 +� +trˆc† +j+rˆcj + ∆rˆc† +j+rˆc† +j + h.c. +� +− h +N +� +j=1 +� +1 − 2ˆc† +jˆcj +� +, +(2.1) +where ˆc† +j and ˆcj are creation and annihilation operators for fermions at site j, while tr and +∆r are the hopping and pairing amplitudes, respectively. We choose their dependence on +the intersite distance r according to the power laws +tr = +1 +Nα1 +J +rα1 , +∆r = +1 +Nα2 +∆ +rα2 , +(2.2) +with the hopping exponent α1 > 0, the pairing exponent α2 > 0, and Nα = �N/2 +r=1 r−α the +Kac scaling factor [79], which guarantees extensivity of the energy in the case αi < 1, with +i = 1, 2. This model, often referred to as long-range Kitaev chain [76], is emerging as a +minimal model for the study of the effects of long-range couplings on a quantum system +[2]. +Indeed, its integrable nature makes it amenable to both analytical and numerical +treatment. Moreover, as observed in Refs. [80–82], when the pairing and hopping power +law decay exponents are equal α1 = α2 = α the model can be related to the quantum +Ising model. In particular, in the short-range case with α → ∞, the relation becomes exact +through the Jordan-Wigner mapping [83]. +The quadratic nature of the Hamiltonian (2.1) allows its exact diagonalization in Fourier +space via the Bogolyubov transformation +ˆck = cos θk +2 ˆγk + sin θk +2 ˆγ† +−k, +(2.3) +– 4 – + +where we have introduced the momentum space fermionic operators +ˆck = e−i π +4 +√ +N +N +� +j=1 +eikjˆcj, +(2.4) +with k = 2πn/N, with n = −N/2+1, . . . , N/2. While the Bogoliubov angles are defined by +the conditions tan θk = ∆k/(h − tk), where Fourier transforms of the hopping and pairing +amplitudes are defined as +tk = +1 +Nα1 +N/2−1 +� +r=1 +cos(kr) +rα1 +, +∆k = +1 +Nα2 +N/2−1 +� +r=1 +cos(kr) +rα2 +. +(2.5) +In terms of the Bogoliubov fermions, the Hamiltonian then takes the diagonal form +H = +� +k +ωk(h) +� +ˆγ† +kˆγk − 1/2 +� +, +(2.6) +with the quasiparticle spectrum +ωk(h) = 2 +� +(h − tk)2 + ∆2 +k. +(2.7) +Since ωk(h) ≥ 0, the ground state corresponds to the Fock space vacuum for the Bogoliubov +modes, defined by the condition ˆγk|gs⟩ = 0, ∀k. +When studying the critical properties associated with the spectrum (2.7), we must +distinguish two main regimes: the weak long-range regime when α1, α2 > 1, i.e., the power +law decay exponents are larger than the system dimensionality, and the strong long-range +regime when α1, α2 < 1. In the weak long-range case, the Kac scaling is a constant in the +thermodynamic limit: Nα>1 → ζ(α), where ζ(α) is the Riemann zeta function. Moreover, +when the system size goes to infinity, we can safely perform a continuum limit in the k +variable. In particular, Eq. (2.5) may be written as +tk = Re +� +Liα1(eik) +� +/ζ(α1), +∆k = Im +� +Liα2(eik) +� +/ζ(α2), +(2.8) +where Liα(z) denotes the polylogarithm function. +This leads to a continuum spectrum +ωk characterized, at the critical points, by a dispersion relation that depends on α1 and +α2. In particular, for α1, α2 > 1, the system possesses two different phases separated by +two quantum critical points hc = 1, −1 + 21−α1, in correspondence of which the dispersion +relation becomes gapless near to the critical mode kc = 0, π, respectively [2, 84]. +The +critical modes of the spectrum are shown in Fig. 1a where ω0(π)(blue(red) lines in the plot) +is plotted as a function of h for different values of α1 = α2. The nature of the transition +is topological and the two topological phases can be distinguished by the value of the bulk +topological invariant [85] +w = +� dθk +2π = +� +1 +if +h ∈ [−1 + 21−α1, 1] +0 +overwise +, +(2.9) +– 5 – + +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +h +0 +2 +4 +6 +ω0,π +(a) +α1 = 0.1 +α1 = 0.5 +α1 = 1.5 +α1 = 8 +0 +2 +4 +6 +8 +α1 +−1 +0 +1 +h +q = −1 +q = +1 +(b) +h = t0 +h = tπ +Figure 1. a) Critical modes k = 0, π of the quasiparticle spectrum as a function of the chemical +potential h for different values of α1 = α2, two critical points emerge at h = t0, tπ, where in the +thermodynamic limit t0 = 1 and tπ = 1 if α1 > 1, tπ = −1 + 21−α1 if 1 < α1 < 2, and tπ = 0 if +0 < α1 < 1. b) Phase diagram of the long-range Kitaev chain in the plane (α1, h), for the pairing +decay exponent α2 = α1, α1 is the hopping decay exponent and h is the chemical potential. The +topological order parameter is q = −1 in the topological phase (blue shaded region) and q = +1 +in the trivial phase (red shaded region). The phase space boundaries correspond to the solid lines +h = t0 and h = tπ. +where the Bogoliubov angles are defined as θk = arctan(∆k/(h − tk)). +In the strong long-range regime 0 < α1, α2 < 1 the scenario is more complicated. +Indeed, in this case, the Kac normalization factor Nα diverges at large N as Nα ≈ N1−α, +and the thermodynamic limit of Eq. (2.5) has to be carefully considered. In particular, as +pointed out in Ref. [86], while the Fourier modes variable k = 2πn/N becomes continuous +as N → ∞, the hopping and pairing amplitudes tk, ∆k, remain discrete and labeled by the +integer n, reading +lim +N→∞ tk = cα1 +� 1/2 +0 +dscos(2πns) +sα1 += tn, +(2.10) +lim +N→∞ ∆k = cα2 +� 1/2 +0 +dssin(2πns) +sα2 += ∆n, +(2.11) +with cα = (1 − α)21−α. Therefore, the presence of long-range couplings leads to a discrete +spectrum ωk → ωn = 2 +� +(h − tn)2 + ∆2n also at N → ∞. The persistence of the discrete +spectrum in the thermodynamic limit does not allow us to define a continuous theory and +hinders the conventional definition of quantum critical points in the Kitaev chain. In par- +ticular, the winding number in Eq. (2.9) is ill-defined as a consequence of the discontinuity +in the Bogolyubov angle distribution [85]. Yet, the transition can still be characterized by +the quantity +q = sign[(h − t0)(h − tπ)] = +� +1 +if +h ∈ [tπ, t0] +−1 +overwise +. +(2.12) +This quantity has proven to be a good topological invariant in cases in which the winding +number turns out to be ill-defined [85, 87]. Then, also in the strong long-range regime, +– 6 – + +the behavior of the order parameter q is still consistent with a change of phase at the +critical points h = t0, tπ [86]. However, as shown in [88], the bulk boundary correspondence +turns out to be weakened by the presence of strong long-range couplings. Consequently, +the change of q at the critical points is not guaranteed to be in one-to-one correspondence +with the appearance of boundary topological edge states. Nevertheless, we expect bulk +properties to remain consistent with a change of phase. Figure 1b shows the model phase +diagram as characterized by the value of q = ±1 as a function of the chemical potential h +and of the hopping power law decay exponent α1. Two quantum critical lines appear when +varying the α1 parameter. In particular, we notice that the location of the critical point +corresponding to ω0 = 0 is fixed to h = t0 = 1 for any value of α1 (blue bold line in Fig. +1b). On the contrary, the critical point corresponding to ωπ = 0 (red bold line in Fig. 1b) +is α1 dependent with two different behaviors in the weak and strong long-range regimes, in +particular in the thermodynamic limit we find +lim +N→∞ tπ = +� +� +� +� +� +� +� +−1 +if +α1 > 2 +−1 + 21−α1 +if +1 < α1 < 2 +0 +if +0 < α1 < 1 +. +(2.13) +Finally, the completely mean-field case with α1 = α2 = 0 needs to be treated separately. +Indeed, in this case, the spectrum becomes strongly degenerate and this may alter the nature +of the ground state. In particular, for completely flat couplings the sums in Eq. (2.5) can +be exactly computed and, in the thermodynamic, they read +tn(α1 = 0) = δn,0, +∆n(α2 = 0) = 1 + (−1)n+1 +πn +. +(2.14) +Accordingly, the single-particle spectrum becomes +ω0 +n = +� +� +� +� +� +� +� +|h| +if |n| even +� +h2 + 4/(πn)2 +if |n| odd +|h − 1| +if n = 0 +, +(2.15) +where we have introduced the shortcut notation ω0 +n = ωn(α1 = 0, α2 = 0). It follows that an +extensive number of single-particle energy levels corresponding to all the even modes become +degenerate. In particular, when the chemical potential is zero h = 0 all the even modes +become zero modes since at this point we have ω0 +2n(h = 0) = 0, ω0 +2n+1(h = 0) = 2/|πn| and +ω0 +0(h = 0) = 1. This fact deeply affects the nature of the many-body ground state which is +no more given by the Bogoliubov vacuum, on the contrary, it allows for a finite population +of Bogoliubov fermions in an extensive number of zero modes. More precisely, the ground +state for α1,2 = 0 and h = 0 is given by a generic superposition of the form +|gs⟩α=0,h=0 = +N0 +� +n0=0 +Cn0|n0⟩, +(2.16) +– 7 – + +1 +L +N +A +B +Figure 2. Schematic representation of a bipartition of a long-range Kitaev chain with periodic +boundary conditions in two subsystems A and B of length L and N − L respectively. +where n0 is the number of fermions occupying the N0 available zero modes. This ground +state is highly degenerate indeed each |n0⟩ state can be realized in +�N0 +n0 +� +ways, leading to +the exponential degeneracy +Deg[|gs⟩α=0,h=0] = +N0 +� +n0=0 +�N0 +n0 +� += 2N0. +(2.17) +As a concluding remark for this section, we stress the importance of the Kac scaling +in the stabilization of the topological order in the strong long-range regime. Indeed, had +we considered not properly rescaled couplings, the presence of long-range hopping α1 < 1 +would have moved the critical point to hc = O(N1−α1) → ∞, thus destroying the transition. +3 +Entanglement scaling in free fermioninc systems +We consider a bipartition of the fermionic chain described by the Hamiltonian in Eq. (2.1), +in two subsystems A and B, where A is a continuous interval of chain sites of length L and +B is its complementary set, see Fig. 2. Given the Hilbert spaces HA and HB associated +to A and B, respectively, then the total Hilbert space of the system can be written as the +tensor product H = HA ⊗ HB. If the total system is in a pure state |ψ⟩, then the reduced +density matrix, describing the state of subsystem A(B) is obtained by taking the partial +trace with respect to HA(B): ρA(B) = TrA(B)|ψ⟩⟨ψ|. The amount of entanglement between +the two subsystems can be characterized by the so-called Rényi entropies of A, defined as +Sν,L(A) = +1 +1 − ν ln Tr[ρν +A], +(3.1) +where ν ≥ 1. These are known to provide an accurate measure for the entanglement of a +bipartite system in a pure state [19]. In particular, the limit ν → 1 of the above expression +– 8 – + +corresponds to the celebrated Von Neumann or entanglement entropy +SL(A) = S1,L(A) = −Tr[ρA ln ρA]. +(3.2) +The main goal of this paper is to study the Rényi entanglement entropy for the ground +state of a Hamiltonian of the kind analyzed in the previous Section. In particular, we are +interested in determining the dependence of Sν,L(A) on the subsystem size L in the scaling +limit N → ∞, L → ∞ with fixed l = L/N and how this is affected by the presence of long- +range hopping and pairing couplings in the Hamiltonian. This task may be achieved by +taking advantage of the fact, that since the Hamiltonian in Eq. (2.1) is quadratic, then all +its eigenstates satisfy the Wick decomposition theorem [26, 89]. Accordingly, the reduced +density matrix can be obtained from the two-point correlation functions. In particular, +given the 2L × 2L correlation matrix defined as +Vij = +� +δij − 2⟨c† +jci⟩ +2⟨cicj⟩ +2⟨c† +ic† +j⟩ +2⟨c† +icj⟩ − δij +� +, +(3.3) +it can be shown [26, 89] that this is related to the Rényi entropies through the formula +Sν,L(A) = +1 +2(ν − 1)Tr ln +��I + V +2 +�ν ++ +�I − V +2 +�ν� +. +(3.4) +It is important to notice that, from the computational point of view, this formula constitutes +a dramatic simplification since the problem complexity is reduced from the diagonalization +of a reduced density matrix of size 2L × 2L to the diagonalization of the correlation ma- +trix (3.3) of size 2L × 2L, thus allowing to reach larger sizes L. From the analytic side, it is +useful to write Eq. (3.4) as an integral on the complex plane along a contour C surrounding +the eigenvalues vj ∈ [−1, 1] of V. Using Cauchy’s residue theorem in order to perform the +integral, one gets [90, 91] +Sν,L(A) = lim +ϵ→0+ +� +C +sν(1 + ϵ, z)d ln DL(z) +dz +dz, +(3.5) +where we have introduced the function +sν(x, y) = +1 +1 − ν ln +��x + y +2 +�ν ++ +�x − y +2 +�ν� +, +(3.6) +and the determinant +DL(z) = det(zI − V). +(3.7) +Due to the translational invariance of the Hamiltonian (2.1) and given the choice of sub- +system A, which is composed of contiguous sites, the correlation matrix is a block Toeplitz +matrix, which can be written in the Fourier basis as +Vlj = 1 +N +� +k +Gkeik(l−j), +(3.8) +– 9 – + +where we have introduced the two dimensional symbol Gk which, as detailed in Appendix +A, can be written as +Gk = (1 − (fk + fk)) +�h − tk +ωk +σz − ∆k +ωk +σy +� +− (fk − f−k)I, +(3.9) +where σa, with a = x, y, z, are the Pauli sigma matrices, I is the 2 × 2 identity, and +fk = ⟨ˆγ† +kˆγk⟩ are the occupation numbers of the Bogoliubov fermionic modes, which for a +generic state satisfy the condition 0 ≤ fk ≤ 1. +Using the techniques introduced in Refs. [26, 89] the asymptotic behavior for L → ∞ of +the Toeplitz determinant DL(z), entering the expression for the Rényi entropies (3.4), can +be determined applying the Szegő-Widom theorem [92, 93] and an extension of the Fisher- +Hartwig conjecture [77, 78] to non-scalar symbols [57, 58]. The leading order contributions +to the logarithm of DL(z) in the L → ∞ limit then read +ln DL(z) = L +2π +� π +−π +dk ln det(zI − Gk) ++ ln L +� +p +bp(z) + O(1), +(3.10) +where the coefficients bp(z) of the logarithmic contribution are associated to the disconti- +nuities of Gk. More precisely, if there is a discontinuity at some k = p, this means that +G+ +p = lim +k→p+ Gk ̸= lim +k→p− Gk = G− +p , +(3.11) +then the coefficient corresponding to such discontinuity can be computed as [58] +bp(z) = +1 +4π2 Tr[ln(zI − G− +p )(zI − G+ +p )−1]2. +(3.12) +Inserting Eq. (3.10) into the integral for the Rényi entropy (3.5) one obtains +Sν,L = +1 +1 − ν +� +k +ln [(1 − fk)ν + fν +k ] + Bν ln L + O(1), +(3.13) +where the coefficient of the logarithmic contribution can be computed as +Bν = +� +p +lim +ϵ→0+ +� +C +sν(1 + ϵ, z)dbp(z) +dz +dz. +(3.14) +As shown in Section 2, whenever α1,2 > 0 or α1 = α2 = 0 and h ̸= 0, the many-body ground +state of the system is the Bogoliubov vacuum with fk = 0 ∀k, therefore we are left with +a leading order contribution given by a constant term O(1) corresponding to the standard +area law in the one-dimensional case, or a logarithmic contribution which is associated to +the discontinuity of the correlation matrix symbol Gk. On the other hand in the specific +case α1 = α2 = 0 and h = 0 the many-body ground state becomes highly degenerate +allowing for a finite fermionic population fk ̸= 0 for an extensive number of Bogoliubov +modes, i.e., all the even modes. As a consequence, the first term in Eq. (3.13) becomes +– 10 – + +the leading contribution to the large L entanglement scaling corresponding to a volume law +behavior Sν,L(α1,2 = 0, h = 0) ≈ L. +Summarizing, the machinery introduced in this section allows us to compute the leading +order contribution to the scaling of Rényi entropies with the subsystem size by simply +analyzing the symbol continuity properties in the different regimes. +4 +Weak long-range regime +Let us start with the weak long-range regime corresponding to α1, α2 > 1. In this case, as +we have seen in Section 2, the quasiparticle spectrum is continuous in the thermodynamic +limit, and the ground state is always given by the Bogoliubov vacuum with zero fermionic +populations fk = 0, ∀k. Accordingly, the first term of the Fisher-Hartwig expansion (3.13) +vanishes and then the leading order contribution to the entanglement scaling comes from +the logarithmic term associated with the matrix symbol discontinuity. +Within the weak long-range regime, we can distinguish three different cases: α1 > +α2, α1 < α2 and α1 = α2 = α. +Therefore, in order to proceed we must identify the +location of the jumps of Gk and compute the lateral limits in these three different situations. +Possible sources of discontinuities for Gk are the discontinuities or the zeros of the spectrum +ωk(h), which appear at the two quantum critical points h = 1, −1 + 21−α1, where the +spectrum becomes gapless at the soft modes k = 0, π, respectively. More precisely, Gk has +no discontinuities when h ̸= 1, −1+21−α1, since in this case the lateral limits at the critical +modes read +G± +0 = lim +k→0± Gk = sgn(h − 1)σz, +(4.1) +G± +π = lim +k→π± Gk = sgn(h + 1 − 21−α1)σz. +(4.2) +This leads to a constant scaling of the entanglement entropy Sν,L = O(1) with the subsystem +size when the system is not at quantum criticality and therefore the spectrum is gapped. +This is nothing but a manifestation of the standard area law for one-dimensional gapped +systems [20, 21]. On the other hand, quantum criticality leads to logarithmic deviations +from the area law. Let us start from the homogeneous critical point (h = 1), when the +spectrum has an α1,2 dependent dispersion relation (see Appendix C), which leads to the +different lateral limits +G± +0 = +� +� +� +� +� +� +� +−sgn(A(α1))σz +if +α1 < α2 +− sin(απ/2)σz ± cos(απ/2)σy +if +α1 = α2 +±sgn(B(α2))σy +if +α1 > α2 +, +(4.3) +where A(α) = sin(απ/2)Γ(1 − α)/ζ(α), and B(α) = cos(απ/2)Γ(1 − α)/ζ(α), with Γ(x) +and ζ(x) the Gamma and the Riemann zeta functions [94], respectively. Accordingly, no +discontinuity is present when the power law decay of the hopping amplitude is slower than +that of the pairing, leading again to a constant entanglement entropy. In the α1 > α2 case +– 11 – + +500 +1000 +1500 +2000 +2500 +L +0.48 +0.49 +0.50 +0.51 +0.52 +SL +(a) +α1 < α2 +c1 + c2L−c3 +S2,L +0 +1000 +2000 +L +0.8 +1.0 +1.2 +SL +(b) +α1 > α2 +1 +6 log L +SL +SL − c1 − c2L−c3 +0 +1000 +2000 +L +0.25 +0.30 +0.35 +0.40 +0.45 +S2,L +(c) +α1 = α2 = α +B2,α log L +S2,L +S2,L − c1 − c2L−c3 +Figure 3. Numerical check of the entanglement scaling as a function of the subsystem size L at +the quantum critical point with chemical potential h = 1 for different values of couplings power +law decay exponents 1 < α1, α2. a) Entanglement entropy (ν = 1), with α1 = 1.5 and α2 = 1.8, +blue squares represent the numerical data while the black solid line is a fit of a constant and +a subleading contribution c1 + c2L−c3. +b) Entanglement entropy (ν = 1), with α1 = 1.8 and +α2 = 1.5, blue squares represents the numerical data, the black solid line correspond to the curve +(1/6)) ln L, red dots have been obtained from the numerics by subtracting the fit of the subleading +corrections of the form c1 + c2L−c3. c) Rényi-2 entropy (ν = 2) with α1 = α2 = 1.5, blues squares +represents the numerics, the black solid line represents the curve B2,α ln L, red dots are obtained +subtracting the subleading corrections to the numerical data as in panel b). +instead, we have a discontinuity in the symbol, with commuting lateral limits. Inserting +the expression for G± +0 in Eq. (3.12) we obtain +b0(z) = +1 +2π2 +� +ln +�z + 1 +z − 1 +��2 +. +(4.4) +Then, inserting this result into the expression for the entanglement entropy, and performing +the integration in Eq. (3.5) we obtain the logarithmic scaling +Sν,L = ν + 1 +12ν ln L + O(1). +(4.5) +This logarithmic scaling is analogous to the one obtained for a conformal field theory with +central charge c = 1/2 [24]. This result is in agreement with previous findings [56, 57] +concerning the entanglement scaling in a Kitaev chain with long-range paring and nearest +neighbors hopping α1 → ∞, here we show that the same scaling holds also for finite α1 +as long as α1 > α2. Figure 3b shows the numerical check of the scaling behavior of the +entanglement entropy SL = S1,L for α1 > α2 and h = 1. We obtain an excellent agreement +once the subleading corrections are taken into account. In particular, we need to subtract +from the numerical data the finite size corrections of the form +SL − 1 +6 ln L = c1 + c2L−c3, +(4.6) +where the ci = ci(α1, α2, h), i = 1, 2, 3, coefficients can be estimated from a fit with the +numerical data. +The most interesting case corresponds to the condition α1 = α2 = α which, as previ- +ously stated, is closely related to the long-range interacting quantum Ising chain. Moreover, +– 12 – + +0 +1000 +2000 +L +0.8 +1.0 +1.2 +SL +(a) +α1 < α2 +1 +6 log L +SL +SL − c1 − c2L−c3 +0 +1000 +2000 +L +0.8 +1.0 +1.2 +SL +(b) +α1 > α2 +1 +6 log L +SL +SL − c1 − c2L−c3 +0 +1000 +2000 +L +0.6 +0.7 +0.8 +0.9 +S2,L +(c) +α1 = α2 = α +1 +8 log L +S2,L +S2,L − c1 − c2L−c3 +Figure 4. Numerical check of the entanglement scaling as a function of the subsystem size L at +the quantum critical point with chemical potential h = −1 + 21−α1 for different values of couplings +power law decay exponents: a) α1 = 1.5, α2 = 1.8, b) α1 = 1.8, α2 = 1.5, c) α1 = α2 = 1.5. As +in Fig.3, blue squares represents the numerical data, the black solid line represents our analytical +prediction for the scaling in the L ≫ 1 limit, red dots are obtained from the numerics by subtracting +the subleading corrections. +we notice that in this regime the matrix symbol Gk, hosts non-commuting lateral limits +as k → 0± (see Eq. (4.3)). This leads to the non-trivial dependence of the logarithmic +contribution coefficient on α +b0(z) = 2 +π2 +� +ln +� +z2 − sin2(απ/2) + cos(απ/2) +√ +z2 − 1 +�2 +. +(4.7) +Inserting b0(z) in Eq. (3.5) and performing the integration (see Appendix B), we obtain the +logarithmic scaling behavior of the Rényi entropy +Sν,L = Bν,α ln L + O(1), +(4.8) +where +Bν,α = +1 +π2(ν − 1) +ν +� +k=1 +arctan2 +� +� +cos(απ/2) +� +sin2(απ/2) + |zk,ν|2 +� +� , +(4.9) +with zk,ν = i tan(π(2k − 1)/2ν). +In particular, for ν = 2, 3, the sum in the previous +expression reduces to +B2,α = 2 +π2 arctan2 +� +cos(απ/2) +� +sin2(απ/2) + 1 +� +, +(4.10) +B3,α = 1 +π2 arctan2 +� +cos(απ/2) +� +sin2(απ/2) + 1/3 +� +. +(4.11) +This analytical scaling of S2,ν at h = 1 and for α1 = α2 = α is compared with the +numerical result in Fig. 3c. Also in this case, a good agreement is found once the subleading +corrections (4.6) are taken into account. +– 13 – + +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +α +0.00 +0.05 +0.10 +Bν,α +(a) +ν = 2 +ν = 3 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +α +0.0 +0.2 +0.4 +ceff +(b) +c = 1/2 +Figure 5. a) Coefficient Bν,α of the logarithmic scaling of the ν-Rényi entropy as a function of +the power law decay exponent α = α1 = α2, for ν = 2 (green solid line) and ν = 3 (purple solid +line). The dashed lines correspond to the short-range values of the coefficients which are matched +by the long-range ones for α = 2. b) effective central charge, obtained as ceff = 6νBν,α/(ν +1), as a +function of α for ν = 2, 3. The black dashed line represents the central charge for nearest neighbor +couplings c = 1/2. +It is important to observe that at variance with the α1 ̸= α2 cases, the scaling coefficient +Bν,α cannot be written in the form +Bν,α ̸= Bν,CFT = ν + 1 +6ν c, +(4.12) +where c is the central charge of some conformal field theory describing the model at the +quantum critical point. This observation supports our previous claim that the case α1 = α2 +is special and, somehow, closer to the one of a strongly interacting system such as the long- +range Ising model. Indeed, while the case α1 ̸= α2 continues to obey the r.h.s. of Eq. (4.12) +and, so, is more likely to be described by a CFT, the case 1 < α1 = α2 < 2 goes beyond this +description as the scaling of the ground state entanglement at the critical point cannot be +related to the universal properties of a conformal field theory. A similar result is expected +for the Ising model in a transverse field, where the inclusion of long-range interactions +is expected to increase the effective dimension of the model and, so, disrupt any CFT +description. +Figure 5a shows the coefficients Bν,α for ν = 2, 3 as a function of α ∈ [1, 2], we notice +that the value of the logarithmic scaling coefficients starts from zero at α = 1 and then +grows with α reaching the short-range value for α = 2. Moreover, Fig. 5b shows the α +dependence of the effective central charge defined as ceff(α) = 6νBν,α/(ν + 1) as a function +of α. We notice that, apart from the extrema ceff(1) = 0 and ceff(2) = 1/2, the effective +charge also depends on the Rényi entropy order ν, thus confirming the fact that it cannot +be considered as the proper central charge of a conformal field theory. These results are in +agreement with the findings of Ref. [95], where the breakdown of conformal symmetry in a +long-range fermionic chain was established. +Finally, we consider the non-homogeneous critical point h = −1 + 21−α1. In this case, +the power of the dispersion relation near the soft mode k = π is not affected by the presence +– 14 – + +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +h +0.0 +0.5 +1.0 +1.5 +2.0 +B2,α +(a) +α = 0.1 +α = 0.25 +α = 0.5 +α = 0.75 +5 × 100 +6 × 100 +7 × 100 +8 × 100 +ln L +100 +101 +102 +S2,L(h = 0) +(b) +Figure 6. a) Rényi-2 scaling coefficient B2,α as a function of the chemical potential h for different +values of the power law decay coefficient 0 < α = α1 = α2 < 1. +The red and blue vertical +lines correspond to the h = 1 and h = 0 critical points, respectively. b) Numerical check for the +entanglement subvolume law scaling at h = 0 for different values of 0 < α < 1, plotted as a function +of the logarithmic of the subsystem size ln L. Scattered points correspond to the numerical data +while solid lines represent our prediction B2,α ln L. +of long-range couplings (see Appendix C). Accordingly, also the symbol discontinuity is +independent of the value of α1,2, in particular, we find +G± +π = lim +k→π± Gk = ±σy, +∀α1, α2 > 1. +(4.13) +This leads to a logarithmic contribution coefficient +bπ(z) = +1 +2π2 +� +ln +�z + 1 +z − 1 +��2 +. +(4.14) +The corresponding scaling of the entanglement entropy is then the one obtained in Eq. (4.5), +which is equivalent to the entanglement scaling in the nearest neighbor Kitaev chain, at a +quantum critical point characterized by a conformal field theory with central charge c = 1/2. +Figure 4 shows the entanglement scaling behavior at the non-homogeneous critical point +h = −1 + 21−α1 with α1 < α2 (Fig. 4a), α1 > α2 (Fig. 4b) and α1 = α2 (Fig. 4c). Also +in this case a nice agreement with the theoretical prediction in the thermodynamic limit is +found once finite size corrections are taken into account. +5 +Strong long-range regime +The situation in the strong long-range regime is more involved. +In particular previous +studies on fermionic systems with strong long-range pairing interactions [56–58] reported +logarithmic violations of the entanglement area law even away from criticality. However, in +those cases, the noncritical logarithmic scaling of the ground state entanglement was asso- +ciated with divergences in the long-range couplings due to the fact that no Kac scaling was +introduced in the model Hamiltonian. Therefore, one may think such anomalous scalings +to be trivially related to the loss of the system extensivity. On the other hand, as shown +– 15 – + +in Sec. 2, the introduction of a Kac scaling in the Hamiltonian allows us to define a model +with strong long-range interaction still preserving the energy extensivity. +In particular, when a Kac scaling is introduced, the coupling divergences for α1, α2 < 1 +are canceled, and accordingly also the symbol discontinuity associated with them disap- +pears. However, an infinite number of new nontrivial discontinuities arise due to the fact +that the spectrum becomes discrete also in the thermodynamic limit. More precisely, as +a consequence of the spectrum discontinuity, the symbol becomes discontinuous for any +k = 2πn/N. Indeed, in the thermodynamic limit, Gk reads +lim +N→∞ Gk = Gn = h − tn +ωn +σz − ∆n +ωn +σy. +(5.1) +Then it can be labeled by a discrete integer number n, while the k variable becomes contin- +uous. More precisely, any real physical implementation of the model has necessarily a finite +size. Therefore, the actual physical meaning of the continuum limit as N → ∞ is that the +difference between two consecutive values of k is of order O(N−1). However, in the strong +long-range case, a difference of order O(N−1) in the k variable results in a finite jump of the +spectrum ωn which remains discrete even in the thermodynamic limits, thus resulting in a +discontinuity of the matrix symbol Gk for any k independently of the value of the chemical +potential h. In particular, since for any α1,2 > 0 or α1 = α2 = 0 and h ̸= 0 the many-body +ground state is still the Bogoiubov vacuum, then the two lateral limits corresponding to a +given k± = 2πn/N, 2π(n + 1)/N can be written as +G± +k = +� +Gn+1 = cos φn+1σz + sin φn+1σy +Gn = cos φnσz + sin φnσy +, +(5.2) +where we have introduced the angles φn defined by the conditions cos φn = (h−tn)/ωn and +sin φn = −∆n/ωn. Then, following the analytic procedure introduced in Section 3, for any +value of h, we obtain a logarithmic scaling of the ground state Rényi entropies of the form +Sν,L = Bν(h) ln L + O(1), +(5.3) +where the Bν(h) coefficient is a function of ν, α1, α2 and h. Then Bν,α1,α2(h) is given by +the sum of N contributions corresponding to the N discontinuities of the symbol, reading +Bν(h) = +N/2 +� +n=−N/2+1 +B(n) +ν (h), +(5.4) +where, as shown in Appendix B, each contribution reads +B(n) +ν (h) = +1 +π2(ν − 1) +ν +� +l=1 +arctan2 +� +sin((φn+1 − φn)/2) +� +cos2((φn+1 − φn)/2) + |zl|2 +� +. +(5.5) +where |zl|2 = tan2(π(2l − 1)/2ν), with l = 1, . . . , ν and l ̸= (1 + ν)/2. In particular, for +ν = 2 the above sum can be written explicitly as +B(n) +2 (h) = 2 +π2 +� +arctan +� +ωn+1ωn − (h − tn+1)(h − tn) − ∆n+1∆n +3ωn+1ωn + (h − tn+1)(h − tn) + ∆n+1∆n +�2 +. +(5.6) +– 16 – + +As we have already seen in the previous Sections, the most interesting situation is the +one with equally long-range hopping and pairing amplitudes, i.e., with α1 = α2 = α, while +we expect only minor differences to appear when α1 ̸= α2, as long as they are both smaller +than the system dimension (here d = 1). Therefore, for the sake of simplicity, we will limit +our treatment to the α1 = α2 = α case in the following analysis of the strong long-range +regime. +Figure 6a shows B2(h) as a function of the chemical potential h for different values +of α1 = α2 = α. +First of all, we notice that for any values of the chemical potential +h ̸= 0 and of α > 0 the scaling coefficient is of order B2(h ̸= 0) = O(1), then leading to a +logarithmic violation of the area law even away from the quantum critical points. Moreover, +two singularities appear at the quantum critical points h = t0, tπ = 1, 0. In particular, we +have a discontinuity for h = 1 and a divergence with the subsystem size for h = 0, leading +to a subvolume law entanglement scaling. +These facts can be understood as follows. The spectrum is labeled by the discrete +index n leading to a finite gap between the ground state and the first excited levels which +are associated with discontinuities of the symbol. +However, for n ≫ 1 all the modes +accumulate around ω∞ = |h|. +This means that an extensive number of single-particle +states is almost degenerate. Consequently, as long as h ̸= 0, we may expect only the first +few modes around n = 0 to provide a significant contribution to the symbol discontinuity +leading to a coefficient Bν(h ̸= 0) = O(1). Accordingly, we may expect many features +of the entanglement scaling coefficients for values of the chemical potential sufficiently far +from the h = 0 point, to be qualitatively reproduced by considering a single discontinuity +approximation in which only the first discontinuity between the n = 0 and the first two +degenerate levels n = ±1 is considered, i.e., Bν(h ̸= 0) ≈ B(0) +ν ++ B(−1) +ν +. Then, as detailed +in Appendix D within this approximation the discontinuity coefficient reads +Bν(h ̸= 0) ≈ +2 +π2(ν − 1) +ν +� +l=1 +arctan2 +� +cos(φ1/2) +|zl|2 + sin2(φ1/2) +� +if +h < 1, +(5.7) +Bν(h ̸= 0) ≈ +2 +π2(ν − 1) +ν +� +l=1 +arctan2 +� +sin(φ1/2) +|zl|2 + cos2(φ1/2) +� +if +h > 1. +(5.8) +This approximation then allows us to capture the origin of the scaling coefficient discontinu- +ity at h = 1. This originates from the fact that the zero mode gives different contributions +at the two sides of the transition, indeed (see Appendix D) +φ0 = arccos[sign(h − 1)] = +� +π +if +h < 1 +0 +if +h > 1 +. +(5.9) +The single discontinuity approximation turns out to correctly reproduce the qualitative +features as long as the chemical potential h is sufficiently far from h = 0 and for sufficiently +large power law decay exponent α > 1/2. On the other hand, this simple approximation +is no more accurate as the chemical potential approaches the h = 0 point. Indeed, in the +zero chemical potential case ω∞ = 0, and more precisely ωn, tn and ∆n approach their +asymptotic values differently if we consider the even or the odd modes (see Appendix D for +– 17 – + +more details). As a consequence, for sufficiently small α, the number of relevant symbol +discontinuities grows as a power law of the subsystem size L, leading to a fractal subvolume- +law entanglement scaling. In particular, using the asymptotic expansion of ωn, tn and ∆n +in the n → ∞ limit we can extract the leading order dependence of Bν(h = 0) from L, +which, as shown in Appendix D, reads +Bν(h = 0) = +� +O(L1−2α) +if +α < 1/2 +O(1) +if +α > 1/2 +. +(5.10) +Accordingly, the leading order contribution to the entanglement Rényi entropy of the system +ground state at zero chemical potential takes the nontrivial form +Sν,L(h = 0) = +� +O(L1−2α ln L) +if +α < 1/2 +O(ln L) +if +α > 1/2 +. +(5.11) +This analytic result matches the numerics in the large L limit. This is shown in Fig. 6b, +where the numerical and analytical results for S2,L are plotted as a function of ln L and +for different values of α. It is important to notice that approaching the thermodynamic +limit in the h = 0 case the spectrum becomes increasingly more degenerate approaching the +α = 0 case. Then, for each finite N, a large number of states nearly degenerate with the +ground state exists, making the estimate of the subleading corrections scaling technically +challenging. +Finally, as already stated in Sections 2 and 3, the mean-field case with α1 = α2 = 0 +and h = 0 must be treated separately. Indeed, in this case the ground state degeneracy +allows for a finite fermionic population of the even Bogoliubov modes, fn ̸= 0 ∀n(even), +this leads to the entanglement scaling +Sν,L(α = 0, h = 0) = +1 +1 − ν +� +n(even) +ln [(1 − fn)ν + fν +n] + O(ln L). +(5.12) +In particular the maximal Rényi entropy is reached when fn = 1/2 ∀n(even) +Smax +ν,L (α = 0, h = 0) = N0 ln 2 + O(1) = L +2 ln 2 + O(1), +(5.13) +where N0 is the number of zero modes, which in this case corresponds to the number of even +modes N0 ≃ L/2 and the subleading corrections are at most of order O(1). Indeed, as shown +in Appendix B, the discontinuity coefficients Bν which would lead to logarithmic corrections +turn out to be exactly zero when all the even fermionic populations are fn(even) = 1/2. +Moreover, we notice that the maximal Rényi entropy that we have obtained employing the +Fisher-Hartwig expansion corresponds to the largest possible entropy allowed by the ground +state degeneracy +Smax +ν,L (α = 0, h = 0) = ln Deg[|gsα=0,h=0⟩] = N0 ln 2. +(5.14) +This tells us that the Fisher-Hartwig result, obtained as a large subsystem size expansion, +actually becomes exact in this maximally entangled case. +– 18 – + +6 +Conclusion and outlooks +In this paper, we have further extended the understanding of the peculiar properties of +entanglement in quantum systems featuring long-range interactions. +At this scope, we +have investigated, as a paradigmatic example, the ground state entanglement scaling of a +spinless fermionic chain with long-range hopping and pairing amplitudes. The simplicity of +the model and its truly non-additive nature allowed us to unveil an extremely rich and non- +trivial phenomenology, which we have fully characterized both numerically and analytically +in the different regions of the relevant parameters, i.e., the power law decay exponents of +the hopping and pairing couplings α1, α2 and the chemical potential h. In particular, two +main regimes may be distinguished: the weak long-range regime with 1 < α1, α2 < 2 and +the strong long-range regime with 0 < α1, α2 < 1. +In the weak long-range case, the system quasiparticle spectrum becomes continuous in +the thermodynamic limit and the main effect of the non-local couplings is to change the +dispersion relation near the gapless critical modes. Accordingly, the standard area law, +typical of gapped local Hamiltonians, is satisfied in this regime apart from the logarithmic +violations which appear in correspondence of the two quantum critical points located at +h = 1, −1+21−α1. Such logarithmic scaling of the ground state Rényi entropies is related to +discontinuities in the symbol of the correlation matrix which is a block Toeplitz matrix. The +fact that the contribution to the entanglement scaling of each discontinuity only depends +on the value of the symbol [57, 58] at each side of the jump, allowed us to exactly compute +its coefficients. +Most significantly, when the hopping and pairing couplings are equally +long-range, i.e., α1 = α2 = α, the coefficient in front of the critical logarithmic divergence +at h = 1 turns out to have a non-trivial dependence on α (4.9). +Interestingly, the coefficient Bν,α is of non-universal nature, since it originates from the +precise form of the spectrum in the proximity of the critical modes, and not only from the +dispersion relation power law exponent. As a consequence, the critical entanglement scaling +is not compatible with the result obtained from any conformal field theory and our result +may be seen as a benchmark of the fact that the presence of long-range couplings explicitly +breaks the critical conformal symmetry [95]. These findings demonstrate the peculiarity of +the α1 = α2 case, whose physics is expected to be, and indeed is, closer to the one of a +strongly interacting system such as the quantum Ising model, where long-range couplings +are expected to increase the effective dimension and, so, disrupt integrability [96]. +Moreover, for α1 ̸= α2, the critical entanglement scaling becomes α independent. In +particular, when α1 > α2, i.e., the pairing coupling has a slower decay with respect to +the hopping, the entanglement scaling is compatible with that of conformal field theory +with central charge c = 1/2. This is in agreement with the results of Ref. [57, 58], where +a Kitaev chain with long-range pairing and nearest neighbors hopping is considered, the +validity of such results is then here extended to any long-range hopping with power law +decay exponent α1 > α2. The strong anisotropy between the case of dominating hopping +α1 < α2 and the case of dominating paring α1 > α2 is typical of the long-range Kitaev +chain [97]. +In the strong long-range regime, the situation is more involved, indeed the quasiparticle +– 19 – + +spectrum can no more be considered continuous in the thermodynamic limit. Consequently, +the matrix symbol of the block Toeplitz correlation matrix formally becomes discontinuous +at every point of the spectrum. However, as shown in Section 5, in most situations only +a few of such discontinuities truly contribute to the entanglement scaling, leading to a +logarithmic dependence on the subsystem size even outside criticality. Also in this case the +coefficients of such logarithmic divergence can be computed analytically for different values +of the parameters α1,2 and h. +The most interesting situation turns out to be the zero chemical potential point h = 0 +in the strong long-range regime. Indeed, in this case, the coefficient in front of the critical +logarithmic entanglement scaling diverges as a power law of the subsystem size, leading to +a fractal subvolume-law entanglement scaling. More precisely, we were able to analytically +extract the leading entanglement dependence from the subsystem size, which turns out to +be of the form Sν,L ≈ L1−2α ln L, with 0 < α = α1 = α2 < 1/2, where Sν,L is any ν-Renyi +entropy with ν > 1. Similar sub-volume laws have already been observed in different (more +complex) scenarios and, in particular, in the entanglement scaling of measurement induced +phase transitions [50], where they arise due to the suppression of entanglement caused by +repeated measurements in a long-range systems. +Here, this phase emerges naturally in +the equilibrium scaling, but it needs stronger interactions to appear with respect to the +dynamical case. +Finally, in the completely mean-field case, the system presents an extensive number +of degenerate modes with zero energy. These zero modes can be populated also in the +many-body ground state whose degeneracy then grows exponentially with the number of +zero modes. Consequently, the ground state entanglement shows a volume law behavior +proportional to the size of the considered subsystem Sν,L(α = 0) ≈ L. +Our studies evidence that long-range couplings can greatly improve the scaling of en- +tanglement at equilibrium and, therefore, that long-range interacting quantum systems +represent the ideal candidate for reliable and robust quantum computation. Nevertheless, +such fostered entanglement properties may not persist out-of-equilibrium, since long-range +interactions have been shown to suppress the dynamical spread of entanglement in certain +systems [39]. For the future, we intend to investigate these issues by performing quantum +simulations of the model on actual quantum computers. This demands a careful engineering +of the artificial non-local couplings on local quantum devices, a task which we are currently +tackling on IBM Quantum devices [98]. +The rich phenomenology hosted by the minimal long-range model we considered, al- +ready at equilibrium, suggests that many of the intriguing dynamical phenomena which are +recently emerging in the quantum community, such as the non-trivial fractal entanglement +scalings in the contest of measurement-induced entanglement transitions [3], can be simply +ascribed to the presence of sufficiently long-range couplings among the microscopic compo- +nents of the model, without any need of further complexity in the physical system under +consideration. Further work is needed in order to investigate the dynamical properties of +entanglement in the Kitaev chain with long-range pairing and hopping couplings subjected +to a unitary or a non-unitary (measurement-like) evolution. These interesting problems are +beyond the scope of this work and we leave them as an outlook for future projects. +– 20 – + +Acknowledgements +We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research +Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg +STRUCTURES Excellence Cluster). This work is part of the MIUR-PRIN2017 project +Coarse-grained description for nonequilibrium systems and transport phenomena (CO- +NEST) No. 201798CZL. AS and SS acknowledge acknowledge financial support from Na- +tional Centre for HPC, Big Data and Quantum Computing (CN00000013). Access to the +IBM Quantum Computers was obtained through the IBM Quantum Hub at CERN. +– 21 – + +A +Derivation of the matrix symbol +In this Appendix we provide the details for the derivation of the matrix symbol in Eq. (3.9) +of the main text. We start from the definition of the correlation matrix of a stationary state +|ψ⟩, then passing to the Fourier basis we obtain +Gk = 2⟨ψ| +� +ˆck +ˆc† +−k +� � +ˆc† +k ˆc−k +� +|ψ⟩ − I. +(A.1) +Introducing the Bogoliubov transformation +� +ˆγk +ˆγ† +−k +� += Uk +� +ˆck +ˆc† +−k +� +, +Uk = +� +cos θk/2 +i sin θk/2 +−i sin θk/2 − cos θk/2 +� +, +(A.2) +we can write the symbol in terms of the Bogoliubov modes as +Gk = 2U † +k⟨ψ| +� +ˆγk +ˆγ† +−k +� � +ˆγ† +k ˆγ−k +� +|ψ⟩Uk − I. +(A.3) +We now compute the expectation value in a stationary stationary state associated to the +fermionic populations of the Bogoliubov modes fk = ⟨ˆγ† +kˆγk⟩, so that +⟨ψ| +� +ˆγk +ˆγ† +−k +� � +ˆγ† +k ˆγ−k +� +|ψ⟩ − I = +� +1 − 2fk +0 +0 +2fk − 1 +� +. +(A.4) +Finally, inserting this expectation value in Eq. (A.3) and using the definition of the Bogoli- +ubov angles tan θk = ∆k/(h − tk) we obtain +Gk = (1 − (fk + fk)) +�h − tk +ωk +σz − ∆k +ωk +σy +� +− (fk − f−k)I, +(A.5) +which is the expression for the matrix symbol used in the main text. +B +Coefficients of the Fisher-Hartwig expansion +The general form of the matrix symbol in Eq. (A.3) can be used to compute the different +terms in the Fisher-Hartwig expansion of the Rényi entropies for large subsystem size in +every situation considered in the main text. For this purpose, it is useful to rewrite Gk as +Gk = ak [cos φkσz + sin φkσy] + bkI, +(B.1) +where we have introduced the coefficients ak = 1 − (fk + f−k) and bk = f−k − fk and the +angle φk such that cos φk = (h − tk)/ωk and sin φk = −∆k/ωk. +Let us start from the first term of the expansion in Eq. (3.13) this is obtained by first +computing the determinant +det [zI − Gk] = (z − bk)2 − a2 +k, +(B.2) +– 22 – + +bn + an cos δϕn +bn + an +bn − an cos δϕn +bn − an +1 + ϵ +−1 − ϵ +Figure 7. Contour of integration and cuts of the integrand in Eq. (B.8). The cuts from ±(1+ϵ) to +the infinity correspond to dsν(1+ϵ, z)/dz while the cuts inside the contour, [bn −an, bn −an cos δφ] +and [bn + an cos δφ, bn + an, ], are due to the other factor of the integrand. +Then, the contribution to first term in the entanglement scaling coming from each k-mode +is obtained from the integral +Sk = lim +ϵ→0+ +� +C +dz +2πisν(1 + ϵ, z) +(z − bk) +(z − bk)2 − a2 +k +(B.3) += 1 +2 [sν(1, bk + ak) + sν(1, bk − ak)] += +1 +2(1 − ν) +� +ln(fν +k + (1 − fk)ν) + ln(fν +−k + (1 − f−k)ν) +� +, +where Cauchy’s residue theorem and the expression (3.6) for sν(x, y) have been used. Fi- +nally, summing over all the modes and using the k → −k symmetry we obtain +� +k +Sk = +1 +1 − ν +� +k +ln(fν +k + (1 − fk)ν). +(B.4) +The logarithmic contribution to the entanglement scaling can be computed by con- +sidering the discontinuity coefficients. +Here, we present their calculation in the general +situation in which Gk is discontinuous at a generic mode k = 2πn/N. We start from the +definition (3.12) of the bk coefficients corresponding to each discontinuity. First of all, we +consider the matrix +Mk = (zI − G− +k )(zI − G+ +k )−1, +(B.5) +where G± +k = limp→k± Gp. The eigenvalues µ± +k (z) of this matrix can be written in the form +µ± +k (z) = +� +� +� +(bk − z)2 − a2 +k cos2(δφk/2) ± ak sin(δφk/2) +� +(bk − z)2 − a2 +k +� +� +2 +, +(B.6) +– 23 – + +with δφk = φ+ +k − φ− +k . Notice also that we have µ+ +k (z) = 1/µ− +k (z), therefore +bk(z) = +1 +2π2 +� +ln µ+ +k (z) +�2 +(B.7) += 2 +π2 +� +�ln +� +� +� +(bk − z)2 − a2 +k cos2(δφk/2) + ak sin(δφk/2) +� +(bk − z)2 − a2 +k +� +� +� +� +2 +, +From this expression we compute the coefficient B(k) +ν +of the contribution of this discontinuity +to the logarithmic term of the Rényi entropy. For this purpose we plug bk(z) into the contour +integral for Sν,L then, performing an integration by parts, we obtain +B(k) +ν += lim +ϵ→0+ +� +C +dz +2πisν(1 + ϵ, z)dbk(z) +dz +(B.8) += − lim +ϵ→0+ +� +C +dz +2π3i +dsν(1 + ϵ, z) +dz +� +�ln +� +� +� +(bk − z)2 − a2 +k cos2(δφk/2) + ak sin(δφk/2) +� +(bk − z)2 − a2 +k +� +� +� +� +2 +. +The integral over the contour C depicted in Fig. 7 can be divided into two integrals along +curves enclosing respectively the cuts [bk − ak, bk − ak cos δφk] and [bk + ak, bk + ak cos δφk], +which in turn can be reduced to two real integrals by performing the integration along the +cuts taking into account the change in the phase of the logarithm when we go around the +branch points bk ± ak and bk ± ak cos δφk. On the other hand, we notice that for integer +ν > 1, dsν/dz is a meromorphic function with poles located at the points of the imaginary +axis [57, 58] +zl = i tan π(2l − 1) +2ν +, +l = 1, . . . , ν, +l ̸= 1 + ν +2 +, +(B.9) +and that the another factor of the integrand is analytic in the whole region outside the +contour C. We can send this contour to infinity and reduce the calculation of Bν to the +computation of the corresponding residues. In this way, we obtain the explicit expression +B(k) +ν += +1 +ν − 1 +ν +� +l=1 +� +�ln +� +� +� +(bk − zl)2 − a2 +k cos2(δφk/2) + ak sin(δφk/2) +� +(bk − zl)2 − a2 +k +� +� +� +� +2 +. +(B.10) +This general formula can be specified in the different cases considered in the main text. In +particular in weak long-range case, 1 < α1, α2 < 2, the ground state corresponds to the +Bogoliubov vacuum, therefore fk = 0, ak = 1 and bk = 0, ∀k. Accordingly, the first term +of the expansion vanishes. Moreover the matrix symbol is continuous for generic values of +the chemical potential leading to an O(1) entanglement. The only discontinuities arise at +the two quantum critical points h = hc = 1, −1 + 21−α1 in correspondence of the critical +– 24 – + +modes k = kc = 0, π. This leads to a logarithmic scaling with coefficient +B(kc) +ν += +1 +ν − 1 +ν +� +l=1 +� +ln +�� +|zl|2 + cos2(δφkc/2) − i sin(δφkc/2) +� +|zl|2 − 1 +��2 += +1 +ν − 1 +ν +� +l=1 +� +arctan +� +sin(δφkc/2) +� +|zl|2 + cos2(δφkc/2) +��2 +, +(B.11) +where in the last step we have used the identity arctan(x) = i[ln(i + x) − ln(i − x)]/2 in +order to make the expression of the coefficient explicitly real. The value of δφkc depends +on the critical point considered and the relative order of the power law decaying exponents +α1 and α2. In particular for h = 1 and k = 0 we find +δφ0 = +� +� +� +� +� +� +� +0 +if +α1 < α2 +π(1 − α) +if +α1 = α2 = α +π +if +α1 > α2. +(B.12) +Leading to the coefficients +B0 +ν(h = 1) = +� +� +� +� +� +� +� +� +� +� +� +0 +if +α1 < α2 +1 +ν−1 +�ν +l=1 +� +arctan +� +cos(απ/2) +√ +|zl|2+sin2(απ/2) +��2 +if +α1 = α2 = α +ν+1 +12ν +if +α1 > α2. +(B.13) +On the other hand, for h = −1 + 21−α1 and k = π, δφπ = π independently from the values +of α1 and α2. This leads to the scaling coefficient +B0 +ν(h = −1 + 21−α1) = ν + 1 +12ν +∀ α1, α2 > 1. +(B.14) +In the strong-long range regime 0 < α1, α2 < 1, the quasiparticle spectrum is discrete +also in the thermodynamic limit, this formally leads to an infinite number of discontinuities +for any mode k = 2πn/N, which are labeled by the integer n = −N/2, . . . N/2. In particular +whenever α1,2 > 0 or α1 = α2 = 0 and h ̸= 0, the many-body ground state is still the +Bogoliubov vacuum characterized by fk = 0, ∀k. Accordingly, the matrix symbol in the +thermodynamic limit takes the form in Eq. (5.1). The coefficients of the logarithmic scaling +is then given by the sum of the contributions coming from all the discontinuity, i.e., +Bν = +N/2 +� +n=−N/2 +B(n) +ν , +(B.15) +where +B(n) +ν += +1 +ν − 1 +ν +� +l=1 +� +ln +�� +|zl|2 + cos2(δφn/2) − i sin(δφn/2) +� +|zl|2 − 1 +��2 += +1 +ν − 1 +ν +� +l=1 +� +arctan +� +sin(δφn/2) +� +|zl|2 + cos2(δφn/2) +��2 +, +(B.16) +– 25 – + +with δφn = φn+1 − φn. +Finally, in the mean-field case α1 = α2 = 0 with zero chemical potential h = 0 the +quasiparticle spectrum develops an extensive number of degenerate zero modes ωn = 0 +corresponding to all the even modes with n = 2m. As a consequence, the ground state +is characterized by a finite even mode fermionic population f2m ̸= 0. The leading order +term in the entanglement scaling in this case is then given by the first term of the Fisher- +Hartwig expansion corresponding to a volume law. In particular the maximum amount of +entanglement allowed by the ground state degeneracy is obtained for f2m = 1/2 for every +even mode. In this case, the logarithmic corrections become zero since an = bn = 0, and +therefore B(n) +ν (fn = 1/2) = 0. +C +Dispersion relation around the critical modes +In this Appendix we provide the explicit expression for the Taylor expansion of the quasi- +particle spectrum (2.7), in the weak long-range regime 1 < α1,2 < 2, at lowest order in +|k − kc|, where kc = 0 at the critical point h = 1, while kc = π at h = −1 + 21−α1. In +particular, in the proximity of k = 0 we find [97] +tk = 1 + sin(α1)Γ(1 − α) +ζ(α) +kα1−1 + O(k2), +(C.1) +∆k = sin(α1)Γ(1 − α) +ζ(α) +kα2−1 + O(k). +(C.2) +Accordingly, the single particle spectrum takes the form [8] +ωk = +� +|h − 1| + O(kα − 1) +if +h ̸= 1 +C(α)|k|α−1 + O(k2α−2) +if +h = 1 +, +(C.3) +where α = min{α1, α2}, and we have introduced the constant prefactor +C(α) = +� +� +� +� +� +� +� +| sin(α1π/2)Γ(1 − α1)/ζ(α1)| +if +α1 < α2 +|Γ(1 − α)/ζ(α)| +if +α1 = α2 +| cos(α1π/2)Γ(1 − α1)/ζ(α1)| +if +α1 > α2 +. +(C.4) +On the other hand, near to the k = π mode we find [97] +tk = −1 + 21−α1 − (23−α1 − 1)ζ(α1 − 2) +2ζ(α1) +(π − k)2 ++ O((π − k)3), +(C.5) +∆k = (1 − 22−α2)ζ(α2 − 1) +ζ(α2) +(π − k) + O((π − k)3). +(C.6) +Leading to the α1,2 independent dispersion relation +ωk = +� +|h + 1 − 21−α1| + O((k − π)2) +if +h ̸= −1 + 21−α1 +K(α2)|π − k| + O((k − π)3) +if +h = −1 + 21−α1 , +(C.7) +where K(α2) = (1 − 22−α2)ζ(α2 − 1)/ζ(α2), ∀α1, α2 > 1. +– 26 – + +D +Discontunities in the strong long-range regime +In this Appendix we provide a detailed analysis of the discontinuities of the matrix symbol +Gk in the strong long-range regime 0 < α1, α2 < 1 for different values of the chemical +potential. As discussed in the Section 5 of the main text, in this regime the matrix symbol +formally develops and infinite number of discontinuities which originate from the discrete +nature of the quasiparticle spetrum. However, it is important to notice that, even if the +spectrum is labeled by the discrete index n leading to a finite gap between the ground +state and the first excited levels, still for n ≫ 1 all the modes accumulate around ω∞ = +|h|. This means that an extensive number of single-particle states is almost degenerate. +Consequently, as long as h ̸= 0, we may expect only the first few modes around n = 0 +to provide a significant contribution to the symbol discontinuity, leading to a coefficient +Bν(h ̸= 0) = O(1). Then, in order to understand the qualitative behavior of Bν(h ̸= 0), it +is useful to consider the approximation in which only the first discontinuities between the +n = 0 and the first two degenerate levels n = ±1 are considered +Bν(h ̸= 0) ≈ B(0) +ν ++ B(−1) +ν +. +(D.1) +In order to compute this two contributions we have to compute the angles φ0 and φ±1 +defined by the conditions +cos φn = h − tn +ωn +, +sin φn = −∆n +ωn +. +(D.2) +For n = 0 we find that, independently of the value of α, the angle reads +cos φ0 +� +−1 +if +h < 1 +0 +if +h > 1 +, +φ0 = +� +π +if +h < 1 +0 +if +h > 1. +(D.3) +This discontinuity is at the quantum critical point h = 1 is due to the fact that at this +point the spectrum becomes gapless for n = 0, and it is at the origin of the discontinuity +in the scaling coefficient which can be seen in Fig. 6a of the main text. The angles for +n = ±1 cannot be computed exactly in close form for generic power law decaying exponent, +however as a consequence of the fact that tn = t−n, ωn = ω−n while ∆n = −∆−n, we have +that +cos φn = cos φ−n +sin φn = − sin φ−n, +(D.4) +and then φn = −φ−n. Combining these properties with Eq.(5.5) we obtain +B(0) +ν += B(−1) +ν += +1 +π2(ν − 1) +ν +� +l=1 +arctan2 +� +cos(φ1/2) +1 + sin2(φ1/2) +� +if +h < 1, +(D.5) +B(0) +ν += B(−1) +ν += +1 +π2(ν − 1) +ν +� +l=1 +arctan2 +� +sin(φ1/2) +1 + cos2(φ1/2) +� +if +h > 1. +(D.6) +Figure 8 shows the comparison between exact values of the logarithmic scaling coefficients +– 27 – + +−1 +0 +1 +2 +h +0.0 +0.1 +0.2 +0.3 +B2,α +(a) +α = 0.5 +−1 +0 +1 +2 +h +0.0 +0.1 +0.2 +0.3 +B2,α +(b) +α = 0.7 +−1 +0 +1 +2 +h +0.0 +0.1 +0.2 +0.3 +B2,α +(c) +α = 0.9 +Figure 8. Comparison between the exact values of the logarithmic scaling coefficients of the Rényi- +2 entropy, and the single discontinuity approximation (dashed lines) results. The coefficients are +plotted as function of the chemical potential h for different values of the decay exponent α. +of the Rényi-2 entropy B2, computed considering the contribution of a formally extensive +number of discontinuities (see Eq. (5.5)), and the results obtained in the single discontinuity +approximation. We notice that the single discontinuity approximation correctly reproduces +the qualitative behavior of the scaling coefficients for sufficiently high α > 0.5 and for +values of the chemical potential h which are sufficiently far from h = 0. In particular, the +discontinuity of the coefficients at the quantum critical point h = 1 is captured by the +approximated result. +On the other hand, when the chemical potential approaches the h → 0 limit and for +sufficiently small decay exponents α < 1/2, the single discontinuity approximation turns +out to be no more accurate. Indeed, in this case the number of relevant discontinuities +grows with the subsystem size, leading to a subvolume law entanglement scaling. This fact +can be understood by considering the h = 0 point. In this case, the spectrum accumulation +point becomes ω∞ = 0. More precisely, it is important to notice that, while at the leading +order as n → ∞ the spectrum goes to zero as ωn = O(nα−1), independently of the parity +of the mode, on the contrary next to leading order corrections differ if n is even or odd. +In particular, if we perform a next to leading order expansion of the terms entering the +coefficient B(m) +2 +(see Eq. (5.6)), corresponding to the discontinuity between the modes +m = 2n and m + 1 = 2n + 1, we find +t2n+1t2n = +s2 +α +n2−2α + O(n2α−3), +(D.7) +∆2n+1∆2n = +c2 +α +n2−2α − a2 +α +n2 + O(n2α−3), +(D.8) +ω2n+1ω2n = s2 +α + c2 +α +n2−2α + bα +n2 + O(n2α−3), +(D.9) +where we have introduced the expansion coefficients +sα = sin(απ/2)Γ(2 − α)(2π)α−1, +cα = cos(απ/2)Γ(2 − α)(2π)α−1, +aα = (1 − α)/(2π), +bα = a2 +α(1/2 − cos2(απ/2)) = a2 +α cos(απ)/2. +(D.10) +– 28 – + +Now, inserting the large n expansions of Eqs. (D.7), (D.8) and (D.9) into Eq. (5.6), we see +that the denominator is always of order O(n2α−2), while in the numerator the leading order +cancels out and we are left with a contribution of order O(n−2) if α < 1/2 or O(n2α−3) if +α > 1/2. Finally, putting everything together and summing over all the modes we obtain +Bν(h = 0) = +� +n +B(n) +ν += +�� +n O(n−2α) = O(L1−2α) +α < 1/2 +� +n O(n−1) = O(1) +α > 1/2 +. +(D.11) +This result leads to the scaling of the Rényi entropy in Eq. (5.11) of the main text. +References +[1] A. Campa, T. Dauxois, D. Fanelli and S. Ruffo, Physics of Long-Range Interacting Systems, +Oxford University Press (Aug., 2014), 10.1093/acprof:oso/9780199581931.001.0001. +[2] N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo and A. Trombettoni, Long-range +interacting quantum systems, arXiv: 2109.01063 (2021) [2109.01063]. +[3] S. Xu, Long-range coupling affects entanglement dynamics, Physics 15 (2022) 2. +[4] A. Russomanno, F. Iemini, M. Dalmonte and R. Fazio, Floquet time crystal in the +lipkin-meshkov-glick model, Phys. Rev. B 95 (2017) 214307. +[5] F.M. Surace, A. Russomanno, M. Dalmonte, A. Silva, R. Fazio and F. Iemini, Floquet time +crystals in clock models, Phys. Rev. B 99 (2019) 104303. +[6] A. Pizzi, J. Knolle and A. Nunnenkamp, Higher-order and fractional discrete time crystals in +clean long-range interacting systems, Nature Communications 12 (2021) 2341. +[7] G. Giachetti, A. Solfanelli, L. Correale and N. Defenu, High-order time crystal phases and +their fractal nature, arXiv: 2203.16562 (2022) . +[8] A. Solfanelli, G. Giachetti, M. Campisi, S. Ruffo and N. Defenu, Quantum heat engine with +long-range advantages, arXiv: 2208.09492 (2022) . +[9] J.W. Britton, B.C. Sawyer, A.C. Keith, C.-C.J. Wang, J.K. Freericks, H. Uys et al., +Engineered two-dimensional ising interactions in a trapped-ion quantum simulator with +hundreds of spins, Nature 484 (2012) 489. +[10] C. Monroe, W.C. Campbell, L.-M. Duan, Z.-X. Gong, A.V. Gorshkov, P.W. Hess et al., +Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93 +(2021) 025001. +[11] R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner and T. Esslinger, Roton-type +mode softening in a quantum gas with cavity-mediated long-range interactions, Science 336 +(2012) 1570 [https://www.science.org/doi/pdf/10.1126/science.1220314]. +[12] F. Mivehvar, F. Piazza, T. Donner and H. Ritsch, Cavity qed with quantum gases: new +paradigms in many-body physics, Advances in Physics 70 (2021) 1 +[https://doi.org/10.1080/00018732.2021.1969727]. +[13] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B.L. Lev and T. Pfau, Dipolar +physics: A review of experiments with magnetic quantum gases, arXiv: 2201.02672 (2022) . +[14] C.H. Bennett and D.P. DiVincenzo, Quantum information and computation, Nature 404 +(2000) 247. +– 29 – + +[15] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information: 10th +Anniversary Edition, Cambridge University Press (2010), 10.1017/CBO9780511976667. +[16] X.-S. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek, W. Naylor et al., Quantum +teleportation over 143 kilometres using active feed-forward, Nature 489 (2012) 269. +[17] N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74 +(2002) 145. +[18] V. Giovannetti, S. Lloyd and L. Maccone, Advances in quantum metrology, Nature Photonics +5 (2011) 222. +[19] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, +Reviews of modern physics 81 (2009) 865. +[20] M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666. +[21] J. Eisert, M. Cramer and M.B. Plenio, Colloquium: Area laws for the entanglement entropy, +Rev. Mod. Phys. 82 (2010) 277. +[22] M.B. Hastings, An area law for one-dimensional quantum systems, Journal of Statistical +Mechanics: Theory and Experiment 2007 (2007) P08024. +[23] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of +Statistical Mechanics: Theory and Experiment 2004 (2004) P06002. +[24] P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, Journal of +Physics A: Mathematical and Theoretical 42 (2009) 504005. +[25] C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal +field theory, Nuclear Physics B 424 (1994) 443. +[26] G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, +Phys. Rev. Lett. 90 (2003) 227902. +[27] J.I. Latorre, E. Rico and G. Vidal, Ground state entanglement in quantum spin chains, +Quantum Info. Comput. 4 (2004) 48–92. +[28] J.I. Latorre and A. Riera, A short review on entanglement in quantum spin systems, Journal +of Physics A: Mathematical and Theoretical 42 (2009) 504002. +[29] L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, Quantum source of entropy for black holes, +Phys. Rev. D 34 (1986) 373. +[30] X. Dong, The gravity dual of rényi entropy, Nature communications 7 (2016) 1. +[31] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de +sitter space/conformal field theory correspondence, Phys. Rev. Lett. 96 (2006) 181602. +[32] M. Van Raamsdonk, Building up spacetime with quantum entanglement, General Relativity +and Gravitation 42 (2010) 2323. +[33] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschritte der +Physik 61 (2013) 781 +[https://onlinelibrary.wiley.com/doi/pdf/10.1002/prop.201300020]. +[34] L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. +Mod. Phys. 80 (2008) 517. +[35] P. Calabrese, J. Cardy and B. Doyon, Entanglement entropy in extended quantum systems, J. +Phys. A 42 (2009) 500301. +– 30 – + +[36] N. Laflorencie, Quantum entanglement in condensed matter systems, Physics Reports 646 +(2016) 1. +[37] Z.-X. Gong, M. Foss-Feig, F.G.S.L. Brandão and A.V. Gorshkov, Entanglement area laws for +long-range interacting systems, Phys. Rev. Lett. 119 (2017) 050501. +[38] T. Kuwahara and K. Saito, Area law of noncritical ground states in 1d long-range interacting +systems, Nature Communications 11 (2020) 4478. +[39] A. Lerose and S. Pappalardi, Origin of the slow growth of entanglement entropy in +long-range interacting spin systems, Phys. Rev. Research 2 (2020) 012041. +[40] G. Giachetti and N. Defenu, Entanglement propagation and dynamics in non-additive +quantum systems, arXiv:2112.11488 (2021) . +[41] L. Capizzi, G. Giachetti, A. Santini and M. Collura, Spreading of a local excitation in a +quantum hierarchical model, Phys. Rev. B 106 (2022) 134210. +[42] Y. Li, X. Chen and M.P.A. Fisher, Quantum zeno effect and the many-body entanglement +transition, Phys. Rev. B 98 (2018) 205136. +[43] B. Skinner, J. Ruhman and A. Nahum, Measurement-induced phase transitions in the +dynamics of entanglement, Phys. Rev. X 9 (2019) 031009. +[44] A. Chan, R.M. Nandkishore, M. Pretko and G. Smith, Unitary-projective entanglement +dynamics, Phys. Rev. B 99 (2019) 224307. +[45] X. Cao, A. Tilloy and A.D. Luca, Entanglement in a fermion chain under continuous +monitoring, SciPost Phys. 7 (2019) 024. +[46] M.J. Gullans and D.A. Huse, Dynamical purification phase transition induced by quantum +measurements, Phys. Rev. X 10 (2020) 041020. +[47] X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte and M. Schiró, Measurement-induced +entanglement transitions in the quantum ising chain: From infinite to zero clicks, Phys. Rev. +B 103 (2021) 224210. +[48] O. Alberton, M. Buchhold and S. Diehl, Entanglement transition in a monitored free-fermion +chain: From extended criticality to area law, Phys. Rev. Lett. 126 (2021) 170602. +[49] T. Minato, K. Sugimoto, T. Kuwahara and K. Saito, Fate of measurement-induced phase +transition in long-range interactions, Phys. Rev. Lett. 128 (2022) 010603. +[50] M. Block, Y. Bao, S. Choi, E. Altman and N.Y. Yao, Measurement-induced transition in +long-range interacting quantum circuits, Phys. Rev. Lett. 128 (2022) 010604. +[51] T. Müller, S. Diehl and M. Buchhold, Measurement-induced dark state phase transitions in +long-ranged fermion systems, Phys. Rev. Lett. 128 (2022) 010605. +[52] S. Sharma, X. Turkeshi, R. Fazio and M. Dalmonte, Measurement-induced criticality in +extended and long-range unitary circuits, SciPost Phys. Core 5 (2022) 023. +[53] P. Zhang, C. Liu, S.-K. Jian and X. Chen, Universal Entanglement Transitions of Free +Fermions with Long-range Non-unitary Dynamics, Quantum 6 (2022) 723. +[54] M. Ippoliti, T. Rakovszky and V. Khemani, Fractal, logarithmic, and volume-law entangled +nonthermal steady states via spacetime duality, Phys. Rev. X 12 (2022) 011045. +[55] D. Vodola, L. Lepori, E. Ercolessi and G. Pupillo, Long-range ising and kitaev models: +phases, correlations and edge modes, New Journal of Physics 18 (2015) 015001. +– 31 – + +[56] D. Vodola, L. Lepori, E. Ercolessi, A.V. Gorshkov and G. Pupillo, Kitaev chains with +long-range pairing, Phys. Rev. Lett. 113 (2014) 156402. +[57] F. Ares, J.G. Esteve, F. Falceto and A.R. de Queiroz, Entanglement in fermionic chains with +finite-range coupling and broken symmetries, Phys. Rev. A 92 (2015) 042334. +[58] F. Ares, J.G. Esteve, F. Falceto and A.R. de Queiroz, Entanglement entropy in the +long-range kitaev chain, Phys. Rev. A 97 (2018) 062301. +[59] F. Ares, S. Murciano and P. Calabrese, Symmetry-resolved entanglement in a long-range +free-fermion chain, Journal of Statistical Mechanics: Theory and Experiment 2022 (2022) +063104. +[60] G. Gori, S. Paganelli, A. Sharma, P. Sodano and A. Trombettoni, Explicit hamiltonians +inducing volume law for entanglement entropy in fermionic lattices, Phys. Rev. B 91 (2015) +245138. +[61] L. Lepori, S. Paganelli, F. Franchini and A. Trombettoni, Mutual information for fermionic +systems, Phys. Rev. Research 4 (2022) 033212. +[62] S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum heisenberg +magnet, Physical review letters 70 (1993) 3339. +[63] J. Maldacena and D. Stanford, Remarks on the sachdev-ye-kitaev model, Physical Review D +94 (2016) 106002. +[64] P. Zhang, Quantum entanglement in the sachdev—ye—kitaev model and its generalizations, +Frontiers of Physics 17 (2022) 43201. +[65] C. Liu, X. Chen and L. Balents, Quantum entanglement of the sachdev-ye-kitaev models, +Phys. Rev. B 97 (2018) 245126. +[66] W. Fu and S. Sachdev, Numerical study of fermion and boson models with infinite-range +random interactions, Phys. Rev. B 94 (2016) 035135. +[67] Y. Huang and Y. Gu, Eigenstate entanglement in the sachdev-ye-kitaev model, Phys. Rev. D +100 (2019) 041901. +[68] P. Zhang, C. Liu and X. Chen, Subsystem Rényi entropy of thermal ensembles for SYK-like +models, SciPost Phys. 8 (2020) 094. +[69] A. Haldar, S. Bera and S. Banerjee, Rényi entanglement entropy of fermi and non-fermi +liquids: Sachdev-ye-kitaev model and dynamical mean field theories, Phys. Rev. Research 2 +(2020) 033505. +[70] M.G. Nezhadhaghighi and M.A. Rajabpour, Entanglement entropy in long-range harmonic +oscillators, Europhysics Letters 100 (2013) 60011. +[71] J.I. Latorre, R. Orús, E. Rico and J. Vidal, Entanglement entropy in the lipkin-meshkov-glick +model, Phys. Rev. A 71 (2005) 064101. +[72] J. Vidal, S. Dusuel and T. Barthel, Entanglement entropy in collective models, Journal of +Statistical Mechanics: Theory and Experiment 2007 (2007) P01015. +[73] R. Orús, S. Dusuel and J. Vidal, Equivalence of critical scaling laws for many-body +entanglement in the lipkin-meshkov-glick model, Phys. Rev. Lett. 101 (2008) 025701. +[74] J.A. Carrasco, F. Finkel, A. González-López, M.A. Rodríguez and P. Tempesta, Generalized +isotropic lipkin–meshkov–glick models: ground state entanglement and quantum entropies, +Journal of Statistical Mechanics: Theory and Experiment 2016 (2016) 033114. +– 32 – + +[75] M. Kumari and Á.M. Alhambra, Eigenstate entanglement in integrable collective spin models, +Quantum 6 (2022) 701. +[76] A.Y. Kitaev, Unpaired majorana fermions in quantum wires, Physics-Uspekhi 44 (2001) 131. +[77] M.E. Fisher and R.E. Hartwig, Toeplitz determinants: Some applications, theorems, and +conjectures, in Advances in Chemical Physics, pp. 333–353, John Wiley & Sons, Ltd (1969), +DOI [https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470143605.ch18]. +[78] E.L. Basor and K.E. Morrison, The fisher-hartwig conjecture and toeplitz eigenvalues, Linear +Algebra and its Applications 202 (1994) 129. +[79] M. Kac, G.E. Uhlenbeck and P.C. Hemmer, On the van der waals theory of the vapor-liquid +equilibrium. i. discussion of a one-dimensional model, Journal of Mathematical Physics 4 +(1963) 216 [https://doi.org/10.1063/1.1703946]. +[80] D. Jaschke, K. Maeda, J.D. Whalen, M.L. Wall and L.D. Carr, Critical phenomena and +kibble–zurek scaling in the long-range quantum ising chain, New Journal of Physics 19 +(2017) 033032. +[81] L. Vanderstraeten, M. Van Damme, H.P. Büchler and F. Verstraete, Quasiparticles in +quantum spin chains with long-range interactions, Phys. Rev. Lett. 121 (2018) 090603. +[82] N. Defenu, T. Enss and J.C. Halimeh, Dynamical criticality and domain-wall coupling in +long-range hamiltonians, Phys. Rev. B 100 (2019) 014434. +[83] E. Fradkin, Field Theories of Condensed Matter Physics, Cambridge University Press, 2 ed. +(2013), 10.1017/CBO9781139015509. +[84] P. Uhrich, N. Defenu, R. Jafari and J.C. Halimeh, Out-of-equilibrium phase diagram of +long-range superconductors, Phys. Rev. B 101 (2020) 245148. +[85] A. Alecce and L. Dell’Anna, Extended kitaev chain with longer-range hopping and pairing, +Phys. Rev. B 95 (2017) 195160. +[86] N. Defenu, Metastability and discrete spectrum of long-range systems, Proceedings of the +National Academy of Sciences 118 (2021) e2101785118. +[87] W. DeGottardi, M. Thakurathi, S. Vishveshwara and D. Sen, Majorana fermions in +superconducting wires: Effects of long-range hopping, broken time-reversal symmetry, and +potential landscapes, Phys. Rev. B 88 (2013) 165111. +[88] L. Lepori and L. Dell’Anna, Long-range topological insulators and weakened bulk-boundary +correspondence, New Journal of Physics 19 (2017) 103030. +[89] I. Peschel, Calculation of reduced density matrices from correlation functions, Journal of +Physics A: Mathematical and General 36 (2003) L205. +[90] B.-Q. Jin and V.E. Korepin, Quantum spin chain, toeplitz determinants and the +fisher—hartwig conjecture, Journal of statistical physics 116 (2004) 79. +[91] A. Its, F. Mezzadri and M. Mo, Entanglement entropy in quantum spin chains with finite +range interaction, Communications in mathematical physics 284 (2008) 117. +[92] H. Widom, Asymptotic behavior of block toeplitz matrices and determinants, Advances in +Mathematics 13 (1974) 284. +[93] H. Widom, Asymptotic behavior of block toeplitz matrices and determinants. ii, Advances in +Mathematics 21 (1976) 1. +– 33 – + +[94] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, +Graphs, and Mathematical Tables, Applied mathematics series, Dover Publications (1965). +[95] L. Lepori, D. Vodola, G. Pupillo, G. Gori and A. Trombettoni, Effective theory and +breakdown of conformal symmetry in a long-range quantum chain, Annals of Physics 374 +(2016) 35. +[96] N. Defenu, A. Trombettoni and S. Ruffo, Criticality and phase diagram of quantum +long-range o(n) models, Phys. Rev. B 96 (2017) 104432. +[97] N. Defenu, G. Morigi, L. Dell’Anna and T. Enss, Universal dynamical scaling of long-range +topological superconductors, Phys. Rev. B 100 (2019) 184306. +[98] A. Solfanelli, S. Ruffo, S. Succi and N. Defenu, In preparation, +(2023) . +– 34 – + diff --git a/ztFQT4oBgHgl3EQfCjXs/content/tmp_files/load_file.txt b/ztFQT4oBgHgl3EQfCjXs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c167cd56cd32db6bb13ba84c70c2af1fc9799ef3 --- /dev/null +++ b/ztFQT4oBgHgl3EQfCjXs/content/tmp_files/load_file.txt @@ -0,0 +1,1351 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf,len=1350 +page_content='Prepared for submission to JHEP Logarithmic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fractal and Volume-Law Entanglement in a Kitaev chain with long-range hopping and pairing Andrea Solfanelli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 Stefano Ruffoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='d Sauro Succic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e Nicolò Defenuf aSISSA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' via Bonomea 265,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' I-34136 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Italy bINFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sezione di Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Via Valerio 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 34127 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Italy cCenter for Life Nano-Neuro Science @ La Sapienza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Italian Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 00161 Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Italy dIstituto dei Sistemi Complessi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Via Madonna del Piano 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' I-50019 Sesto Fiorentino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Italy ePhysics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Oxford Street 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' USA fInstitut für Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' ETH Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Wolfgang-Pauli-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 27 Zürich, Switzerland E-mail: asolfane@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='it Abstract: Thanks to their prominent collective character, long-range interactions pro- mote information spreading and generate forms of entanglement scaling, which cannot be observed in traditional systems with local interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this work, we study the asymp- totic behavior of the entanglement entropy for Kitaev chains with long-range hopping and pairing couplings decaying with a power law of the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We provide a fully-fledged analytical and numerical characterization of the asymptotic growth of the ground state en- tanglement in the large subsystem size limit, finding that the truly non-local nature of the model leads to an extremely rich phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Most significantly, in the strong long- range regime, we discovered that the system ground state may have a logarithmic, fractal, or volume-law entanglement scaling, depending on the value of the chemical potential and on the strength of the power law decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13231v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='[quant-ph] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='30 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Contents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Introduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Kitaev chain with long-range couplings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Entanglement scaling in free fermioninc systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Weak long-range regime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Strong long-range regime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Conclusion and outlooks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A Derivation of the matrix symbol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='B Coefficients of the Fisher-Hartwig expansion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C Dispersion relation around the critical modes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='D Discontunities in the strong long-range regime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Introduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='The interest of the quantum community in long-range physics has steadily risen in recent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='years since long-range interacting quantum systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', systems in which the coupling en- ergy between couples of microscopic constituents Vi,j decays as a power law of their distance r = |i − j|: Vi,j ∝ r−α, with α > 0 [1, 2] are emerging as promising platforms for quantum technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This is due to their stability against external perturbations, which allows keeping the impact of dynamically generated excitations under control, there- fore mitigating their detrimental effects [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' An example of the rigidity of long-range interacting platforms against external drivings and of its utility for quantum technological applications is the possibility for such systems to host clean discrete Floquet time crystal phases [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Another example is the recently introduced advantage in the finite time per- formance of quantum heat-engines with a working substance hosting long-range couplings [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, this technological and theoretical interest is also supported from the ex- perimental side by the possibility to implement long-range interacting systems in typical quantum simulation platforms, such as atomic molecular and optical (AMO) systems [9– 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Interestingly, trapped ions setups allow tuning the power law exponent α, dictating the decay of the interaction energy with distance, from α ≃ 0 to α ≃ 3 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The most important feature a system should have to be a good candidate for quantum technologies is the capability of hosting highly entangled states in its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, this – 1 – crucial property is the essential ingredient to perform tasks that are classically impossible or very inefficient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, entanglement is the property that makes quantum computation overtake classical one providing the computational speed-up in quantum algo- rithms as compared to algorithms based on the processes of classical physics [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, it is crucial for many quantum technological applications such as quantum teleportation [16], quantum cryptography [17] or quantum metrology [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' A set of key quantities entering the characterization of entanglement is provided by the entanglement Rényi entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' For their definition, one takes a partition of a given system in two subsystems A and B (the complement of A), determines the reduced density matrix of a subsystem (say, of A) ρA by tracing out the degrees of freedom of B, and then computes its Rényi entropies: Sν = ln Tr[ρν A]/(1 − ν) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' One of the most fundamental properties of entanglement Rényi entropies is their behavior with the size of the subsystem considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The celebrated area law [20, 21] refers to the fact that typically entanglement grows as the boundary of the subsystem considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', for a system in d dimensions and a subsystem of size L having volume ∼ Ld and area ∼ Ld−1, then the entanglement entropy of the subsystem scales as ∼ Ld−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, the area law has been proven to be satisfied in the ground state of one-dimensional systems with mass gap and short-range couplings when the size of the subsystem is much larger than the correlation length [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' At a quantum critical point, where the correlation length diverges, the area law is known to be violated by a logarithmic term proportional to the central charge of the conformal field theory (CFT) that describes the low-energy spectrum of the model [23–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These facts motivated initially the study of this quantity due to its similarity to the black hole entropy [20, 29], and have eventually revealed the important role that entanglement plays in high-energy physics [30–33] as well as in the investigation of condensed matter systems [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The previous discussion changes and becomes more involved for systems with long-range couplings [2, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed the prominent collective character of such non-local systems promotes entanglement spreading and leads to novel forms of equilibrium and dynamical scaling, which cannot be observed in traditional systems with local interactions [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, the anomalous scaling of entanglement in the presence of long-range couplings has recently attracted great interest in the context of the so-called measurement-induced transitions [42–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this case, the dynamical generation of entanglement is weakend by the presence of local measures applied randomly during the system evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, if the measurement rate is high enough, the steady state entanglement saturates to an area law value independent of the considered subsystem size, if only nearest neighbor interactions are present [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, in the presence of long-range couplings, subvolume law scalings [3, 49–52], also referred to as fractal entanglement phases [53, 54], appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These interesting dynamical phenomena have no clear equilibrium counterpart showing that their origin is directly related to the presence of long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The entangle- ment properties of the ground state of a fermionic chain with long-range pairing couplings and nearest neighbors hopping amplitudes were fully characterized in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [55–59] which reported standard logarithmic violations of the area law in the weak long-range regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 2 – Moreover, an anomalous logarithmic growth was found even if the mass gap is not zero, associated to the divergence of unnormalized couplings, in the strong long-range regime characterized by a power law decay exponent smaller than the system dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, the authors of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [60, 61] considered a model of fermions with strong long- range hopping amplitudes and no pairing discovering a volume law entanglement scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, the entanglement properties of the Sachdev-Ye-Kitaev (SYK) model [62, 63], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' a fully connected fermionic model with random interactions, have been extensively studied [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Also in this case, the eigenstates of the SYK Hamiltonian display a volume law entanglement scaling whose coefficient has been computed numerically using exact di- agonalization techniques [65, 66] and analytically assuming the eigenstate thermalization hypothesis [67] or using a path-integral approach which becomes exact in the large-N limit [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, also in long-range bosonic [70] and in fully connected spin systems [71–75] only logarithmic violations of the area law were reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Despite the extensive amount of literature on the topic summarized above, none of the considered long-range models display a fractal entanglement scaling at equilibrium unless additional ingredients are added such as modifications of the couplings which violate time translational symmetry or the presence of a fractal Fermi surface [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Here, we are going to show that the subvolume law observed in measurement induced transitions [3, 49–54] is directly caused by long-range interactions and also appears at equilibrium, provided certain conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' To prove our claim, we study the ground state entanglement scaling in a prototypical model of fermions with power-law decaying hopping and pairing amplitudes, also known as the long-range Kitaev chain [2, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This model is sufficiently simple to allow us to perform analytic calculations but at the same time it turns out to host an extremely rich phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Using the well-known Fisher-Hartwig expansion [77, 78], we were able to analytically determine the leading order dependence of the ground state entanglement on the subsystem size L in the scaling limit of an infinite chain of N → ∞ sites and infinite subsystem L → ∞ with fixed l = L/N, for different values of the available parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, we can distinguish two main regimes: the weak long-range regime in which the coupling’s power law decaying exponents are larger than the system dimension and the strong long-range regime in which they are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In the former case, the system shows standard logarithmic deviations from the entanglement area law in correspondence with the quantum critical points, however, in the most interesting case of equal long- range hopping and pairing the coefficients in front of these logarithmic divergences show a nontrivial dependence on the power law decay exponent α which is not compatible with the standard scaling predicted by critical conformal field theory [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, in the strong long-range case, the system becomes genuinely non-additive, therefore showing a logarithmic deviation from the area law even away from criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Most significantly, when the system chemical potential is zero, no local terms are present in the Hamiltonian (as we will see this simple fact strongly affects the nature of the ground state which becomes highly degenerate) thus resulting into a subvolume law entanglement scaling, S ∼ L1−2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Summarizing, our work correctly reproduces previously known results in different limits, thus bringing several disparate results present in the literature into a coherent picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 3 – Moreover, we are able to detect a fractal entanglement scaling phase which is entirely due to the non-additive nature of the model and does not need the dynamical setting of measurement induced transitions to be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In Section 2 we introduce the long-range Kitaev model and we describe its phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In Section 3 we briefly review the techniques which allow us to study the entanglement scaling of generic quadratic fermionic models (the expert reader may safely skip this part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, Section 4 and 5 are devoted to the detailed characterization of the ground state entanglement scaling of the model in the weak and strong long-range regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 2 Kitaev chain with long-range couplings We consider a generic model of spinless fermions hopping across the N sites of a one- dimensional chain in the presence of pairing interactions, and with a chemical potential h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Assuming periodic boundary conditions, the system Hamiltonian reads H = − N � j=1 N/2−1 � r=1 � trˆc† j+rˆcj + ∆rˆc† j+rˆc† j + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' � − h N � j=1 � 1 − 2ˆc† jˆcj � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) where ˆc† j and ˆcj are creation and annihilation operators for fermions at site j, while tr and ∆r are the hopping and pairing amplitudes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We choose their dependence on the intersite distance r according to the power laws tr = 1 Nα1 J rα1 , ∆r = 1 Nα2 ∆ rα2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) with the hopping exponent α1 > 0, the pairing exponent α2 > 0, and Nα = �N/2 r=1 r−α the Kac scaling factor [79], which guarantees extensivity of the energy in the case αi < 1, with i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This model, often referred to as long-range Kitaev chain [76], is emerging as a minimal model for the study of the effects of long-range couplings on a quantum system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, its integrable nature makes it amenable to both analytical and numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, as observed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [80–82], when the pairing and hopping power law decay exponents are equal α1 = α2 = α the model can be related to the quantum Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, in the short-range case with α → ∞, the relation becomes exact through the Jordan-Wigner mapping [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The quadratic nature of the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) allows its exact diagonalization in Fourier space via the Bogolyubov transformation ˆck = cos θk 2 ˆγk + sin θk 2 ˆγ† −k, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) – 4 – where we have introduced the momentum space fermionic operators ˆck = e−i π 4 √ N N � j=1 eikjˆcj, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) with k = 2πn/N, with n = −N/2+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' , N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' While the Bogoliubov angles are defined by the conditions tan θk = ∆k/(h − tk), where Fourier transforms of the hopping and pairing amplitudes are defined as tk = 1 Nα1 N/2−1 � r=1 cos(kr) rα1 , ∆k = 1 Nα2 N/2−1 � r=1 cos(kr) rα2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) In terms of the Bogoliubov fermions, the Hamiltonian then takes the diagonal form H = � k ωk(h) � ˆγ† kˆγk − 1/2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) with the quasiparticle spectrum ωk(h) = 2 � (h − tk)2 + ∆2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) Since ωk(h) ≥ 0, the ground state corresponds to the Fock space vacuum for the Bogoliubov modes, defined by the condition ˆγk|gs⟩ = 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' When studying the critical properties associated with the spectrum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7), we must distinguish two main regimes: the weak long-range regime when α1, α2 > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', the power law decay exponents are larger than the system dimensionality, and the strong long-range regime when α1, α2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In the weak long-range case, the Kac scaling is a constant in the thermodynamic limit: Nα>1 → ζ(α), where ζ(α) is the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, when the system size goes to infinity, we can safely perform a continuum limit in the k variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) may be written as tk = Re � Liα1(eik) � /ζ(α1), ∆k = Im � Liα2(eik) � /ζ(α2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) where Liα(z) denotes the polylogarithm function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This leads to a continuum spectrum ωk characterized, at the critical points, by a dispersion relation that depends on α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, for α1, α2 > 1, the system possesses two different phases separated by two quantum critical points hc = 1, −1 + 21−α1, in correspondence of which the dispersion relation becomes gapless near to the critical mode kc = 0, π, respectively [2, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The critical modes of the spectrum are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 1a where ω0(π)(blue(red) lines in the plot) is plotted as a function of h for different values of α1 = α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The nature of the transition is topological and the two topological phases can be distinguished by the value of the bulk topological invariant [85] w = � dθk 2π = � 1 if h ∈ [−1 + 21−α1, 1] 0 overwise , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) – 5 – −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 h 0 2 4 6 ω0,π (a) α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 α1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 α1 = 8 0 2 4 6 8 α1 −1 0 1 h q = −1 q = +1 (b) h = t0 h = tπ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' a) Critical modes k = 0, π of the quasiparticle spectrum as a function of the chemical potential h for different values of α1 = α2, two critical points emerge at h = t0, tπ, where in the thermodynamic limit t0 = 1 and tπ = 1 if α1 > 1, tπ = −1 + 21−α1 if 1 < α1 < 2, and tπ = 0 if 0 < α1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' b) Phase diagram of the long-range Kitaev chain in the plane (α1, h), for the pairing decay exponent α2 = α1, α1 is the hopping decay exponent and h is the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The topological order parameter is q = −1 in the topological phase (blue shaded region) and q = +1 in the trivial phase (red shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The phase space boundaries correspond to the solid lines h = t0 and h = tπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' where the Bogoliubov angles are defined as θk = arctan(∆k/(h − tk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In the strong long-range regime 0 < α1, α2 < 1 the scenario is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in this case, the Kac normalization factor Nα diverges at large N as Nα ≈ N1−α, and the thermodynamic limit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) has to be carefully considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, as pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [86], while the Fourier modes variable k = 2πn/N becomes continuous as N → ∞, the hopping and pairing amplitudes tk, ∆k, remain discrete and labeled by the integer n, reading lim N→∞ tk = cα1 � 1/2 0 dscos(2πns) sα1 = tn, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) lim N→∞ ∆k = cα2 � 1/2 0 dssin(2πns) sα2 = ∆n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) with cα = (1 − α)21−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Therefore, the presence of long-range couplings leads to a discrete spectrum ωk → ωn = 2 � (h − tn)2 + ∆2n also at N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The persistence of the discrete spectrum in the thermodynamic limit does not allow us to define a continuous theory and hinders the conventional definition of quantum critical points in the Kitaev chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In par- ticular, the winding number in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) is ill-defined as a consequence of the discontinuity in the Bogolyubov angle distribution [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Yet, the transition can still be characterized by the quantity q = sign[(h − t0)(h − tπ)] = � 1 if h ∈ [tπ, t0] −1 overwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) This quantity has proven to be a good topological invariant in cases in which the winding number turns out to be ill-defined [85, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then, also in the strong long-range regime, – 6 – the behavior of the order parameter q is still consistent with a change of phase at the critical points h = t0, tπ [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, as shown in [88], the bulk boundary correspondence turns out to be weakened by the presence of strong long-range couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Consequently, the change of q at the critical points is not guaranteed to be in one-to-one correspondence with the appearance of boundary topological edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nevertheless, we expect bulk properties to remain consistent with a change of phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Figure 1b shows the model phase diagram as characterized by the value of q = ±1 as a function of the chemical potential h and of the hopping power law decay exponent α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Two quantum critical lines appear when varying the α1 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, we notice that the location of the critical point corresponding to ω0 = 0 is fixed to h = t0 = 1 for any value of α1 (blue bold line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the contrary, the critical point corresponding to ωπ = 0 (red bold line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 1b) is α1 dependent with two different behaviors in the weak and strong long-range regimes, in particular in the thermodynamic limit we find lim N→∞ tπ = � � � � � � � −1 if α1 > 2 −1 + 21−α1 if 1 < α1 < 2 0 if 0 < α1 < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) Finally, the completely mean-field case with α1 = α2 = 0 needs to be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in this case, the spectrum becomes strongly degenerate and this may alter the nature of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, for completely flat couplings the sums in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) can be exactly computed and, in the thermodynamic, they read tn(α1 = 0) = δn,0, ∆n(α2 = 0) = 1 + (−1)n+1 πn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='14) Accordingly, the single-particle spectrum becomes ω0 n = � � � � � � � |h| if |n| even � h2 + 4/(πn)2 if |n| odd |h − 1| if n = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='15) where we have introduced the shortcut notation ω0 n = ωn(α1 = 0, α2 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' It follows that an extensive number of single-particle energy levels corresponding to all the even modes become degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, when the chemical potential is zero h = 0 all the even modes become zero modes since at this point we have ω0 2n(h = 0) = 0, ω0 2n+1(h = 0) = 2/|πn| and ω0 0(h = 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This fact deeply affects the nature of the many-body ground state which is no more given by the Bogoliubov vacuum, on the contrary, it allows for a finite population of Bogoliubov fermions in an extensive number of zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, the ground state for α1,2 = 0 and h = 0 is given by a generic superposition of the form |gs⟩α=0,h=0 = N0 � n0=0 Cn0|n0⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='16) – 7 – 1 L N A B Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Schematic representation of a bipartition of a long-range Kitaev chain with periodic boundary conditions in two subsystems A and B of length L and N − L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' where n0 is the number of fermions occupying the N0 available zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This ground state is highly degenerate indeed each |n0⟩ state can be realized in �N0 n0 � ways, leading to the exponential degeneracy Deg[|gs⟩α=0,h=0] = N0 � n0=0 �N0 n0 � = 2N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='17) As a concluding remark for this section, we stress the importance of the Kac scaling in the stabilization of the topological order in the strong long-range regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, had we considered not properly rescaled couplings, the presence of long-range hopping α1 < 1 would have moved the critical point to hc = O(N1−α1) → ∞, thus destroying the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 3 Entanglement scaling in free fermioninc systems We consider a bipartition of the fermionic chain described by the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1), in two subsystems A and B, where A is a continuous interval of chain sites of length L and B is its complementary set, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Given the Hilbert spaces HA and HB associated to A and B, respectively, then the total Hilbert space of the system can be written as the tensor product H = HA ⊗ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' If the total system is in a pure state |ψ⟩, then the reduced density matrix, describing the state of subsystem A(B) is obtained by taking the partial trace with respect to HA(B): ρA(B) = TrA(B)|ψ⟩⟨ψ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The amount of entanglement between the two subsystems can be characterized by the so-called Rényi entropies of A, defined as Sν,L(A) = 1 1 − ν ln Tr[ρν A], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) where ν ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These are known to provide an accurate measure for the entanglement of a bipartite system in a pure state [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, the limit ν → 1 of the above expression – 8 – corresponds to the celebrated Von Neumann or entanglement entropy SL(A) = S1,L(A) = −Tr[ρA ln ρA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) The main goal of this paper is to study the Rényi entanglement entropy for the ground state of a Hamiltonian of the kind analyzed in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, we are interested in determining the dependence of Sν,L(A) on the subsystem size L in the scaling limit N → ∞, L → ∞ with fixed l = L/N and how this is affected by the presence of long- range hopping and pairing couplings in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This task may be achieved by taking advantage of the fact, that since the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) is quadratic, then all its eigenstates satisfy the Wick decomposition theorem [26, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, the reduced density matrix can be obtained from the two-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, given the 2L × 2L correlation matrix defined as Vij = � δij − 2⟨c† jci⟩ 2⟨cicj⟩ 2⟨c† ic† j⟩ 2⟨c† icj⟩ − δij � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) it can be shown [26, 89] that this is related to the Rényi entropies through the formula Sν,L(A) = 1 2(ν − 1)Tr ln ��I + V 2 �ν + �I − V 2 �ν� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) It is important to notice that, from the computational point of view, this formula constitutes a dramatic simplification since the problem complexity is reduced from the diagonalization of a reduced density matrix of size 2L × 2L to the diagonalization of the correlation ma- trix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) of size 2L × 2L, thus allowing to reach larger sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' From the analytic side, it is useful to write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) as an integral on the complex plane along a contour C surrounding the eigenvalues vj ∈ [−1, 1] of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Using Cauchy’s residue theorem in order to perform the integral, one gets [90, 91] Sν,L(A) = lim ϵ→0+ � C sν(1 + ϵ, z)d ln DL(z) dz dz, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) where we have introduced the function sν(x, y) = 1 1 − ν ln ��x + y 2 �ν + �x − y 2 �ν� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) and the determinant DL(z) = det(zI − V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) Due to the translational invariance of the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) and given the choice of sub- system A, which is composed of contiguous sites, the correlation matrix is a block Toeplitz matrix, which can be written in the Fourier basis as Vlj = 1 N � k Gkeik(l−j), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) – 9 – where we have introduced the two dimensional symbol Gk which, as detailed in Appendix A, can be written as Gk = (1 − (fk + fk)) �h − tk ωk σz − ∆k ωk σy � − (fk − f−k)I, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) where σa, with a = x, y, z, are the Pauli sigma matrices, I is the 2 × 2 identity, and fk = ⟨ˆγ† kˆγk⟩ are the occupation numbers of the Bogoliubov fermionic modes, which for a generic state satisfy the condition 0 ≤ fk ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Using the techniques introduced in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [26, 89] the asymptotic behavior for L → ∞ of the Toeplitz determinant DL(z), entering the expression for the Rényi entropies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4), can be determined applying the Szegő-Widom theorem [92, 93] and an extension of the Fisher- Hartwig conjecture [77, 78] to non-scalar symbols [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The leading order contributions to the logarithm of DL(z) in the L → ∞ limit then read ln DL(z) = L 2π � π −π dk ln det(zI − Gk) + ln L � p bp(z) + O(1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) where the coefficients bp(z) of the logarithmic contribution are associated to the disconti- nuities of Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, if there is a discontinuity at some k = p, this means that G+ p = lim k→p+ Gk ̸= lim k→p− Gk = G− p , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) then the coefficient corresponding to such discontinuity can be computed as [58] bp(z) = 1 4π2 Tr[ln(zI − G− p )(zI − G+ p )−1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) into the integral for the Rényi entropy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) one obtains Sν,L = 1 1 − ν � k ln [(1 − fk)ν + fν k ] + Bν ln L + O(1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) where the coefficient of the logarithmic contribution can be computed as Bν = � p lim ϵ→0+ � C sν(1 + ϵ, z)dbp(z) dz dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='14) As shown in Section 2, whenever α1,2 > 0 or α1 = α2 = 0 and h ̸= 0, the many-body ground state of the system is the Bogoliubov vacuum with fk = 0 ∀k, therefore we are left with a leading order contribution given by a constant term O(1) corresponding to the standard area law in the one-dimensional case, or a logarithmic contribution which is associated to the discontinuity of the correlation matrix symbol Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand in the specific case α1 = α2 = 0 and h = 0 the many-body ground state becomes highly degenerate allowing for a finite fermionic population fk ̸= 0 for an extensive number of Bogoliubov modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', all the even modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' As a consequence, the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) becomes – 10 – the leading contribution to the large L entanglement scaling corresponding to a volume law behavior Sν,L(α1,2 = 0, h = 0) ≈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Summarizing, the machinery introduced in this section allows us to compute the leading order contribution to the scaling of Rényi entropies with the subsystem size by simply analyzing the symbol continuity properties in the different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 4 Weak long-range regime Let us start with the weak long-range regime corresponding to α1, α2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this case, as we have seen in Section 2, the quasiparticle spectrum is continuous in the thermodynamic limit, and the ground state is always given by the Bogoliubov vacuum with zero fermionic populations fk = 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, the first term of the Fisher-Hartwig expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) vanishes and then the leading order contribution to the entanglement scaling comes from the logarithmic term associated with the matrix symbol discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Within the weak long-range regime, we can distinguish three different cases: α1 > α2, α1 < α2 and α1 = α2 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Therefore, in order to proceed we must identify the location of the jumps of Gk and compute the lateral limits in these three different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Possible sources of discontinuities for Gk are the discontinuities or the zeros of the spectrum ωk(h), which appear at the two quantum critical points h = 1, −1 + 21−α1, where the spectrum becomes gapless at the soft modes k = 0, π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, Gk has no discontinuities when h ̸= 1, −1+21−α1, since in this case the lateral limits at the critical modes read G± 0 = lim k→0± Gk = sgn(h − 1)σz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) G± π = lim k→π± Gk = sgn(h + 1 − 21−α1)σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) This leads to a constant scaling of the entanglement entropy Sν,L = O(1) with the subsystem size when the system is not at quantum criticality and therefore the spectrum is gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This is nothing but a manifestation of the standard area law for one-dimensional gapped systems [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, quantum criticality leads to logarithmic deviations from the area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Let us start from the homogeneous critical point (h = 1), when the spectrum has an α1,2 dependent dispersion relation (see Appendix C), which leads to the different lateral limits G± 0 = � � � � � � � −sgn(A(α1))σz if α1 < α2 − sin(απ/2)σz ± cos(απ/2)σy if α1 = α2 ±sgn(B(α2))σy if α1 > α2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) where A(α) = sin(απ/2)Γ(1 − α)/ζ(α), and B(α) = cos(απ/2)Γ(1 − α)/ζ(α), with Γ(x) and ζ(x) the Gamma and the Riemann zeta functions [94], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, no discontinuity is present when the power law decay of the hopping amplitude is slower than that of the pairing, leading again to a constant entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In the α1 > α2 case – 11 – 500 1000 1500 2000 2500 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='52 SL (a) α1 < α2 c1 + c2L−c3 S2,L 0 1000 2000 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 SL (b) α1 > α2 1 6 log L SL SL − c1 − c2L−c3 0 1000 2000 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='45 S2,L (c) α1 = α2 = α B2,α log L S2,L S2,L − c1 − c2L−c3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Numerical check of the entanglement scaling as a function of the subsystem size L at the quantum critical point with chemical potential h = 1 for different values of couplings power law decay exponents 1 < α1, α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' a) Entanglement entropy (ν = 1), with α1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 and α2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8, blue squares represent the numerical data while the black solid line is a fit of a constant and a subleading contribution c1 + c2L−c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' b) Entanglement entropy (ν = 1), with α1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 and α2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5, blue squares represents the numerical data, the black solid line correspond to the curve (1/6)) ln L, red dots have been obtained from the numerics by subtracting the fit of the subleading corrections of the form c1 + c2L−c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' c) Rényi-2 entropy (ν = 2) with α1 = α2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5, blues squares represents the numerics, the black solid line represents the curve B2,α ln L, red dots are obtained subtracting the subleading corrections to the numerical data as in panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' instead, we have a discontinuity in the symbol, with commuting lateral limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Inserting the expression for G± 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) we obtain b0(z) = 1 2π2 � ln �z + 1 z − 1 ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) Then, inserting this result into the expression for the entanglement entropy, and performing the integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) we obtain the logarithmic scaling Sν,L = ν + 1 12ν ln L + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) This logarithmic scaling is analogous to the one obtained for a conformal field theory with central charge c = 1/2 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This result is in agreement with previous findings [56, 57] concerning the entanglement scaling in a Kitaev chain with long-range paring and nearest neighbors hopping α1 → ∞, here we show that the same scaling holds also for finite α1 as long as α1 > α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Figure 3b shows the numerical check of the scaling behavior of the entanglement entropy SL = S1,L for α1 > α2 and h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We obtain an excellent agreement once the subleading corrections are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, we need to subtract from the numerical data the finite size corrections of the form SL − 1 6 ln L = c1 + c2L−c3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) where the ci = ci(α1, α2, h), i = 1, 2, 3, coefficients can be estimated from a fit with the numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The most interesting case corresponds to the condition α1 = α2 = α which, as previ- ously stated, is closely related to the long-range interacting quantum Ising chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, – 12 – 0 1000 2000 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 SL (a) α1 < α2 1 6 log L SL SL − c1 − c2L−c3 0 1000 2000 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 SL (b) α1 > α2 1 6 log L SL SL − c1 − c2L−c3 0 1000 2000 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9 S2,L (c) α1 = α2 = α 1 8 log L S2,L S2,L − c1 − c2L−c3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Numerical check of the entanglement scaling as a function of the subsystem size L at the quantum critical point with chemical potential h = −1 + 21−α1 for different values of couplings power law decay exponents: a) α1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5, α2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8, b) α1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8, α2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5, c) α1 = α2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3, blue squares represents the numerical data, the black solid line represents our analytical prediction for the scaling in the L ≫ 1 limit, red dots are obtained from the numerics by subtracting the subleading corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' we notice that in this regime the matrix symbol Gk, hosts non-commuting lateral limits as k → 0± (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This leads to the non-trivial dependence of the logarithmic contribution coefficient on α b0(z) = 2 π2 � ln � z2 − sin2(απ/2) + cos(απ/2) √ z2 − 1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) Inserting b0(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) and performing the integration (see Appendix B), we obtain the logarithmic scaling behavior of the Rényi entropy Sν,L = Bν,α ln L + O(1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) where Bν,α = 1 π2(ν − 1) ν � k=1 arctan2 � � cos(απ/2) � sin2(απ/2) + |zk,ν|2 � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) with zk,ν = i tan(π(2k − 1)/2ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, for ν = 2, 3, the sum in the previous expression reduces to B2,α = 2 π2 arctan2 � cos(απ/2) � sin2(απ/2) + 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) B3,α = 1 π2 arctan2 � cos(απ/2) � sin2(απ/2) + 1/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) This analytical scaling of S2,ν at h = 1 and for α1 = α2 = α is compared with the numerical result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Also in this case, a good agreement is found once the subleading corrections (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 13 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10 Bν,α (a) ν = 2 ν = 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4 ceff (b) c = 1/2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' a) Coefficient Bν,α of the logarithmic scaling of the ν-Rényi entropy as a function of the power law decay exponent α = α1 = α2, for ν = 2 (green solid line) and ν = 3 (purple solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The dashed lines correspond to the short-range values of the coefficients which are matched by the long-range ones for α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' b) effective central charge, obtained as ceff = 6νBν,α/(ν +1), as a function of α for ν = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The black dashed line represents the central charge for nearest neighbor couplings c = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' It is important to observe that at variance with the α1 ̸= α2 cases, the scaling coefficient Bν,α cannot be written in the form Bν,α ̸= Bν,CFT = ν + 1 6ν c, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) where c is the central charge of some conformal field theory describing the model at the quantum critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This observation supports our previous claim that the case α1 = α2 is special and, somehow, closer to the one of a strongly interacting system such as the long- range Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, while the case α1 ̸= α2 continues to obey the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) and, so, is more likely to be described by a CFT, the case 1 < α1 = α2 < 2 goes beyond this description as the scaling of the ground state entanglement at the critical point cannot be related to the universal properties of a conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' A similar result is expected for the Ising model in a transverse field, where the inclusion of long-range interactions is expected to increase the effective dimension of the model and, so, disrupt any CFT description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Figure 5a shows the coefficients Bν,α for ν = 2, 3 as a function of α ∈ [1, 2], we notice that the value of the logarithmic scaling coefficients starts from zero at α = 1 and then grows with α reaching the short-range value for α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 5b shows the α dependence of the effective central charge defined as ceff(α) = 6νBν,α/(ν + 1) as a function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We notice that, apart from the extrema ceff(1) = 0 and ceff(2) = 1/2, the effective charge also depends on the Rényi entropy order ν, thus confirming the fact that it cannot be considered as the proper central charge of a conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These results are in agreement with the findings of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [95], where the breakdown of conformal symmetry in a long-range fermionic chain was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, we consider the non-homogeneous critical point h = −1 + 21−α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this case, the power of the dispersion relation near the soft mode k = π is not affected by the presence – 14 – −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 B2,α (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='25 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='75 5 × 100 6 × 100 7 × 100 8 × 100 ln L 100 101 102 S2,L(h = 0) (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' a) Rényi-2 scaling coefficient B2,α as a function of the chemical potential h for different values of the power law decay coefficient 0 < α = α1 = α2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The red and blue vertical lines correspond to the h = 1 and h = 0 critical points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' b) Numerical check for the entanglement subvolume law scaling at h = 0 for different values of 0 < α < 1, plotted as a function of the logarithmic of the subsystem size ln L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Scattered points correspond to the numerical data while solid lines represent our prediction B2,α ln L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' of long-range couplings (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, also the symbol discontinuity is independent of the value of α1,2, in particular, we find G± π = lim k→π± Gk = ±σy, ∀α1, α2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) This leads to a logarithmic contribution coefficient bπ(z) = 1 2π2 � ln �z + 1 z − 1 ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='14) The corresponding scaling of the entanglement entropy is then the one obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5), which is equivalent to the entanglement scaling in the nearest neighbor Kitaev chain, at a quantum critical point characterized by a conformal field theory with central charge c = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Figure 4 shows the entanglement scaling behavior at the non-homogeneous critical point h = −1 + 21−α1 with α1 < α2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 4a), α1 > α2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 4b) and α1 = α2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Also in this case a nice agreement with the theoretical prediction in the thermodynamic limit is found once finite size corrections are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 5 Strong long-range regime The situation in the strong long-range regime is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular previous studies on fermionic systems with strong long-range pairing interactions [56–58] reported logarithmic violations of the entanglement area law even away from criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, in those cases, the noncritical logarithmic scaling of the ground state entanglement was asso- ciated with divergences in the long-range couplings due to the fact that no Kac scaling was introduced in the model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Therefore, one may think such anomalous scalings to be trivially related to the loss of the system extensivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, as shown – 15 – in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 2, the introduction of a Kac scaling in the Hamiltonian allows us to define a model with strong long-range interaction still preserving the energy extensivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, when a Kac scaling is introduced, the coupling divergences for α1, α2 < 1 are canceled, and accordingly also the symbol discontinuity associated with them disap- pears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, an infinite number of new nontrivial discontinuities arise due to the fact that the spectrum becomes discrete also in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, as a consequence of the spectrum discontinuity, the symbol becomes discontinuous for any k = 2πn/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in the thermodynamic limit, Gk reads lim N→∞ Gk = Gn = h − tn ωn σz − ∆n ωn σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) Then it can be labeled by a discrete integer number n, while the k variable becomes contin- uous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, any real physical implementation of the model has necessarily a finite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Therefore, the actual physical meaning of the continuum limit as N → ∞ is that the difference between two consecutive values of k is of order O(N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, in the strong long-range case, a difference of order O(N−1) in the k variable results in a finite jump of the spectrum ωn which remains discrete even in the thermodynamic limits, thus resulting in a discontinuity of the matrix symbol Gk for any k independently of the value of the chemical potential h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, since for any α1,2 > 0 or α1 = α2 = 0 and h ̸= 0 the many-body ground state is still the Bogoiubov vacuum, then the two lateral limits corresponding to a given k± = 2πn/N, 2π(n + 1)/N can be written as G± k = � Gn+1 = cos φn+1σz + sin φn+1σy Gn = cos φnσz + sin φnσy , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) where we have introduced the angles φn defined by the conditions cos φn = (h−tn)/ωn and sin φn = −∆n/ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then, following the analytic procedure introduced in Section 3, for any value of h, we obtain a logarithmic scaling of the ground state Rényi entropies of the form Sν,L = Bν(h) ln L + O(1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) where the Bν(h) coefficient is a function of ν, α1, α2 and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then Bν,α1,α2(h) is given by the sum of N contributions corresponding to the N discontinuities of the symbol, reading Bν(h) = N/2 � n=−N/2+1 B(n) ν (h), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) where, as shown in Appendix B, each contribution reads B(n) ν (h) = 1 π2(ν − 1) ν � l=1 arctan2 � sin((φn+1 − φn)/2) � cos2((φn+1 − φn)/2) + |zl|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) where |zl|2 = tan2(π(2l − 1)/2ν), with l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' , ν and l ̸= (1 + ν)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, for ν = 2 the above sum can be written explicitly as B(n) 2 (h) = 2 π2 � arctan � ωn+1ωn − (h − tn+1)(h − tn) − ∆n+1∆n 3ωn+1ωn + (h − tn+1)(h − tn) + ∆n+1∆n �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) – 16 – As we have already seen in the previous Sections, the most interesting situation is the one with equally long-range hopping and pairing amplitudes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', with α1 = α2 = α, while we expect only minor differences to appear when α1 ̸= α2, as long as they are both smaller than the system dimension (here d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Therefore, for the sake of simplicity, we will limit our treatment to the α1 = α2 = α case in the following analysis of the strong long-range regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Figure 6a shows B2(h) as a function of the chemical potential h for different values of α1 = α2 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' First of all, we notice that for any values of the chemical potential h ̸= 0 and of α > 0 the scaling coefficient is of order B2(h ̸= 0) = O(1), then leading to a logarithmic violation of the area law even away from the quantum critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, two singularities appear at the quantum critical points h = t0, tπ = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, we have a discontinuity for h = 1 and a divergence with the subsystem size for h = 0, leading to a subvolume law entanglement scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These facts can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The spectrum is labeled by the discrete index n leading to a finite gap between the ground state and the first excited levels which are associated with discontinuities of the symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, for n ≫ 1 all the modes accumulate around ω∞ = |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This means that an extensive number of single-particle states is almost degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Consequently, as long as h ̸= 0, we may expect only the first few modes around n = 0 to provide a significant contribution to the symbol discontinuity leading to a coefficient Bν(h ̸= 0) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, we may expect many features of the entanglement scaling coefficients for values of the chemical potential sufficiently far from the h = 0 point, to be qualitatively reproduced by considering a single discontinuity approximation in which only the first discontinuity between the n = 0 and the first two degenerate levels n = ±1 is considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', Bν(h ̸= 0) ≈ B(0) ν + B(−1) ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then, as detailed in Appendix D within this approximation the discontinuity coefficient reads Bν(h ̸= 0) ≈ 2 π2(ν − 1) ν � l=1 arctan2 � cos(φ1/2) |zl|2 + sin2(φ1/2) � if h < 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) Bν(h ̸= 0) ≈ 2 π2(ν − 1) ν � l=1 arctan2 � sin(φ1/2) |zl|2 + cos2(φ1/2) � if h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) This approximation then allows us to capture the origin of the scaling coefficient discontinu- ity at h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This originates from the fact that the zero mode gives different contributions at the two sides of the transition, indeed (see Appendix D) φ0 = arccos[sign(h − 1)] = � π if h < 1 0 if h > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) The single discontinuity approximation turns out to correctly reproduce the qualitative features as long as the chemical potential h is sufficiently far from h = 0 and for sufficiently large power law decay exponent α > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, this simple approximation is no more accurate as the chemical potential approaches the h = 0 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in the zero chemical potential case ω∞ = 0, and more precisely ωn, tn and ∆n approach their asymptotic values differently if we consider the even or the odd modes (see Appendix D for – 17 – more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' As a consequence, for sufficiently small α, the number of relevant symbol discontinuities grows as a power law of the subsystem size L, leading to a fractal subvolume- law entanglement scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, using the asymptotic expansion of ωn, tn and ∆n in the n → ∞ limit we can extract the leading order dependence of Bν(h = 0) from L, which, as shown in Appendix D, reads Bν(h = 0) = � O(L1−2α) if α < 1/2 O(1) if α > 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) Accordingly, the leading order contribution to the entanglement Rényi entropy of the system ground state at zero chemical potential takes the nontrivial form Sν,L(h = 0) = � O(L1−2α ln L) if α < 1/2 O(ln L) if α > 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) This analytic result matches the numerics in the large L limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 6b, where the numerical and analytical results for S2,L are plotted as a function of ln L and for different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' It is important to notice that approaching the thermodynamic limit in the h = 0 case the spectrum becomes increasingly more degenerate approaching the α = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then, for each finite N, a large number of states nearly degenerate with the ground state exists, making the estimate of the subleading corrections scaling technically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, as already stated in Sections 2 and 3, the mean-field case with α1 = α2 = 0 and h = 0 must be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in this case the ground state degeneracy allows for a finite fermionic population of the even Bogoliubov modes, fn ̸= 0 ∀n(even), this leads to the entanglement scaling Sν,L(α = 0, h = 0) = 1 1 − ν � n(even) ln [(1 − fn)ν + fν n] + O(ln L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) In particular the maximal Rényi entropy is reached when fn = 1/2 ∀n(even) Smax ν,L (α = 0, h = 0) = N0 ln 2 + O(1) = L 2 ln 2 + O(1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) where N0 is the number of zero modes, which in this case corresponds to the number of even modes N0 ≃ L/2 and the subleading corrections are at most of order O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, as shown in Appendix B, the discontinuity coefficients Bν which would lead to logarithmic corrections turn out to be exactly zero when all the even fermionic populations are fn(even) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, we notice that the maximal Rényi entropy that we have obtained employing the Fisher-Hartwig expansion corresponds to the largest possible entropy allowed by the ground state degeneracy Smax ν,L (α = 0, h = 0) = ln Deg[|gsα=0,h=0⟩] = N0 ln 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='14) This tells us that the Fisher-Hartwig result, obtained as a large subsystem size expansion, actually becomes exact in this maximally entangled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 18 – 6 Conclusion and outlooks In this paper, we have further extended the understanding of the peculiar properties of entanglement in quantum systems featuring long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' At this scope, we have investigated, as a paradigmatic example, the ground state entanglement scaling of a spinless fermionic chain with long-range hopping and pairing amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The simplicity of the model and its truly non-additive nature allowed us to unveil an extremely rich and non- trivial phenomenology, which we have fully characterized both numerically and analytically in the different regions of the relevant parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', the power law decay exponents of the hopping and pairing couplings α1, α2 and the chemical potential h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, two main regimes may be distinguished: the weak long-range regime with 1 < α1, α2 < 2 and the strong long-range regime with 0 < α1, α2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In the weak long-range case, the system quasiparticle spectrum becomes continuous in the thermodynamic limit and the main effect of the non-local couplings is to change the dispersion relation near the gapless critical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, the standard area law, typical of gapped local Hamiltonians, is satisfied in this regime apart from the logarithmic violations which appear in correspondence of the two quantum critical points located at h = 1, −1+21−α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Such logarithmic scaling of the ground state Rényi entropies is related to discontinuities in the symbol of the correlation matrix which is a block Toeplitz matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The fact that the contribution to the entanglement scaling of each discontinuity only depends on the value of the symbol [57, 58] at each side of the jump, allowed us to exactly compute its coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Most significantly, when the hopping and pairing couplings are equally long-range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', α1 = α2 = α, the coefficient in front of the critical logarithmic divergence at h = 1 turns out to have a non-trivial dependence on α (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Interestingly, the coefficient Bν,α is of non-universal nature, since it originates from the precise form of the spectrum in the proximity of the critical modes, and not only from the dispersion relation power law exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' As a consequence, the critical entanglement scaling is not compatible with the result obtained from any conformal field theory and our result may be seen as a benchmark of the fact that the presence of long-range couplings explicitly breaks the critical conformal symmetry [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These findings demonstrate the peculiarity of the α1 = α2 case, whose physics is expected to be, and indeed is, closer to the one of a strongly interacting system such as the quantum Ising model, where long-range couplings are expected to increase the effective dimension and, so, disrupt integrability [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover, for α1 ̸= α2, the critical entanglement scaling becomes α independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, when α1 > α2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', the pairing coupling has a slower decay with respect to the hopping, the entanglement scaling is compatible with that of conformal field theory with central charge c = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This is in agreement with the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [57, 58], where a Kitaev chain with long-range pairing and nearest neighbors hopping is considered, the validity of such results is then here extended to any long-range hopping with power law decay exponent α1 > α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The strong anisotropy between the case of dominating hopping α1 < α2 and the case of dominating paring α1 > α2 is typical of the long-range Kitaev chain [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In the strong long-range regime, the situation is more involved, indeed the quasiparticle – 19 – spectrum can no more be considered continuous in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Consequently, the matrix symbol of the block Toeplitz correlation matrix formally becomes discontinuous at every point of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, as shown in Section 5, in most situations only a few of such discontinuities truly contribute to the entanglement scaling, leading to a logarithmic dependence on the subsystem size even outside criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Also in this case the coefficients of such logarithmic divergence can be computed analytically for different values of the parameters α1,2 and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The most interesting situation turns out to be the zero chemical potential point h = 0 in the strong long-range regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in this case, the coefficient in front of the critical logarithmic entanglement scaling diverges as a power law of the subsystem size, leading to a fractal subvolume-law entanglement scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, we were able to analytically extract the leading entanglement dependence from the subsystem size, which turns out to be of the form Sν,L ≈ L1−2α ln L, with 0 < α = α1 = α2 < 1/2, where Sν,L is any ν-Renyi entropy with ν > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Similar sub-volume laws have already been observed in different (more complex) scenarios and, in particular, in the entanglement scaling of measurement induced phase transitions [50], where they arise due to the suppression of entanglement caused by repeated measurements in a long-range systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Here, this phase emerges naturally in the equilibrium scaling, but it needs stronger interactions to appear with respect to the dynamical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, in the completely mean-field case, the system presents an extensive number of degenerate modes with zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These zero modes can be populated also in the many-body ground state whose degeneracy then grows exponentially with the number of zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Consequently, the ground state entanglement shows a volume law behavior proportional to the size of the considered subsystem Sν,L(α = 0) ≈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Our studies evidence that long-range couplings can greatly improve the scaling of en- tanglement at equilibrium and, therefore, that long-range interacting quantum systems represent the ideal candidate for reliable and robust quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nevertheless, such fostered entanglement properties may not persist out-of-equilibrium, since long-range interactions have been shown to suppress the dynamical spread of entanglement in certain systems [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' For the future, we intend to investigate these issues by performing quantum simulations of the model on actual quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This demands a careful engineering of the artificial non-local couplings on local quantum devices, a task which we are currently tackling on IBM Quantum devices [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The rich phenomenology hosted by the minimal long-range model we considered,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' al- ready at equilibrium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' suggests that many of the intriguing dynamical phenomena which are recently emerging in the quantum community,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' such as the non-trivial fractal entanglement scalings in the contest of measurement-induced entanglement transitions [3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' can be simply ascribed to the presence of sufficiently long-range couplings among the microscopic compo- nents of the model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' without any need of further complexity in the physical system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Further work is needed in order to investigate the dynamical properties of entanglement in the Kitaev chain with long-range pairing and hopping couplings subjected to a unitary or a non-unitary (measurement-like) evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' These interesting problems are beyond the scope of this work and we leave them as an outlook for future projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 20 – Acknowledgements We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This work is part of the MIUR-PRIN2017 project Coarse-grained description for nonequilibrium systems and transport phenomena (CO- NEST) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 201798CZL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' AS and SS acknowledge acknowledge financial support from Na- tional Centre for HPC, Big Data and Quantum Computing (CN00000013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Access to the IBM Quantum Computers was obtained through the IBM Quantum Hub at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 21 – A Derivation of the matrix symbol In this Appendix we provide the details for the derivation of the matrix symbol in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We start from the definition of the correlation matrix of a stationary state |ψ⟩, then passing to the Fourier basis we obtain Gk = 2⟨ψ| � ˆck ˆc† −k � � ˆc† k ˆc−k � |ψ⟩ − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) Introducing the Bogoliubov transformation � ˆγk ˆγ† −k � = Uk � ˆck ˆc† −k � , Uk = � cos θk/2 i sin θk/2 −i sin θk/2 − cos θk/2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) we can write the symbol in terms of the Bogoliubov modes as Gk = 2U † k⟨ψ| � ˆγk ˆγ† −k � � ˆγ† k ˆγ−k � |ψ⟩Uk − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) We now compute the expectation value in a stationary stationary state associated to the fermionic populations of the Bogoliubov modes fk = ⟨ˆγ† kˆγk⟩, so that ⟨ψ| � ˆγk ˆγ† −k � � ˆγ† k ˆγ−k � |ψ⟩ − I = � 1 − 2fk 0 0 2fk − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) Finally, inserting this expectation value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) and using the definition of the Bogoli- ubov angles tan θk = ∆k/(h − tk) we obtain Gk = (1 − (fk + fk)) �h − tk ωk σz − ∆k ωk σy � − (fk − f−k)I, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) which is the expression for the matrix symbol used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B Coefficients of the Fisher-Hartwig expansion The general form of the matrix symbol in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) can be used to compute the different terms in the Fisher-Hartwig expansion of the Rényi entropies for large subsystem size in every situation considered in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' For this purpose, it is useful to rewrite Gk as Gk = ak [cos φkσz + sin φkσy] + bkI, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) where we have introduced the coefficients ak = 1 − (fk + f−k) and bk = f−k − fk and the angle φk such that cos φk = (h − tk)/ωk and sin φk = −∆k/ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Let us start from the first term of the expansion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) this is obtained by first computing the determinant det [zI − Gk] = (z − bk)2 − a2 k, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) – 22 – bn + an cos δϕn bn + an bn − an cos δϕn bn − an 1 + ϵ −1 − ϵ Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Contour of integration and cuts of the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The cuts from ±(1+ϵ) to the infinity correspond to dsν(1+ϵ, z)/dz while the cuts inside the contour, [bn −an, bn −an cos δφ] and [bn + an cos δφ, bn + an, ], are due to the other factor of the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then, the contribution to first term in the entanglement scaling coming from each k-mode is obtained from the integral Sk = lim ϵ→0+ � C dz 2πisν(1 + ϵ, z) (z − bk) (z − bk)2 − a2 k (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) = 1 2 [sν(1, bk + ak) + sν(1, bk − ak)] = 1 2(1 − ν) � ln(fν k + (1 − fk)ν) + ln(fν −k + (1 − f−k)ν) � , where Cauchy’s residue theorem and the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) for sν(x, y) have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fi- nally, summing over all the modes and using the k → −k symmetry we obtain � k Sk = 1 1 − ν � k ln(fν k + (1 − fk)ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) The logarithmic contribution to the entanglement scaling can be computed by con- sidering the discontinuity coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Here, we present their calculation in the general situation in which Gk is discontinuous at a generic mode k = 2πn/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We start from the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) of the bk coefficients corresponding to each discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' First of all, we consider the matrix Mk = (zI − G− k )(zI − G+ k )−1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) where G± k = limp→k± Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The eigenvalues µ± k (z) of this matrix can be written in the form µ± k (z) = � � � (bk − z)2 − a2 k cos2(δφk/2) ± ak sin(δφk/2) � (bk − z)2 − a2 k � � 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) – 23 – with δφk = φ+ k − φ− k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Notice also that we have µ+ k (z) = 1/µ− k (z), therefore bk(z) = 1 2π2 � ln µ+ k (z) �2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) = 2 π2 � �ln � � � (bk − z)2 − a2 k cos2(δφk/2) + ak sin(δφk/2) � (bk − z)2 − a2 k � � � � 2 , From this expression we compute the coefficient B(k) ν of the contribution of this discontinuity to the logarithmic term of the Rényi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' For this purpose we plug bk(z) into the contour integral for Sν,L then, performing an integration by parts, we obtain B(k) ν = lim ϵ→0+ � C dz 2πisν(1 + ϵ, z)dbk(z) dz (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) = − lim ϵ→0+ � C dz 2π3i dsν(1 + ϵ, z) dz � �ln � � � (bk − z)2 − a2 k cos2(δφk/2) + ak sin(δφk/2) � (bk − z)2 − a2 k � � � � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The integral over the contour C depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 7 can be divided into two integrals along curves enclosing respectively the cuts [bk − ak, bk − ak cos δφk] and [bk + ak, bk + ak cos δφk], which in turn can be reduced to two real integrals by performing the integration along the cuts taking into account the change in the phase of the logarithm when we go around the branch points bk ± ak and bk ± ak cos δφk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, we notice that for integer ν > 1, dsν/dz is a meromorphic function with poles located at the points of the imaginary axis [57, 58] zl = i tan π(2l − 1) 2ν , l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' , ν, l ̸= 1 + ν 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) and that the another factor of the integrand is analytic in the whole region outside the contour C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We can send this contour to infinity and reduce the calculation of Bν to the computation of the corresponding residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this way, we obtain the explicit expression B(k) ν = 1 ν − 1 ν � l=1 � �ln � � � (bk − zl)2 − a2 k cos2(δφk/2) + ak sin(δφk/2) � (bk − zl)2 − a2 k � � � � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) This general formula can be specified in the different cases considered in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular in weak long-range case, 1 < α1, α2 < 2, the ground state corresponds to the Bogoliubov vacuum, therefore fk = 0, ak = 1 and bk = 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, the first term of the expansion vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Moreover the matrix symbol is continuous for generic values of the chemical potential leading to an O(1) entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The only discontinuities arise at the two quantum critical points h = hc = 1, −1 + 21−α1 in correspondence of the critical – 24 – modes k = kc = 0, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This leads to a logarithmic scaling with coefficient B(kc) ν = 1 ν − 1 ν � l=1 � ln �� |zl|2 + cos2(δφkc/2) − i sin(δφkc/2) � |zl|2 − 1 ��2 = 1 ν − 1 ν � l=1 � arctan � sin(δφkc/2) � |zl|2 + cos2(δφkc/2) ��2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) where in the last step we have used the identity arctan(x) = i[ln(i + x) − ln(i − x)]/2 in order to make the expression of the coefficient explicitly real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The value of δφkc depends on the critical point considered and the relative order of the power law decaying exponents α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular for h = 1 and k = 0 we find δφ0 = � � � � � � � 0 if α1 < α2 π(1 − α) if α1 = α2 = α π if α1 > α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='12) Leading to the coefficients B0 ν(h = 1) = � � � � � � � � � � � 0 if α1 < α2 1 ν−1 �ν l=1 � arctan � cos(απ/2) √ |zl|2+sin2(απ/2) ��2 if α1 = α2 = α ν+1 12ν if α1 > α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='13) On the other hand, for h = −1 + 21−α1 and k = π, δφπ = π independently from the values of α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This leads to the scaling coefficient B0 ν(h = −1 + 21−α1) = ν + 1 12ν ∀ α1, α2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='14) In the strong-long range regime 0 < α1, α2 < 1, the quasiparticle spectrum is discrete also in the thermodynamic limit, this formally leads to an infinite number of discontinuities for any mode k = 2πn/N, which are labeled by the integer n = −N/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular whenever α1,2 > 0 or α1 = α2 = 0 and h ̸= 0, the many-body ground state is still the Bogoliubov vacuum characterized by fk = 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Accordingly, the matrix symbol in the thermodynamic limit takes the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The coefficients of the logarithmic scaling is then given by the sum of the contributions coming from all the discontinuity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', Bν = N/2 � n=−N/2 B(n) ν , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='15) where B(n) ν = 1 ν − 1 ν � l=1 � ln �� |zl|2 + cos2(δφn/2) − i sin(δφn/2) � |zl|2 − 1 ��2 = 1 ν − 1 ν � l=1 � arctan � sin(δφn/2) � |zl|2 + cos2(δφn/2) ��2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='16) – 25 – with δφn = φn+1 − φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, in the mean-field case α1 = α2 = 0 with zero chemical potential h = 0 the quasiparticle spectrum develops an extensive number of degenerate zero modes ωn = 0 corresponding to all the even modes with n = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' As a consequence, the ground state is characterized by a finite even mode fermionic population f2m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The leading order term in the entanglement scaling in this case is then given by the first term of the Fisher- Hartwig expansion corresponding to a volume law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular the maximum amount of entanglement allowed by the ground state degeneracy is obtained for f2m = 1/2 for every even mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this case, the logarithmic corrections become zero since an = bn = 0, and therefore B(n) ν (fn = 1/2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' C Dispersion relation around the critical modes In this Appendix we provide the explicit expression for the Taylor expansion of the quasi- particle spectrum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7), in the weak long-range regime 1 < α1,2 < 2, at lowest order in |k − kc|, where kc = 0 at the critical point h = 1, while kc = π at h = −1 + 21−α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, in the proximity of k = 0 we find [97] tk = 1 + sin(α1)Γ(1 − α) ζ(α) kα1−1 + O(k2), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) ∆k = sin(α1)Γ(1 − α) ζ(α) kα2−1 + O(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) Accordingly, the single particle spectrum takes the form [8] ωk = � |h − 1| + O(kα − 1) if h ̸= 1 C(α)|k|α−1 + O(k2α−2) if h = 1 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) where α = min{α1, α2}, and we have introduced the constant prefactor C(α) = � � � � � � � | sin(α1π/2)Γ(1 − α1)/ζ(α1)| if α1 < α2 |Γ(1 − α)/ζ(α)| if α1 = α2 | cos(α1π/2)Γ(1 − α1)/ζ(α1)| if α1 > α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) On the other hand, near to the k = π mode we find [97] tk = −1 + 21−α1 − (23−α1 − 1)ζ(α1 − 2) 2ζ(α1) (π − k)2 + O((π − k)3), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) ∆k = (1 − 22−α2)ζ(α2 − 1) ζ(α2) (π − k) + O((π − k)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) Leading to the α1,2 independent dispersion relation ωk = � |h + 1 − 21−α1| + O((k − π)2) if h ̸= −1 + 21−α1 K(α2)|π − k| + O((k − π)3) if h = −1 + 21−α1 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) where K(α2) = (1 − 22−α2)ζ(α2 − 1)/ζ(α2), ∀α1, α2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 26 – D Discontunities in the strong long-range regime In this Appendix we provide a detailed analysis of the discontinuities of the matrix symbol Gk in the strong long-range regime 0 < α1, α2 < 1 for different values of the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' As discussed in the Section 5 of the main text, in this regime the matrix symbol formally develops and infinite number of discontinuities which originate from the discrete nature of the quasiparticle spetrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' However, it is important to notice that, even if the spectrum is labeled by the discrete index n leading to a finite gap between the ground state and the first excited levels, still for n ≫ 1 all the modes accumulate around ω∞ = |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This means that an extensive number of single-particle states is almost degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Consequently, as long as h ̸= 0, we may expect only the first few modes around n = 0 to provide a significant contribution to the symbol discontinuity, leading to a coefficient Bν(h ̸= 0) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Then, in order to understand the qualitative behavior of Bν(h ̸= 0), it is useful to consider the approximation in which only the first discontinuities between the n = 0 and the first two degenerate levels n = ±1 are considered Bν(h ̸= 0) ≈ B(0) ν + B(−1) ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1) In order to compute this two contributions we have to compute the angles φ0 and φ±1 defined by the conditions cos φn = h − tn ωn , sin φn = −∆n ωn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2) For n = 0 we find that, independently of the value of α, the angle reads cos φ0 � −1 if h < 1 0 if h > 1 , φ0 = � π if h < 1 0 if h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3) This discontinuity is at the quantum critical point h = 1 is due to the fact that at this point the spectrum becomes gapless for n = 0, and it is at the origin of the discontinuity in the scaling coefficient which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 6a of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The angles for n = ±1 cannot be computed exactly in close form for generic power law decaying exponent, however as a consequence of the fact that tn = t−n, ωn = ω−n while ∆n = −∆−n, we have that cos φn = cos φ−n sin φn = − sin φ−n, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='4) and then φn = −φ−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Combining these properties with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) we obtain B(0) ν = B(−1) ν = 1 π2(ν − 1) ν � l=1 arctan2 � cos(φ1/2) 1 + sin2(φ1/2) � if h < 1, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5) B(0) ν = B(−1) ν = 1 π2(ν − 1) ν � l=1 arctan2 � sin(φ1/2) 1 + cos2(φ1/2) � if h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6) Figure 8 shows the comparison between exact values of the logarithmic scaling coefficients – 27 – −1 0 1 2 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3 B2,α (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 −1 0 1 2 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3 B2,α (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7 −1 0 1 2 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='3 B2,α (c) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Comparison between the exact values of the logarithmic scaling coefficients of the Rényi- 2 entropy, and the single discontinuity approximation (dashed lines) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' The coefficients are plotted as function of the chemical potential h for different values of the decay exponent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' of the Rényi-2 entropy B2, computed considering the contribution of a formally extensive number of discontinuities (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5)), and the results obtained in the single discontinuity approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' We notice that the single discontinuity approximation correctly reproduces the qualitative behavior of the scaling coefficients for sufficiently high α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='5 and for values of the chemical potential h which are sufficiently far from h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, the discontinuity of the coefficients at the quantum critical point h = 1 is captured by the approximated result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' On the other hand, when the chemical potential approaches the h → 0 limit and for sufficiently small decay exponents α < 1/2, the single discontinuity approximation turns out to be no more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Indeed, in this case the number of relevant discontinuities grows with the subsystem size, leading to a subvolume law entanglement scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' This fact can be understood by considering the h = 0 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In this case, the spectrum accumulation point becomes ω∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' More precisely, it is important to notice that, while at the leading order as n → ∞ the spectrum goes to zero as ωn = O(nα−1), independently of the parity of the mode, on the contrary next to leading order corrections differ if n is even or odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' In particular, if we perform a next to leading order expansion of the terms entering the coefficient B(m) 2 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6)), corresponding to the discontinuity between the modes m = 2n and m + 1 = 2n + 1, we find t2n+1t2n = s2 α n2−2α + O(n2α−3), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7) ∆2n+1∆2n = c2 α n2−2α − a2 α n2 + O(n2α−3), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) ω2n+1ω2n = s2 α + c2 α n2−2α + bα n2 + O(n2α−3), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) where we have introduced the expansion coefficients sα = sin(απ/2)Γ(2 − α)(2π)α−1, cα = cos(απ/2)Γ(2 − α)(2π)α−1, aα = (1 − α)/(2π), bα = a2 α(1/2 − cos2(απ/2)) = a2 α cos(απ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='10) – 28 – Now, inserting the large n expansions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='7), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='8) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='9) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='6), we see that the denominator is always of order O(n2α−2), while in the numerator the leading order cancels out and we are left with a contribution of order O(n−2) if α < 1/2 or O(n2α−3) if α > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finally, putting everything together and summing over all the modes we obtain Bν(h = 0) = � n B(n) ν = �� n O(n−2α) = O(L1−2α) α < 1/2 � n O(n−1) = O(1) α > 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) This result leads to the scaling of the Rényi entropy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Campa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dauxois, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fanelli and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ruffo, Physics of Long-Range Interacting Systems, Oxford University Press (Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', 2014), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1093/acprof:oso/9780199581931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Donner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Macrì, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pagano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ruffo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Trombettoni, Long-range interacting quantum systems, arXiv: 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='01063 (2021) [2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='01063].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Xu, Long-range coupling affects entanglement dynamics, Physics 15 (2022) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Russomanno, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Iemini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dalmonte and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fazio, Floquet time crystal in the lipkin-meshkov-glick model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 95 (2017) 214307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Surace, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Russomanno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dalmonte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Silva, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fazio and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Iemini, Floquet time crystals in clock models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 99 (2019) 104303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pizzi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Knolle and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nunnenkamp, Higher-order and fractional discrete time crystals in clean long-range interacting systems, Nature Communications 12 (2021) 2341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Giachetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Solfanelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Correale and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, High-order time crystal phases and their fractal nature, arXiv: 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='16562 (2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Solfanelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Giachetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Campisi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ruffo and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, Quantum heat engine with long-range advantages, arXiv: 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='09492 (2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Britton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sawyer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Keith, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Freericks, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Uys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', Engineered two-dimensional ising interactions in a trapped-ion quantum simulator with hundreds of spins, Nature 484 (2012) 489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Monroe, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Campbell, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Duan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gorshkov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', Programmable quantum simulations of spin systems with trapped ions, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 93 (2021) 025001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mottl, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Brennecke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Landig, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Donner and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Esslinger, Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions, Science 336 (2012) 1570 [https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1220314].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mivehvar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Piazza, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Donner and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ritsch, Cavity qed with quantum gases: new paradigms in many-body physics, Advances in Physics 70 (2021) 1 [https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1080/00018732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1969727].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chomaz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ferrier-Barbut, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ferlaino, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Laburthe-Tolra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lev and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pfau, Dipolar physics: A review of experiments with magnetic quantum gases, arXiv: 2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='02672 (2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Bennett and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' DiVincenzo, Quantum information and computation, Nature 404 (2000) 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 29 – [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nielsen and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge University Press (2010), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1017/CBO9780511976667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [16] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Herbst, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Scheidl, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kropatschek, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Naylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=', Quantum teleportation over 143 kilometres using active feed-forward, Nature 489 (2012) 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gisin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ribordy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Tittel and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Zbinden, Quantum cryptography, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 74 (2002) 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [18] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Giovannetti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lloyd and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Maccone, Advances in quantum metrology, Nature Photonics 5 (2011) 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Horodecki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Horodecki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Horodecki and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Horodecki, Quantum entanglement, Reviews of modern physics 81 (2009) 865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Srednicki, Entropy and area, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 71 (1993) 666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Eisert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Cramer and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Plenio, Colloquium: Area laws for the entanglement entropy, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 82 (2010) 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Hastings, An area law for one-dimensional quantum systems, Journal of Statistical Mechanics: Theory and Experiment 2007 (2007) P08024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Calabrese and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment 2004 (2004) P06002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Calabrese and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Cardy, Entanglement entropy and conformal field theory, Journal of Physics A: Mathematical and Theoretical 42 (2009) 504005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Holzhey, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Larsen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Wilczek, Geometric and renormalized entropy in conformal field theory, Nuclear Physics B 424 (1994) 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [26] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Latorre, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rico and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kitaev, Entanglement in quantum critical phenomena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 90 (2003) 227902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Latorre, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rico and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vidal, Ground state entanglement in quantum spin chains, Quantum Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 4 (2004) 48–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Latorre and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Riera, A short review on entanglement in quantum spin systems, Journal of Physics A: Mathematical and Theoretical 42 (2009) 504002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Bombelli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Koul, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lee and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sorkin, Quantum source of entropy for black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' D 34 (1986) 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [30] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dong, The gravity dual of rényi entropy, Nature communications 7 (2016) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ryu and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Takayanagi, Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 96 (2006) 181602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Van Raamsdonk, Building up spacetime with quantum entanglement, General Relativity and Gravitation 42 (2010) 2323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Maldacena and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Susskind, Cool horizons for entangled black holes, Fortschritte der Physik 61 (2013) 781 [https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1002/prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='201300020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [34] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Amico, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fazio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Osterloh and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vedral, Entanglement in many-body systems, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 80 (2008) 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [35] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Calabrese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Cardy and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Doyon, Entanglement entropy in extended quantum systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' A 42 (2009) 500301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 30 – [36] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Laflorencie, Quantum entanglement in condensed matter systems, Physics Reports 646 (2016) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [37] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Foss-Feig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Brandão and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gorshkov, Entanglement area laws for long-range interacting systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 119 (2017) 050501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [38] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kuwahara and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Saito, Area law of noncritical ground states in 1d long-range interacting systems, Nature Communications 11 (2020) 4478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lerose and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pappalardi, Origin of the slow growth of entanglement entropy in long-range interacting spin systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Research 2 (2020) 012041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Giachetti and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, Entanglement propagation and dynamics in non-additive quantum systems, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='11488 (2021) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [41] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Capizzi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Giachetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Santini and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Collura, Spreading of a local excitation in a quantum hierarchical model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 106 (2022) 134210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [42] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chen and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fisher, Quantum zeno effect and the many-body entanglement transition, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 98 (2018) 205136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Skinner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ruhman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' X 9 (2019) 031009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [44] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nandkishore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pretko and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Smith, Unitary-projective entanglement dynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 99 (2019) 224307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [45] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Cao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Tilloy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 7 (2019) 024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gullans and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Huse, Dynamical purification phase transition induced by quantum measurements, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' X 10 (2020) 041020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [47] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Turkeshi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Biella, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fazio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dalmonte and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Schiró, Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 103 (2021) 224210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [48] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Alberton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Buchhold and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Diehl, Entanglement transition in a monitored free-fermion chain: From extended criticality to area law, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 126 (2021) 170602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [49] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Minato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sugimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kuwahara and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Saito, Fate of measurement-induced phase transition in long-range interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 128 (2022) 010603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Block, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Bao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Choi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Altman and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Yao, Measurement-induced transition in long-range interacting quantum circuits, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 128 (2022) 010604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [51] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Müller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Diehl and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Buchhold, Measurement-induced dark state phase transitions in long-ranged fermion systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 128 (2022) 010605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [52] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sharma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Turkeshi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fazio and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dalmonte, Measurement-induced criticality in extended and long-range unitary circuits, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Core 5 (2022) 023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [53] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Jian and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chen, Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics, Quantum 6 (2022) 723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [54] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ippoliti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rakovszky and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Khemani, Fractal, logarithmic, and volume-law entangled nonthermal steady states via spacetime duality, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' X 12 (2022) 011045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [55] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vodola, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lepori, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ercolessi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pupillo, Long-range ising and kitaev models: phases, correlations and edge modes, New Journal of Physics 18 (2015) 015001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 31 – [56] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vodola, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lepori, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ercolessi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gorshkov and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pupillo, Kitaev chains with long-range pairing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 113 (2014) 156402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [57] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ares, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Esteve, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Falceto and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' de Queiroz, Entanglement in fermionic chains with finite-range coupling and broken symmetries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' A 92 (2015) 042334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [58] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ares, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Esteve, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Falceto and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' de Queiroz, Entanglement entropy in the long-range kitaev chain, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' A 97 (2018) 062301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [59] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ares, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Murciano and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Calabrese, Symmetry-resolved entanglement in a long-range free-fermion chain, Journal of Statistical Mechanics: Theory and Experiment 2022 (2022) 063104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [60] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Paganelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sharma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sodano and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Trombettoni, Explicit hamiltonians inducing volume law for entanglement entropy in fermionic lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 91 (2015) 245138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [61] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lepori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Paganelli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Franchini and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Trombettoni, Mutual information for fermionic systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Research 4 (2022) 033212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [62] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sachdev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ye, Gapless spin-fluid ground state in a random quantum heisenberg magnet, Physical review letters 70 (1993) 3339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [63] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Maldacena and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Stanford, Remarks on the sachdev-ye-kitaev model, Physical Review D 94 (2016) 106002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [64] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Zhang, Quantum entanglement in the sachdev—ye—kitaev model and its generalizations, Frontiers of Physics 17 (2022) 43201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [65] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chen and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Balents, Quantum entanglement of the sachdev-ye-kitaev models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 97 (2018) 245126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [66] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sachdev, Numerical study of fermion and boson models with infinite-range random interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 94 (2016) 035135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [67] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Huang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gu, Eigenstate entanglement in the sachdev-ye-kitaev model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' D 100 (2019) 041901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [68] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Liu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Chen, Subsystem Rényi entropy of thermal ensembles for SYK-like models, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 8 (2020) 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [69] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Haldar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Bera and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Banerjee, Rényi entanglement entropy of fermi and non-fermi liquids: Sachdev-ye-kitaev model and dynamical mean field theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Research 2 (2020) 033505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [70] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Nezhadhaghighi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rajabpour, Entanglement entropy in long-range harmonic oscillators, Europhysics Letters 100 (2013) 60011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [71] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Latorre, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Orús, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rico and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vidal, Entanglement entropy in the lipkin-meshkov-glick model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' A 71 (2005) 064101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [72] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vidal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dusuel and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Barthel, Entanglement entropy in collective models, Journal of Statistical Mechanics: Theory and Experiment 2007 (2007) P01015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [73] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Orús, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dusuel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vidal, Equivalence of critical scaling laws for many-body entanglement in the lipkin-meshkov-glick model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 101 (2008) 025701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [74] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Carrasco, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Finkel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' González-López, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rodríguez and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Tempesta, Generalized isotropic lipkin–meshkov–glick models: ground state entanglement and quantum entropies, Journal of Statistical Mechanics: Theory and Experiment 2016 (2016) 033114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 32 – [75] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kumari and Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Alhambra, Eigenstate entanglement in integrable collective spin models, Quantum 6 (2022) 701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [76] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kitaev, Unpaired majorana fermions in quantum wires, Physics-Uspekhi 44 (2001) 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [77] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fisher and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Hartwig, Toeplitz determinants: Some applications, theorems, and conjectures, in Advances in Chemical Physics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 333–353, John Wiley & Sons, Ltd (1969), DOI [https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1002/9780470143605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='ch18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [78] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Basor and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Morrison, The fisher-hartwig conjecture and toeplitz eigenvalues, Linear Algebra and its Applications 202 (1994) 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Kac, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Uhlenbeck and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Hemmer, On the van der waals theory of the vapor-liquid equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' discussion of a one-dimensional model, Journal of Mathematical Physics 4 (1963) 216 [https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1703946].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [80] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Jaschke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Maeda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Whalen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Wall and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Carr, Critical phenomena and kibble–zurek scaling in the long-range quantum ising chain, New Journal of Physics 19 (2017) 033032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [81] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vanderstraeten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Van Damme, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Büchler and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Verstraete, Quasiparticles in quantum spin chains with long-range interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' 121 (2018) 090603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [82] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Enss and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Halimeh, Dynamical criticality and domain-wall coupling in long-range hamiltonians, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 100 (2019) 014434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [83] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Fradkin, Field Theories of Condensed Matter Physics, Cambridge University Press, 2 ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' (2013), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='1017/CBO9781139015509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [84] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Uhrich, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Jafari and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Halimeh, Out-of-equilibrium phase diagram of long-range superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 101 (2020) 245148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [85] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Alecce and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dell’Anna, Extended kitaev chain with longer-range hopping and pairing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 95 (2017) 195160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [86] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, Metastability and discrete spectrum of long-range systems, Proceedings of the National Academy of Sciences 118 (2021) e2101785118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [87] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' DeGottardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Thakurathi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vishveshwara and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Sen, Majorana fermions in superconducting wires: Effects of long-range hopping, broken time-reversal symmetry, and potential landscapes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 88 (2013) 165111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [88] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lepori and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dell’Anna, Long-range topological insulators and weakened bulk-boundary correspondence, New Journal of Physics 19 (2017) 103030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [89] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Peschel, Calculation of reduced density matrices from correlation functions, Journal of Physics A: Mathematical and General 36 (2003) L205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [90] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Jin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Korepin, Quantum spin chain, toeplitz determinants and the fisher—hartwig conjecture, Journal of statistical physics 116 (2004) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [91] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Its, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mezzadri and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Mo, Entanglement entropy in quantum spin chains with finite range interaction, Communications in mathematical physics 284 (2008) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [92] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Widom, Asymptotic behavior of block toeplitz matrices and determinants, Advances in Mathematics 13 (1974) 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [93] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Widom, Asymptotic behavior of block toeplitz matrices and determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' ii, Advances in Mathematics 21 (1976) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 33 – [94] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Abramowitz and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Applied mathematics series, Dover Publications (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [95] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Lepori, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Vodola, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Pupillo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Gori and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Trombettoni, Effective theory and breakdown of conformal symmetry in a long-range quantum chain, Annals of Physics 374 (2016) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [96] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Trombettoni and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ruffo, Criticality and phase diagram of quantum long-range o(n) models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 96 (2017) 104432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [97] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Morigi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Dell’Anna and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Enss, Universal dynamical scaling of long-range topological superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' B 100 (2019) 184306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' [98] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Solfanelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Ruffo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Succi and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' Defenu, In preparation, (2023) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'} +page_content=' – 34 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFQT4oBgHgl3EQfCjXs/content/2301.13231v1.pdf'}